iSEgcii 
 
 ''..- ' - :;.?:-- ; 
 
 >:~ ' / S3 fell 
 ; ^ .-.V 
 
 *ffi'ts& 
 
 -,.k ;--.,... , ' . , j ifi 
 
 : , .'.'.-' 
 
 "' - 
 
 -'...' - W 
 
9 - 
 
 <y *s> 
 
 *S - 
 
 QJ/^XD 
 LIBRARY OF THE UNIVERSITY OF CALIFORNIA 
 
 __ *,_ 
 
 "^SsiteS^* 
 
 LIBRARY OF THE UNIVERSITY OF CALI 
 
 (5V 
 
 LIBRARY OF THE UNIVERSITY OF CALIFORNIA 
 
 LIBRARY OF THE UNIVERSITY OF CALI 
 
 2i^ 
 
 LIBRARY OF THE UNIVERSITY OF CALIFORNIA 
 
 LIBRARY OF THE UNIVERSITY OF CALI 
 
 jo | - 
 
9|fc 
 
 ^vS^^^fg 
 
 ^^Wi 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THI 
 
 5 ^^^EXX^^S' I 
 
 = s* Y y\^ Y \<fV x 
 
 : THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF TH! 
 
THE THEORY 
 
 OF 
 
 ELECTRICITY 
 
CAMBRIDGE UNIVERSITY PRESS 
 C. F. CLAY, MANAGER 
 
 LONDON 
 FETTER LANE, E.G. 4 
 
 EDINBURGH 
 
 100 PRINCES STREET 
 
 NEW YORK : G. P. PUTNAM'S SONS 
 BOMBAY, CALCUTTA, MADRAS: MACMILLAN AND CO., LTD. 
 
 TORONTO : J. M. DENT AND SONS, LTD. 
 TOKYO: THE MARUZEN-KABUSHIKI-KAISHA 
 
 All rights reserved 
 
THE THEORY 
 
 OF 
 
 ELECTRICITY 
 
 BY 
 G. H. LIVENS, M.A. 
 
 SOMETIME FELLOW OF JESUS COLLEGE, CAMBRIDGE 
 LECTURER IN MATHEMATICS IN THE UNIVERSITY OF SHEFFIELD 
 
 Cambridge : 
 
 at the University Press 
 1918 
 
 
THIS VOLUME IS IN GBATITUDE DEDICATED TO 
 
 THE MASTER AND FELLOWS OF JESUS COLLEGE 
 
 CAMBEIDGE 
 
CORRECTIONS AND ADDITIONS. 
 
 pp. 39, 158, 661. Until quite recently I have not had an opportunity of examining Einstein's 
 generalised theory of relativity and was in consequence unable to take up any other 
 point of view as regards gravit&tional phenomena. 
 
 p. 84, 88. The argument of the e^jponential function under the integral sign in the formulae 
 for if) should read ( - kz) not ( - kr). 
 
 p. 338. The values for the electronic charge givi on this page are those first obtained by the 
 method under discussion: subsequent improvements and corrections have increased 
 this value by about .Ml ",, to that '_ f iven on p. .'541 and in other parts of the work. 
 
 p. 345, 388. In the expression for the potential of a magnetic shell the solid angle is measured 
 by that area of the unit sphere round the field point which is such that if it were coated 
 with a double sheet with a similar <][*, osition to that on the given shell, then the direction 
 of tin- outwap nljus at ajr.\ point would correspond to the positive direction of 
 
 magnetisation in this 
 
 
CONTENTS 
 
 MATHEMATICAL INTRODUCTION 
 
 STATICS AND KINETICS 
 
 CHAP. 
 
 I. THE PRODUCTION AND DEFINITION OF THE ELECTROSTATIC FIELD . 
 
 II. THE CHARACTERISTIC PROPERTIES OF THE ELECTRIC FIELD 
 
 III. THE ELECTRICAL AND MECHANICAL RELATIONS OF A SYSTEM OF CONDUCTORS 
 
 IV. THE FARADAY-MAXWELL THEORY OF ELECTROSTATIC ACTION . 
 
 V. THE THEORY OF POLARISED MEDIA 
 
 VI. MAGNETO -STATICS . . 
 
 VII. ELECTRIC CURRENTS IN METALLIC CONDUCTORS 
 
 VIII. ELECTRIC CURRENTS IN LIQUID AND GASEOUS CONDUCTORS 
 
 DYNAMICS 
 
 IX. THE ELECTROMAGNETIC FIELD . . 3$|H 
 
 X. SOME SPECIAL ELECTROMAGNETIC FIELDS . 
 
 XI. ELECTRODYNAMICS OF LINEAR CURRENTS . ' . 
 
 XII. ELECTROMAGNETIC OSCILLATIONS AND WAVKS; . 
 
 XIII. REFLEXION, REFRACTION AND CONDUCTION OF ELECTRIC WAVES 
 
 XIV. GENERAL ELECTRODYNAMIC THJEORY 
 
 XV. THE ELECTRODYNAMICS OF MOVING MEDIA 
 
 APPENDIX: EXAMPLES .... 
 INDEX OF NAMES . ... 
 GENERAL INDEX . * . 
 
 PAGE 
 
 1 
 
 33 
 66 
 126 
 156 
 188 
 235 
 279 
 324 
 
 343 
 371 
 408 
 459 
 515 
 549 
 614 
 
Ls- 
 
 PHY. ., 
 UBisARY 
 
 PREFACE 
 
 nnHE following work is offered as a general text book on the mathematical 
 - aspects of modern electrical theory, and incidentally also as an attempt 
 to present the complete subject in a consistent form. There seems room 
 for a comprehensive work of this kind, for in the standard text books on 
 this subject the treatment, besides being incomplete, is often far from 
 convincing and at times not free from error. 
 
 The treatment is based mainly on the original Faraday-Maxwell theory, 
 generalised and extended to the case of moving systems by Sir Joseph 
 Larmor. This form of the theory has been almost entirely abandoned in 
 recent accounts of the subject, but it remains the only one which appears 
 to be completely satisfactory from the point of view of mathematical and 
 physical consistency, and in its generality it is unapproached by any other 
 form. 
 
 Although the present exposition is essentially a mathematical one, much 
 of the purely analytical mathematics usually associated with the subject 
 has been omitted. Particular attention has however been given to the 
 rigorous formulation of underlying physical principles and to their transla- 
 tion into a mathematical theory. The dynamical aspects of the subject 
 have been specially emphasised throughout. 
 
 In the development of the general plan and of the details of the book 
 I have derived great assistance from my notes of lectures delivered by 
 Sir Joseph Larmor at Cambridge during the academical year 1909-10, 
 afterwards supplemented from his various published works, more particularly 
 the papers, ' On a dynamical theory of the electric and luminif erous medium ' 
 [Phil, Trans. 1894-1897] and the book, Aether and Matter [Cambridge, 1900]. 
 I am extremely grateful to Professor Larmor for his kind permission to 
 make free use of these notes. 
 
 405544 
 
vi Preface 
 
 For assistance in the preparation of the work I am indebted also to the 
 various other standard works on this subject. In addition to the essential 
 Treatise on Electricity and Magnetism of Maxwell I -may mention especially : 
 Recent Researches on Electricity and Magnetism,, Oxford, 1893, and Element* 
 of the Mathematical Theory of Electricity and Magnetism, 3rd ed., Cambridge, 
 1904, by Sir J. J. Thomson; The Theory of Electricity and Magnetism, Cam- 
 bridge, 1907, by J. H. Jeans; Modern Electrical Theory, 1st ed., Cambridge, 
 1907, by N. R. Campbell; Electrical and Optical Wave Motion, Cambridge, 
 1915, by H. Bateman; The Electron Theory of Matter, Cambridge, 1915, by 
 0. W. Richardson ; Das elektromagnetische Feld, Leipzig, 190Q, by E. Cohn ; 
 Die Theorie der Elektrizitdt, 2nd ed., Leipzig, 1907, by M. Abraham and 
 A. Foppl; The Theory of Electrons, Leipzig, 1910, by*H. A. Lorentz; and 
 finally the various appropriate articles in the Encyclopaedia Britannica and 
 Die Encyklopadie der mathematischen Wissenschaften, Bd. v. 
 
 My friend Mr H. Spencer Jones, Chief Assistant at the Royal Observatory, 
 Greenwich, has laid me under the deepest obligation by his generous help 
 in the production of this book. At a time of great pressure in his own work 
 he kindly offered to read all the proofs and his criticisms and suggestions 
 thereon have been of the greatest service to me. 
 
 I have lastly to offer my thanks to the officials of the University Press 
 for their kindness and courtesy in all matters concerning the printing. 
 
 G. H. L. 
 
 SHEFFIELD, 
 
 July llth, 1917. 
 
MATHEMATICAL INTRODUCTION 
 
 1. Scalars and Vectors. We shall find it most convenient and certainly 
 conducive to greater mathematical precision to adopt the system of notation 
 and analysis which is now associated with the word ' vector-analysis *.' 
 Owing however to the fact that there is as yet no suitable work of reference 
 in this subject accessible to the average English student it was deemed 
 advisable to outline in this introduction the main propositions of this analysis 
 so far as it is needed for our future work. We may also take the opportunity 
 of developing in full certain complex mathematical formulae, the derivation 
 of which in the text would hinder the progress of the reader through the train 
 of physical reasoning to be there exposed. 
 
 In physics we distinguish between two kinds of quantities which we now 
 usually designate as scalars and vectors. A scalar quantity is completely 
 determined by the number expressing the ratio of its magnitude to that of 
 a definitely chosen unit. A vector, on the other hand, has in addition to a 
 numerical value, also a definite direction, so that vectors are distinguished 
 from one another not only by their numerical values but also by their direction ; 
 the numerical value is described as the magnitude of the vector. 
 
 Following the usual methods we shall always find it convenient to represent 
 graphically a vector by a straight line, whose length on a definite scale is equal 
 to the magnitude of the vector and whose direction and sense (considered as 
 drawn from one end, the origin, to the other) are the same as of the vector. 
 
 Throughout this book we shall always use thick letters of the Clarendon f 
 type, e.g. A, P, c, n, etc. to represent vector quantities, scalar quantities being 
 expressed by letters of the ordinary Latin or Greek types. The magnitude 
 of a vector may occasionally be denoted by the corresponding Latin letter, 
 e.g. A, P, c, n, etc., or in the usual notation of function theory by enclosing 
 the vector thus A ; v . We shall always represent a line or curve by s, 
 
 * First systematically expounded by Hamilton, Elements of Quaternions. See also Kelland 
 and Tait, Introduction to Quaternions; Tait, Elementary Treatise on Quaternions; E. B. Wilson. 
 Vector Analysis founded on the lectures of J. W. Gibbs. The treatment given in the text is on the 
 lines followed by W. von Ignatowsky, Die Vektoranalysis (Leipzig, 1910). A very illuminating 
 critical exposition is given by Burali-Forti and Marcolongo, Elements de calcul veclorielle (Paris, 
 1910). 
 
 t This is the notation employed by Heaviside, Electromagnetic Theory (London, 1891-93) 
 and subsequently by Gibbs, Lorentz and others. 
 
 L. 1 
 
2 Mathematical Introduction [1, 2 
 
 a surface by / and a volume by v, differential elements of these quantities 
 being respectively denoted by ds, df and dv; it is to be noticed that the 
 element ds of a curve may also be required with its direction, it will then 
 be denoted in the vector sense by ds. In the case of a surface we shall often 
 have to consider the normal to it ; this is denoted by n. It is always to be 
 drawn towards a definite side and we shall agree to draw it towards the 
 outside if we are dealing with a closed surface. 
 
 The geometrical space with which we deal will generally be referred to 
 a system of rectangular coordinate axes with the right-handed screw con- 
 ventions more usual in such expositions. The coordinates (x, y, z) of any 
 point of space are then the distances measured parallel to the perpendicular 
 axes from the respective coordinate planes. In certain special problems 
 however we shall find it convenient to adopt other types of coordinates. We 
 shall in fact use both the spherical polar and cylindrical polar coordinate 
 systems. In the former system the coordinates of a point are its distance 
 r from a fixed pole and the declination and azimuth angles 6 and </> of this 
 distance referred to fixed polar axis and azimuth plane through the pole. 
 In the cylindrical or columnar system the coordinates of a point are its 
 distances (z, r) from a fixed plane and from a fixed axis perpendicular to the 
 plane and the azimuth angle 6 of the latter distance. 
 
 2. Elementary vector operations, (i) Addition and subtraction of 
 vectors : suppose we are given two vectors A and B ; at the end of the line 
 representing A draw a line repre- 
 senting B. We then say that the 
 sum of the two vectors A and B is 
 the vector C which is represented by 
 the line joining the origin of A to the 
 end of B, and in this sense. We 
 express this by the equation 
 
 A + B = C. 
 
 Fig 1 
 The difference (A B) between two 
 
 vectors A and B is defined as the vector D which is such that 
 
 D + B = A. 
 This involves the conclusion that the sum of the vectors B and ( B) is zero 
 
 and also that 
 
 (A - B) = A + (- B) 
 
 since each of these is equal to the vector 
 
 (A - B) + B + (- B). 
 
 We have thus sufficient rules for the addition and subtraction of vectors ; we 
 see at once that the ordinary commutative and associative laws of algebra 
 are true for vector quantities as well as for scalar. 
 
2, 3] Mathematical Introduction 3 
 
 It is also quite clear that we may regard a vector as the sum of any 
 number of other vectors. In particular we can represent a vector A as the 
 sum of three vectors whose directions are not all parallel to one plane ; such 
 a method of resolution is of importance because the knowledge of three such 
 vectors is sufficient to completely determine the vector A : the magnitudes 
 of three such vectors are called the components of the vector A along the 
 three directions respectively. If we take the three directions mutually 
 perpendicular to one another and respectively parallel to the axes of a rect- 
 angular coordinate system, then the three components are denoted by A x , 
 A y , A. respectively. 
 
 All this will be quite familiar to the student who is acquainted with the 
 elementary ideas of statics and kinetics and it is not necessary to dwell at 
 any greater length on the significance of the definitions thus involved. 
 
 We now introduce the fundamental conception of the unit vector. The 
 unit vector in any direction is that vector whose magnitude is unity, so that 
 if we denote it by u, any other vector A is equivalent to the vector An ; 
 
 f A = Au. 
 
 If we use x, y, z for the unit vectors along the directions of the coordinate 
 axes, we get from the rule for the addition of vectors 
 
 A = A x x + A v y + A z z; 
 
 and similarly for any other vector B 
 
 B = B w x + B^y + B 2 z, 
 
 so that 
 
 A + B = (A, + B,) x + (A + BJ y + (A, + B z ) z 
 
 and generally the component of the geometric sum of any number of vectors 
 in any definite direction is equal to the algebraic sum of the components of 
 the separate vectors in this direction. 
 
 3. (ii) Multiplication of vectors; scalar products. From considerations of 
 the rules of the previous paragraph and of the fundamental concepts under- 
 lying the idea of a vector it is easy to deduce the general rule for the 
 multiplication of a vector A by a scalar quantity a, the resulting vector B 
 defined by 
 
 B = aA 
 
 being such that its direction is the same as that of A, but its magnitude 
 a times as large. Thus also 
 
 ah . A = a . &A = b . aA 
 and also 
 
 a (A + B) = aA + aB, 
 
 (a + 6) A = aA + 6A, 
 so that the ordinary rules of algebra are still applicable. 
 
 12 
 
4 Mathematical Introduction [4, 5 
 
 4. The scalar product. We must next define the scalar product of two 
 vectors A and B. This is defined as the scalar quantity whose value is equal 
 to the product of the magnitudes of the two given vectors multiplied by the 
 cosine of the angle included between their directions. We usually denote 
 this quantity by (A, B) so that 
 
 (A, B) - A . B cos AB, 
 
 where AB denotes the angle between the positive directions of the vectors 
 A and B. We deduce then that 
 
 (A, B) - (B, A), 
 (A + B, C) = (A, C) + (B, C), 
 (A, A) - A 2 = (A 2 ). 
 
 A 
 
 If A, B are at right angles, then cos AB = and thus 
 
 (A, B) - 0. 
 
 From this we deduce the important properties of the fundamental unit 
 vectors x, y, z, viz. (xy) _ (yz) = (o) = Q 
 
 and (x 2 ) = (y 2 ) = (z 2 ) = I. 
 
 Thus (A, B) = (A^x + A,y + A z z, B x x + E y y + B 2 z) 
 = A X B X + A y By + A 2 B.j, 
 
 expressing the vector products of two vectors in terms of their components. 
 
 5. (iii) The vector product. Suppose we are given a plane surface element 
 df : call one side of it the positive side and the other side the negative. On 
 the positive side draw a unit vector n x normal to the surface 
 
 through its mean centre. This surface element will be bounded 
 by a curve s and we choose the sense in this curve so that the 
 positive direction of its description corresponds to the direction 
 of the vector iij in the same way as translation to rotation in 
 a left-handed screw (Fig. 2). We can now specify this surface 
 element as a vector df whose magnitude is df and direction that 
 
 ofn *' thus tt=M/. 
 
 Such a vector is called an axial vector as it has associated with 
 
 Fig. 2 
 
 it a definite sense round its direction as axis. 
 
 Since the sum of any number of vectors is still a vector we may have 
 associated with any unclosed surface / a vector C defined by 
 
 C = I df, 
 
 -I, 
 
 the integral being extended over the whole of the surface/. In the particular 
 case in which the surface / is the parallelogram whose sides are the two 
 
Mathematical Introduction 
 
 vectors A and B, this third vector C is called the vector Drodnct of the two 
 vectors A and B. The implied direction about the axis of C is ?iven by the 
 order in which the vectors A and B are taken and in the standard case 
 exhibited in the figure and in which A is taken first 
 
 C - nAB sin AB, 
 
 the angle AB being that described in rotation round the axis C from the 
 plane (CA) to the plane (CB) in the positive direction. This is usually 
 
 expressed in the form 
 
 C - [A, B] . 
 
 f 
 
 We see at once that 
 
 and 
 
 And thus also 
 
 Fig. 3 
 
 [A, B]= -[B, A], [A, A] = 
 
 [A + B, D] = [A, D] + [B, D]. 
 
 [x, y] - z, [y, z] = x, [z, x] = y, 
 
 [x, x] - [y, y] = [z, z] - 0. 
 
 E y y + B 2 z] 
 - A X E Z ) -f z 
 
 Therefore also 
 
 [A, B] - [A x x + A^y + A z z, 
 
 = x (A,B S - A z E y ) + y 
 or in determinant form 
 
 [A, B] = x, y, z 
 
 Z J Ay , Aj. 
 
 Z , "By , Bj, 
 
 The components of the vector product are the minors of x, y, z and are 
 denoted by [A, B] x , etc. 
 
6 Mathematical Introduction [6, 7 
 
 6. (iv) Products of three vectors. There are several 'important results 
 involving three vectors which it would be as well to set out in detail here, 
 for future reference. 
 
 (a) The product of a vector and the scalar product of two other vectors ; 
 A (B, C). 
 
 Since (B, C) is a scalar quantity the product is a vector parallel to A 
 whose magnitude is equal to ABC cos BC. 
 
 (b) The scalar product of a vector and the vectorial product of two 
 other vectors; (A, [B, C]). 
 
 This is equal to 
 A x [B, CL + A, [B, C], + A z [B, C] z = A, (B,C, - B.C.) + .'.. + ... 
 
 A A A 
 
 **a;> *i/5 **z > 
 
 j ft ri ri 
 
 so that (A, [B, C]) - (B, [C, A]) - (C, [A, B]) 
 
 and each is equal to the volume of the pa allelepiped whose three edges are 
 A, B, C. 
 
 (c) The vector product of a vector and the vector product of two other 
 vectors ; [A, [B, C]] . Call this the vector D, then its cc-component is 
 
 D a = A, (B.C, - B,C) - A 2 (B.C. - B.C,) 
 
 = B K (A, C) - G x (A, B). 
 Thus D = B (C, A) - C (A, B). 
 
 7. Differentiation of scalars and vectors. In the general application 
 of the present analysis we shall have continually to deal with what are 
 called scalar and vector fields. That is instead of single isolated scalars and 
 vectors we have to discuss infinite groups of these quantities inasmuch as 
 each point of a finite or infinite space has associated with it certain scalar 
 and vector quantities the magnitudes, and in the latter case also the directions 
 of which in general vary continuously not only from point to point in the 
 field but also in time. In such cases the rates of variation of the magnitudes 
 of the scalar quantities and both the magnitudes and directions of the vector 
 quantities are matters of fundamental importance. We must therefore 
 establish rules for the differentiation of scalar and vector quantities with 
 respect to their scalar (time) or vector (space position) arguments. 
 
 The differentiation of a scalar quantity with respect to a scalar argument 
 follows the usual rules of the calculus and need not now be discussed. 
 
 The rules for the ! differentiation of a vector function with respect to a 
 scalar argument are easily established. Suppose that a vector A regarded 
 
7, 8] Mathematical Introduction 7 
 
 as a vector function of the scalar quantity t undergoes a small variation and 
 changes to the slightly different vector A' corresponding to the value t + 8t 
 of the argument t ; the vector 
 
 SA = A' - A 
 
 is the differential variation of the vector A with respect to the argument t ; 
 8A of course refers to a variation not only of the magnitude of A but also of 
 
 dA 
 its direction. The differential quotient -j~ of the vector A with respect to 
 
 the scalar quantity t is now defined as the vector represented by the limiting 
 
 value of the quotient 
 
 SA 
 
 8t ' 
 
 when 8t is made indefinitely small. The implied existence of a limit to this 
 quotient is involved in the definition. * 
 
 Our rules for vector operation are identical with those of ordinary algebraic 
 analysis so that it is evident that we may employ the elementary rules of 
 the differential calculus for the differentiation of vectors and functions of 
 vectors. Thus if 6 is any scalar function of the variable t 
 
 , 6A 7 dA . db 
 
 d "IT* - = b j7 + A j7 > 
 at dt at 
 
 dA dA dt 
 
 and also -T- = -,.- j- . 
 
 da dt da 
 
 We can similarly deduce that 
 
 and also 
 
 d FA BI 
 
 ^ [A - B] 
 
 and many other results are directly deducible. 
 
 8. We have next to discuss the differentiation of scalar and vector 
 quantities with respect to other vectors: the most frequent operation of 
 this nature and the only one with which we shall now deal involves the 
 differentiation of a scalar or vector quantity along a line element ds, the 
 result depending essentially on the direction of this element. 
 
 The rate of increase of any scalar quantity </> along the line element ds 
 is as usual 
 
 x d(f> dy d(f> dz 
 ds dx ds dy ds 'dz ds ' 
 
 and this may be regarded as the component along ds of the vector 
 
 dd> dd> d<b 
 
 A = x ~r + y a + z -f 
 ox oy oz 
 
8 Mathematical Introduction [8, 9 
 
 The components of the vector A in the three principal directions are 
 
 (A,, A,, A z ) 
 so that 
 
 A = Y/ + 
 
 V 
 
 and its direction is parallel to the direction 
 
 dcf> d(f> d(f> 
 
 dx ' dy ' dz' 
 We call this vector A the gradient of the scalar and express the relation in 
 
 the form 
 
 A = grad </>, 
 
 or A = V</>, 
 
 where V is the Hamiltonian vector operator 
 
 a a a 
 
 V = x^ + y^ + z^-. 
 ox dy oz 
 
 As we shall see later the great importance of this latter form of operator lies 
 in the fact that V may be treated by the ordinary rules of this analysis just 
 as if it were a vector with components 
 
 v -1 v -1 v-1 
 
 x dx' * ty' dz' 
 
 The gradient of a vector quantity B may also be expressed in the same 
 way but the result is rather different and is better approached in the indirect 
 way through the important analytical theorem of the next section. 
 
 9. Green's Lemma. Let A x be any function of the three rectangular 
 coordinates (x, y, z) which is continuous inside any finite space v. The region 
 v is bounded by the closed surface / which has at each point of it a definite 
 normal n (or if we wish to imply direction we use a vector n). Now consider 
 
 the integral 
 
 fdA x , fl[dA x , , , 
 -~^ dv= -s- 2 dxdydz 
 
 J dx JJJ ox 
 
 taken throughout the whole space v. If we assume that a line parallel to 
 the x-axis cuts the surface /in two points only, then integration with respect 
 to x gives 
 
 dydz [ yrr - dydz (A x " - A^}, 
 
 J x' OX 
 
 where A x ", A x ' are the values of the function A x at the points (x r , y, z), (x", y, z) 
 on the surface/. The surface element dydz in the Oyz plane is the common 
 projection of the surface elements df, df" which a small prism with its axis 
 parallel to the x-axis cuts on the surface. Since the normal n' forms an 
 obtuse angle with Ox and n" an acute angle, we have 
 dydz = - n lx 'df - + n lx "df" 
 
9] Mathematical Introduction 9 
 
 n/ and ii/' denoting respectively the unit vectors along the normals n' and 
 n". Thus 
 
 dydz I -~ dx = + A x 'n lx 'df + A x "n lx "df". 
 
 , If the surface / is more complicated so that a line parallel to the se-axis 
 cuts it in more than two points x', x", x'", ..., these points will be alternately 
 points of entry and exit from the space v. The number is even and the values 
 of n la /, n lC ", n la /", ... are alternately negative and positive and thus 
 
 dydz f X d ^ x dx = dydz (- A x ' + A x " - A x '" + ...) 
 
 ^-< I 7 f 
 
 flffi 
 
 .-1 
 
 -v 
 
 Fig. 4 
 
 If now we take the sum over all the elements dydz every element df of the 
 surface occurs once in the total sum and thus we get 
 
 O^jl -i / i / 
 
 the second integral being taken over the whole of the closed surface /. 
 Similar formulae can be obtained by changing x into y and z respectively : 
 combining them together we have Green's Lemma* 
 - dA v dA. 
 + dy + 3? 
 
 * 'An Essay on the Application of Mathematical Analysis to the Theory of Electricity and 
 Magnetism' (Nottingham, 1828) reprinted in Mathematical Papers (Paris, 1903). 
 
10 Mathematical Introduction [9-12 
 
 expressing a surface integral taken over a closed surface as a volume 
 integral taken through the volume enclosed. A y , A z are any other functions 
 of (x, y, 2). 
 
 10. Let us firstly suppose that 
 
 A X = Ay = A, = <f> 
 
 we then see that our theorem proves that 
 
 f V<f>dv = I n l( f>df, 
 
 >v 'f 
 
 thus if all the functions concerned are continuous we shall have in the limit 
 when the volume v is reduced indefinitely in every direction that 
 
 * V</, = Lt i I n^df, 
 
 v--0 v f 
 
 and this property can be used to define the operator V. 
 
 11. If as is usually the case in our theory A X) A y , A z are the components 
 of some vector A, the integral on the right can be expressed in the form 
 
 I A n df or I (A . nj) df, 
 
 where A n is used to denote the normal component of A at the surface 
 element df. 
 
 The expression forming the integrand on the left, viz. 
 
 dx dy dz 
 
 is called the divergence of A and is usually written in the form div A ; so 
 that Green's lemma is expressible in the form 
 
 / div A dv = I A n df. 
 
 -' v J f 
 
 It follows immediately that if div A is a continuous function of the space 
 coordinates (x, y, z) that 
 
 div A = Lt l I (Anj) df, 
 
 and again this property may be used to define the operator div A. More- 
 over if we now notice that the scalar quantity div A may be regarded as 
 obtained as the scalar product of the vector operator V and the vector A, 
 
 or that 
 
 div A = (VA) 
 
 then the analogy with the result obtained above for the definition of V is 
 obvious. 
 
 12. Suppose finally that we put 
 
 A x = 0, Ay = A z , A g = A^ 
 
12, 13] Mathematical Introduction 11 
 
 then we see that 
 
 Generalising this result we see that if B is the vector whose components are 
 
 8A Z dA y dA x dAg dA y dA x 
 
 dy dz ' dz dx ' dx dy ' 
 
 this vector B so derived from the vector A is called the curl of the vector A 
 
 and is written 
 
 Curl A. 
 
 On the basis, of the ideas sketched above in the two former cases we see that 
 this vector Curl A may be defined as the limiting value of the function 
 
 as the volume v is diminished indefinitely. 
 It is important to notice that 
 
 Curl A - [VA] 
 where the vector product on the right is formed in the usual way. 
 
 The real significance of these derived scalar and vector quantities will 
 appear in the physical theories to be expounded in the sequel ; their analytical 
 import is to some extent elucidated in the general theorems developed 
 immediately below. Before however concluding the present discussion we 
 must present another aspect of the present definitions. 
 
 13. The fundamental differential operations just defined have a meaning 
 only when the field is referred to an ordinary frame of reference. It is however 
 sometimes more convenient to use other types of coordinates and we must 
 therefore extend our definitions to apply to these cases. Let us first consider 
 the case where the field is defined by a spherical polar coordinate system; 
 there are then three principal perpendicular directions at any point (r Q , 6 Q , </> ) 
 in the field to which it is convenient to refer any vector associated with the 
 point in the ordinary way. These directions are the directions of the radius 
 vector and of the tangents to the meridian and latitude circles defined by the 
 intersection of the sphere r = r Q with the cone 6 = 6 and the plane <f> = </> 
 respectively. 
 
 The differential elements of length in the principal directions just defined 
 are easily seen to be 
 
 dr, rdd, r sin 
 
 respectively. Thus if we place the cartesian axes parallel to the polar co- 
 ordinates at the point we see at once that the vector operator V has com- 
 ponents 
 
 9 1 3_ 1 c 
 
 dr' rW rsin08</>' 
 
12 Mathematical Introduction [13, 14 
 
 parallel to the three principal polar directions. The vector V<I> has com- 
 ponents 
 
 80 1 8O 1 8O 
 dr ' "r 30 ' rsm0 dtf> ' 
 in the same directions. 
 
 Again if a vector A has components A r , A 6 , A^ parallel to the respective 
 polar directions and if df is an element of surface normal to the unit vector 
 u lf then 
 
 (Ajij) df = A x dydz + A y dzdx + A z dxdy 
 
 = A r rd6 . r sin d<f> + A e r sin dd<f> . dr + A^dr . rdd 
 
 and thus if we now treat (r, 6, </>) as analytical coordinates in an ordinary 
 cartesian space we can apply Green's lemma in its simple form and then find 
 that 
 
 (An x ) df = [A r r 2 sin BdddJ) + A e r sin 6drd<f> + A^rdrdO] 
 -f 
 
 (r 2 sin 0A r ) + ^(r sin 6A g ) + ~ 
 
 We can now deduce a definition of the operator div in these coordinates by 
 an application of the form derived above. In fact, if we make the surface/ 
 diminish indefinitely in all directions we derive at once, assuming continuity 
 
 Lt ^An,) df= (r- sin 6A r} + (r sin 6A e} + 
 
 and thus 
 
 div A EE . =- (r 2 sin 0A r ) + ( r sin 0A ) + = 
 r 2 sm [8r 80 8 
 
 8< 
 
 The operator cwrZ may be similarly interpreted but it is most convenient to 
 derive it from the analytical theorem of the next section. 
 
 14. If the coordinate frame of reference had been that suited to columnar 
 fields the results could be equally readily obtained. The elements of length 
 in the three principal directions at any point in the field are 
 
 dr, r dd, dz 
 and the operator V has components 
 
 8 1 8_ d_ 
 dr' rd9' dz' 
 
 and the operator div is such that 
 
 div A = - \j- (rA r ) + I, (A,) + A (rA z )] , 
 r lor 80 dz 
 
 A r , A e , A z denoting the principal components of A at the point under con- 
 sideration. 
 
15, 16] Mathematical Introduction 13 
 
 15. Miscellaneous Rules for Vector Calculations. Before concluding this 
 preliminary account of the analytical foundations of the formulae we will 
 collect a few miscellaneous rules for the calculation and transformation of 
 certain vector functions which will be of subsequent use. 
 
 (i) di 
 
 = (C, Curl B) - (B, Curl C), 
 and thus 
 
 | [B, C] n df=[ (C, Curl B) dv - I (B, Curl C) dv, 
 f > 
 
 the former integral being taken over any closed surface and the two latter 
 over the interior space. 
 
 (ii) Curl aA = a Curl A + [Va, A], 
 
 a being any scalar function. 
 
 If b is another scalar function and 
 
 A= V6, 
 
 then Curl A = Curl (V6) - 0, 
 
 so that Curl aV6 - [ Va, V6] . 
 
 (iii) We often waht to use the operation 
 
 (A, V) 
 
 on other scalar and vector functions, (A, V) <f> denotes in general the rate of 
 increase of </> per unit length in the direction of the vector A. We may 
 even use the operator on a second vector B and the interpretation of the 
 result is similar; thus 
 
 (A V^ B - A 8B * 4- A dEx 4- A aB * 
 ') B *- A * dx +Av ^dy +Az ~W 
 
 (A, VVB V = A X +... + ..., 
 
 (A, V) B z = A, * +... + ..., 
 
 and many results can be deduced in this way. We need not however enter 
 into this subject any farther at present. 
 
 16. Stokes' Theorem. The definition of the differential operator curl 
 obtained above hardly emphasises the most important property of this 
 operator. This is however easily obtained from a theorem given by Stokes* 
 and which expresses the line integral of a vector A taken along a closed curve 
 by a surface integ al taken over any surface bounded by the curve, viz. 
 
 - (t - ) + "- ( - ) + * 
 
 the direction of the normal n a in the second integral being generally related 
 
 * Smith's Prize Examination (1854). Math, and Phys. Papers, v, p. 320. 
 
14 
 
 Mathematical Introduction 
 
 [16 
 
 to the sense in which the first integral is taken round the circuit in the same 
 manner as translation to rotation in a left-handed screw relation. 
 
 In order to prove this theorem it is first necessary to notice that it will 
 be merely sufficient to prove it for a small elementary plane circuit ; because 
 if we consider any finite barrier surface 
 divided up into a large number of such small 
 surfaces by means of a network of lines, then 
 the total sum of the line integrals taken for 
 each little elementary circuit separately is 
 equivalent to the integral taken round the 
 outer circuit alone, any interior element of a 
 circuit being counted on the whole twice over Fig. 5 
 
 with opposite signs in each case. 
 
 We shall therefore content ourselves with proving the theorem for a small 
 plane area. 
 
 Fig. 6 
 
 Consider the integral 
 
 taken round the boundary of such an element. If A,,, is the value of this 
 quantity at the mean centre of the element, the value at a point on the 
 boundary whose small coordinates relative to this centre are (x 1 ', y', z') is 
 
 . 9A , 9A,, . 9A- 
 
Mathematical Introduction 15 
 
 which is practically 
 
 = ^f y ^' + d*?f 2 '<fo.' f 
 dy . " oz 
 
 but taking account of the signs 
 
 | y'dx' = - n lz df, I z'dx' = n ly df, 
 
 so that 
 
 / dA x dA z \ ,. 
 A x dx = (TL IV -=-' - ~ n l2 df. 
 s V oz ox j 
 
 Similar expressions are obtained for the other parts of the line integral 
 
 f 
 
 A. y dy and A. z dz, 
 
 and adding them together we have 
 
 f 
 
 and summation over the whole area establishes the theorem for any circuit 
 with the corresponding barrier surface. 
 
 We usuallv write this relation in the form 
 
 (Ads) - [ B n df, 
 ./ 
 
 where B w is the normal component of the vector B = curl A so that Stokes' 
 theorem expresses that 
 
 I A s ds = \ curl n Adf, 
 s f 
 
 where A s denotes the component of A along the directio^n of ds and curl n A 
 the normal component of the vector curl A. 
 
 17. We shall find it frequently necessary to quote this theorem of Stokes 
 in spherical polar coordinates instead of the ordinary cartesian coordinates as 
 above. If the axis Ox is taken as polar axis with the origin still the same, 
 and 6 denotes the angular distance of the radius vector r from this axis and 
 (f) azimuth round it measured from the plane Ozx, the transformation may 
 be effected by the substitution 
 
 x = r cos 6, y = r sin 6 cos </>, z = r sin 6 sin </>, 
 
 with the appropriate transformation for the other vector components. It is 
 however more easily obtained if we notice that the orthogonal elements of 
 length in polar coordinates are 
 
 dr, rdd, r sin Od<f>, 
 
16 Mathematical Introduction [17, 18 
 
 for then 
 
 I (Ads) = I (A x dx + Aydy + A,dz) 
 
 S 8 
 
 = I (A r dr + A e rd8 + A^r sin Qdfy. 
 s 
 
 where A r , A H . A<t, are used to denote the components of the vector A in the 
 directions of the respective small displacements. Moreover 
 
 E n df = (Ba, dy dz + B y dz dx + B 2 dx dy) 
 f f 
 
 = I (B r r 2 sin 6d6d(f> + E e r sin 0drd<f> + E^rdrdO), 
 
 and if therefore we regard (r, 9, <f>) as coordinates of the (x, y, z) type we may 
 apply the analytical transformation of Stokes and we thus get 
 
 O o 
 
 r z sin 6E r = - ^ (rA g ) + ^(r sin 0A+), 
 
 O p 
 
 r sin 6E = - (r sin ^A^,) + ^ (A,.), 
 or 0(p 
 
 o o 
 
 rE*= - g (A r ) + ^ (rA), 
 
 and the transformation is thus effected : the vector B is completely denned 
 as regards its components in the three principal spherical directions. 
 
 18. Green's Lemma and Stokes' Theorems for moving circuits*. In 
 
 our treatment of the electrodynamic phenomena in moving systems we 
 shall find it convenient to be able to apply the general integral theorems of 
 Green and Stokes to non-stationary surfaces or curves. The results for such 
 cases are easily obtained. 
 
 (1) The time rate of variation of the line integral 
 
 -2 
 
 A s ds 
 i 
 
 taken along any unclosed curve between the points 1 and 2 is 
 
 I ( ^ + V (u, A) - [u . curl A]) ds, 
 i\at J s 
 
 when the points of the line are in motion so that the velocity of the ' s ' point 
 on it is u. 
 
 The time variation of the integral consists of two parts the first of which 
 would exist if the curve remained at rest and this amounts to 
 
 and the second part, resulting from the motion of the curve, which may be 
 * Cf. Abraham u. Foppl, Theorie der Elektr.izitat, Bd. i. 33, 34. 
 
18, 19] Mathematical Introduction 17 
 
 calculated as if the field itself were stationary. We denote the position of 
 the curve at the time t + dt by its end points (!', 2') : the variation of the 
 line integral during dt so far as it arises from the motion of the curve is then 
 
 12' '2 /* 2' r 1 
 
 (Ads) - I (Ads) = (Ads) + (Ads). 
 r .' i J r J 2 
 
 Now consider the integrals of the tangential component of the vector A along 
 the two small elements of curves (1, 1') and (2, 2') described by the end points 
 of the given curve during dt : these two integrals are 
 1' r2 
 
 (Ada) = (Au)i dt, (Ads) = - (Au) 2 dt, 
 
 1 Jy 
 
 and when they are added to the above two integrals we get a single integral 
 
 (Ads), 
 
 taken round the closed curve (1, 2, 2', 1', 1) : this integral transforms by 
 Stokes' theorem as above into the integral of the normal component of curl A 
 over a surface bounded by this curve. If we take the surface to be that 
 actually described by the elements of the line in its motion it will consist 
 
 simply of elements 
 
 [nds] dt, 
 
 described by the separate elements of the curve. Each element of this type 
 thus contributes a part 
 
 dt (curl A, [uofe]) = dt (ds [curl A, u]) 
 to the integral considered. We thus find that 
 
 (Ads) + (Ada) + (Au)i dt - (Au) 2 dt = dt\ (ds [curl A, u]), 
 
 1' .'2 J 1 
 
 and if we write 
 
 (Au)i (Au) 2 = (dsV) (Au), 
 
 we get 
 
 f (Ads) - i (Ads) = dt (ds, V (u . A) - [u . curl A]), 
 .' i' .' i .1 
 
 and this is the part of the variation of the line integral which arises from the 
 motion of the curve. The result quoted is obtained by adding this part to 
 the former one due to the variation of the field. 
 
 19. (2) The time rate of variation of the surface integral 
 
 taken over any part of a surface in motion as above is 
 
 - + curl [B, u] + u div B ) . 
 / 
 
18 Mathematical Introduction [19, 20 
 
 As before the total variation consists of the part 
 
 due to the variation of the field and which can be calculated as if the surface 
 were at rest, and a part which arises entirely from the motion of the surface 
 and in the calculation of which the time variation of the field itself can be 
 neglected. We denote the position of the surface at time (t + dt) by/', that 
 at the time t being/. The variation of the integral on account of the motion 
 is then 
 
 I E n df-JE n df. 
 
 Now close up the two unclosed surfaces / and /' by adding the small 
 band of surface described by the bounding curve during the motion between 
 the times t and t + dt. In this motion each element ds of the curve describes 
 a small area [ds, u] dt so that we have by Green's theorem 
 
 f E n df- I" E n df + dt!(E. [ds, u]) = f divBdv, 
 
 Jf Jf J J 
 
 where dv is the element of the space enclosed by the surfaces / and /' and 
 the bounding strip : since this volume element is described by the element 
 df during the time dt we must have 
 
 dv = dtdfu n . 
 If we also write (R ^ u]) _ _ ((fe [fi u]) ^ 
 
 we find that 
 
 f E n df- I B n df=dt\tdfu n tivB+l(ds, [B,n])j. 
 
 Jf Jf ( J ) 
 
 The last integral on the right can be transformed by Stokes' theorem and we 
 get 
 
 f / E n df - t E n df = dt I df (u div B + curl [B, u]), , 
 
 j f J f J 
 
 whence by addition the result is obtained as stated. 
 
 20. Green's Theorem. The derivation of the characteristic analytical 
 properties of electrostatic fields is facilitated by the use of an important 
 theorem due to Green. 
 
 Let (f>, iff be any two scalar functions. From them we can deduce a 
 
 vector A of the form 
 
 A = cV/r - 0V< 
 
 and thus div A - </>V V - 0V 2 </>, 
 
 where, as always, we understand by V 2 the differential operator 
 
 52 ^ 92 ^ 
 
 a^ 2 + 8^ 2 + 8z 2 '' 
 
 it is to be noticed that it is precisely the square of the Hamiltonian operator 
 treated according to the ordinary rules. 
 
20, 21] Mathematical Introduction 19 
 
 Now suppose that the functions </> and / are continuous and have contin- 
 uous derivatives inside the space v enclosed by the surface /. A simple 
 application of Green's lemma then gives 
 
 where we use ~ as the component of the gradient of <f> along the outward 
 
 normal at the element df of the surface. This is Green's Theorem. 
 
 The more important forms of this theorem are however obtained by 
 adopting a special form for one of the functions. 
 
 If P is a variable point in the region v with coordinates (x, y, z) and P l 
 any fixed point with coordinates (x 1} y 1} z t ), then if we use r^ for the distance 
 
 j we have 
 
 r 2 = (x - xtf + (y- ytf + (z - z 
 and consequently 
 
 
 
 dx 2 (r) r 3 "' r 5 ' 
 
 d 2 /1\ d 2 /1\ 
 
 and similarlv for TT-T I - 1 and x ( - I . We therefore see that 
 dy z \rj dz 2 \rj 
 
 V 2 (-} = 0. 
 If now the point P x lies outside the region v we can put 
 
 and the above theorem takes the form 
 
 [ V 2 rA I $tk(- 
 
 j v r f ( dn \r 
 
 21 . If however Pj lies inside the region v then - regarded as a function of 
 the position of P will be infinite at P 1 , and if we wish to use our formula with 
 ifj = - we must exclude P 1 by putting a small surface round it. We shall 
 
 do this by taking a small sphere of radius r round the point P 1 as centre. 
 The space between this and the surface/ we call v'. We can now apply our 
 theorem to this region v', so that 
 
 v -' 
 
 where/' includes, in addition to the surface/, also the surface of the small 
 sphere. 
 
 22 
 
20 Mathematical Introduction [21, 22 
 
 If now we denote by dot the element of solid angle, the element of spherical 
 surface of radius r is r 2 da> and the element of volume is r z drda). It is also 
 to be noticed that on the surface of the small sphere the normal n coincides 
 with the direction of r (but is in the opposite sense) and so 
 
 O I I " 9 5 
 
 dn\rj rj 2 
 and therefore the part of the surface integral due to the surface of the sphere is 
 
 fj.j Wj 
 
 (bdw r-, I -~- da), 
 
 J Y J dr 
 
 where the integrations are over the surface of the unit sphere. If at the 
 point P l the function (/> and its differential coefficients are continuous, then 
 
 is finite and thus if we make the sphere infinitely small the second integral 
 
 or 
 
 ( > j 
 -~ da> 
 
 or 
 
 tends to zero, and if the value of (f> has the value </>! at P x the first integral 
 tends to 
 
 In the volume integral the part due to the inside of the sphere is excluded : 
 but this is 
 
 I | V 2 <f>rdrda>, 
 
 and obviously vanishes with r x if V 2 </> is finite. We have thus in all 
 
 I" U2J. dv " fld<f> ,8 fl\\ , 
 4-770! = - V 2 (f> --- h i - ~r 9 ^~ I r dj , 
 J v r . f (ran r dn\rj) ' 
 
 where the integral with respect to v is now over the whole region inside / 
 and the surface integral is over / only ; all traces of the cavity drawn about 
 the point P x have disappeared. 
 
 22. The analysis so far is limited to the case in which the functions in- 
 volved are continuous over the whole region v. We can however immediately 
 extend it to include the most important cases involving discontinuity. 
 Supposing that (f> and its first differentia] coefficients are discontinuous over 
 the -surf ace/' lying in this region, otherwise having determinate continuous 
 values throughout the region. Draw a normal n at each point of the surface 
 /' and regard directions in it as positive when in some definite chosen sense. 
 The side of the surface/' on the side of increasing n we call the positive side 
 and the other the negative side and we distinguish the values of functions 
 on the two sides by suffices -f- and , e.g. <t> + and <_. 
 
 We can then apply our previous formula if we include in the boundary 
 of v the two sides of the surface/' as well as the surface/. On the positive 
 
22-24] Mathematical Introduction 21 
 
 side of /' dn is negative and on the negative side it is positive. The point 
 P T is assumed not to lie on the surface/'. We thus get 
 
 a formula which will hold even if/' consists of several separate surfaces. 
 
 This is the general result. In applications however one often has to 
 apply it to indefinitely extended fields from which certain finite spaces are 
 exclujjfcd. In such cases a detailed discussion of the behaviour of the infinite 
 integrals becomes necessary and each case must be treated on its merits. 
 The following general result is however easily deduced if the theorem is applied 
 as though the field were bounded by a very large enclosing surface which is 
 ultimately extended indefinitely in all directions, and it provides a sufficient 
 criterion in most cases. 
 
 23. Suppose the function (f), now given throughout all space, is such that 
 
 Lt = 
 
 r-*-oo 
 3jr 
 
 and Lt r 2 x- is finite, 
 
 on 
 
 then the part of the surface integral corresponding to the infinite boundary 
 will be zero in the limit and the formula can be written as 
 
 f T7.Ji dt> - 
 
 4:77-01 = V 2 rf> - 
 
 J r 
 
 |~/d<\ fd(f)\ ~\df f - . 9 /1\ 
 
 (a - ( a ) , *-. < / ) + - ^-) a~ I J 
 
 f[\dnj + \8n/_J r Jy T on\rj 
 
 where the volume integral is extended over the whole of space outside certain 
 specified regions, the first surface integral over the boundaries of these regions 
 and the second over all discontinuity surfaces in the region investigated. 
 
 A function cf> limited by the usual conditions of continuity as well as the 
 above conditions at infinity will be said to be regular in the space investigated. 
 
 This genera] result shows that a function ^ which fulfils the conditions 
 and which apart from the surfaces /' is with its derivatives continuous in 
 the whole of space, is uniquely determined in the whole region if the values 
 of V 2 (/> are given at each point and also the discontinuities in (f> and its normal 
 gradient on all the surfaces/'. 
 
 The continuity of V 2 </> is not involved. 
 
 24. If we write _ 9 . 
 
 V 2 </> = 
 
22 Mathematical Introduction [24 
 
 and </> + </>_ = 47TT, 
 
 then our formula shows that 
 
 J r .! r r .!/ 0n 
 where we now consider the whole of space without any excluded regions. 
 
 If we consider the values of p, a and T to be those as denned above, then 
 this formula is merely the expression of an identity. It contains an expres- 
 sion by definite integrals of a general function </> subject merely to the specified 
 continuity conditions. 
 
 If however we regard the question from the other point of view and 
 consider the quantities p, a and T as given a priori, then we want to know 
 whether the function c/> defined in the same way satisfies the same conditions. 
 It is easily proved that it does. 
 
 We define </> at any point P*by the relation 
 
 fj* * + r a/iw, 
 
 r r dn \rj J 
 
 the first integral being taken over the whole of space and the second and third 
 over those surfaces on which a and T have finite values. 
 
 If the point P is at an external point, i.e. at a point in space in the 
 immediate neighbourhood of which p = 0, then we can differentiate each of 
 these integrals with respect to the coordinates of P under the sign of 
 integration. We get 
 
 - / PV* (1) A, + j ^ (1) if + / r V* (i) 4T , 
 
 and since V 2 ( ) = we have 
 
 \rj 
 
 Vty = 0. 
 
 If however P is at any other point where the value of p is not zero we must 
 proceed in a different manner. Notice that it is only possible for discon- 
 tinuities in </> or its derivatives to occur near one of the surfaces /' ; we may 
 therefore from our previous theorem write 
 
 and thus on elimination of (j> we get 
 
 !\V72JL , \ dv nW\ fi<f>\ 
 (V 2 </> + 477/3) - Mis 2 -) - U^ 
 
 r .1 (\dnj + \dnj_ 
 
24, 25] Mathematical Introduction 23 
 
 Thus we see that, on account of the arbitrariness of the position of the 
 
 point P, we must have * 
 
 V 2 (/> + 4rrp = 0, 
 
 $L 
 
 and </>+ (/>_ 4-7TT = 0, 
 
 which are precisely the same as the previous conditions. 
 
 25. Kirchhoff's Theorem. In the previous section we obtained a solution 
 of the fundamental equation 
 
 V 2 </> = - 4777), 
 
 subject to certain continuity conditions in the form of a volume integral 
 throughout the whole of space together with appropriate surface integrals 
 over the surfaces of discontinuity. We now proceed to the more general case 
 of the fundamental wave equation 
 
 A general method of solution given by KirchhofEf in a paper on the theory 
 of rays of light is based on Green's theorem and on the proposition that if 
 r is the distance from a fixed point, and F an arbitrary function, the expression 
 
 has the property expressed by 
 
 V2 !_ o 2 x 
 x ~ c 2 dt 2 
 
 This follows at once from the formula 
 
 r dr r d 
 
 which is true for any function of r, not explicitly containing the coordinates, 
 and in virtue of which (i) assumes the form 
 
 1 V (r x ) 
 
 dr 2 c z dt* 
 It is well known that 
 
 c and r x- 
 
 are solutions of this equation. 
 
 * At any point of the field we can choose an infinite number of variations of the position of 
 P such that along them any two of the integrals together are constant. 
 
 f Berlin. Ber. (1882), p. 641; Wied. Ann. xvra. (1883); Ges. Abh. n. p. 22. The present 
 proof is given by Lorentz, The Theory of Electrons, p. 233. 
 
24 Mathematical Introduction [26, 27 
 
 26. Let now / be the bounding surface of a space v throughout which 
 i/ is subjected to the equation (i), P the point of v for which we want to 
 determine the function, dv an element of volume situated at the distance r 
 from P, f a small spherical surface having P' as centre, n and ri the normals 
 to / and/', both drawn towards the outside*. 
 
 Introducing the auxiliary expression 
 
 where F is a function to be specified later on, we shall consider the integral 
 
 extended to the space between /and/'. 
 
 In the first place we have by Green's theorem 
 
 and in the second place on account of the equations satisfied by /r and 
 
 Hence, combining the two results 
 
 This equation must hold for all values of t. After being multiplied by dt, 
 it may therefore be integrated between arbitrary limits ^ and t 2 , giving 
 
 From this equation we can derive the solution of our problem by means of 
 a proper choice of the function F, which has thus far been left indeterminate. 
 
 
 
 27. We shall now suppose that F (e) differs from zero only for values of 
 lying between and a certain positive quantity S, this latter being so small 
 that we may neglect the change which any of the other quantities occurring 
 
 * We need not here attend to all the saving restrictions that would be necessary in formal 
 pure mathematics : such limitations are, in the main, sufficiently obvious, and are satisfied by 
 the nature of the case, in continuous physical analysis. 
 
27] Mathematical Introduction 25 
 
 in the problem undergoes during an interval of time equal to e. As to the 
 function F itself we shall suppose its values between e = and e = d to be 
 so great that 
 
 f F (e) dt = 1. 
 3 
 
 Since for a fixed value of r 
 
 - 
 F(e)dc, 
 
 t +- 
 
 C 
 
 t+ _ 
 '' c 
 
 r , 
 t= _ r -' 
 
 it is clear that on the above assumptions 
 
 =l and 
 provided t l and t 2 are such values of t that 
 
 L + < and (. + -) > 8, 
 
 c V c/ 
 
 and : is one of the functions of with which we are concerned. This latter 
 formula enables us to select as it were the values of tfi corresponding to definite 
 moments. 
 
 Let ^ have a fixed positive value and ^ a negative one, so great that even 
 
 T 
 
 for the points of / most distant from P, t^ + - < 0. Then all values of x 
 occur in the last term of (2). Indeed 
 
 and this vanishes for t = ^ and t = t 2 . because F'(e), like F(e) itself vanishes 
 for all values of e outside the interval (0, 8). The last term on the right-hand 
 side of (2) is thus seen to be zero. 
 
 The term involving p may be written 
 
 n f^ / r\ 
 
 iTT \-dv pF (t+ -}dt 
 
 Jr J tt \ cJ 
 
 and this is = 4?r - p r dv. 
 
 J r r t= c 
 
 Again f '' it l x f d f=l'dtt l -F(t + r -} d / if 
 
 it, J/dn *' j tl Jf? \ cJ dn J 
 
26 Mathematical Introduction [28, 29 
 
 9y 
 28. . We have next to consider the integral containing ~. This 
 
 differential coefficient being equal to 
 
 we have dt * 4T-"' * E ^ ' + 
 
 fc J/ 8w J, Jdn\rj r \ c 
 
 r 
 
 - 
 
 c 
 
 The first integral is 
 
 and the second expression may be integrated by parts 
 
 -E* 
 
 r on J 
 
 [1 3r /d^N T/- 
 }rWn\Tt)t=- r - J> 
 
 c 
 
 (T\ I T\ 
 
 ti + ~ ) an d & ( ^2 + ~ ) vanish. Combining these results we 
 c J \ c / 
 
 find for the right-hand side of (2) 
 
 ri ri f/a<A a /IN , i ar /a0\ ) , 
 
 T I r p . <W + I - { ( sr- r~o~ " J # r~i ^~l^ rf^/- 
 
 J r S = _l Jr Ri/* t dn\rJ T t=-- or tin \dtJt i] J 
 
 /* x f*. (* f* ' 
 
 29. We now suppose the radius R of the sphere/' to diminish indefinitely. 
 By this process the above terms are unchanged but we must now calculate 
 the limiting value of the terms on the left-hand side of (2). As the integral 
 over the sphere has the same form as that over the surface/ just considered, 
 we may write 
 
 - 
 
 t =- r - cr dn' 
 c c c 
 
 or since the normal n' has the direction of r and since at the sphere r = R 
 this is 
 
 [J 1 /^ l i l ( 
 
 )( t= -K + fctteJL + CR ( 
 
29, 30] Mathematical Introduction 27 
 
 Now when R tends towards 0, the integrals with ^ vanish, so that the expres- 
 sion reduces to 
 
 which in the limit becomes 
 
 t=o) being the value of ifj at P for the instant t = 0. We thus have 
 
 i fa /a/r\ a /i\ , i dr /a<A\ ) , . n 
 
 which determines the value of at the chosen point P for the instant t = 0. 
 We are, however, free in the choice of this instant, and therefore the formula 
 may serve to calculate the value of ijjp for any instant t ; for this we have 
 
 ~) I O / 
 
 only to replace the values of i/r, ~ and ~- on the right-hand side by those 
 
 T 
 
 relating to the time t . 
 
 30. On the convergence and differentiability of potential integrals*. The 
 
 analyses of the preceding paragraphs and certain considerations which subse- 
 quently present themselves in our physical discussions necessitate a rather 
 close investigation of the validity of the usual processes of the calculus as 
 applied to certain integrals of the type f 
 
 >dv r. , a /i\ r , a 2 
 
 , A! = odv~- - , X 2 = \pdv~ 
 r r dx^ \rj J r ox-? 
 
 the notation being the same as before ; p denoting any function of (x, y, z) ; 
 dv being the element of volume dxdydz and r the distance of the point (x, y, z) 
 defining the position of this element from the fixed point (x 1} y lf 2j). 
 
 When the point (x lt y l} %) is at a finite distance from all the points at 
 which p is not zero there is no question as to convergence but it is necessary 
 to prove that 
 
 d T D V 
 
 F Y x 
 
 We consider the general integral 
 
 / = 
 
 the function / (xj) involving also the parameters (x, y, z) of integration as 
 well as y 1 and z l . We have then 
 
 | = Lim . I I P f(x 1 + Ascj) dv - (p/fa) dv\ . 
 
 P*l Aid -* "*J ( J J 
 
 * The considerations of the present paragraph are derived mainly from Poincare's treatise 
 Thdorie du potentiel newtonien (Paris, 1899), to which the reader is referred for a more detailed 
 exposition of many points which have been slurred in the present discussion. 
 
 f The types here chosen are those that specifically occur in the physical discussions. Other- 
 wise they are probably of little or no import. 
 
28 Mathematical Introduction [30, 31 
 
 If the second differential of / (x-^) exists at and near the point (x t , y t , z x ) we 
 may write this in the form 
 
 dd> [ fdf , Az 3 2 /' 
 = Lim \ p M- -J- *. 
 
 [ /df Ax 8 2 /\ , 
 
 \p M- + -ir 5I ^ v ' 
 J \9zi 2 aajjV 
 
 where however we write 0% + flAxi (0 < < 1) in jr^- 1 ,, after the differentiation 
 
 fy'T* 
 
 has been carried out. 
 
 We may proceed at once to the limit when Aa^ = and get 
 
 dx : 
 
 9 2 f 
 provided only that the function ~\ is limited at all points in the 
 
 hood of the point (x 1} y l} z x ) and for every combination of values (x, y, z) 
 for which the function p is different from zero. 
 Now the function 
 
 f- - ( l 
 
 J 3~. s I 
 
 satisfies the condition stated if (aj l9 y ly z^) is outside the region of integration 
 so that we have 
 
 Y dl v dX-L 
 
 Zl = 3xV ^ %* 
 
 When however the point (x 1} y 1} z-J coincides with one point of the field 
 of integration the integrands become infinite and the question at once arises 
 as to whether the expressions have any meaning at all. We can approach 
 the matter by the consideration of certain simple cases. 
 
 31 . Let us consider the definite integral 
 
 1 = 
 
 b 
 
 f (x) dx. (a < b) 
 
 a 
 
 If the function/ becomes infinite for x a the integral has in itself no meaning: 
 to give it one we consider the integral 
 
 /.= (* f(x)dx, 
 .'*+ 
 
 which is quite definite. If then I f tends to a limit / when e tends to zero 
 the integral above is said to be convergent and its value is represented by /. 
 If on the contrary I f increases indefinitely as e diminishes to zero the integral 
 is said to be divergent and the symbol has no meaning. 
 
 E.g. if f(x)\< , r- where s < 1 the integral is convergent. 
 ( c a } 
 
 Next consider the double integral 
 
 </ (*, y) df 
 
31, 32] 
 
 Math ematical Introduction 
 
 29 
 
 extended over the area / enclosed by a given curve S in the (x, y) plane. If 
 </> (x, y) becomes infinite at the point (x 1} y-^ in the area, the integral has no 
 meaning : to give it one surround the point by a small closed curve S and 
 let / be the area between the two curves S and S . The integral 
 
 7 =[ <{>df 
 
 fa 
 
 has then a definite meaning, but its value changes as the curve S alters. Now 
 suppose that / has a limit / when the curve S Q diminishes in all directions to 
 the point : we then say that the integral is convergent and its value is I. 
 If on the other hand 7 increases indefinitely as this curve is diminished the 
 integral is divergent. 
 
 32. 
 
 Fig. 7 
 
 If at each point of the surface / 
 
 M 
 < - and 
 
 r a 
 
 a<2, 
 
 the integral is convergent. To prove this we first suppose the function <f> to 
 be positive at all points in /. Now draw two circles $j and S 2 round as 
 
 Fig. 8 
 
 centre, radii r^ and r 2 , and denote the areas between these circles by/ 12 and 
 between them and the outer curve S by/! and/ 2 respectively. 
 
30 Mathematical Introduction [32, 33 
 
 The integral 
 
 f 
 /i 
 
 has a sense ; it is > and increases when r l diminishes. Moreover 
 
 [ <f>df< ! df 
 hi hi*" 
 
 [ M , f , ( M , , 
 < -df+\ -df. 
 
 Jf,^ ' Jfu ra 
 If r 2 is constant the first integral here remains fixed while the second is equal to 
 
 and also tends to a limit if a < 2. 
 Thus the integral 
 
 L 
 
 is always increasing as r l diminishes but always remains less than a certain 
 fixed quantity so that it tends to a finite limit. 
 
 This result is not changed if we replace the circle S^ by any curve S Q 
 surrounding and gradually decreasing to the point 0. W T e can see this 
 easily by tracing round as centre two circles $/ and S z comprising the 
 curve S between them and diminishing with it to the point 0. 
 
 33. Suppose now that the function / has any sign whatever; the 
 preceding conclusions still apply, for if we put 
 
 and choose ^ and (f> z so that 
 
 <f> <j> t and (f) z = 
 for all points where </> > and 
 
 <p = (f>2 and <pi = 
 where </> < 0, and then apply the theorem to the functions </>j and </> 2 , we find 
 
 that the integrals 
 
 r ' f 
 
 Jfo J Jf 
 
 are both convergent : their difference is also convergent and the proposition 
 is proved. 
 
 We can go further : in fact the integral 
 
 f 
 
 > fo 
 
 is also convergent for it is equal to /j + 7 2 . For this reason the integral 
 7 is said to be absolutely convergent. In this case the limits / are independent 
 
33, 34] Mathematical Introduction 31 
 
 of the sequence of forms which the curve / takes when it vanishes into the 
 point 0. 
 
 It may happen that by particular choice of the curves S Q it may be possible 
 
 to prove that the integral 
 
 r 
 Z = <f>df 
 
 '/o 
 
 is convergent but that the integral 
 
 f \t\df 
 Jf. 
 
 is divergent. In this case the integral I is said to be semi-convergent and 
 it can be proved that its value depends fundamentally on the sequence of 
 curves S chosen to define the limit ; and in fact if it is possible to divide the 
 region near the point into a finite number of constituent parts in each of 
 which the function </> has the same sign, it can be proved that a choice of 
 sequence for the curves S can always be made so as to make the integral 
 7 equal to any assignable quantity positive or negative. 
 
 34. Let us now define the convergence of triple integrals : this can be 
 done just as for the case of double integrals. The integral 
 
 I rft(x, y, z)dv 
 
 J V 
 
 extended over any finite volume has no meaning if i/f becomes infinite at the 
 point (ajj , y , %) in the field of integration : to give it a meaning consider 
 the integral 
 
 / = [ ifidv 
 
 J 
 
 extended over the same volume but excluding a small volume around the 
 point 0. If this integral tends to a limit / when this small excluded volume 
 is diminished indefinitely in all directions to the point 0, the integral is said 
 to be convergent and its value is I. 
 
 The test of convergence corresponding to that given above for double 
 integrals is that 
 
 | j/r (x, y, z) < and a < 3 
 
 at all points of the surface : r is of course the distance from the point to 
 the point (x, y, z). In this case the integral is absolutely convergent and the 
 limit is independent of the sequence of forms assumed by the excluded 
 volume as it diminishes to the point 0. 
 
 We can have analogously semi-convergent volume integrals which are 
 such that 
 
 f 
 
 dv 
 
32 Mathematical Introduction [34, 35 
 
 is divergent but such that a definite limit can be obtained for the integral 
 
 ifidv 
 
 by a suitable choice of the diminishing sequence of small excluded volumes. 
 This limit will depend fundamentally on the particular sequence of volumes 
 taken. 
 
 The integral 
 
 f a ' f l \ A 
 ^* \-}dv 
 J dx-i 2 \rj 
 
 provides the simplest example of a semi- convergent integral, and it is a 
 fundamentally important one in the physical theory. 
 
 35. The question of the integration and differentiation of functions 
 represented by improper integrals of the type under consideration is not one 
 that we can here enter into in any detail but we may state the following 
 results which will be fairly obvious if the corresponding theorems concerning 
 functions represented by uniformly convergent series are borne in mind. 
 
 It is permissible under all circumstances to integrate an improper integral 
 with respect to a parameter under the sign of integration, provided that the 
 range of values taken for the parameter is such that the given integral is 
 absolutely convergent for all values of the parameter within it. 
 
 It is permissible to differentiate an absolutely convergent improper 
 integral under the sign of integration provided that the new integral obtained 
 is itself absolutely convergent. 
 
 If follows therefore that in the above integrals 
 
 Y dl 
 
 AI ~ 9^ ' 
 but it does not follow, and is not necessarily true, that 
 
 mainly because the integral representing X 2 has no definite meaning unless 
 the sequence of diminishing excluded volumes is specified in its definition. 
 
 Proofs of most of these theorems will be found in Poincare's book already 
 mentioned. Reference may also be made to the small tract by Mr Leathern 
 on Volume and Surface Integrals used in Physics (C.U. Press). 
 
CHAPTER I 
 
 % 
 
 ON THE PRODUCTION AND DEFINITION OF THE ELECTROSTATIC FIELD 
 
 36. On electrification by friction*. If a piece of glass and a piece of resin 
 be rubbed together they will be found on separation to attract one another. 
 Also if a second piece of glass be rubbed by a second piece of resin and if the 
 pieces be then separated and suspended in the neighbourhood of the former 
 pieces of glass and resin it will be observed that (i) the two pieces of glass 
 repel each other, (ii) each piece of glass attracts each piece of resin, and 
 (iii) the two pieces of resin repel one another. 
 
 These phenomena of attraction and repulsion are called electrical pheno- 
 mena and the bodies which exhibit them are said to be electrified or charged 
 with electricity. Bodies may, as we shall soon see, be electrified in many 
 other ways than by friction. 
 
 The electrical properties of two pieces of glass are similar to one another 
 but opposite to those of the two pieces of resin. If a body electrified in any 
 manner behaves as glass does, i.e. it repels the glass and attracts the resin, 
 that body is said to be positively electrified, and if it attracts the glass and 
 repels the resin it is said to be negatively electrified. The exactly opposite 
 properties of the two kinds of electrification justify us in thus indicating them 
 by opposite signs, but the application of the positive sign to one rather than 
 the other kind must be considered as a matter of arbitrary convention. 
 
 No force either of attraction or repulsion can be observed between an 
 electrified body and a body not electrified. When in any case bodies not 
 previously electrified are observed to be acted on by an electrified body it 
 is because they have been electrified by induction, a process which will be 
 more fully explained in the next paragraph. 
 
 This property of electrified bodies may be used to examine roughly the 
 degree of electrification of a body and also to discover whether it is positively 
 or negatively electrified. In fact if a glass ball of small dimensions is sus- 
 pended on a long silk thread and then electrified by rubbing with a piece 
 of resin it will become positively electrified and thus if it is brought into the 
 
 * The experiments here described are quoted with slight modifications by Maxwell from 
 Faraday's Experimental Researches. It is not pretended that they can be carried out under 
 modern conditions suitably accurately enough to form the basis of a theory ; but their description 
 serves to illustrate in a remarkable manner the properties of electricity. 
 
 L. 3 
 
34 The production of the electrostatic field [CH. i 
 
 neighbourhood of any other electrified body it will be attracted or repelled 
 by that body according as it is negatively or positively charged and with 
 a force depending on the degree of electrification of the body. 
 
 Two similar electrified bodies placed in the same position relative to the 
 suspended ball would produce the same deflection if their electrifications are 
 the same. 
 
 37. Electrification by induction. Let us suspend a metallic rod by silk 
 threads and suppose that it is originally uncharged. If then we bring up 
 a small body charged with electricity near to one end of the rod, without 
 however allowing it to touch the rod, it will be found that this end of the rod 
 has become charged with electricity opposite in sign to that on the small 
 body, whilst the other end is found to be charged with electricity of the same 
 sign as that on the given body. On the removal of the small electrified body 
 it will be found that it has itself lost none of its charge and that the rod 
 has lost all signs of electrification. If the rod is arranged so that it can be 
 divided into two parts, we can separate the two parts before removing the 
 inducing charge, and it will then be found on examination of each part 
 separately that the one is positively charged and the other negatively charged. 
 
 This electrification of the metal rod which depends on the presence in its 
 neighbourhood of an electrified body and which vanishes when the body is 
 removed, is called electrification by induction. 
 
 Now let a hollow metallic vessel be hung up by silk threads and let a 
 similar thread be attached to the lid of the vessel so that it may be opened 
 or closed without touching it. Let also pieces of glass and resin be similarlv 
 suspended and electrified by rubbing together. 
 
 Suppose the vessel to be originally un electrified : then if the electrified 
 piece of glass is hung up within it and the lid closed the outside of the vessel 
 will be found to be positively electrified by induction, the interior being 
 negatively electrified. Now it may be shown that the electrification on the 
 outside of the vessel is exactly the same in whatever part of the interior space 
 the glass is suspended, and that it is also not changed by altering the shape of 
 the piece of glass, as might, for instance, be accomplished by having it in detachable 
 portions. It follows that the electrification of the outside of the vessel in 
 this experiment must measure some definite property of the electrified body 
 which remains constant under similar circumstances. This property is called 
 the electric charge of the body. 
 
 The experiment thus provides us with a method of comparing the electric 
 charges on different bodies without altering the electrification on them. In 
 fact two bodies will have equal charges if they produce the same electrification 
 outside the vessel when inserted separately into it. A body will have a charge 
 equal to twice that on a given body, if when introduced into the vessel it 
 
36-38] Induction and conduction 35 
 
 produces on the outside an electrification equal to that produced by two 
 bodies whose charges are equal to that of the given body, when inserted 
 together. Finally two bodies will have equal and opposite charges ; if when 
 introduced simultaneously into the metal vessel they produce no electrification 
 outside the metal vessel. Proceeding in this way we can test what multiple 
 the charge on any given electrified body is of the charge on another body, 
 so that if we take the latter charge as the unit charge we can express any 
 charge in terms of this unit. 
 
 The charge of a body is therefore a physical quantity capable of measure- 
 ment and two or more charges can be experimentally combined with a result 
 of the same kind as when two quantities are added algebraically. We are 
 therefore entitled to use language fitted to deal with electrification as a 
 quantity as well as a quality and to speak of an electrified body as charged 
 with a certain quantity of positive or negative electricity. 
 
 It can now be proved that the charges on the two portions of the separated 
 rod electrified by induction as in. the first experiment of this paragraph are 
 equal and opposite by inserting them together in the metal vessel as described 
 above and examining the electrification produced outside. 
 
 38. Electrification by conduction. Let the metal vessel of the las"t 
 experiment be electrified by induction as there explained and let a second 
 metallic body be suspended by silk threads near it, and let a metal wire 
 similarly suspended be brought up so as to touch simultaneously the electrified 
 vessel and the second body. 
 
 The second body will now be found to be positively electrified and the 
 positive electrification of the vessel will have diminished. The electrical 
 condition has been transferred from the vessel to the second body by means 
 of the wire. The wire is on this account called a conductor of electricity and 
 the second body is said to be electrified by conduction. 
 
 If a glass rod, a stick or resin or gutta percha, or even a silk thread had 
 been used instead of the metal wire, no transfer of electricity would have taken 
 place. These latter substances are therefore called non-conductors of 
 electricity. Such substances are .used in electrical experiments to support 
 electrified bodies without carrying off their electricity : they are then called 
 insulators. Although it is convenient for the present to draw this distinction 
 between conductors and non-conductors, we shall find that in reality all 
 substances resist the passage of electricity and all substances allow it to pass, 
 though in exceedingly different degrees. 
 
 Of all substances the metals are by very much the best conductors. Next 
 come solutions of salts and acids and lastly as very bad conductors (and 
 therefore good insulators) come oils, waxes, silk, glass and such substances as 
 sealing wax, resin, shellac, indiarubber, etc. Gases under ordinary conditions 
 
 32 
 
36 The production of the electrostatic field [CH. I 
 
 are good insulators. Flames however conduct well, and, for reasons which 
 will be explained later, all gases become good conductors when in the 
 presence of radium or so-called radio-active substances. Distilled water is 
 an almost perfect insulator, but any other sample of water will contain 
 impurities which generally cause it to conduct tolerably well, and hence a 
 wet body is generally a bad insulator. 
 
 When a body is in contact with insulators only, it is said to be insulated. 
 
 In the second experiment of the previous paragraph an electrified body 
 produced electrification in the metal vessel while separated from it by air, 
 a non-conductor medium. Such a medium, considered as transmitting the 
 electrical effects without conduction, has been called by Faraday a dielectric 
 medium, or simply a dielectric. 
 
 39. On charging a body with electricity. A combination of the results 
 of the experiments of the previous paragraphs provides us with a ready 
 means of charging any body with any desired quantity of electricity. 
 
 Let A and B be two metallic vessels of the type of those used above : 
 let pieces of glass and resin be suspended as before and electrified by rubbing 
 with one another. Now let the electrified piece of glass be put into the vessel 
 A and the resin in the vessel B. Let the two vessels be then put into com- 
 munication by a metal wire. All signs of electrification outside the vessels 
 A and B will disappear. 
 
 Next let the wire be removed and let the pieces of glass and of resin be 
 taken out of the vessels without touching them. It will then be found that 
 A is electrified negatively and B positively. If now the glass and the vessel 
 A be introduced into a large insulated metallic vessel C, it will be found that 
 there is no electrification outside C. This shows that there is a charge on the 
 vessel A exactly equal and opposite to that on the piece of glass ; it is found 
 similarly that the charge on the vessel B is exactly equal and opposite to that 
 on the piece of resin. 
 
 We have thus a method of producing on any conducting body a charge 
 exactly equal and opposite to that of an electrified body without altering the 
 electrification of the latter, and we may in this way charge any number of 
 such bodies with exactly equal quantities of electricity of either kind, which 
 we may take for provisional units. 
 
 Now let the vessel B, charged with a quantity of positive electricity, 
 which we shall call, for the present, unity, be introduced into the larger 
 insulated vessel C without touching it. It will produce a positive electrifica- 
 tion on the outside of C. Now let B be made- to touch the inside of C. No 
 change of external electrification will be observed. If B is now taken out 
 of C without touching it, and removed to a sufficient distance, it will be found 
 that B is completely discharged and that C has become charged with a unit 
 of positive electricity. 
 
38-40] The laws of electrical phenomena 37 
 
 We have thus a method of transferring the charge from B to C. 
 
 Let B be now charged again with another unit of positive electricity, 
 introduced into C, already charged, made to touch the inside of C and 
 removed. It will be found again that B is completely discharged and that 
 the charge of C is doubled. 
 
 If this process is repeated, it w T ill be found that however highly C is 
 previously charged, and in whatever way B is charged, when B is first 
 entirely enclosed in C, then made to touch C, and finally removed without 
 touching it, the charge of B is entirely transferred to C. This experiment 
 thus provides a method of charging a body with any number of units of 
 electricity. We shall find in discussing the mathematical theory that the 
 result of this experiment affords an accurate test of the truth of the theory. 
 
 40. The laws of electrical phenomena. In the experiments of 37 it was 
 shown that if a piece of glass, electrified by rubbing it with resin, is being 
 hung up in an insulated metal vessel, the electrification observed outside does 
 not depend on the position of the piece of glass. If we now introduce the 
 piece of resin, with which the glass was rubbed into the same vessel, without 
 touching either the glass or the vessel, it will be found that there is no 
 electrification outside the vessel. From this we may conclude that the electric 
 charge imparted to the resin by rubbing it with the glass is exactly equal, but 
 opposite in sign, to that imparted to the glass by the same process. We may 
 conclude that in generating electricity by friction equal quantities of both positive 
 and negative electricity are generated. 
 
 Now suppose that two bodies, both charged with a certain quantity of 
 electricity, are inserted together into a metallic vessel in the manner described 
 above. Electrification will be induced on the outside of the vessel equal to 
 that produced by a single body charged with a quantity of electricity equal 
 to the algebraic sum of the quantities on the two bodies. Suppose now that 
 while thus inside the vessel the two bodies are connected by a wire so that 
 a transfer of electricity takes place between them by conduction through the 
 wire. It will be found that the electrification on the outside of the vessel 
 remains unaltered; we may therefore conclude that the total quantity of 
 electricity on the two bodies remains the same. Thus 
 
 When one body electrifies another by conduction the total charge on the two 
 remains the same, that is, the one loses as much positive or gains as much negative 
 electricity as the other gains of positive or loses of negative electricity. 
 
 We have also seen that when a body electrifies another one by induction 
 the amount of positive charge induced on the one part of the second body is 
 equal to the amount of negative charge induced on the other part, so that 
 the total charge of the second body remains constantly zero. Thus 
 
 The total charge of a body, or a system of bodies, always remains the same, 
 except in so far as it receives electricity from or gives electricity to other bodies. 
 
38 
 
 The production of the electrostatic field [CH. I 
 
 These are the fundamental laws of electrostatic processes as formulated 
 by Faraday from the result of a long series of experiments of the type of those 
 described above. To render the results deduced from them susceptible of 
 mathematical specification, one further fact is required and that is the law 
 of action between electrified bodies. 
 
 41. The law of action in electrical theory. The actual law for the action 
 between electrified bodies was first discovered experimentally by Coulomb 
 (1785)*, who measured the force by means 
 of a torsion balance. This apparatus consists 
 essentially of two light balls A, C fixed at the 
 two ends of a rod which is suspended at its 
 middle point B by a very fine thread of silver 
 quartz or other material. The upper end of 
 the thread is fastened to a moveable head Z), 
 so that the thread and rod can be made to 
 rotate by screwing the head. If the rod is 
 acted on only by its weight the condition for 
 equilibrium is that there shall be no torsion 
 in the thread. If however we fix a third small 
 ball E in the same horizonal plane as the other 
 two, and if the three balls are electrified, the 
 forces between the fixed ball and the moveable 
 ones will exert a couple on the moving rod, 
 and the condition for equilibrium is that this 
 couple shall balance that due to the torsion. Coulomb found that the couple 
 exerted by the torsion of the thread was exactly proportional to the angle 
 through which one end of the thread had been turned relative to the other, 
 and in this way he was enabled to measure his electric forces. In Coulomb's 
 experiments only one of the two moveable balls was electrified, the second 
 serving merely as a counterpoise, and the fixed ball was at the same distance 
 from the torsion thread as the two moveable balls. 
 
 A similar instrument had previously been used by Cavendish to measure 
 the gravitational force between two small bodies. 
 
 The result obtained by Coulomb may be stated in the following terms. 
 
 If we suppose the dimensions of the two bodies on which charges f q and 
 q' are placed to be so small compared with the distance between them that 
 the result is not much affected by any inequality of distribution of electrifica- 
 tion on either body and if also the bodies be supposed to be suspended in 
 
 * Histoire et Memoir -es de VAcademie Royale. Paris, 1785-1787. 
 J" Measured in any provisional units as suggested above. 
 
40-42] The law of action 39 
 
 air at a considerable distance from other bodies, then the force between them 
 is radial and of amount 
 
 when they are at a distance r apart. The constant y is a physical constant 
 depending on the unit of distance chosen and also on the provisional unit 
 adopted for measuring the charges q and q'. 
 
 We now define the absolute unit of electric charge in Gauss' manner so 
 as to make y numerically equal to unity. It is that quantity of electricity 
 which if condensed at unit distance from a similar charge would exert unit 
 force on it. This is the absolute electrostatic unit of charge. 
 
 The idea of the concentration of a charge at a point involved in this 
 definition is of course merely a theoretical device introduced to simplify the 
 mathematical expression of the law of action. If we wish to be precise we 
 must speak of the point charge as above as the charge on a small body whose 
 dimensions are infinitesimal compared with the distance at which we investi- 
 gate its action. In this sense we regard a point charge in our mathematical 
 theory in the same way as we do a mass-particle in ordinary dynamics, and 
 the proved real existence of the electron justifies such a procedure. 
 
 42. This law of force is essentially an empirical one and no absolute proof 
 is therefore possible, although very strong theoretical evidence can be 
 advanced in its favour. We may therefore very well ask whether the law is 
 absolutely true or is it merely true within the limits of experimental error? 
 
 The experimental proofs of the law originated by Laplace and Cavendish, 
 which will be soon discussed, have established the fact that the force between 
 two point charges q l and q 2 must be 
 
 where p is certainly less than 10~ 5 . Of course we may now assert that if 
 p is so exceedingly small, the chances are that it is zero rigorously. 
 
 There is however some powerful but indirect evidence in favour of the 
 exactness of this law : this law of action as the inverse square of the distance 
 is precisely that which holds in gravitational theory ; and in this case it must 
 certainly be true to an enormous degree of refinement as it explains the 
 motions of the heavenly bodies so well. 
 
 In any case the law for both electrical and gravitational theories is always 
 exact as far as direct experiment can follow it and the conclusion seems there- 
 fore to be that it is exactly true. The evidence in favour of its exactness 
 in electrical theory is less complete than the evidence provided by astro- 
 nomical facts in gravitational theory; but this is counterbalanced by the 
 
40 The production of the electrostatic field [OH. I 
 
 fact that our knowledge of the mechanism underlying electrical actions is 
 much more precise. 
 
 The real fundamental reason of the exactness of this law is not yet 
 evident. We can show that transmission of force by an elastic medium 
 involves a law like this, but this does not provide us with a reason for the 
 validity of the law in the present case. 
 
 , 
 43. We have said that the constant y which enters into our relation 
 
 is a physical constant depending on the units adopted. We may therefore 
 enquire as to the way in which it depends on these units. To answer this 
 it is necessary to find the dimensions* of y. 
 
 The expression of a physical law must be independent of the units of 
 measurement of the quantities involved. The physical law expressing the 
 force between two electric point charges q^ and q 2 , viz. that the force is equal to 
 
 r> 9i?2 
 Jl ^ = *y 
 
 is of this type : and we therefore conclude that the dimensions of the quantities 
 equated in this relation must be the same. 
 
 If we use [m], [I], [t], [q] to denote the respective dimensions of the chosen 
 arbitrary units of mass, length, time and quantity of electricity the dimen- 
 sional equation for the above law is 
 
 'mil , fa 2 " 
 
 [I 3 m 
 
 so that [y] = -^ 
 
 [_q 2 t 2 
 
 defines the dimensions of y. Now the definition of dimensions implies that 
 the unit of any quantity is increased in the ratio of its dimensions when these 
 are increased, and the numerical value of the quantity is consequently reduced 
 in the same ratio. Thus if we had chosen different arbitrary units for the 
 fundamental quantities, which have respectively measures M, L, T, Q in 
 terms of the former units the new value F of the constant of the physical law 
 of action between electric charges would be given by 
 
 and this relation defines completely the way in which this constant depends 
 on the fundamental units. 
 
 * .The doctrine of dimensions was really first started by the section of Newton's Principia 
 entitled ' Principle of Dynamical Similarity ' ; but the theory was not very definitely understood 
 until Fourier crystallised Newton's rules in his Theory of Heat. 
 
42-45] The constitution of electricity 41 
 
 44. We have so far assumed a knowledge of some arbitrary but definite 
 unit of electric charge which does not depend on the other fundamental 
 units employed. We can however simplify the matter by regarding the 
 physical law of action as defining an electric charge and then the constant 
 y can be chosen at will, provided that the electric charge is measured properly. 
 The simplest plan is to take, after Gauss, y = 1 so that the equation for the 
 law of action is 
 
 - < 
 
 r 2 
 In this case the dimensions of an electric charge must clearly be 
 
 \q] = p*f-i*3 
 
 and the unit charge is such that if condensed at unit distance in vacua from 
 a similar quantity it would exert unit force on it. This is Gauss's absolute 
 unit of charge in which we shall henceforth assume all charges to be measured. 
 We have then no further concern with the constant y. 
 
 We have now sufficient information to enable us to formulate mathe- 
 matically the general theory of electrostatic phenomena, and it was in fact 
 on this basis that the earlier mathematical physicists of the French school 
 (Poisson, Laplace and others) developed the theory. It will however be 
 convenient for us to give first a short statement of the position of the more 
 fundamental problem regarding the constitution of electricity itself before 
 proceeding to our main problem. 
 
 45. The constitution of electricity. Previous to the researches of Faraday 
 the generally accepted explanation of the phenomena of electrical action was 
 based ultimately on Coulomb's experimental law for the interaction between 
 electrical charges and involved the fundamental concept of the two electric 
 fluids : these two fluids were assumed to be composed of very small particles 
 of a non-gravitative subtile matter of a more refined and penetrating kind 
 than ordinary liquids and gases : two particles of the same fluid repel each 
 other with the law of action determined by Coulomb, whilst two particles of 
 different fluids attract one another by the same law. It was then supposed 
 that all bodies in their ordinary conditions contained equal amounts of both 
 positive and negative electricity and that in rubbing two bodies together as 
 described in 36 a difference in the quantities of the two fluids in each of 
 the bodies is produced, so that the one has an excess of positive fluid and is 
 therefore positively charged, while the other has an excess of negative fluid 
 and is negatively charged. This theory therefore effectively explains the 
 phenomenon of electrification by friction and also the forces of attraction or 
 repulsion between electrified bodies : it would also account for the phenomena 
 of conduction and induction if it is assumed that either one or both of the 
 fluids are freely moveable through good conducting media, being however 
 more or less rigidly fixed to the elements in an insulating medium. 
 
 
42 The production of the electrostatic field [CH. I 
 
 This two fluid theory may be said to have held the field until the time 
 when Faraday began his researches on electricity. After educating himself 
 by the study of the phenomena connected with his discoveries on electro- 
 magnetic induction, he applied his knowledge to electrostatic problems and 
 finally came to the conclusion that the so-called charge of electricity on a 
 conductor was not in reality anything of a material nature on the conductor 
 or in the conductor, but consisted in a state of strain or polarisation, or 
 a physical change of some kind in the particles of the dielectric medium 
 surrounding the conductor and that it was this physical state in the dielectric 
 which constituted electrification. 
 
 But while thus attempting to dispense with the necessity of the concept 
 of the electric fluids, Faraday himself provided in his electrochemical 
 researches the starting point for the next great development in electrical 
 theory which culminated in J. J. Thomson's discovery and practical isolation 
 and examination of the 'atom of negative electricity,'' the now famous ''electron,' 1 
 and the consequent formulation of the 'electron theory' of electrical (and 
 optical) phenomena, which in many respects is very similar, although more 
 explicit, than the old fluid theory. While thus crediting Thomson with the 
 discovery of the electron it must not be forgotten that its existence had long 
 previously been surmised by Helmholtz* and Maxwell f in a general way, and 
 by Crookes in more explicit terms : and it had formed the basis for the 
 fundamental theoretical researches of Larmor and Lorentz published long 
 before Thomson's discovery J. 
 
 The electron itself is an extremely minute electrically charged particle 
 with an inertia mass of 9 . 10~ 26 gms. and a charge of about 4'69 . lO" 10 electro- 
 static units which is enormously large compared with its mass : in fact two 
 grams of electrons placed at a distance of one metre apart would repel one 
 another with a force equal to the weight of about 10 21 tons. The really 
 extraordinary thing about them is that however they are obtained, they are 
 apparently always identical at least as regards their charges and masses, 
 which is all that we are concerned with. It is of course beyond the powers 
 of a physical science to say what is the ultimate cause of this exact identit} T 
 of all the electrons. 
 
 No one has yet succeeded in isolating positive electrons, so that the term 
 electron is temporarily applied only to the negatively charged particles 
 
 * Faraday Lecture, 1881. 
 
 t 'A dynamical theory of the electromagnetic field,' Phil. Trans. (1864). 
 
 % For full information the student may consult : J. J. Thomson, Conduction of Electricity 
 in Oases; E. Rutherford, Radioactivity; Townsend, Electricity in Gases; which are the standard 
 works on the physics of the electron and radioactivity disintegrations. The mathematical side 
 is discussed with references by Larmor, Aether and Matter ; Lorentz. Theory of Electrons ; Richard- 
 son, Electron Theory of Matter. See also Campbell, Modern Electrical Theory ; J. J. Thomson, 
 The Corpuscular Theory of Matter. 
 
 I 
 
45-47] The constitution of electricity 43 
 
 discovered by Thomson, but experimental evidence is gradually tending to 
 the view that the positively charged hydrogen atom is the ultimate element 
 of the positive electricity which exists in all substances. The element of 
 positive electricity if assumed to have the mass of a hydrogen atom also 
 carries the same charge, viz. ^. 10~ 10 units ; but its mass is 1700 times that 
 of an electron. I? 3-7 
 
 46. There is now an enormous mass of experimental evidence, to which 
 contributions are made, not only by the phenomena of electrostatics, but also 
 by the phenomena of almost every branch of physics and chemistry tending 
 to show that each chemical atom of matter contains as an essential part of 
 its constitution a certain number of electrons grouped together in various 
 more or less stable congeries; each atom also possesses in some as yet 
 undetermined form the necessary positive electric charge to make it electrically 
 neutral on the whole. In every solid body there is a continual process of 
 atomic dissociation going on, the electronic configuration inside the atom 
 being sufficiently unstable in many cases to be capable of breaking up on 
 small provocation, with the consequent liberation of one or more electrons 
 and occasionally of positive elements as well : the result of this is that mixed 
 up with the atoms of chemical matter composing a body we have a greater 
 or less percentage of negative electrons, and a few positive elements freely 
 moveable in the interstices between the atoms. It is in fact to these free 
 electrons and positive charges that the phenomena of electric conduction is 
 due. An electrically charged body is one in which there is an excess or 
 deficit of (negative) electrons. The action between the charges of the electrons 
 and the charges in the atoms is precisely that specified by Coulomb's law 
 provided the charges are at rest and at distances from one another large 
 compared with ordinary molecular dimensions. The distinction between 
 insulators and conductors as regards the phenomena of induction and con- 
 duction depends essentially on the fact that in the conductors there is a large 
 number of the free dissociated electrons which can be pulled about from one 
 part of the medium to another under the action of forces from other electrified 
 bodies; whereas in insulators there is such an extremely small number of 
 these free electrons, that the phenomena depending on them can under most 
 circumstances be neglected. 
 
 47. We shall in our future investigations discuss many facts which have 
 led up to this conception of the essential electronic constituent of matter ; but 
 we may here just mention one important point in its favour to which we shall 
 not have any need to refer to in our future work. In 1896 Becquerel dis- 
 covered the so-called radioactive substances, which are continually and 
 spontaneously emitting a complicated type of radiation, of which two of the 
 main constituents have been proved to be composed, the one of rapidly 
 moving electrons (ft particles) and the other of more slowly moving positive 
 
44 The definition of the electrostatic field [OH. i 
 
 particles (a particles), the properties of which however suggest that they are 
 attached to a molecule of helium. These phenomena were found by Becquerel 
 to be associated with the element uranium, but Mme Curie working with her 
 husband soon succeeded by a wonderful process of fractional distillation in 
 isolating the chloride of a much more powerfully active element, then 
 unknown to chemical science, and which they called Radium. Many other 
 similar substances have subsequently been discovered, but radium is still 
 the most active of them all. 
 
 The velocity with which the electrons are thrown off from radium is 
 really astounding, in some cases it amounts to as much as 3 . 10 7 metres per 
 second, i.e. one-tenth of the velocity of light. Moreover the phenomenon 
 is still more remarkable on account of the great development of heat that 
 accompanies the process of disintegration. This suggests, of course, that 
 not only are the atoms emitting the electrons with this enormous velocity, but 
 that what remains is also undergoing violent transformation ; and it is now 
 definitely established that this is actually the case. In fact, a detailed 
 investigation of the phenomena associated with radioactivity has led to the 
 result that a substance which is continually radiating electrons or positive 
 particles or both spontaneously changes into another substance differing 
 chemically from the former substance from which it arises; moreover the 
 atomic weight of the new substance bears a definite relation to that of the 
 old and the number of elements of charge lost in the transformation. This 
 discovery probably provides the most direct evidence we have of a relation 
 between the electrical constitution of an atom and its chemical composition, 
 a relation which necessarily implies the essential existence inside any atom 
 of matter of certain positive and negative charges. 
 
 48. Although the evidence thus deduced appears to be conclusive only as 
 regards the radioactive substances there are a large number of facts which 
 point to the conclusion that it is universally valid for all substances known 
 to us. Of these facts the most direct and conclusive have recently been 
 obtained by Profs. J. J. Thomson* and Rutherford f and their collaborators 
 in an extensive examination of the electronic constitution of matter. The 
 method employed consists in firing a stream of rapidly moving a- or /3-particles 
 into a piece of matter and examining the deflection and loss of energy of the 
 individual particles caused by their collision with the atoms of matter. From 
 the nature of the scattering of the heavier a-particles it is concluded that the 
 positive charge in a simple atom, which is the active nucleus in producing 
 the effect observed, is equal to the charge on an electron multiplied by the 
 atomic number J of the substance ; but that it is concentrated in an extremely 
 
 * Camb. Phil. Soc. Proc. xv. p. 465 (1910) 
 t Phil. Mag. xxi. p. 669 (1911). 
 
 $ The ordinal number of the substance in the periodic table. In most cases this number is 
 equal to half the atomic weight. 
 
47-49] Point charge distributions 45 
 
 small volume, much smaller than the electron is presumed to occupy. The 
 number of electrons in the neutral atom will therefore be equal to the atomic 
 
 number of the substance. 
 i 
 
 The results obtained from the scattering of the smaller ^8-particles are 
 not nearly so definite and are also not consistent with those just explained, 
 but the discrepancy may be due to other causes*. 
 
 With the question as to whether there is anything else beyond the electric 
 charges in the atom we are not at present concerned : the fact that it is 
 impossible to obtain the positive particle unassociated with an atom of 
 hydrogen or helium at the least, which are known for other reasons to be 
 particularly complex systems, rather suggests that there is something else 
 in an atom than mere electricity, but it is impossible to say what it is, chiefly 
 because we are thus prevented from determining the true nature of the 
 positive charge. 
 
 In the majority of our future discussions we shall have no special necessity 
 to use this definite conception of electricity which now underlies the modern 
 electrical theory, and we shall occasionally offer tentative illustrative 
 explanations which are based on a less explicit conception. We shall not 
 however refrain from resorting to the conception of an electron and, in fact, 
 it will often be conducive to clearness if we elaborate various details in the 
 usual expositions of the electron theory of the phenomena which arise. 
 
 49. The definition of the electric field of a system of point charges f. 
 
 If an electrified body is brought into the space surrounding any system of 
 charges it will in general produce a sensible disturbance in tire electrification 
 of the system by induction. But if the body is very small and its charge 
 also very small the electrification of the other bodies will not sensibly be 
 disturbed. The force acting on the body as a result of the action from the 
 charges on the other bodies will then be proportional to its charge and will 
 be reversed if the sign of the charge is reversed. That is if SF be the force 
 and 8q the charge, then when 8q is an infinitesimal SF is proportional to Sq or 
 
 SF - ESq, 
 
 where E is a vector function of the position of 8q only, which is determinate 
 when the system of charges is given. We may thus regard E as a property 
 of the point. 
 
 The space in and around the given system of charges is called the electric 
 field of those charges and the vector E, which represents the force 'per unit 
 charge ' at the point in the field is called the intensity, of the electric force at 
 that point in the field. 
 
 In the neighbourhood of a single point charge q and at a distance r from 
 it the electric force intensity is along the direction of r and of amount q/r 2 . 
 
 * Cf. N. Bohr, Phil. Mag. xxx. p. 581 (1915). 
 
 t J. Lagrange, Par. sav. (dtr.) 1, 1773 (Oeuvres, 6, p. 349). 
 
46 The definition of the electrostatic field [CH. i 
 
 If we refer to ordinary rectangular coordinates with the point charge q^ at 
 the point (a^, y 1} z^) then the components of the force intensity along the 
 coordinate axes at the point (x, y, z) are 
 
 vCl I/ ^/1 Z %' 
 
 ^ ' 3 ' 3 
 
 where rf = (x x-^ 2 + (y yj 2 + (z Z L ) Z . 
 
 These are simply 
 
 \dx ' dy ' dzj r-^ ' 
 
 In a similar manner it can be seen that if we have any system of point 
 charges q 1} q 2) ... q n at the points (x lt y i} %; x 2 , y 2 , z. 2 ; ...; x n , y n , z n ) 
 respectively, then the components of the total electric force at the point 
 (x, y, z) of the field are 
 
 re v v\ v [9>( x - x ) 9s(y-y s ) q s (z-z s )' 
 
 (El x J ^2/5 '"Z/ " " 
 
 8=1 
 
 where r 2 = (x - x s ) 2 + (y - y s ) 2 + (z - z a ) 2 . 
 
 n Q 
 
 The function (f> = S , 
 
 from which the components of the force intensity at any point of the field 
 are obtained by simple differentiation along the axes, is called the potential* 
 of the electric* field at the point (x, y, z). It has an important physical 
 significance which we shall discuss later : for the present it is merely defined 
 
 so that 
 
 / 1 o *-) 
 
 (E . ti ,, , E~) = I ^ , s~ , '^~ 
 \pT uij cz 
 
 and thus 
 
 dx dy _ dz _ dcf> dx d(f> dy d(f> dz 
 x ds v ds z ds dx ds dy ds dz ds 
 
 ds ' 
 
 The component of the force intensity in any direction at a point is the space 
 rate of fall or the negative gradient of the potential at that point and in that 
 direction. 
 
 50. If we choose rectangular axes with the origin conveniently near 
 the system of charges and if we write 
 
 : This function was used first in the theory of attractions by Laplace. The name potential 
 was given to it by Green and independently by Gauss, 'Allgemeine Lehrsatze iiber...anziehungs 
 und Abstossungskrafte, ' 3. (Collected Works, 5. p. 200.) 
 
49, 50] Point charge distributions 47 
 
 and denote by S the angle between the radii r and r s0 from the origin then 
 
 ~ 2 r 2 _i_ 2 O r r prxj f) 
 
 r s r O "" r sO Zr O r sO COS ^s ; 
 
 so that (/> = 2 -7= = , 
 
 Vr 2 _i_ r 2 _ 9 r r rnt , ^ 
 v ' o i 'so -^'o'so uub p a 
 
 /7 
 
 and so also r (/> = 
 
 The greatest interest attaches to the approximate values of ^ at a considerable 
 distance from the origin*. In this case f J is small for each of the charges 
 
 (assumed all to be at a finite distance) and we can expand each term of the 
 above sum in an absolutely convergent series. We thus get in fact 
 
 r sn r s0 2 /3 cos 2 0, 1 
 
 Thus for points at a very large distance the first approximation to the 
 potential is 
 
 A* 2?s 
 
 unless S< S = when it is 
 
 If we had chosen at the centroid of the charges then "q s r s0 cos 6 S = and 
 the second term in the expansion vanishes. The third term has as coefficient 
 
 2 9s r s0 2 (3cos 2 s -l), 
 
 and in this form it is easily identified with Gauss' term in the attraction of 
 a gravitating system of masses. There is of course no centroid at a finite 
 distance if S<? s = 0. 
 
 We have also 
 
 1 cos 
 
 rfrh *" \ 
 
 so that 
 
 and therefore also 
 
 y g 9<p _ y, 
 
 The action of the charges at a" large distance is thus to a first approximation 
 as if they were all collected at a point, which is their mean centre. These 
 
 . * The expansion of the potential at a distant point is originally due to Poisson but was put 
 into a, convenient form by MacCullagh. R. Irish Trans. (1855). 
 
48 The definition of the electrostatic field [CH. i 
 
 remarks will help us in the elucidation of certain difficulties which crop up 
 when 'we attempt to extend our definitions so as to apply to continuous 
 distributions. 
 
 51. The definition of the electric field at points outside a continuous 
 distribution of charge*. The discussion of the previous paragraph applies 
 only to a system of discrete point charges or electrons and it is only in this 
 sense that the analytical functions have any meaning. In actual practice 
 however the distributions of charge with which we deal include such an 
 enormous number of electrons that the expression of their field by functions 
 of the type discussed above, even if it were possible, would be quite untract- 
 able. Fortunately however any such complete atomic analysis is useless 
 in a physical theory, whose results can only be tested by observation and 
 experiment on matter in bulk, for we are unable to take cognisance of the 
 single molecule of matter, much less of the separate electrons inside it Jo 
 which this analysis has regard. The development of the theory which is to 
 be in line with experience must instead concern itself with an effective 
 differential element of volume containing a crowd of molecules numerous 
 enough to be expressible continuously, as regards their average relations 
 as a volume density of matter. 
 
 Thus in any physical theory all that we are directly concerned with as 
 regards the charge in any ' physically ' small element of volume di\ is its 
 total amount dq^ and the ratio of these two magnitudes defines the density 
 of the charge at the point, viz. p lt where 
 
 The distinction here introduced between physically, as distinct from 
 mathematically, small differentia] elements of volume is important and must 
 be emphasised. In the speculations of pure mathematics there is no limit 
 to the fineness of the subdivision of a region into volume elements, but in 
 the physical theory there comes a limit when the element is so small that the 
 number ofr-elements of mass or charge in it is so small that the total mass 
 included in the element depends appreciably upon its shape, so that the 
 definition of density as the ratio of this total mass or charge to the volume 
 ceases to have any meaning. The passage to the limit involved in a strict 
 mathematical definition is thus not possible in a physical theory. 
 
 52. Now the element of charge p^dv acts effectively at all points which 
 are at a distance from it which is large compared with the linear dimensions of 
 the element of volume dv-^ containing it, just like a charged particle so that 
 
 * The points here briefly dealt with are discussed at length by Leathern, Volume and Surface 
 Integrals used in Physics (Camb. Univ. Press, 1st ed. 1905). Cf. also J. Boussinesq, Journ. de 
 math. (3) 6, p. 89 (1880); H. Poincare, Amer. Journ. of Math. 12, p. 284 (1890). 
 
50-53] Continuous charge distributions 49 
 
 its field at such points is defined by the force vector SE and potential </> 
 which are determined b the relations 
 
 In these expressions r l denotes the distance of the volume element di\ at the 
 point (x l , y l} zj from the point (x, y, z) at which the functions are calculated. 
 Thus for the whole system of charges grouped together in this way the field 
 is defined by 
 
 E= - 
 
 whilst cf> - 
 
 the integrals in each case being extended over the whole charge distri- 
 bution. 
 
 The use of the definite integral expressions necessarily implies the possibility 
 of endless subdivision in the strict mathematical sense of the electric charge 
 and attributes to the density p at any point the value obtained by passing 
 to a limit in the usual way. This inconsistency is however removed by the 
 simple device of replacing the actual distribution of electric charge by a 
 hypothetical perfectly continuous distribution with the same density at each 
 point and referring the integral expressions to this distribution. Such a 
 continuous distribution is effectively the same as the actual one, at least 
 as regards its effect at all points which are not too near the distribution, the 
 actual distribution of charge in any physically small volume element being 
 then quite irrelevant. 
 
 53. So far the field-point at which the force and potential are calculated is 
 restricted to be at a distance from the nearest charge element which is large 
 compared with the linear dimensions of the t physically small element of the 
 charge distribution; it is however easy to see that the definitions remain 
 valid up to a distance comparable with the dimensions of the physically small 
 element. Let us consider the potential integral : the difference between the 
 
 sum S * for the elements of charge in any physically small element of volume 
 
 of linear dimensions I and the integral l - taken throughout the same 
 
 element will be of the same order of magnitude as either quantity separately 
 so long as r for all points of the element is of the same order of magnitude 
 as Z, but that the difference will diminish to a quantity smaller in the ratio 
 
 when r becomes great compared with 1. Thus for purposes of estimating 
 
 * J. Lagrange, I.e. p. 45. 
 
50 The definition of the electrostatic field [OH. i 
 
 the order of magnitude it is reasonable to represent the difference between 
 these two expressions for the element under consideration as 
 
 'alp , 
 
 2 dv > 
 r z 
 
 where a is a finite number. Thus the difference between the representation 
 by a sum or by an integral of the potential at any point due to the charge 
 between the spherffe of radii I and a ( > I) is 
 
 f a 
 
 < kTTo.pl dr 
 
 Ji 
 
 < 4wap'l (a I), 
 
 where p is the maximum value of p in the region : this difference is negligibly 
 small if a is not too big (say 1 cm.) on account of the smallness (physical) 
 of I. A similar proof also applies to the other integral. 
 
 It thus appears that the definitions of the potential and force intensity by 
 means of integrals extended throughout the hypothetical continuous distri- 
 bution of charge, which replaces the actual or discrete one, are completely 
 effective and valid without sensible error not only for points well outside the 
 charge, but also for points whose distance from the nearest portion of charge 
 is small of the order of the physically small length I. This includes the case 
 when the point is so close to the apparent outer surface of the charged body 
 as to be sensibly just not in contact with it and also the case where the point 
 is in a small but not imperceptibly small cavity of such a size that the piece 
 excavated would have the properties of matter in bulk rather than the 
 properties of a few molecules or electrons. Moreover the integrals involved 
 in these definitions give rise to no mathematical difficulties. The subjects of 
 integration are finite at all points of the region of integration and the integrals 
 themselves are finite and differentiable with respect to the coordinates 
 (x, y, z) of the external point by the method known as differentiation under 
 the sign of integration. It thus follows that we still have on the modified 
 
 definitions 
 
 E = grad(/>, 
 
 so that the electric force intensity at an external point in the field is still 
 equal to the negative gradient of the potential at that point. 
 
 54. The definition of the electric field at points inside the continuous 
 charge distribution. The generalised specification of the electric field of a 
 continuous charge distribution given in the previous paragraph is perfectly 
 definite, but applies only to points external to the charge distribution. As 
 however we shall want to extend our analysis also to points inside the con- 
 tinuous charge distributions we must see whether the definitions still hold 
 for such points*. 
 
 * Cf. Gauss, ' Allg. Lehrsatze, etc.' 6 (Footnote 7) ( Works, 5, p. 202) ; 0. Holder, Dissertation, 
 Tubingen, 1882, p. 6; .T. Weingarten, Acta math. 10, p. 303 (1887); C. Neumann, Leipz. Ber. 42, 
 p. 327 (1890). 
 
53, 54] Continuous charge distributions 51 
 
 Let us first assume, without preliminary justification, that the hypothetical 
 continuous distribution with which the real distribution of charge was replaced, 
 effectively represents this actual charge at any point of the field however 
 near to the actual charge it may be. The force and potential at a point inside 
 the distribution will then be defined by the same integral expressions if these 
 have any meaning at all : we see that the integrands in both cases become 
 infinite so that if the integrals are not convergent the expressions have no 
 meaning whatever. We have however already indicated in the introduction 
 that the two integrals are absolutely convergent in all cases if p is everywhere 
 finite, so that there is no difficulty in the application of these expressions in 
 this case. 
 
 We can regard the matter physically in the following manner. Imagine 
 a small volume cut out of the charge distribution around the internal point 
 at which it is desired to calculate the functions. The potential and force 
 due to the remaining distribution have then definite values which may 
 however be large. The question is now : do the values of the functions at 
 this point depend appreciably on the size and shape of the cavity, if it is 
 made very small? If they do, the integrals given are either divergent or 
 semi-convergent and no meaning can be attached to the functions they 
 represent. If on the other hand, as is the case in the present instance, they 
 do not depend on the shape or size of the cavity, if it is only made small 
 enough, the integrals, although of the type called improper, have distinct 
 values and the definitions remain. 
 
 Thus if we can assume that the continuous charge distribution effectively 
 replaces the real one at all points of the field the definitions of the force 
 intensity and potential at an internal point are consistent and definite and 
 moreover the removal of a physically small portion of this charge round the 
 point does not appreciably affect the values of the functions at the point, so 
 that in their definition it is immaterial whether this small portion of the 
 charge is present or not. But any attempt to justify the use of this effective 
 distribution in calculating the field at a point whose distance from the nearest 
 element of charge is of a higher order of smallness than a physically small 
 differential length (I, of our previous analysis) can only result in failure. 
 For now the single electron contributes to the potential, for example, a term 
 q/r which in spite of the smallness of q may become very great as r diminishes, 
 so that the presence of a few such electrons might easily become so important 
 as to make the potential quite different from the value obtained from the 
 continuous distribution and expressed by 
 
 to which, as we have just mentioned, the part of the distribution near the 
 point contributes only a negligible amount. But there is from the physical 
 
 42 
 
52 The definition of the electrostatic field [CH. i 
 
 point of view no real motive for pursuing the enquiry further as we have in 
 fact obtained a formulation of our field which is completely effective in a 
 physical theory, and for the following reasons. 
 
 55. The real charge distribution may, as regards its action at any point 
 inside the medium, be divided into two distinct portions by a physically small 
 closed surface drawn round the point. The first part of the charge, viz. that 
 outside the elementary surface, may be replaced by the continuous distribution 
 as above which is, as regards its action at the point under consideration, 
 effectively equivalent to it : to this we may also add, without appreciable 
 modification, the continuation of this distribution throughout the interior of 
 the small volume round the point. The second is the purely local distribution 
 of charge elements inside the surface drawn. The contributions to the force 
 and potential in the field at the internal point due to the former part of the 
 charge are perfectly definite and are in fact expressed by the convergent 
 integrals given above; but the local contribution from the elements of charge 
 near the point is entirely unknown and may be continually changing. The 
 only possible expressions for these functions are therefore the ones that omit 
 altogether the contribution of these neighbouring elements. 
 
 If it were not possible thus to separate the physical functions into a molar 
 and a molecular part a dependence would be involved between mechanical 
 change and molecular structure, so that mechanical causes would alter the 
 constitution of the medium and might even undermine its stability ; whereas 
 it is a postulate in ordinary mechanical theory that the physical properties of 
 the medium are not affected by small forces*. 
 
 Thus for the purposes of a physical theory the force and potential at 
 points inside the medium are properly defined as the corresponding quantities 
 belonging to the field of the hypothetical distribution of charge which is thus 
 concluded to be a completely effective representation of the real distribution. 
 We have therefore both at external and internal points 
 
 , 1 
 E = - 
 
 / 
 
 and </> = 
 
 Moreover since the integrals in these expressions are both absolutely con- 
 
 * Cf. Larmor, Aether and Matter (particularly the footnote on p. 265), also Phi!. Trans. 190 A 
 (1897). "The principle of D'Alembert, which is the basis of the dynamics of finite material 
 bodies necessarily involves this order of ideas. That part of the aggregate forcive on the molecules 
 in the element of volume which is spent in accelerating the motion o'f that element as a tvhole 
 is written off; and the regular part of the remainder must mechanically equilibrate. But the 
 wholly irregular parts of the molecular motions and forces are left to take care of themselves, 
 which they are known to do for the simple reason that the constitution of the material body is 
 observed to remain permanent." 
 
54-57] Surface charge distributions 53 
 
 vergent it is legitimate in all cases to derive the former from the latter by 
 the process of differentiation under the sign of integration so that we have 
 
 E = - grad</>. 
 
 Thus the components of the force intensity at any point of the field in any 
 direction is the space rate of fall of the potential in that direction. 
 
 56. On surface distributions and double-sheets. We must now pass to 
 the consideration of certain important types of discontinuity in the volume 
 charge distribution with which we shall have to deal in our future work. 
 Such cases actually occur in nature and it seems necessary to consider what 
 modifications are needed in the above definitions in order that they may 
 apply to them. 
 
 The first example leads us to the idea of a surface density. If the volume 
 density becomes very large in the neighbourhood of a surface / in the field 
 we may separate the comparatively infinite values from the rest by drawing 
 two surfaces parallel and very close to /one on each side. The layer between 
 these surfaces is then of very small thickness Aw but the volume density 
 p is so large that 
 
 a = pdn 
 
 is finite when integrated across the common normal at any point of the surface. 
 We then regard this part of the charge distribution as a surface distribution 
 of density a (i.e. amount per unit area) on the surface /. It is of course 
 merely a big volume density concentrated in a shell of small thickness, but 
 as we do not as a rule wish to be bothered about the constitution of the layer 
 we treat it in this way. 
 
 The potential function associated with this part of the charge distribution is 
 
 JL _ f fjf (* n pdn 
 
 .if .0 f ' 
 
 and since the shell is very thin this is practically 
 
 f df ( n , f a 
 _ | ^M pdn = 
 
 h + J* 'f r 
 
 extended over the surface /, r denoting the distance of the element df from 
 the point at which the potential is calculated. 
 
 The components of force are expressed in an analogous manner. 
 
 57. The second case of infinities in the volume density appears at first 
 sight rather an artificial one, but as a matter of fact it actually exists in 
 nature and the analysis associated with it is of immense importance for other 
 branches of the work. Imagine a surface/' placed parallel and infinitely near 
 to a surface / so that the small normal distance between them is Aw. Now 
 
 * G. Green, Essay, etc. Art. 4. 
 
54 The definition of the electrostatic field [CH. i 
 
 suppose -the surface/' has a charge distribution of surface density a' and the 
 surface / one of density a. The potential of this distribution would be 
 
 a'df adf, 
 
 now make o-'d/' = adfso that there are equal and opposite charges on opposing 
 elements of the surfaces and also put 
 
 1 1 3 /1\ . 
 -, = ~ + *- ( - An, 
 r / o Vr/ 
 
 so that < = [ aAjr-(-)<2/V 
 
 J f U iv \ I / 
 
 We are therefore no longer concerned with the surface distributions separately, 
 but must treat them together. They form what is called a double sheet 
 distribution. The quantity which mathematically specifies the sheet is the 
 product crAn, which is called the moment of the sheet and is denoted by r. 
 We must have a very large a to get a finite r for r Aw . a : the surface 
 densities involved are therefore large compared with the usual ones which 
 occur separately. 
 
 The potential of this double sheet is 
 
 am ,, 
 
 r 5- I - } df , 
 dn\rj ' 
 
 and the components of force analogously. 
 
 These represent the only types of distribution with which we have to 
 deal in our theories. Other types may occur in nature, but they are of little 
 importance, if they occur at all f . 
 
 58. If there are surface densities and double sheets in the field the 
 ordinary considerations as to convergence and continuity of the integrals 
 expressing the force and potentials still apply provided the point under con- 
 sideration does not lie on any of the surface infinities. If the point is on an 
 ordinary distribution of surface density the theorem stated in the introduction 
 ( 32) shows that the potential is quite definite, the integral expressing it 
 being convergent, but that there is a certain indefiniteness in the expression 
 of the force, which is given by a conditionally convergent integral. The 
 significance of these results in the physical theory is however obvious from 
 our former discussions and need not now be further elaborated. 
 
 There are certain discontinuities introduced as we approach the surface 
 distributions thus specified but these are best attacked by the indirect method 
 as discussed in the next chapter. 
 
 * Helmholtz, Ann. Phys. Chemie, 89 (1853); Collected Works, i. p. 491. 
 
 f Cf. however W. Voigt, Lehrbuch der Kristallphysik, ch. in. (Leipzig, 1910). 
 
57-59] The potential 55 
 
 59. On the nature of the potential. It now seems necessary to enquire 
 into the physical significance of the important analytical function called the 
 potential; why should there be such a function? 
 
 Lagrange* and von Helmholtzf have given easy mathematical proofs of 
 the existence of the potential function in electrostatic and allied theories, 
 associating it with the potential energy function of any ordinary dynamical 
 system. Their analysis is however based on the assumption that the action 
 forces involved consist merely of attractions or repulsions in the direct lines 
 of the particles and according to some function of the distance. We may 
 however take up the standpoint that there is no reason at all why all mutual 
 physical actions of this kind should be built up of direct attractions, instancing, 
 for example, the action of a magnetic pole on a current filament discovered 
 by Oersted, where the forces are certainly not radial attractions J. 
 
 All these difficulties may be avoided and at the same time a far wider 
 idea of the meaning of the potential obtained by connecting it with the 
 doctrine of the conservation of energy. Let us examine a general problem 
 from this point of view. 
 
 Suppose we have a number of physical systems M 1} M 2 , ... acting upon 
 one another across space and suppose also that we have a small element Sm 
 of a similar system. 
 
 We shall first suppose either that 8m is so small that it does not disturb 
 the finite systems when moved about in their neighbourhood, or that these 
 finite systems are held rigid during the motion of 8m, so that they are not 
 thereby disturbed. When the element 8m receives a small displacement 8s 
 the system does work on it of amount f s 8s, where T s is the component of 
 the force exerted on 8m in the direction of its displacement. The work done 
 in any finite displacement from the initial" position 1 to a second position 2 is 
 
 2 -2 
 
 5 T s 8s = T s ds. 
 i . i 
 
 But this work ought to be the same fgr all paths, if the general path is 
 reversible, i.e. if the work gained in any displacement is equal to the work 
 lost in the same displacement taken the reverse way. If the paths are all 
 reversible and the work not equal for all of them we could take the element 
 8m down one path and bring it back along another, so that the work lost in 
 taking it down is less than that gained in bringing it back and thus on the 
 whole there would be a gain of work. Where could this work come from? 
 M 1 , M 2 , ... are all effectively rigid and so the work must have been created 
 from nothing ! This might be so but for the fact that we could repeat the 
 
 * Par. sav. (etr.) 1 (1773). (Oeuvres, 6, p. 349.) 
 
 | Uber die Erhaltung der Kraft (Berlin, 1847). Cf. Planck, Das Prinzip der Erhaltung de.r 
 Energie (Leipzig, 1913). 
 
 J See chapter ix, where this particular case is discussed in detail. 
 
56 The definition of the electrostatic field [CH. i 
 
 process . as often as we please and thus get an unlimited quantity of work, 
 and all out of nothing. This is reasonably taken to be incredible as it 
 involves the idea of perpetual motion : the essence of the matter is the 
 unlimited extent. 
 
 Thus the argument from perpetual motion shows that for any natural 
 
 law of action across space 
 
 /i 
 I T s ds 
 
 is independent of the path described in going from position 1 to position 2. 
 The only other assumption involved is that of the reversibility of each path. 
 In general this condition is satisfied. 
 
 The above argument is restricted to the case in which the systems M l , 
 M 2 , ... were supposed to be uninfluenced by the motion of 8m. We can 
 however easily remove this restriction and consider for example the case 
 where Mj, M 2 , ... are systems of charged conductors, when the moving of 
 a small charge 8q about alters the distribution on the conductors by influence. 
 However even in this case if we take 8q round a closed path so that at the 
 end the distribution is everywhere the same as the original one, the work 
 done in the complete cycle must still be zero. There cannot be any loss of 
 work for if it is reversible we could promptly turn it into a gain by reversing it. 
 
 Thus in the most general case 
 
 2 
 
 T s ds 
 i 
 
 is independent of the path from position 1 to 2. That is there must be a 
 function < of the position of 8m such that 
 
 is true for any two positions of the points 1 and 2. If we can find this function 
 <b then 
 
 The function O represents for a definite position of the element 8m the 
 work done on it by the systems M lf M z , ... in bringing the element from a 
 standard position assumed to correspond to the value 3) = and this work 
 may be utilised by external agents for any ulterior purpose that may be 
 desired. This is interpreted in the usual way by saying that the element 
 8m possesses a store of potential energy in virtue of its position relative to 
 the systems M 1} M 2 , ..., the amount in the typical position being <I> more 
 than that in the standard position for which O = 0. 
 
 60. If we now regard the physical systems as the system or systems of 
 charged bodies acting across space in the manner specified, or in fact in any 
 
59-61] The energy of the field 57 
 
 manner, and if 8m is a small charged body carrying the quantity 8q of 
 electricity then we know that 
 
 and thus O = (f> 8m + const. 
 
 and the existence of the potential energy function <I> implies the existence 
 of the analytical potential function tj> and vice versa. 
 
 This proof of the existence of the potential function of our analytical 
 theory rests on a physical basis. It assumes nothing about the method of 
 transmission of the force, but merely that the effects are reversible and 
 that perpetual motion does not exist. If there were friction effects or if the 
 charge Sq were moved about rapidly electric currents and perhaps also electro- 
 magnetic waves would be excited which would result in heat production and 
 the essential condition of reversibility would then be lost. 
 
 In the whole of this discussion we have neglected altogether the store of 
 energy possessed by every system in the form of heat. Why should not 
 work come out of this store of heat? Such a question is easily answered 
 by an appeal to Carnot's principle. The Carnot Axiom states that if we have 
 systems in thermal equilibrium then it is impossible for work to be done at 
 the expense of the heat they contain. The test of thermal equilibrium is 
 equality of temperature. Thus to make the above argument correct we 
 must put in the criterion of equal temperatures. If two bodies were at 
 different temperatures they could be used as a heat engine from which work 
 could be obtained. 
 
 61. The energy in the electrostatic field. We have found that in order 
 to establish an electric field by bringing the charges which define it. into 
 position (by friction, conduction, etc.) a certain amount of work has to be 
 done but that when once the field is established its maintenance requires 
 the expenditure of no work, provided of course there is no leakage to be 
 counteracted. Thus there is a certain amount of work associated with each 
 electric field, and this amount must be independent of the method of estab- 
 lishing the field in order that the energy principle may be verified : it measures 
 the amount of energy in the electrostatic field relative to the same group 
 of masses in their uncharged state, which would be transformed into other 
 forms of energy if the charges were removed or cancelled. 
 
 A theory of the present type regards the electrical conditions in any field 
 as characteristic of the electric charges in the field, the distribution of these 
 charges being the most essential thing required for the specification of the 
 system. In such a theory we therefore require a definition of the electric 
 energy which makes it depend on the charges and potentials. This is readily 
 obtained in the following way. 
 
58 The definition of the electrostatic field (~CH. i 
 
 The work required to bring up a small charge Sq to any place where the 
 potential is </> is, by the generalised definition of the potential function, equal 
 
 to 
 
 f&q. 
 
 This statement is valid and consistent whatever the complexity of the field. 
 
 Thus the work required to increase the density of the volume charge at 
 any point of the field by $p and the density of any surface charge by Scr is 
 
 the first integral being extended to all points of space where there is a volume 
 charge p and the second over all surfaces/ on which there is a surface charge a*. 
 
 This is the fundamental differential equation of the subject, representing 
 as it does, in a differential form, the characteristic equation of energy for 
 the system. If we can by any process succeed in integrating this equation 
 we shall be in a position to know the complete mechanical circumstances of 
 the system. 
 
 But in bringing up these charges the potential at each point of the field 
 is increased by </> so that the potential energy of the charges already existing 
 in the field is increased by the amount $W 2 where 
 
 SJF = 
 
 2 
 
 f 
 
 But if the mechanical process of establishing the charges in the field is a 
 reversible one so that all operations involved in it can be reversed and this 
 is essential to the existence of a potential function the work which is done 
 in charging the system must all be stored up as potential energy of a purely 
 electrical nature in the field so that 
 
 and thus each of these is equal to the half of their sum or 
 8W 1 = STF, = i (8W ! + -BW 2 ) 
 
 dv 
 
 We have therefore in such a case 
 
 * Double sheets are excluded as they are of relatively small importance in the present aspects 
 of the theory. It is however quite easy to generalise the discussion to include them. 
 
 f Kelvin, Glasgow Phil. Soc. Proc. m. (1853); Helmholtz, I.e. p. 55; Clausius, Die Potential 
 Funktion (Leipzig, 1859), 63 and 64. 
 
61-63] The energy of the field 59 
 
 Thus if we multiply each element of charge by half the potential at its position 
 and add up over the whole distribution we get the general value for the 
 potential energy of the electrical system referred to the state in which 
 p = a = everywhere as the zero state. 
 
 62. The result here deduced in a physical manner can also be obtained 
 analytically as follows. We know that 
 
 r : / r 
 
 Thus if we use dashed letters to denote these integrals in </> we see that 
 xw ifp'Spdvdv' f >'a'8pdvdf ff p'dadfdv' j" I'a'dadfdf 
 
 Wl= " ~V~ "V r ' f .'.' / / f.y~~7~ J 
 where r' denotes the distance between the typical elements of the double 
 integrations over the volume of the field and the surfaces of discontinuity 
 in it. 
 
 It follows immediately from this form of the expression, by integrating 
 with respect to the undashed elements first a process that is fully justified 
 in view of the absolute convergence of the integrals concerned that 
 
 SH 7 ! = I p'fy'dv' + I a'fy'df, 
 where 8<f> = 
 
 r 
 
 is the increment of the potential at the typical field point consequent on the 
 addition of the charges Sp and Scr. We thus see that 
 
 and the result deduced above now follows immediately by the same argument. 
 
 63. We have thus succeeded in establishing the existence of a mechanical 
 potential energy function associated with our electrically charged system 
 of a type similar to that with which we are familiar in ordinary mechanics 
 and the usual properties of this function now follow as a matter of course. 
 A system of charges on a rigid system of masses will adjust themselves in 
 such a way as to make the electrical potential energy function W^ stationary 
 subject only to the constancy of the total charge and the conditions implied 
 by the natural restraints of the system. In fact the mutual forces exerted 
 between the various charge elements tend to produce motion of these elements 
 and if the energy of such motions can be obtained at the expense of the 
 internal store of potential energy they are almost certain to take place. 
 Thus equilibrium is possible only in those configurations in which the potential 
 energy has a stationary value as regards small displacements from the con- 
 figuration, because it is only then that the small initial displacement from 
 the configuration results in no appreciable change in the store of potential 
 
60 The definition of the electrostatic field [en. i 
 
 energy from which the kinetic energy of any further motions that take place 
 must be derived. 
 
 Moreover it appears that if any configuration of the system is a con- 
 figuration of stable equilibrium as regards the charge distribution throughout 
 it, the stationary value of the potential energy must be an absolute minimum 
 value, for it is only in such a case that the system will not, if slightly disturbed, 
 depart widely from the configuration by the action of its own internal forces. 
 Thus it is only when the natural restraints* of the system prevent any 
 further running down of the electrical potential energy that equilibrium 
 among the charges is permanently possible. 
 
 64. If we apply this condition for the equilibrium of a system of charges 
 some or all of the elements in which are capable of free movement within 
 certain limited spaces we see that the potential function <f> must be constant 
 throughout any space in which the charges are freely movable. This follows 
 at once because the condition for equilibrium is that the variation of the 
 potential energy consequent on a slight rearrangement of the charges, which is 
 
 '/ 
 
 must vanish subject to the condition that the total charge in each partial 
 space must be constant, or that 
 
 '/ 
 
 where the volume integral is taken throughout the partial space and the 
 surface integral over the boundary of that space and any surfaces of dis- 
 continuity inside it. This leads to the result that </> has a constant value 
 throughout the partial space. 
 
 65. The importance of this last result lies in its application to the field 
 of a number of charged conductors. The elements of a charge on a conductor 
 are, effectively speaking, freely movable throughout the material of the 
 conductor but not beyond its outer surface. Thus in order that the charge 
 distribution may be one of stable equilibrium it must be such that the potential 
 </> is constant throughout the volume of each conductor. We shall see later 
 that this means that the charge on the conductor exists entirely on its outer 
 surface ; this of course also results from the fact that the greatest dispersion 
 then exists in the group of elements constituting the charge and the least 
 value of the potential energy is attained. 
 
 In the sequel these fundamental results here deduced as consequences of 
 the energy principle, will be discussed from a rather different standpoint and 
 
 * The insulation of a conductor acts as a ' restraint ' to prevent the charge on it getting across 
 to another conductor. 
 
63-66] Gauss' reciprocal theorem 61 
 
 certain additional details will be obtained respecting them. For the present 
 it is sufficient once more to emphasise the fact that so long as we confine 
 ourselves to static systems everything is summed up in the doctrine of energy. 
 
 66. Gauss' Reciprocal Theorem. There is an important reciprocal 
 theorem due to Gauss* which is worth quoting at this stage : we shall first 
 give it in terms of discrete charge elements and then indicate its extension to 
 the more general case. 
 
 Let q be an element of charge in any distribution and </> the potential of 
 that distribution at any point and in a second system let these be q' and </>' ; 
 
 we have then _, . . . , 
 
 2gc/> = 20 q, 
 
 where in <?</>' every element of the first distribution is multiplied by the 
 potential of the second distribution at the position of the element, and 
 similarly in 2<7'</>. 
 
 The usual proof of this theorem in the present case is that each sum is 
 equal to the double sum 
 
 y W' 
 r ' 
 
 wherein the summation extends to each element of charge of the one system 
 
 with each element of the other system and r is the distance between them. 
 
 A more fundamental interpretation of the relation is obtained by noticing 
 
 that 
 
 is the work required to bring up the first charged system supposed rigidly 
 fixed in its final relative configuration into its position relative to the second, 
 and that therefore this must be equal to the work required to bring up the 
 second charge system into its position relative to the first which is expressed 
 by the second sum ^q'(f> ; these are in fact merely two different ways of 
 establishing the combined field. 
 
 In this sense we see that it must also be true in the more general case 
 with effectively continuous charge distributions, or in other words if the two 
 systems of charges are specified by their charge densities p and p' at the 
 typical point of space and if the potentials of their respective fields are <f> 
 and (f>' then 
 
 | p(f)'dv = I p'(f>dv, 
 
 the integrals in each case being taken throughout the entire charge distribu- 
 tion. 
 
 If the second system of charges is the first increased by very small 
 increments Sp then <f>' differs from (f> only by a differential amount 8<f> and 
 
 thus we may put , 
 
 p = p + 8p, <f> = tf> + btf>, 
 
 * ' Allgeiru-ini; Lehrsatz^ iiber...Anziehung:- und Abstcmuneskiafte,' Art. 19 (Collected Works, 
 v. p. 200). 
 
62 The definition of the electrostatic field [CH. i 
 
 and on substituting these values in the above relation we see at once that 
 
 (f>8pdv, 
 a relation deduced above from the energy principle. 
 
 67. The mechanical forces on the matter in the field. The analytical 
 theory up to the present stage has been concerned merely with the electro- 
 motive forces of the field, the question of the ponderomotive forces has so far 
 not arisen : we have only explained how charges are separated and not how 
 electrified bodies attract one another. 
 
 It is however at once evident that such forces must exist. The electro- 
 motive force acting on a charge connected with a material body and in 
 equilibrium must be counterbalanced by an equal and opposite force resulting 
 from the action of the material body on the same charge : and the reaction 
 to this latter force will be an equal and opposite force exerted from the 
 charge on the material medium with which it is rigidly connected. Thus 
 any material body carrying a charge and in equilibrium will be acted upon 
 by a force equal to and in the same direction as the resultant electromotive 
 force on the system of charges contained in it. Thus if p denote the density 
 of the charge distribution at the point (x, y, z) in the body where the intensity 
 of the electric force is E, then there will be a ponderomotive force on the 
 body determined by 
 
 F! - I pEdv, 
 
 the integral being extended throughout the volume of the body at no point 
 of which is p infinite. 
 
 This is the general form ; but it is convenient to have the specialisation 
 of it applicable when there are surface distributions associated with the 
 body. This is easily obtained by considering such a surface distribution as 
 the limiting case of a volume density concentrated in a thin layer. On this 
 view the force on the small element df of the surface is practically 
 
 Epdv, 
 
 this integral being taken throughout the small volume of the sheet standing 
 on df. Now E varies continuously throughout the sheet from a value E x 
 on one side to a value E 2 on the other and thus the average value throughout 
 the sheet is | (E x + E 2 ) * and with this the force on df is 
 
 df = I (B x + E 2 ) \pdv = i (E! + E 2 ) adf, 
 
 * This statement and the whole proof depending on it is perhaps not as rigorous as might 
 be desired. The usual argument divides the force close up to the surface into a local and a general 
 part ; the local part is due to the small portion of the surface charge near the point and the general 
 
66-68] The ponderomotive forces 63 
 
 and thus for the whole surface charge 
 
 F 2 = | | (EjL + E 2 ) a (If, 
 
 and a proper combination of the forces F x and F 2 will give the correct form of 
 the force on any electrified body or at least the linear components of the 
 resultant force. The angular components can be obtained in an analogous 
 manner or they may be calculated in a general way from the results deduced 
 below where the question of these forces is regarded from another and more 
 general point of view. 
 
 68. The mere existence of mechanical forces on the matter in the 
 field is involved in the idea of the energy of the charged system. When 
 two charged bodies are moved relative to each other the total electrical 
 energy in the field is altered and if the charges are kept constant the loss (or 
 gain) of energy is due to some other system linked with the electrical one. 
 It reappears in fact as a gain (or loss) in the mechanical energy of the charged 
 bodies, which determines the mechanical forcive between them. Thus to 
 obtain the forces we need only give the bodies small virtual displacements 
 and include the virtual work in these displacements in the general expression 
 of the work done on the system during a general virtual change in its 
 configuration. 
 
 If the positions of the material bodies of the system are determined by 
 
 the generalised coordinates l5 6. 2 , ... in the usual Lagrangian sense and if 
 
 the internal force components corresponding to the coordinates are 1: 2 , ... 
 
 respectively, then the work done by external agency (which is applying forces 
 
 - u 2 , ... in the various coordinates) during a displacement is 
 
 jOC/! 2OC/2 ' 
 
 This is the generalised Lagrangian definition of a force component in statics : 
 it is defined so that the force multiplied into the small change in the coordinate 
 is the work done, provided none of the other coordinates vary. 
 
 The work done in increasing the charge distribution of the system is 
 
 if the bodies are fixed; if however in addition the matter receives a small 
 virtual displacement as above we must add the work done against the forces 
 acting in these displacements. The general form for the work done on the 
 system during the most general virtual change in its configuration (a complete 
 
 part to the remainder. For two near points equidistant from the surface but on opposite sides 
 the local parts are equal in magnitude but opposite in direction whilst the general parts are the 
 same at both points. The general part of the force which alone is mechanically effective is then 
 equal to the mean of the total forces at the two points. 
 
64 The definition of the electrostatic field [CH. i 
 
 definition of a configuration involving a knowledge of the charge distribution 
 and the positions of the material bodies) is thus 
 
 >8adf- Q l S0 l - a 80 2 - ...; 
 
 and again the usual argument based on the assumption of reversibility and 
 the negation of perpetual motion requires that &W should be a complete 
 differential of some function W which ultimately measures relative to some 
 standard configuration the potential energy which the electrified system 
 possesses in virtue of its charge. In other words W is the electrical potential 
 energy of the system. 
 
 We know however from the discussions of the previous section that the 
 electrical potential energy of the system can be expressed in the form 
 
 so that as soon as these integrals can be effected W is known and the complete 
 mechanical relations of the system are theoretically determinate. 
 
 69. If the charge distribution on the system of masses is maintained 
 constant during the slight displacement of the system the first part of the 
 total expression for STF does not occur and the increase in the electrical 
 potential is simply given by 
 
 &W C = -0^0! -0*802..., 
 
 where W c denotes exactly the same quantity as W above but the suffix 
 implies that the charge distribution throughout each body is maintained 
 constant during a displacement of that body. In this case the work of the 
 internal forces, viz. 
 
 0^0! + 0Z80JJ+ ... 
 
 which can be used to drive a machine outside the system is derived solely 
 from the store of internal energy which the system of masses possesses in 
 virtue of the charges rigidly attached to them. 
 
 We have also in this case 
 
 3W_ C _ 
 
 "90, " 
 
 so that the force in any one of the material coordinates under the specified 
 conditions is determinate. 
 
 70. In some important cases the total charge and its distribution are 
 altered in such a way as to maintain the potential distribution throughout 
 the various masses constant. In this case the work done on the system during 
 a small virtual displacement will involve the complete expression given above 
 but this is not now in a convenient form as it requires a knowledge of the 
 distribution of Sp and So- necessary to secure the maintenance of the potential 
 
68-71] The ponderomotive forces 65 
 
 distribution. To obtain a more suitable form for such cases we have only to 
 rewrite it in the form 
 
 8(\p<f>dv + f a<j>df- W}= I pS<f>dv + ! aS<f)df + 0^^+ ..., 
 
 \J J f / J J f 
 
 and then notice that since in the present instance 
 
 a<f>df=W, 
 
 f 
 
 the term on the left is still 8W, so that 
 
 3F= ! P S<f>dv+ I aS<f>df+ 1 8^ 1 + .... 
 
 J J f 
 
 71. We now see immediately that when the potential distribution 
 throughout the various masses is maintained constant throughout the dis- 
 placement the change of the internal potential energy is 
 
 W v = Q 1 80 1 + 2 80 2 + ... 
 
 where we have used W v to denote the value of W in which the potential 
 distribution in the various masses is maintained constant. 
 Again, as above, Q^ + 9^ + _ 
 
 represents the energy expended by the system in the mechanical work of 
 shifting the masses and ^ 
 
 is the increase in the internal potential energy of the system of charged 
 masses. These quantities are not now of opposite sign so that some outside 
 source must provide both parts. The only available source of energy is, 
 generally speaking, that which is used to maintain the potential distribution 
 and thus the amount of energy derived from it is double the amount of the 
 mechanical work gained from tlie system ; the other half of the total supply 
 goes to increase the internal potential energy of the charge distribution. 
 
 We have also in this case 
 
 so that 
 
 90 S " a0 s 
 
 Thus a variation of the configuration of the system which increases the internal 
 energy when the charge distribution is maintained constant decreases this 
 energy when the potential distribution is maintained constant. 
 
CHAPTER II 
 
 THE CHARACTERISTIC PROPERTIES OF THE ELECTRIC FIELD 
 
 72. Some particular types of electric fields. Having now obtained 
 consistent definitions of the analytical functions determining the field of any 
 distribution of electric charges we may proceed to examine the properties 
 of these functions which are characteristic of the fields to which they appertain. 
 Before however entering on the general examination it seems desirable to 
 consider the form which the definitions assume in the case of certain simple 
 specified distributions, with the view principally to obtaining some insight 
 into the analytical nature of the functions involved. Most distributions of 
 charge may be regarded as more or less approximately composed of a certain 
 number of simple distributions of standard type, so that if we know the 
 nature of the fields associated with these simple distributions we shall be in 
 a position to obtain an approximate estimate at least of the nature of any 
 more complex distribution. 
 
 It is of course not necessary in each case to determine all the integrals 
 discussed in the last chapter. When the integral for the potential is known 
 in any case the components of force in any direction may be most simply 
 derived as the component of the vector 
 
 E = - V</> 
 
 or as the negative gradient of the potential in that direction, and it is not 
 necessary to evaluate separately the integrals for the force components. We 
 shall therefore merely discuss the integral for the potential function in the 
 separate cases. 
 
 73. The potential of a linear distribution of charge. The charge is con- 
 tinuously distributed along a line of continuous curvature. The charge on 
 the element of length ds is dq and 
 
 da 
 
 v = ^r 
 ds 
 
 is the line density ; the conception of which depends on the existence of the 
 differential coefficient expressing it. Physically the charge is so concentrated 
 around a line that for all practical purposes it is convenient to regard it as 
 actually on the line. The potential function is 
 
 [dq [vds 
 
f 
 
 72, 73] The rectilinear distribution 67 
 
 The simplest case is the straight line with constant linear density v. 
 Choose coordinate axes as usual and suppose the line lies in the axis of x 
 between x = a and x= - b ; the point P at which we wish to calculate the 
 potential being at a distance r along the z-axis ; then 
 
 , 
 <?> = 
 
 o J Vr* + x 2 
 
 _ __ a 
 
 log x + Vr 2 + x 2 + v log x + Vx 2 + r 2 
 o 
 
 A a + Va 2 + r 2 b + Vb 2 + r 2 \ 
 
 = v(\og --- - + I g -- - ---- J. 
 
 The equi-potentials i.e. the surfaces over which <f> is constant are in this 
 case confocal spheroids. For we can write 
 
 a + Va 2 + r 2 Vb 2 ^ 2 - b\ 
 
 - -log- -X --- J 
 
 Vo*T r 2 + a Vb 2 '+~r* + b 
 
 = V lOg r^. - = V log -- 
 
 Vb 2 + r 2 - b & Va 2 + r 2 -a 
 
 so that if we use r l = Va 2 + r 2 , r. z = Vb 2 + r 2 , c = a + b, then 
 
 
 
 a + r 1 = l^ (r 2 - 6), 
 
 * 
 
 b + r 2 = l v (r 1 - a), 
 
 * 
 
 or TJ_ + r 2 == c . - c coth $- . 
 
 I* 1 
 
 The interesting result for our purposes is however the behaviour of </. as 
 the point P approaches the line ; we make r very small and thus find that 
 
 Lt + = - 2v, 
 r+o logr 
 
 and also since generally 
 
 r 
 
 (a + Va 2 + r 2 ) Va 2 + r 2 
 we see that 
 
 Lt r d /=-2v 
 r^oo or 
 
 as well. Thus the potential becomes infinite as P approaches the line, but 
 the degree of the infinity is definitely calculable. 
 If the line is of infinite length we have similarly 
 
 (j> = C - 2v log r, 
 _d$_2v 
 dr r ' 
 where C is a very large constant of the order log a. 
 
 52 
 
68 The characteristic properties of the electric field [CH. n 
 
 Kesults exactly similar to these can be proved to hold, whatever the form 
 of the line on which the distribution is given. As the point P approaches 
 
 J4J 
 
 the curve any infinities and discontinuities in </> or ^ can only arise from the 
 
 small portion of the curve in the neighbourhood of the foot of the normal 
 from P, .which piece may, under the assumption of continuous curvature, 
 be considered as practically straight and of very great length compared with 
 the distance of P from it. 
 
 74. The potential of a constant surface density a over a sphere of radius a. 
 In this case the field is obviously symmetrical about the centre of the 
 sphere. To calculate the potential at the point P distant r from the centre 
 C of the sphere we resolve the sphere into small rings with their planes 
 perpendicular to CP. The elements of charge on any one of these rings are 
 at the same distance from P and thus the contribution to the potential due 
 to the ring of radius a sin d is 
 
 2?ra sin 6 . 
 
 o<p = =r- . 
 
 V a 2 + r 2 2ar cos 6 
 
 Now write z 2 = a 2 + r 2 2ar cos 6, 
 
 so that zdz = ar sin 6dd, 
 
 and then this element of the potential is 
 
 zdr 7 
 
 80 = - - dz. 
 
 z 
 
 z - , 2-n-aa 
 Thus = - - dz = - 
 
 f V 
 
 I . Zi 
 
 We must now distinguish the cases when P is inside or outside the sphere. 
 (i) P inside : the limits for z are z v = a r and z. 2 = a + r and thus 
 
 <, = 477-aor. 
 
 (ii) P outside : the limits for z are z l = r a and z 2 = a + r and thus 
 
 or using Q = 4-7ra 2 o- as the total charge on the sphere 
 
 Q 
 
 
 
 The external potential thus appears as if the single charge Q were collected 
 at the centre. 
 
 The internal and external potentials have the same value at the surface 
 of the sphere and are otherwise continuous functions in their respective 
 regions. 
 
73-75] The spherical and circular distributions 
 
 The force intensity in the field is now derived as the negative gradient 
 of the potential. The form of this latter function suggests that it is convenient 
 to employ a spherical polar coordinate system of reference for the field, with 
 the pole at the centre of the sphere. If this is done it is seen that the resultant 
 force intensity is purely radial and of amount 
 
 Q >_ _<ty 
 
 r 2 dr 
 
 outside the sphere and zero inside it. 
 
 The normal differential coefficients or the force intensity at the surface 
 are 
 
 Ef = [Tc-M = 0, 
 
 7T fi<t>0\ , Q 
 
 = ~ \ ~d I 2 ' 
 
 so that at the surface there is a sudden jump in the value of the normal 
 differential rate of variation of the potential 
 
 T7t TJ A 
 
 &i &Q = 47TCT. 
 
 which is independent of the size of the sphere. 
 
 75. The potential and force of a fiat circular disc uniformly charged at a 
 point on its axis. We again make the calculation by dividing the disc into 
 concentric rings : r is the radius of the elementary ring and we calculate 
 the potential at a point distant z along the axis. 
 
 The contribution to the potential from the elementary ring is 
 
 . 
 
 V z 2 + r 2 
 Thus in all 
 
 ra r fj r _ 
 
 <f> = 2 . = 2 (Va 2 + z 2 - z). 
 
 JoVz 2 + r 2 
 
 The force is directed along the axis and is of amount 
 
 E=-f = 2 7 ra(l- =} 
 
 dz \ Va 2 + z 2 / 
 
 quite near the disc 
 
 tf> = 2 (a - z} and E = 2 (l - -}. 
 
 \ aj 
 
 Thus on the disc itself the force is l-na away from the disc and so jumps by 
 47rc7 as we pass through in a definite direction. The potential is continuous 
 from one side to the other. 
 
 For a very large plate the values are approximately 
 
 (f> = C 2-n-crz, 
 
 C being a large constant of the order of the radius of the plate. 
 
70 The characteristic properties of the electric field [~CH. n 
 
 The results for a point not on the axis of the disc are very complex and 
 will be derived by another method in the next section. 
 
 76. The uniform volume distribution of charge through a, sphere. The 
 volume density p of the distribution of charge is constant throughout the 
 sphere r = a so that the conditions in the field are again obviously sym- 
 metrical round the sphere. We calculate the potential at a point P distant 
 r from the centre C of the sphere. To do this we divide the sphere into 
 small concentric spherical shells, which will be uniformly charged ; if the 
 radii of a shell are t and t + dt, then the total charge inside it is farpfibt and 
 this is distributed over the surface like a surface charge of density p$t. We 
 can thus apply the results obtained for a uniform spherical shell. 
 
 We must distinguish the cases where P is inside or outside the sphere. 
 
 (i) If P is outside the sphere it is outside all the elementary shells, and 
 the corresponding contribution to the potential from each shell is 
 
 so that c/> = 
 
 r 
 
 A ^ 
 
 or if we use Q = ~ for the total charge on the sphere 
 
 _Q _ty> = Q 
 
 90 r ' ~ dr r*' 
 
 (ii) If P is inside the sphere it will be an external point for the shells 
 of radii t < r and an internal point for shells of radii t > r, and we shall there- 
 fore have the potential at P consisting of two parts 
 
 . o 
 
 [ a 
 I 
 
 j r 
 
 (a z 
 2 - 
 
 3a 2 r 2 \* 
 
 and - f 
 
 Thus at the surface both force and potentiaj are continuous. 
 
 Also notice that if we use Cartesian coordinates with the origin at C 
 
 r 2 = X 2 + y2+ Z 2 } 
 
 * Laplace, Thforie du mouvement et de la figure des planetes, Paris, 1784, p. 89 ((Euvr. x. 
 p. 349). 
 
The ellipsoidal distributions 
 
 3a 2 x 2 + + 
 
 71 
 
 75-77] 
 and thus 
 
 so that 
 
 a result which will be proved to be more general than the particular conditions 
 here implied would indicate. 
 
 77. The potential of a homogeneous charge distribution throughout the 
 interior of the volume of the ellipsoid bounded by 
 
 X 2 U 2 Z 2 
 
 a 2 b 2 T c 2 
 
 1*. 
 
 We first calculate the potential for an internal point P. Consider the 
 attraction of an elementary cone EE' 
 with its vertex at P and of solid angle 
 doj. The potentials of the two portions 
 of charge at P are 
 
 PE*dw and PE' 2 da>, 
 
 or, say, 
 
 Thus 
 
 i 2 + r a 2 ) 
 
 Now if (cc, y, 2) are the coordinates of 
 
 P relative to the principal axes of the ellipsoid and (I, m, n) the direction 
 cosines of PE, then r x and r 2 are the roots of the quadratic equation 
 
 (x + rl) 2 (y + rm) z (z + rn) 2 _ 
 ~~a^~ 6 2 ~1*~ 
 
 Thus 
 
 U + 7 
 
 where 
 
 Now obviously 
 and thus we have 
 
 I 2 m 2 
 
 77 _ i _ _ v_ 
 
 a 2 b 2 c 2 ' 
 
 fmnyz nlzx lmxy\ 
 Vb^c 2 + cW + a'&V' 
 
 Vda> 
 
 I 2 m 2 n z \ 2 
 
 9 "I 
 
 2 c 
 
 f I 2 
 
 \ ~ 9 
 
 \a 2 
 
 = 0, 
 
 - ^- ~ or- a' 
 
 a da b cb c cc 
 
 * J. Soraoff, St Petersburg Bull. xix. (1873), p. 215. See also Gauss, Werke, v. p. 296. 
 
72 The characteristic properties of the electric field [CH. n 
 
 f da> 
 
 where . <P = -75 5 s . 
 
 I I 2 m n* 
 
 ] a 2 b 2 c 2 
 
 To evaluate this last integral we use ordinary spherical polar coordinates 
 with the x-axis as polar axis and P as origin. We have then 
 
 I = cos 6, m = sin cos </>, n = sin sin </>, da> = sin 6dB d<f), 
 so that 
 
 sin 6d6d(f> 
 
 sin 2 cos 2 0Wsn'0co8 2 
 
 _ , /sin 2 cos 2 
 
 cos n "F + ~s 
 
 -277 
 
 Now put t = a 2 tan 2 and then the integral reduces to 
 
 dt 
 
 o V(a 2 + (6 2 + (c 2 + 
 The potential at the point P is therefore 
 
 or, say, <^ = ^ 
 
 r /Jf 
 
 where A = Trpabc - 
 
 .' o (a 2 + V(a 2 + t) (b 2 + (c 2 + ) 
 
 The components of force are 
 
 (E x , E y , E z }= -2 (Ax, By, Cz). 
 
 The internal equipotential surfaces are ellipsoidal, the squares of the axes 
 being in the inverse ratio of A : B : C. 
 
 78. The potential at an external point is now best obtained by a 
 theorem due to Maclaurin which may be stated in the following form: 
 
 Any two confocal homogeneous solid ellipsoids of equal total charge 
 produce equal attraction throughout all space external to both*. 
 
 To prove this we notice that of two uniformly charged confocal ellipsoids 
 (a, b, c and a', b'. c') a line element PQ of the inner one parallel to the a-axis 
 
 * Maclaurin proved special cases of this theorem. Cf. Grube, Zeitschr. Math. Phys. xiv. (1869), 
 p. 261. Laplace proved the general theorem, I.e. p. 70. 
 
77, 78] The ellipsoidal distributions 73 
 
 will attract a point R r on the outer with a force in the direction of its length 
 
 equal to pdydz (>.,>, ^7^), if dudz is its cross sectional area and p the 
 \R Q R PJ 
 
 density of charge in it. But if P', Q', R are the points corresponding to 
 (P, Q, R') the line charge P'Q' will pull the corresponding point R in the 
 same direction with a force 
 
 but these are in the ratio pdydz : p'dy'dz'. i.e. pbc : p'b'c'. We thus conclude 
 that the attraction of the inner ellipsoid parallel to the x-axis at the point R 
 
 Fig. 11 
 
 outside it (x, y, z) is the same as that of the outer confocal in the same direction 
 at the internal point xyz ( = , -^ , -7] increased in the ratio pbc : p'b'c' ; 
 
 \ Ct C / 
 
 but this latter is 
 
 E ' = + V6V s_ f <ft 
 
 ' J o (a 2 + t) V(a' z + t) (6' 2 + i) (V 2 +7) * 
 and so the former is 
 
 f 00 dt 
 
 ^x = + Trpabcx 
 
 o (a' 2 + i) V(a' 2 + (6' 2 + t) (c' 2 + ) ' 
 or if a' 2 - a 2 + A, 6' 2 - 6 2 + A, c' 2 - c 2 + A this is 
 
 dt 
 + Trpabcx 
 
 J A (a 2 + f ) V(a 2 + (6 2 + (c 2 + 
 with similar results for E v , E z \ Maclaurin's theorem results immediately. 
 
 Thus if we take two such charged confocals and a point external to both 
 of them, the separate attractions at that point are in the same direction and 
 proportional to the total charges. They must therefore have their external 
 potentials in the ratio of their charges. The potential at the external point 
 is therefore* 
 
 \ 
 
 , lui 
 
 <p = TToabc 1 
 
 . x V a 
 
 2 + t b* + t c 2 + t V(a 2 + t) (6 2 + (c 2 + t) 
 
 where of course A is the positive root of the equation 
 _J^ y 2 z 2 
 
 o^X + & a ~+A + ^TA = 
 
 the point (x, y, z) being on this confocal. 
 
 * Chasles, Jour, de math. v. (1840), p. 465. 
 
74 The characteristic properties of the electric field [CH. n 
 
 This is the general result. One or two particular forms of it as applied 
 to special cases are however of particular interest for our future work 
 
 79. The homoeoidal ellipsoidal shell with a uniform volume charge through- 
 out its thickness. The potential of a uniform charge distribution in the shell 
 between two ellipsoids is now easily obtained. We shall assume that the 
 ellipsoids are similar and similarly situated, their axes being (a, b, c) and 
 (Vl + 8) (a, b, c). The potential of this shell at an internal point is obviously 
 
 dt 
 
 . : 
 
 write t (1 + S) for t in the second integral and it reduces to the first except 
 for one term so that 
 
 dt 
 
 , , sj 00 
 
 (p i = irpaoco . , 
 
 J o V(a 2 + t) (b 2 + t) (c 2 + t) 
 
 and the potential is constant throughout the interior of the shell. 
 
 A similar argument shows that the potential at an external point (x, y, z) 
 on the A confocal is 
 
 * r\ t 2 
 
 if 30 \ 
 
 <j> = TTpabc I (1 , TT I 
 
 JA a 2 + t V( 2 
 
 dt 
 
 i 2 + t) (b 2 + t) (c 2 + 
 + 
 where A' is the positive root of 
 
 t 
 
 The most important application of these results is to the limiting case 
 when the shell is very thin but p so very big that we have practically a surface 
 charge distribution on the one ellipsoid. The surface density at any point 
 will be ultimately proportional to the normal distance between the two 
 surfaces. To obtain this distance notice that the tangent plane to the first 
 ellipsoid normal to the direction (I, m, n) is at a distance p from the origin 
 given by 
 
 p* = 
 
 the parallel tangent plane to the other surface being at a distance (p + dp) 
 
 where 
 
 (p + dp) 2 = (a 2 l 2 + 6 2 m 2 + c 2 n 2 ) (1 + S) 
 
 = P 2 (1 + S), 
 so that ultimately dp = ^- is the thickness of the shell. 
 
78-80] The ellipsoidal distributions 75 
 
 The total charge is 
 
 Q = %7rpabc (1 + 8 f - 1) 
 
 = knpabc ^. 
 
 A 
 
 The surface density is 
 
 and varies as /). 
 
 The potential function of such a distribution is : 
 (i) Internal 
 
 V (a 2 + "t) (b 2 + t) (c 2 + 
 (ii) Outside 
 
 i _Q [ dt 
 
 2 J A vV + 0(& 2 + 0(c* + 0' 
 80. We write the last equation but one in the form 
 
 t* = c> 
 
 so that 
 
 o V(a 2 + t) (6 2 + *) (c 2 + 
 The values of this function in special cases will be frequently required. 
 
 (i) Oblate spheroid : b = c the larger axes and a is the smaller : if we 
 
 - 
 use * = - - we easily verify that 
 
 dt 1 
 
 = tan" 1 e, 
 
 o V(a 2 + (b 2 + t) 2 
 
 b 2 a 2 
 or if we use 2 = ' , ae = bt, 
 
 and then r< = M- t 9 - 11 " 1 ^*- 
 
 (ii) For the circular plate a = and 
 
 I * 
 
 C 7T 
 
 (iii) For the sphere a = b = c and 
 
 (7 = a. 
 
 (iv) For the prolate spheroid : c is the axis of rotation and a = b (< c) ; 
 put 
 
 2 ' 
 
 * Cf. Riemann-Weber, Partielle J)ifferentialgleichungen der mathematischen Physik. i. p. 236. 
 R. Gans, Zeitschr.fiir Math. u. P%S..XLIX. (1903), p. 298, LIII. (1906), p. 434. (_' 
 
76 The characteristic properties of the electric field [CH. n 
 
 
 then we- get ~ = log rj + Vl + r 
 
 C at] 
 
 If a is small compared with c we have a very long thin ellipsoid 
 
 a 2 
 =!--- approximately, 
 
 1 1.0 
 
 6<=c log a- 
 
 It is important to notice that in the case of the circular plate the limiting 
 value of the surface density of the charge is 
 
 where r z = y z + z 2 . 
 
 The density becomes infinite at the edge of the plate. 
 
 81. Green's analysis of the electric field. We can now proceed to a 
 discussion of the more important properties characteristic of the general 
 electrostatic field in which the force and potential are defined by the 
 integrals given in the last chapter. The most direct method of approach is 
 that provided by Green's analysis, an account of which has already been 
 given above. In fact we have seen that any function <f> defined analytically 
 in the whole of space by 
 
 fpdv adf f 8 /1\ ,, 
 
 = ~ >- - + T a- ( - ) d f> 
 
 J r J f r J f dn\rj 
 
 where the first integral is taken over the whole of space and the second and 
 third over those surfaces on which a and T exist is subject to the usual con- 
 ditions of continuity and regularity, that 
 
 V*<f> = 
 
 at every point of the field and 
 
 <f> + <f>- = 47TT 
 
 at any point of the surface or surfaces/ : and conversely any regular function 
 <f> satisfying these three differential conditions is uniquely determined in the 
 integral form. 
 
 But we have seen that such a function </> is the potential of the ideal 
 charge distribution specified by a volume density p throughout the whole 
 
80, 81] Green's analysis of the field 77 
 
 of space and a surface density a and double sheet T on the surfaces/. We 
 therefore conclude that the potential function of any electric field defined 
 in the manner previously specified must be subject to the following conditions. 
 
 (i) It must be such that 
 
 V 2 < + 477/> = 
 
 at all points of the field where there is a finite volume density p : and where 
 P = Q 
 
 vv = o. 
 
 T$e first of these equations contains Poisson's extension* of the second, 
 which is Laplace's equation f. It expresses the general characteristic pro- 
 perty of the potential function in its differential form and provides us with 
 a test that any stated function is a potential function. 
 
 (ii) At any point on the surface distributions of charge 
 
 and </> + </>_ 4?rT = 0, 
 
 where a is the density of the surface distribution and T the moment of the 
 double sheet at the point of the surfaces. 
 
 These two conditions are in reality merely the particular forms which 
 the general property expressed by Poisson's equation assumes when applied 
 to the respective limiting forms of distribution. Notice that in crossing 
 a simple surface distribution a the normal force only is discontinuous, the 
 potential and tangential forces being continuous, whereas in crossing a double 
 sheet the potential and tangential forces are discontinuous, but the normal 
 force is continuous. 
 
 (iii) (f) must be an otherwise continuous function regular everywhere in 
 the field; and also if the charge distribution is a finite one it satisfies the 
 conditions that the limiting values of 
 
 R<f> and R 2 l^\\, 
 
 remain finite when R, the distance of the field point from the finite origin 
 of coordinates, increases indefinitely. 
 
 * Nouveau Bulletin des Sciences par la Societe Philomathiquede Paris, ill. (1813), p. 388. Gauss 
 gave the first correct proof, Allgemeine Lehrsatze, ..., 9, 10. 
 
 t Par. Hist. 1782 [85], pp. 135, 252 (CEuvr. x., pp. 302, 278). Cf. also Mecanique Celeste, t. n. 
 J Poisson, Par. Mdm. 1811 [xn.], p. 30. Cauchy, Bull. Soc. Phil. 1815, p. 53. Green, Essay 
 'etc., 4. 
 
 Helmholtz, Ges. Abh. i. p. 489. 
 
 || These conditions are quoted by. Lejeune-Dirichlet, Jour. f. Math. xxxn. (1846), p. 80 ( Werke, 
 II. p. 40). 
 
78 The characteristic properties of the electric field [CH. n 
 
 We might sum this last condition up by saying that the function ^ is 
 regular everywhere and at infinity*. 
 
 Moreover the analysis shows that there is only one solution of these 
 conditions and that is the one we have got. 
 
 82. If therefore we can by any means obtain a solution of these con- 
 ditions for given values of p % a and T, then we have a complete specification 
 of the field of the given charges. The inverse problem in electrostatics 
 is the determination of such solutions. It is easily seen that the solution 
 required assumes the form 
 
 where O is an appropriate solution of the differential equation of Laplace, 
 
 V 2 O = 
 
 which must be so chosen that </> satisfies the specified boundary conditions 
 at the surface infinities in the charge distribution. 
 
 Now it appears that the first part of the complete solution thus obtained 
 is a perfectly continuous function of position with continuous first derivatives. 
 The complementary function <I> has therefore to take full account of the 
 discontinuities in </> and it is in fact the potential of the surface distributions 
 which give rise to these discontinuities. 
 
 The problem thus resolves itself into a determination of the function O 
 which satisfies the equation 
 
 V 2 o = o, 
 
 at all points of the field and 
 
 90 90 
 
 2 ~ \- 47TO- == 0, 
 
 on + on_ 
 
 O + - O_ - 47TT - 0, 
 
 at the surface distributions 
 
 The nature of the solution required will of course depend essentially on 
 the type of surface or surfaces on which the charge infinities exist, and it is 
 in fact only in the cases where these surfaces are of the simplest geometrical 
 form that a solution can be obtained at allf. It is moreover clear that any 
 desired solution will be obtained in its simplest form in terms of those coor- 
 dinates in which the equation to the surface or surfaces to which it is to be 
 appropriate is in its simplest form. Thus in dealing with any particular type 
 
 * I am reminded that this last expression is not a usual one : it appears however a useful 
 and concise method of expressing a definite property of such functions and saves detailed 
 repetition of the conditions in every case where they occur. 
 
 t Cf. Heine, Handbuch der Kugelfunktionen, 2nd ed., Berlin, 1878. Byerly, Fourier's Series 
 and Spherical, Cylindrical and Ellipsoidal Harmonics, Boston (U.S.A. ), 1893. Whittaker, Modern 
 Analysis, 2nd ed., Cambridge, 1915. 
 
81-84] Solutions of Laplace's equation 79 
 
 of surface it is desirable to begin by transferring the fundamental differential 
 equation to coordinates suitable for that surface, and then to tabulate the 
 different types of solution. We can then choose the particular solution 
 appropriate to the problem in hand by bringing in the surface conditions. 
 
 83. In its general cartesian form the equation 
 
 - - 
 
 ~ dx* dy* dz* 
 is obviously satisfied by the function 
 
 O =f(lx+ my + nz; I, m, n), 
 
 where / is any arbitrary* function of its argument and (I, m, n) are any 
 constants subject to the condition that 
 
 I 2 + m 2 + n 2 = 0. 
 We might for instance take 
 
 I = cos to, m sin to, n = * = V 1, 
 so that the function 
 
 O = F (x cos to + y sin to + iz, o>) 
 
 is a solution of the equation. To generalise this solution we can multiply 
 it by any arbitrary function of to and integrate between any two constant 
 limitsf > thus 
 
 <!> = 
 
 F (x cos to -f y sin to + iz, to) da> 
 
 
 
 is a solution w T hich is due to Whittaker J. 
 
 If the solution thus obtained is a homogeneous polynomial in (x, y, z) it 
 is called a solid harmonic of order equal to its degree in these variables. The 
 most general solid harmonic of order n is of the form 
 
 (x cos co + y sin to + iz) n ifr (to) da), 
 
 and particular cases are obtained when n = 1 and 2 of the form 
 X = Ajcc + X 2 y + A 3 z 
 
 O 2 = A! (x 2 y 2 ) + A 2 (y 2 z 2 ) + A 3 (z 2 
 
 84. When any solution of the equation has been obtained other solutions 
 can be derived from it in various ways. For instance the function obtained 
 by differentiating the given solution any number of times with regard to the 
 
 * When we speak of an arbitrary function it is understood that it may be restricted so as 
 to render the processes of differentiation and integration intelligible operations. 
 
 f The limits need not be constant ; they can for instance be any two roots of an equation of 
 the type 
 
 x cos + y sin + iz=g (0). 
 
 J Math. Ann. LVII. p. 333 (1902). 
 
80 The characteristic properties of the electric field [CH. n 
 
 coordinates (x, y, z) is also a solution. By adding together arbitrary constant 
 multiples of the solutions thus obtained we may derive a very general type 
 of solution. Again if any solution involves one or more arbitrary parameters 
 we may obtain others from it by differentiating with regard to the parameters, 
 or by integrating with regard to them after having multiplied the expression 
 by any arbitrary function of the parameters. For instance we know that 
 
 r ~ V(x- xtf + (y - 2/J 2 + (z - ztf 
 is a solution of the equation ; we may thus infer that the function 
 
 &""-&> (fa, dy, dz, 
 
 f 
 is also a solution, whatever the arbitrary function p may be. 
 
 In addition to these linear transformations there are certain other special 
 transformations which enable us to pass from one solution of Laplace's 
 equation to another. The principal transformation of this kind is. that of 
 inversion with respect to the arbitrary origin of coordinates. It is in fact 
 easy to verify that if <D (x, y, z) is a solution of Laplace's equation then the 
 function 
 
 is also a solution if r 2 = x 2 + y 2 + z 2 . 
 
 In particular if O n (x, y, z) is a solid harmonic of order n, then 
 
 is a solution of the potential equation. Thus 
 \x + A 2 y + A 3 z A (x 2 - 
 
 are solutions*. 
 
 85. In dealing with spherical distributions it is usually much more 
 convenient to use spherical polar coordinates. The transformation to this 
 case is easily effected by the substitution 
 
 x r cos 6, y r sin 6 sin </>, z = r sin 6 cos (f>, 
 or it may be more readily obtained by noticing that 
 
 V* = div V* - -, ! , f * (r* sin B f) + * fain fl ?*) + ( JL) ] , 
 
 r 2 sm0[8rV drj 80V cdj d<f> \sm 6 c<f> J J 
 
 so that the equation becomes 
 
 a^O 28O l^^O Jj_ 320 cot08O_ 
 
 dr 2 r "dr + r 2 W + r 2 ^m 2 ~(9 df 2 4 ~T~ W = 
 
 * Cf. Thomson and Tait, Treatise on Natural Philosophy, Vol. n., where several applications 
 of this transformation to definite problems are made. 
 
84, 85] Solutions of Laplace's equation 81 
 
 The general type solution corresponding to that given above for cartesian 
 coordinates is 
 
 ( r cos @ <*> sm <f> + ir cos c/> 3 CD) dco, 
 
 , . , A u cos 6 
 
 particular cases giving , - - 
 
 as solutions. 
 
 Elementary solutions are obtained of the type 
 <&=U (r). V (0). W (6). 
 
 d*U 2 dU 
 
 where -= + - n (n + 1) U = 0, 
 
 dr* r dr 
 
 /72W 
 
 and 
 
 i 
 
 The first of these equations is satisfied by 
 
 and the second by W = cos m (</> (f> m ), 
 
 whilst the solutions of the third are the associated Legendre functions 
 P n m (cos 6} and Q n m (cos 6). In the usual notation for the Gamma and 
 hypergeometric functions* 
 
 m e 
 
 2 / 6\ 
 
 P n m (cos 9) = p /-, ( ~ n ' w + 1 '> 1 ~ m > sm2 9 ) > 
 
 TT r 
 
 $ w w (cos $) = I cot (n + m) TrP n m (cos 
 
 Zi 
 
 2 / 0x 
 
 -- s 7 ; - x fTTi - x F ( n, n+l; I m; cos 2 s ) 
 
 sin (n + m) TrF (1 m) V 2/ 
 
 When m is a positive integer 
 
 flm 
 Qn (X) = (1 - X*)* & (Qn(x)), 
 
 and when n is a positive integer 
 
 P ( X \ - 2n! \ x n n ( n ~ !) ^n-2 , 1 
 
 "(n!) 2 2-! 2(2^ni)* ]* 
 
 * E. W. Hobson, Phil. Trans. A, Vol. CLXXXVII. (1896), pp. 443-531. 
 
82 The characteristic properties of the electric field [on. n 
 
 The general normal solution of the equation is thus of the form 
 
 (A n r n + B n r- n -*} (A n m P n (cos 6) + B n m Q n m (cos 6)} cos m (<f> - < ). 
 
 the part depending on r n being regular at the origin if n > when the other 
 part is regular at infinity. Any other solution of the equation can always 
 be reduced to this form by the application of the theorem of Fourier and 
 the corresponding theorems for the expansion of arbitrary functions in terms 
 of the Legendre functions. 
 
 86. In cylindrical coordinates the equation assumes the form 
 
 to 
 
 The general solution is 
 
 1 8O 1 82<1> 8 2 O _ 
 
 rdr + r z 90 2 + W 
 
 / (r cos 6 a) + iz, o>) da), 
 
 and normal solutions can be obtained in the form 
 
 0>=E/(r). 7(0). W(z), 
 
 d*U 1 dU ( m*\ TJ 
 
 when -j-=- H j~ + (P r I " = 0, 
 
 dr 2 r dr v f J 
 
 _. 4. m zy _ n 
 
 2 ^ 
 
 -3* ~ * 
 
 The second and third equations are satisfied by 
 
 7 = cos m (6 ), 
 W = Ae pr + Be~ pr , 
 
 respectively whilst the solutions of the first are the Bessel functions J m (pr) 
 and K m (pr) of the first and second kind. When m is integral 
 
 / 1 d \ m 
 
 Jm(x) = X m (- x(lx ) JoW, 
 
 K m (x} = tf(- ^ 
 
 \ Ju U/JL 
 
 where for small values of x* 
 
 x 2 x* 
 
 2 2 2 2 4 2 
 and 
 
 2 f x \ x 2 / l\x 4 
 
 K (x)= - - (log^ + 7) Jo M - 22 + ( l + 2 J 2^T1 2 ~ I" ' 
 
 * Whittaker, Modern Analysis, 2nd ed. (1915), ch. xvn. 
 
85-87] Spherical charge distributions 83 
 
 and y = -577... is Euler's constant. For large values we have 
 
 2 
 
 The elementary solutions of the potential equation in cylindrical coordinates 
 are thus of the type 
 
 O = ev* (aJ m (pr) + bK m (pr)) cos m (0 - ), 
 
 the part involving J m being regular near the axis and the other part at a large 
 distance from it. 
 
 87. Let us now consider one or two specific problems illustrating the 
 usefulness of some of the analytical solutions of the potential equation just 
 obtained. We firstly find the potential in the field of a simple surface 
 distribution of charge on concentric spherical surfaces r a and r = b . (a < b) 
 when the surface density at any point on the inner sphere can be expanded- 
 
 in the form ' y. p /,_ m 
 
 a = 2jd n Jr n (COS v), 
 
 and that on the outer sphere in the form 
 
 a = 26 W P W (cos 0). 
 
 The type of functions that occur in these expressions suggest the types of 
 solutions to try for the potential function ; but it must be remembered that 
 the complete function representing the potential must be regular at each 
 point of the field. The most general type of function regular inside the inner 
 sphere and consistent with the values for a is 
 
 <S> 1 ='Za n f rP n (coBO); 
 between the spheres we may take 
 
 <D 2 = S (a B V + 6/r-"-i) P n (cos 0), 
 whilst outside the bigger sphere we can only. have 
 
 8 = S& B '"r ^(cosfl). 
 
 As there is no double sheet distribution the potential is continuous throughout 
 the whole of the field so that these functions must agree at the surfaces 
 separating their regions of validity. Thus we must have 
 
 a n 'a n = a n "a n + b n "a~ n -\ 
 b n '"b- n - 1 = a n "b n + 6/6-"- 1 . 
 
 In addition to this the difference of the normal gradients in the potential at 
 either side of the two spherical surfaces must be equal respectively to the 
 value of 47rcr there. This requires that 
 
 62 
 
84 The characteristic properties of the electric field [CH. n 
 
 There are thus four equations from which the arbitrary constants a n ' , a n ", 
 b n "> b n '" can be determined for each value of n. The potential function in 
 each part of the field is thus completely determined so that our problem is 
 solved. 
 
 If both surface densities are uniform and equal respectively to Q a /4:7ra 2 and 
 on the two spheres we find that 
 
 a b 
 
 _Q,Qa 
 
 '-T + T' 
 
 88. We next examine the fields of certain distributions of charge on a 
 plane (z 0) which are symmetrical about a point in that plane. If this 
 latter point is taken as origin of coordinates the field will be symmetrical 
 about the axis of z, so that it will now be convenient to use cylindrical polar 
 coordinates. The field will moreover be symmetrical on either side of the 
 plane z = so that the condition that 
 
 a 
 
 on + on_ 
 
 may be interpreted as implying that on the positive side of z = 
 
 a< 
 
 -J- = 27TCT. 
 
 OZ 
 
 Now cr is a function of r only and by a well known theorem in Bessel's 
 functions* it may be written in the form 
 
 (00 / 00 
 
 a = \ JQ (kr) kdk a (x) J (kx) xdx. 
 
 J o .' o 
 
 We now see that the solution of the potential equation given by 
 
 ,OC f ' ,00 
 
 <f> = 27T e~ kt J (kr) dk a (x) J (kx) xdx, 
 
 Jo J o 
 
 which is regular at all points on the positive side of the plane 2 = 0, satisfies 
 the condition that 
 
 on this plane. It is therefore the proper potential function of the specified 
 
 distribution. Two particular cases of this general result are worth noticing. 
 
 (i) If a is constant within the circle r = a and zero at all other points we 
 
 have /" f a 
 
 (f> = 27ror 1 e~ kr J Q (kr) dk J (kx) xdx. 
 
 Jo Jo 
 
 * Due substantially to C. Neumann. Cf . Heine, Handbuch etc. I. p. 442. Nielsen, Hand- 
 buch der Theorie der Cylinder -funktionen (Leipzig, 1904), p. 360. Lamb, Hydrodynamics (4th ed. 
 1916), ch. v. 
 
87-89] Distributions of charge on conductors 85 
 
 ra Q 
 
 Now* | J (kx)xdx= ,J 1 (ka), 
 
 JO K 
 
 f dk 
 
 so that (f> = 2rra(T e~ kr J (kr) J r (ka) -=- , 
 
 Jo K 
 
 This is the general value of the potential in the field surrounding the 
 uniformly charged circular plate. It is easy to verify that it reduces to the 
 value 
 
 27ra (Va?+~z z - z), 
 
 on the axis (r = 0) in agreement with the result of the last section. 
 
 (ii) If a - for values of r < a and vanishes for all other values 
 
 Z-nQ "" ,,,,. [ a J Q (kx)xdx 
 
 then (4 = - e~ kr J (kr) dk -. -- = . 
 
 a . o .'o Va z - x z 
 
 Now it is easily verified f by using the substitution x = a sin 9 and the 
 series for J (kx) that 
 
 ' a J (kx) xdx _ sin ka 
 
 f) I 00 J7 
 
 so that </ = -- e- fcr J (kr) sin &a ^ . 
 
 CL % Q A/ 
 
 This is the formula for the potential of a circular plate over which ^ is 
 constant; it is obtained in a different form in the previous section as the 
 limiting case of an ellipsoidal distribution. 
 
 89. The circumstances in the physical problems with which we usually 
 deal are, as a rule, simpler than the previous general discussion would imply. 
 The bodies in the field carrying the charges are usually composed of conducting 
 materials (metals) which admit of the free passage of any elements of charge 
 which may exist in them under the action of any external force. Thus in 
 such a system equilibrium of the charge distribution is possible only when the 
 force intensity is zero at any point in the interior of the charged bodies ; but 
 if the force intensity is everywhere zero, the potential, of which it is the 
 negative gradient, must be constant so that 
 
 V 2 < = 
 
 there; there can therefore be no volume distribution of charge inside the 
 conductor. There is however a distribution on the surface of each conductor 
 and its density at any point is determined simply by the normal gradient of 
 the potential in the external field just outside the point 
 
 J^ 
 
 4rr dn 
 
 * By series. 
 
 t Cf. Rayleigh, Scientific Papers, t. in. p. 98. Hobson, Proc. L.M.S. t. xxv. p. 71 (1893). 
 
 J Coulomb Par. Hist. 1788 (1791), p. 676. 
 
86 The characteristic properties of the electric field [en. IT 
 
 The total charge on the surface is simply 
 
 [ 1 fty 
 
 adf = -j U- a/. 
 
 J/ 47T J dn * 
 
 Moreover ifj. as is usually 'assumed to be the case, there is no double sheet 
 distribution the potential outside must be continuous with the constant 
 interior value at the surface of each conductor. In other words the 
 external potential reduces to a constant value on the surface of the con- 
 ductor; this condition provides a usually sufficient criterion for its deter- 
 mination in all practical cases. 
 
 Thus for instance the potential function 
 
 reduces to the constant value at the surface of the sphere r a ; it is 
 
 a 
 
 therefore the potential in the field of an isolated conducting sphere whose 
 potential is 
 
 Q 
 
 a ' 
 The density of the charge at any point of the sphere is 
 
 W\ 
 
 and is uniformly distributed over its surface. Of course we might have 
 guessed this to be the case, on account of the symmetry, and then we could 
 have used the result of the integration carried out in the previous section. 
 
 90. We can similarly use any of the results obtained by direct integration 
 which fulfil the necessary conditions implied in the physical circumstances of 
 the problem. As a final example let us briefly consider the case of a 
 charged conducting ellipsoid whose boundary surface is given by 
 
 ?! _i_ ! , ?! = i 
 
 2 """ 6 2 "*" c 2 
 
 In this case the distribution of charge on the outer surface must be such that 
 the potential function throughout its interior is constant and outside the 
 ellipsoid it must satisfy the usual conditions and agree with the constant 
 value on the surface. But if we refer to the results obtained in the first 
 section we see at once that the charge distribution which satisfies these 
 conditions is identical with the uniform distribution throughout a very thin 
 homoeoidal shell at the surface of the ellipsoid in question provided only 
 that the total charge is right. Thus if Q is the total charge on the ellipsoid 
 the constant internal potential will be 
 
 , = Q r dt _ ^ 
 
 2 .' o V(d* + t) (6 2 + (c 2 + 0' 
 
89-91] Gauss' theorem 87 
 
 whilst the potential at the point (x, y, z) outside is 
 
 6 = 
 2 
 
 A being the positive root of the equation 
 
 o 
 
 ~~ = 1. 
 
 a 2 + t b 2 
 The density of charge at any point on the ellipsoid is given by 
 
 QP 
 
 
 p being the central perpendicular on the tangent plane at the point. From 
 this we see that the density is greatest where p is largest. If the ellipsoid 
 is very long in one direction the charge will be practically all concentrated 
 at the ends : in the case of a very flat ellipsoid with one very short axis the 
 charge will be concentrated round the edge. These remarks illustrate the 
 general theorem that the charge always tends to disperse itself as much as 
 possible so as to secure the minimum condition for the potential energy in 
 the final field. 
 
 91. Gauss' Theorem: normal induction. So far our basis is purely 
 analytical and as a consequence the physical significance of the results is 
 not very clear. In order to obtain a better insight into this other side of 
 the matter we shall proceed from a different standpoint and along more 
 elementary lines. 
 
 The chief characteristic property of the electric field contained in Poisson's 
 equation is expressed in an integral form by a theorem usually ascribed to 
 Gauss* but which was probably first stated by Faraday in a physical 
 manner. 
 
 If any closed surface / is taken in the electric field and if E n denote the 
 component of the electric force intensity at any point of the surface in the 
 direction of the outward normal dn, then 
 
 './ 
 
 where the integration extends over the whole of the surface and Q is the 
 total charge enclosed by it. 
 
 This theorem is a mere mathematical verification for a single point charge 
 q : for at any point of the closed surface distant r from q 
 
 d ^ 
 
 E n = -^ cos (nr), 
 
 * Allg. Lehrsdtze. The theorem was also given by Kelvin in 1842. Cf. Papers on Electricity 
 and Magnetism. The present demonstration is due to Stokes. 
 
88 The characteristic properties of the electric field [CH. n 
 
 A, 
 
 where, (nr) denotes the angle between the positive directions of n and r; 
 but if da) is the element of solid angle subtended by the surface element df 
 at the point charge q then 
 
 A 
 
 dfcos (nr) = r 2 da>, 
 and thus E n df= qda>, 
 
 so that E n df = q 
 
 da> 
 
 = farq if q is inside / 
 = if q is outside. 
 If there are any number of charges q 1} q 2 , ... q n present in the field, then 
 
 B n = B! + Eo + ... 
 
 L n l n 
 
 is the sum of the normal components of force due to the separate point 
 charges and thus we get by simple addition 
 
 E n df= 477 (total charge inside/). 
 
 Moreover although we have proved this theorem for a system of point charges 
 it remains valid when the charges are merged into continuous volume or 
 surface distributions as the following analysis, which exhibits the theorem 
 as an immediate consequence of Poisson's equation, proves. If we consider 
 
 the integral 
 
 r 
 div Edv 
 
 taken throughout the space v inside the surface / its value is 
 
 r 
 4-n- \p dv ; 
 
 but by Green's lemma it also msists of 
 
 Jf 
 
 together with the surface integrals arising at the surface distributions of 
 charge. These are simply 
 
 " (E. - E B J df, 
 
 f referring to the surfaces on which the charge infinities are distributed. But 
 so that this latter integral is 
 
 E n+ - B B _ = - 47TOT, 
 
 - 47T | adf, 
 
 and we have 4?r I pdv + adf = I E n df, 
 
 Uv . /' Jf 
 
91, 92] Applications of Gauss' theorem 
 
 which is precisely Gauss' theorem. If we use 
 
 En= ~dn' 
 then the equation may be written in the form 
 
 89 
 
 which exhibits it as the integral form of the characteristic property of the 
 potential function of an electrostatic field. 
 
 r 
 
 The integral \"& n df is defined as the total normal induction through the 
 surface /. 
 
 92. Many of the results of the previous section may be obtained very 
 simply by an application of Gauss' theorem of normal induction. 
 
 (a) Uniformly charged infinite circular cylinder. 
 
 By symmetry the field is radial and symmetrical round the axis. 
 
 Choose as the arbitrary surface in Gauss' theorem the portion of a con- 
 centric cylinder (radius r) cut off by two 
 parallel planes unit distance apart and 
 parallel to the axes. If the force at a 
 distance r is E then the total normal 
 induction over this surface is 
 
 2wr . E, 
 
 the flat ends contributing nothing. .... 
 
 If r < a, the radius of the cylinder 
 
 27rrE = 0, E = ; 
 if r > a, then ^ - 
 
 E = W, 
 
 Q being the charge per unit length. 
 
 (6) Uniformly charged infinite plate. 
 
 By symmetry again we see that the 
 force is everywhere normal to the plate. 
 
 Now apply Gauss' theorem to a cylinder 
 of unit sectional area perpendicular to the 
 plane and bisected by it and we easily deduce that 
 
 2# - 47TC7, 
 E = 277(7. 
 
 
 Fig. 12 
 
90 The characteristic properties of the electric field [CH. n 
 
 (c) The uniformly charged conducting sphere. 
 
 The method is the same : apply Gauss' theorem to concentric spherical 
 surfaces. 
 
 (d) We can determine in the same way the field of force for two concentric 
 spherical conductors (radii a^ and 2 ) carrying charges Q l and Q 2 and prove 
 that if Q 1 = Q z the field is confined to the space between the conductors. 
 
 E 
 
 I I 
 V / 
 
 Fig. 13 
 
 The force inside the inner sphere is zero : in the space between the spheres 
 it is radial and symmetrical and at a distance r from the centre it amounts to 
 
 *' 
 
 Outside both spheres the force is also symmetrical and radial but is now of 
 intensity 
 
 at a distance r from the centre. 
 
 93. We now make use of the general theorem to analyse the properties 
 of electrostatic fields in such a way as will lead most directly to a con- 
 sideration of Faraday's speculations on the origin of all electrical actions; 
 but before going on to this it is interesting to notice that when we consider 
 
92-94] 
 
 Lines of force 
 
 91 
 
 as in Gauss' theorem a part only of a closed surface instead of the whole of 
 the surface and take the surface integral of normal induction for it then the 
 
 contribution of each' element q is 
 
 qw, 
 
 where to is the solid angle the boundary of the surface subtends at q. 
 
 The solid angle of a right circular cone of angle 6 is ITT (1 cos 6) ; thus 
 the surface integral of the normal flux through a circle due to a system of 
 point charges ranged on its axis is 
 
 277-22 (1 - cos 6). 
 This result will be of use to us later on. 
 
 
 ,". 
 
 1 
 
 w 
 
 Pig. 14 
 
 94. Lines of Force and Equipotential Surfaces*. The electric field due 
 to any static system of charges is completely specified if we know the 
 magnitude and direction of the electric force intensity at any point of the 
 field. The direction of the force at any point is understood to be the 
 direction in which a small point charge 8q would be displaced if put there. 
 If we follow this direction from point to point we obtain curves which are 
 called lines of force. 
 
 A line of force in an electric field is a curve such that along it the force 
 is always tangential. It follows that the positive direction of the line is 
 always that of decreasing potential. Hence a line of force cannot return 
 into itself, but must have a beginning and an end. We shall prove 
 presently that it can only begin on a positively charged surface and end on 
 a negatively charged one. 
 
 Now suppose an electric field to be given and all the lines of force drawn 
 in it. At any position P in the field where there is no electricity we place 
 
 * Faraday, Roy. Soc. Trans. 141 (1831), p. 2. 
 
92 The characteristic properties of the electric field [CH. n 
 
 a small surface element df, perpendicular to the lines of force at that place. 
 Then all the lines of force which pass through the edges of this surface 
 element will form the sides of a tube called a tube of force. 
 
 The inside of a tube of force is also filled with lines of force, i.e. through 
 each point in its interior we can draw one line of force but only one. If there 
 were several they would cut at this point and this is impossible because at 
 each point the line of force gives the direction of the resultant force intensity 
 of the field and this is uniquely determinate unless the force is zero, an 
 exceptional case which is reserved for future consideration. Thus as a general 
 rule lines of force cannot cut one another and none of them can pass through 
 the sides of a tube of force. 
 
 At another point P' of our tube of force let us draw another surface 
 element df also perpendicular to the 
 direction of the lines of force there. 
 Now apply Gauss' theorem to the 
 portion of the tube PP' . In summing 
 up the induction over the surface we 
 can neglect the curved sides of the tube 
 since the electric force intensity is at 
 each point tangential and thus has no 
 normal component. Let now E, E' 
 be the force intensities at P and P' 
 
 respectively : their directions are along tangents to a mean axis in the tube 
 and may therefore be considered normal to the end surfaces df, df both of 
 which are very small. A simple application of Gauss' theorem thus gives 
 
 provided there is no electricity in the part of the tube between P and P'. 
 
 The intensity of force at every point of the tube is inversely proportional 
 to the cross section of the tube at the point. We have thus a convenient 
 method of graphically representing the intensity of a field of electric force. 
 We fill up the space of the field by drawing tubes of force, choosing their 
 cross sections so that the constant value of EC?/" along each is the same for 
 all. The density of the tubes, or merely their thickness would then give a 
 graphical measure of the strength of the field. The tubes are thickest in 
 the positions of small intensity. 
 
 95. A still simpler method of representing the field is obtained by 
 considering the potential function*. The force intensity is a vector with 
 three components at each point of the field and so it is much easier if we 
 notice that this vector is always the gradient of the scalar quantity, which 
 
 * Maclaurin, Treatise on Fluxions (Edinb. 1742), 640. Clairaut, Figure de la terre (Par. 
 1743). 
 
94-96] 
 
 Equi-potential surfaces 
 
 93 
 
 we have called the potential, so that instead of having three things to 
 consider we have only one. The field of force is completely specified by 
 this one function </>. We could therefore map out the field by plotting the 
 function (f>, the simplest method being to draw the surfaces over which <f> 
 is constant : such surfaces are called equi- potential surfaces, level surfaces or 
 simply equi- potentials. 
 
 Since the potential is as a general rule a one- valued function of position 
 we see that two equi-potentials cannot in general cut one another. Now let 
 us choose two adjacent equi-potentials 
 whose potential difference is A(/> and let 
 As be a small line drawn from a point on 
 the surface of higher potential to a point 
 on the other surface. We know then that 
 
 . is in the limit the force intensity in 
 
 the direction of As : but - A is a maximum 
 
 As 
 
 when As is a minimum. Since we may 
 regard the two adjacent surfaces as 
 parallel, at least when we confine our- 
 selves to small opposing regions on them, 
 As will be a minimum when it is the 
 normal distance Aw between the surfaces, 
 intensity is & 
 
 E = T^ 
 
 An 
 and is normal to the equi-potential surfaces. 
 
 The lines of force are therefore everywhere normal to the equi-potentials. 
 Conversely, of course, we may conclude that if we can find a surface every- 
 where normal to the lines of force, it must be a level surface. 
 
 Thus if we draw all the level surfaces in the field so that the potential 
 difference for any two succeeding ones is the same, the density of the surfaces 
 so drawn is everywhere proportional to the force intensity in the field. We 
 thus obtain another very real representation of the field. If we draw the 
 tubes of force as well we notice that the cross sections of the tubes are 
 proportional to the distances between the level surfaces. This would provide 
 a good test as to whether the field had been correctly mapped. The tubes 
 are thickest where the surfaces are widest apart. 
 
 96. As an illustration oi the usefulness of the conceptions here intro- 
 duced we may discuss in terms of them some very important properties of 
 the fields in question*. 
 
 * Gauss, Allg. Lehrsdtze etc. (1840). Stokes, Camb. and Dublin Math. Journal, iv. (1849). 
 See also papers by Lord Kelvin in the same journal (1842-3) and a memoir by Chasles, Connais- 
 sances des Temps (1845). 
 
 Fig. 16 
 Thus the resultant force 
 
94 The characteristic properties of the electric field [OH. n 
 
 The potential cannot be a maximum or minimum at a point of the field 
 where there is no charge. For supposing that at any point P there is a true 
 maximum value of </>, then at all other points in the immediate neighbourhood 
 of P the value of </> is less than its value at P. Hence P will be surrounded by 
 a series of closed equi-potential surfaces, each outside the one before it, and 
 at all points of any one of these surfaces the electrical force will be directed 
 outwards so that the total normal induction through the surface will be 
 positive and cannot be zero : it follows then that there must be a positive 
 charge inside the surface, and since we may take the surface as near to P 
 as we please, there is a positive charge at P. 
 
 In the same way we may prove that if <f> is a minimum at P there must 
 be a negative charge at P. 
 
 This enables us also to complete the proof of the statement made above 
 that lines of force must start from a place where there is positive electricity 
 and end at a place where there is negative electricity, for it can only begin 
 at a position of maximum potential and end at a position of minimum 
 potential. 
 
 It is of course possible for a line of force to begin on a positive charge 
 and go to infinity, the potential decreasing all the way, in which case the line 
 of force has, strictly speaking, no second end at all. So also a line may come 
 from infinity and end on a negative charge. 
 
 97. To obtain a still closer insight into the significance of the various 
 properties of electric field we may examine one or two cases where it is possible 
 to determine by simple methods the full details of the electric field in terms 
 of the lines of force and equi-potential surfaces. The examples in each case 
 are typical of the general problem where the field of a number of point 
 charges on an axis is under investigation. 
 
 Q' 
 
 Fig. 17 
 
 The equation to the lines of force are easily determined : for the curves 
 in each plane through the axis for which 
 
 Z27r<? (1 cos 6} = const. 
 
 lie on a tubular surface with, the given line of charges as axis : moreover 
 from the theorem quoted at the end of 93 the normal induction across 
 
96-98] 
 
 Point charge distributions 
 
 95 
 
 any section of this tube is constant : it is therefore a tube of force and 
 its curve of section by an axial plane gives the equations of the lines of force 
 in that plane : they are therefore the curves 
 
 S^ cos 6 = const. 
 The equi-potential surfaces are of course those on which 
 
 is constant. 
 
 When a line of force passes through one of the point charges, the radius 
 vector at that point becomes a tangent and 6 is then the angle that tangent 
 makes with the positive direction of the axis. Let a line of force pass through 
 the hth and &th particle, then we have 
 q l + q 2 + ... + q h cosd h -q h+1 - ... 
 
 Thus q h sin 2 $0 h + q h+1 + ... + q k ^ + q k cos 2 $0* = 0. 
 
 If all the charges have the same sign the only line of force which can pass 
 
 from one particle to another is the straight line along the axis. 
 
 Fig. 18 
 
 Now let a line of force pass from the particle q s to a point at an infinite 
 distance in a direction which ultimately makes an angle a with the axis. 
 We have then in the same way 
 
 q l 
 
 + ... + &_! + q, cos 
 
 q g+1 
 
 8=1 
 
 q) cos a. 
 
 If 'Zq = no line of force can in general pass to an infinite distance. 
 
 98. Fig. 18 represents the lines of force due to two equal and opposite 
 charges. In this case all the lines of force start from the positive charge 
 and end on the negative charge. The force in the field is strongest in the 
 
96 The characteristic properties o/ the electric field [OH. n 
 
 neighbourhood of the charges and especially between them ; it decreases 
 rapidly as the point gets farther and farther from the charges. 
 
 When the two point charges are very close together they are collectively 
 described as an electric doublet. 
 
 99. Figs. 19 and 20 represent respectively the lines of force and equi- 
 potential surfaces in a plane section of the field due to two equal positive 
 
 Fig. 19 
 
 Fig. 20 
 
 charges : the lines of force being represented by continuous lines and the 
 sections of the equi-potentials by dotted lines. The properties of the lines 
 of force and equi-potential surfaces are again clearly indicated, except 
 perhaps in the neighbourhood of the point C. The behaviour at this point 
 is reserved for future consideration. 
 
98-100] 
 
 Point charge distributions 
 
 97 
 
 100. Figs. 21 and 22 represent the similar section of the field due 
 to a positive charge of 4 units at A and a negative charge equal to 1 
 at B. In this case all the lines of force which fall on B start from A, but 
 
 Fig. 21 
 
 Fig. 22 
 
 since the charge at A is numerically greater than that at B, lines of force 
 will start from A which do not fall on B but travel off to an infinite 
 distance*. 
 
 * These and several other diagrams of lines of force of point charge systems are given by 
 Maxwell, Treatise, i. 
 
 L. 7 
 
98 The characteristic properties of the electric field [OH. n 
 
 101. We have seen that there may be in an electrostatic field certain 
 points where the resultant force intensity vanishes: and since the existence 
 of such points invalidates the general applicability of certain simple types 
 of argument expounded above it seems necessary to enter into a closer 
 investigation of the field in the neighbourhood of such points, called points 
 of equilibrium, when they do not coincide with any part of the charge 
 distribution. 
 
 We shall refer the discussion to convenient rectangular axes chosen at 
 the point under consideration. Now it can be proved that the potential 
 function ^ at any point external to the charge distribution is a holomorphic 
 function of the position of the point so that in the neighbourhood of the point 
 under special consideration it may be expanded in the form 
 
 </> = (f> + Ix + my + nz + |- (ax 2 + by 2 + cz 2 + 2fyz + 2gzx + 2hxy) + 
 As the point is one of equilibrium we know that 
 
 dcf) d<> d<f> 
 dx dy dz 
 
 i.e. I = m = n 0. 
 
 Thus <f> = < + | (ax 2 + by 2 + cz 2 + 2fyz + 2gzx + 2hxy) + .... 
 
 Also from the condition that V 2 (f> = we conclude that 
 
 a + b + c = 0; 
 
 the asymptotic cone of a surface of equal potential has therefore three 
 mutually perpendicular generators. 
 If we turn the axes round and make 
 
 </> =</>o+ I (<*z 2 + )8y* + -yz 2 ) +.;., 
 then a + /3 + y = 0, 
 
 and the surfaces of equi -potential in the neighbourhood of such a point are 
 similar hyperboloids all with the same asymptotic cone 
 
 ax 2 + fry 2 + yz 2 = 0. 
 
 If there is an axis of symmetry (say the axis x) in the field, as is the case 
 with the three examples examined above then /3 = y and thus 
 
 a + 2j8 = 0. 
 The sections of the level surface in the plane (x, y) are then hyperbolas 
 
 the asymptotes being 
 
 The lines of force in the same plane are the orthogonal trajectories of this 
 system of hyperbolas and they appear as the curves 
 
 y 2 x = const. 
 
101, 102] The field of a system of conductors 
 
 99 
 
 The accompanying diagram illustrates the general course of the lines of 
 force and equi-potentials in the neighbourhood of such a point. 
 
 There thus appears to be no real difficulty such as that anticipated in the 
 previous discussion except perhaps in connection with the single line of 
 force or equi-potential which actually goes through the point of equilibrium ; 
 but even in this case we can regard these particular curves or surfaces as the 
 limiting cases of others exhibited in the figure so that they result as the 
 limiting form of two systems of curves or surfaces moving up to coincidence 
 in part of their length or surface, so that they do not in reality cut across 
 one another. The general argument given above therefore remains valid if 
 care be taken to consider this point when it arises. As a general rule it is 
 
 Fig. 23 
 
 however safer, in discussing any particular example, to determine the point 
 of equilibrium and the line of force which goes through it and then to 
 discuss the portions of the field inside and outside this line separately : all 
 difficulty is then avoided. 
 
 102. The field of a system of conductors. A very important class of 
 electrostatic problem is concerned with the field of a system of charged metallic 
 conductors, as we may gather from the experiments described at the beginning 
 of the previous chapter. The field surrounding such a system possesses 
 
 72 
 
100 The characteristic properties of the electric field [OH. n 
 
 certain, characteristic properties which we can very easily interpret in terms 
 of lines of force and equi-potential surfaces. Metallic conductors as we know 
 them are chiefly characterised by the presence in them of an exceedingly 
 large number of electrons (about 10 23 per c.c.) which are freely moveable in 
 the space between the atoms. Thus as long as any electric force acts on a 
 free electron inside a conductor equilibrium is not possible. 
 
 Thus if such a body be brought into an electric field the electric force will 
 act in opposite directions on the positive electricity (in the atoms of the body) 
 and the negative electrons at each point of the metal and separate them so 
 that they no longer cancel one another's effects. A num'ber of the negative 
 electrons in any volume element A will be driven by the electric force into a 
 neighbouring element B; the consequence of this is that the element A 
 becomes positively charged whilst the element B is negatively charged, the 
 total quantity of electricity remaining the same. 
 
 Between the two neighbouring elements A and B a new field will arise 
 due to their charge. This new field is superposed on the old one, but is in 
 the opposite direction to it, as lines of force always go from a positive to a 
 negative charge. In this way the field in the interior of the metal will tend 
 to annul itself, and moreover this process will go on until the field in the 
 whole of the body is zero. 
 
 We may now enquire as to the whereabouts of the charges which arise 
 from the separation in each element and which give rise to the field which 
 when superposed on the original field makes the resultant intensity zero 
 throughout the whole of the interior of the conductor. Now since the 
 force intensity at every point of the interior of the conductor is zero, the 
 total normal induction, in Gauss' sense, through any closed surface entirely 
 in the metal must also be zero and therefore this surface can contain no 
 electricity. There is therefore no charge in the interior of the conductor. 
 The charge is concentrated on its surface. Thus any conductor introduced 
 into a field of intensity E will have induced on its surface a charge which 
 gives rise to a field of intensity E x which must be such that at every point 
 in the conductor 
 
 E + B! = 0. 
 
 Moreover at a point on the surface of the conductor the total electric force 
 E + E! can have no component tangential to the surface, as otherwise an 
 unending separation of charge would take place in the surface itself. The 
 electric force intensity of the total field just outside the conductor is therefore 
 entirely normal to the surface. 
 
 Even if we introduce a charge to the metallic body from any source this 
 charge must distribute itself over the surface of the body so that the above 
 conditions are still satisfied. 
 
102, 103] The field of a system of conductors 
 
 103. In the interior of the conductor the force intensity of the total field 
 is everywhere zero. Therefore the potential of the field must be constant 
 throughout the interior : this is the potential of the conductor. The surface 
 of the conductor is therefore an equi-potential surface in the field. This is 
 another reason why the lines of force just outside a conductor are normal to 
 its surface and it also shows the great importance of the potential function 
 introduced on general lines as above. 
 
 Since there are no lines of force inside a conductor, the tubes of force of 
 the external field must end on the conductors. We can now show quite 
 simply they must begin on positive electricity and end on negative and also 
 that the quantities of electricity on the 
 two ends of the tube are equal and opposite. 
 Consider the portion of a small tube of 
 force from a conductor up to the cross 
 section df of it and apply Gauss' theorem 
 to the tube closed by a slight extension 
 into the metallic conductor. We see at 
 once that 
 
 Erf/ = brdq, 
 
 where dq is the charge on the portion of 
 the conductor included in the tube. Simi- 
 larly from the other part of the tube 
 
 -Erf/=47rS ? '. * 24 
 
 mi o o / Erf/ 
 
 Inus oq oq = -r^-. 
 
 We may thus regard an electrified system as consisting always of positive 
 and negative electricity in equal amounts, each element of electricity being 
 associated with an equal and opposite element and connected with it by a 
 tube of force running through the dielectric medium. 
 
 Moreover since along a tube of force 
 
 Erf/ = const., 
 
 we see that a line of force is always a line of ascending or of descending 
 potential throughout its length ; the sign of E cannot change along any one 
 line. From this we conclude that 'the potential cannot be an absolute 
 maximum or minimum in free space. The greatest potential in the field must 
 occur on a conductor (among the conductors we must include the earth, or 
 infinity, as we say, if necessary). Moreover this conductor must have its 
 charge all of the positive kind because if there were negative electricity at 
 any point of its surface lines of ascending potential would pass from the 
 conductor and this is impossible. 
 
 Similarly the least potential must occur on a conductor on which the 
 electricity is wholly negative. 
 
"fhe characteristic properties of the electric field [CH. n 
 
 104. We also see that if the potential in a field is constant over any 
 surface not enclosing any charge, it must be the same constant value 
 throughout the interior of that surface, because if it were not we could draw 
 lines of force from one point of the surface to another and we should then 
 have an ascending and descending potential (the initial and final values at 
 the ends are the same) in the same line. There can therefore be no charge 
 on the interior surface of a hollow in a charged conductor unless of course 
 there were a charge placed somewhere inside the hollow. 
 
 As may be gathered from the experiments described in the previous 
 chapter this last exceptional case is of great practical importance and therefore 
 deserves special examination. 
 
 We can easily show that if any number of charged bodies exist in a hollow 
 in a closed conductor the charge on the inner surface of the conductor (i.e. 
 round the hollow) will be equal in magnitude but opposite in sign to the 
 total charge on the system of bodies inside. This can be seen in various 
 ways : we can draw a closed surface entirely inside the material of the con- 
 ductor and surrounding the hollow ; the normal force at every point of this 
 surface being a point in the interior of the metal must be zero so that the 
 total normal induction through the surface is zero ; the total charge inside 
 the surface must therefore also be zero. The total charge is, however, the 
 sum of the charge induced on the inner surface of the conductor and the 
 charges on the bodies in the hollow : these must therefore be equal and 
 opposite. 
 
 But in the experiments mentioned the. total charge of the metallic vessel 
 into which the charged bodies were inserted was zero ; so that the charge 
 on the outer surface of the vessel must be equal and opposite to that on the 
 inner surface : it will therefore be equal to the total charge on the bodies 
 inserted in the interior. This is the result deduced experimentally. 
 
 105. Referring back again to the result just established, that the 
 quantity of electricity at the end of the tube of force multiplied by 4?r is 
 numerically equal to the induction along the tube, we may conclude that 
 the density of the charge at any point of a conductor in any field is given by 
 
 ff = E^ _\_ty 
 far . 4rr dn ' 
 
 O I 
 
 where E n = - determines the normal component of the electric force at 
 
 a point in the field just outside the conductor near the point where the density 
 is examined. This is of course a particular case of the more general result 
 established above that 
 
 d<f>\ fd<f> 
 
 d<f>\ fd<f>\ 
 
 *r) ( a + ^ a = > 
 dnj + \dnj_ 
 
 because I ~~ 1 is zero the potential inside the conductor being constant. 
 
104-106] Two dimensional fields 103 
 
 The total charge on the conductor is 
 
 f^df 
 
 taken over its surface. 
 
 106. Electrostatic problems in two dimensions. In many cases which 
 occur in actual practice the configuration of the field both as regards the 
 distribution of masses on it and the distribution of charge is uniform in a 
 certain direction over a considerable range : this is for instance the case for 
 a very long conducting cylinder carrying any charge and placed near to and 
 parallel to other cylindrical bodies, conductors or dielectrics, charged or 
 uncharged. If the cross dimensions of the field are small compared with 
 the length over which it is uniform (i.e. compared with the lengths of the 
 cylindrical bodies involved) the circumstances are so simple that they are 
 worthy of separate consideration. From the results obtained we can of 
 course infer the general nature of the result to be expected from a more 
 complex case. 
 
 In all these cases the variation of the field is so slight in the direction of 
 the axes of the field that it is entirely negligible as compared with its variation 
 in the cross directions : thus if we choose the z-axis of the rectangular co- 
 ordinate system parallel to the direction of the field axis and use < as the 
 potential in the field the general equation 
 
 dx z dy z 9z 2 
 will reduce practically to* 
 
 In this case the density p of the charge distribution is of course independent 
 practically speaking, of the z-coordinate. The^general integral of the funda- 
 mental equation is known to be of type 
 
 with the usual notation : if we write 
 
 dv = dxdydz = dfdz 
 and then integrate along the cylindrical axis of the field we get 
 
 or writing (x, y, z) for the coordinates of the volume element in the integral 
 and (x p , y p , z p ) for the coordinates of the point in the field at which the 
 function is calculated we have 
 
 r 2 = (x - x P Y + (y - y ft Y + (z - z p ) 2 
 = ri + (z - z p ) 2 . 
 
 * Laplace, M&anique celeste (Paris, 1799), n. 13. 
 
104 The characteristic properties of the electric field [CH. n 
 
 -I O 
 
 Thus - - = =- log (2 - 2 P + r), 
 
 T OZ 
 
 <f>= I pdf log2-Z, 
 
 and this is easily verified to be of the form * 
 
 = C- 
 
 where C is a very large constant of the order log 1 , 1 being the length of the 
 cylinder. 
 
 This is the general form of the logarithmic potential applicable in problems 
 of the present type : in it pdf is to be interpreted as the amount of charge 
 per unit length of cylinder standing on the element of surface df = dxdy in 
 the (x, y) plane which may be conveniently chosen across the central section 
 of the length along which the field is uniform. 
 
 The components of force are derived in the usual way : that along the 
 direction of the arcs of the field is practically zero and the others are 
 
 _a/> _a 
 
 X **\ 3 ?/ O * 
 
 dx oy 
 
 The function (J> is of course constant throughout the interior of all conducting 
 masses and the density of charge on any one of these reckoned as the amount 
 per unit length of cylinders is 
 
 the total charge being 
 
 477 dn' 
 
 taken round the cross section curve of the cylindrical conductor. 
 
 107. In the cases which usually occur there is no volume distribution of 
 charge so that the potential function satisfies an equation of the type 
 
 d*<f> dy>_ 
 
 dx* T dy z 
 
 and this fact enables us to introduce all the beautiful results of Cauchy's 
 theory of analytic functions. In fact we know from the general theory of 
 these functions that the real part of any differentiable analytic function of 
 
 thetype /(*) = /> + y) = o, 
 
 must satisfy the fundamental equation : in fact if we write 
 
 w = <f) + iijj=f(x + iy), 
 
 * C. Neumann [Jour. /. Math. LXIX. (1861), p. 335] calls this the logarithmic potential. Cf 
 also his Untersuchungen ilber das logarithmische und Newtonsche Potential (Leipzig, 1877). 
 
106-108] Two dimensional fields 105 
 
 we see that 
 
 oo 'o> 
 
 Sec ox \oy oyj 
 
 so that a<_ _80 a/r_ety 
 
 QU Uild U ^ o *-\ r\ 5 
 
 ox cy ox cy 
 
 8V , sv 8 2 <A 3 
 
 and thus =- + .-' = ^ + -^ = 0. 
 
 ox 2 Oy 2 dx 2 du 2 
 
 iy u 
 
 Thus either the functions </> or t/j will serve as a solution of the fundamental 
 potential equation. If we take <f> then the curves 
 
 <f> = const. 
 
 'are the sections by the (x, y) plane of the cylindrical equi-potential surfaces 
 in the field : in this case also the curves 
 
 ifi = const., 
 
 which are the orthogonal trajectories of the former set, represent the lines of 
 force in the plane section of the field. 
 
 108. The simplest example of this type is provided by taking 
 
 w = z n , 
 or introducing polar coordinates in the (x, y} plane so that 
 
 x = r cos 6, y = r sin 6, 
 z n = Ar n (cos B + i sin 6) n 
 
 = A(r n cos nd + r n i sin nd>, 
 and thus $ = Ar n cos nd, 
 
 are the equi-potential curves 
 
 j/i = Ar n sin nd, 
 the lines of force, or vice versa : particular cases are worth considering. 
 
 (i) n 1 gives 
 
 </> - Ax, tfi= Ay, 
 and the field is uniform. 
 
 If we take any two of the surfaces (/> = const, and place on them charges 
 to make their potentials assume the appropriate values then the field between 
 will be completely specified as to its potential by the function 
 
 (f) = Ax, 
 
 which satisfies the fundamental equation and all conditions as to regularity 
 besides taking the proper values on the surfaces. 
 
 Of course one of the two surfaces may be taken a very long distance off 
 when the field is that of the single conductor by itself. 
 
 The present example determines the field between two uniformly charged 
 planes. 
 
106 The characteristic properties of the electric field [CH. n 
 
 (ii) n = 2 gives 
 
 (h = Ar 2 cos 20 
 
 so that the equi-potentials are rectangular hyperbolic cylinders, including as 
 a special case two planes intersecting at right angles. 
 
 This transformation gives the field in the immediate neighbourhood of 
 two conducting planes meeting at right 
 angles in any field of force, or freely 
 charged. It also gives the field between 
 
 two coaxal rectangular hyperbolic ^ \ WX s - /"*> / / 
 cylinders. 
 
 m 
 
 = \ gives 
 
 x + iy = (<f> + iii 
 
 Fig. 25 
 
 and thus also 
 
 y* = 4c 2 (x + cf> 2 ). 
 
 Thus the equi-potential curves are con- 
 focal parabolas, including as a special 
 case ((j> = 0) a semi -infinite plane bounded by the line of foci. 
 
 This transformation gives the field in the immediate neighbourhood of 
 a conducting sharp straight edge in any 
 field. 
 
 (iv) w = log z gives 
 <f> + *0 = A log r (cos Q -\- i sin 6) 
 = A log re ie 
 = A log r + iAO, 
 so that the equi-potentials are the cylinders 
 
 r = const., 
 the lines of force are 
 
 6 = const., 
 
 they radiate out in straight lines from the 
 axes. 
 
 This is the field due to a uniform line 
 charge or to a uniformly charged right- 
 circular cylinder : the charge per unit length 
 on the cylinder is equal to 
 
 Fig. 26 
 
108, 109] Two dimensional fields 107 
 
 1 ' 8</> 1 i~(d<f> dy d(f> dx 
 
 "477. dn d ' '47T \\dxds~ dyds 
 
 dy d(f> dx\ , 
 d ' ' ~ ) d 
 
 -I /. 
 
 <fe 
 
 J4 
 
 [j/r] 
 
 = ITT"' 
 
 </> denoting the total change in i/r on going round the boundary curve of the 
 section : in the present case 
 
 [iff] = [A8] = 277^, 
 
 A 
 
 so that Q = -^ , 
 
 and thus <j> = 2Q log r, 
 
 apart from a constant. 
 
 109. The relation w = A log 
 
 & 2 + a 
 
 determines a field equivalent to the superposition of the fields given by 
 
 w = A logfz a ' and w = A log z + a ; 
 
 the transformation is accordingly that appropriate to two equal and opposite 
 line charges along the parallel lines 
 
 z a and z = a. 
 
 We may generalise this because if we introduce two sets of polar coordinates 
 with these lines as axes we get 
 
 w = A log 
 
 ,4- ~ 
 
 = A Io 8 in 
 
 (j) + iifi = A log ^ + ^ (0! - 2 ). 
 
 r z 
 The equi-potentials are the surfaces 
 
 T 
 
 = const., 
 r 
 
 that is they are cylinders standing on two members of a coaxial system of 
 circles with the points on the lines 
 
 z = a, z = a, 
 as limiting points. 
 
 The solution is therefore that appropriate to any two of such cylinders 
 equally charged and influencing one another. The lines of force are the 
 
 orthogonal circles in any plane : the charge per unit length on the cylinders 
 
 ^ 
 is as before = ^, so that 
 
108 The characteristic properties of the electric field [OH. n 
 
 110. If w = cosh" 1 - , then 
 c 
 
 x c cosh <f> cos iff, 
 y = c sinh </> sin /r. 
 The curves </> = const, are the ellipses 
 
 /Y-2 ni2 
 
 c 2 cosh 2 (f> c 2 sinh 2 </> 
 and the curves iff = const, are the hyperbolas 
 
 c 2 cos 2 iff c 2 sin 2 iff 
 
 These conies have the common foci ( c, 0). 
 The field in which <= const. 
 
 are the equi-potential surfaces is therefore that appropriate to the case of 
 either a single elliptic cylinder freely charged or of two such cylinders confocal 
 with one another and carrying equal charges : a particular cylinder (</> = 0) 
 is the flat plate between the line of foci. 
 
 Fig. 27 
 
 We might however also take the surfaces = const, as the equi-potential 
 surfaces : the field is then that appropriate to a freely charged hyperbolic 
 cylinder including as a particular case the infinitely extended plate with 
 a slit in it between the focal lines. 
 
 111. Let w be defined as a function of z by the implicit relation 
 
 z = w + e w , 
 so that x = <f> + e* cos $ y = iff + e* smtjj. 
 
110, 111] 
 
 Two dimensional fields 
 
 109 
 
 The line A = 
 
 coincides with the axis of x : the line 
 
 iff TT 
 
 is the line y = TT between the points x = oo and 1 : the line 
 
 iff = TT 
 
 is the line y = TT, 
 
 also between the same values of a;. 
 
 Thus if we take the surfaces 
 
 ifj = const. 
 
 as the equi-potential surfaces in the field we shall have the field of two 
 similarly charged semi-infinite plates placed symmetrically parallel to one 
 another with a difference of potential equal to ITT. 
 
 Fig. 28 
 
 This solution provides us with the approximate nature of the field in the 
 neighbourhood of the edges of the parallel plate condenser to be discussed 
 at a later stage provided the distance apart of the plates is small compared 
 with the curvature of the edges. 
 
 The methods here illustrated depend of course on a knowledge of suitable 
 forms of relations between z and w and they exhibit no systematic method 
 for obtaining the relation when the particular form of the surfaces is given. 
 
110 The characteristic properties of the electric field [CH. n 
 
 There is no method of doing this for the most general cases, but when 
 the curves of intersection of the cylinders on a cross section plane are linear 
 polygons a general method suggested by Schwarz* will always, theoretically 
 at least, effect the transformation. A consideration of this method would 
 however take us beyond the sc6pe of this work; it will be quite familiar 
 to those students acquainted with the methods of conformal representation. 
 
 112. On the mathematical theory of electrification by induction. We 
 
 have in the previous chapter briefly explained the phenomenon of electri- 
 fication by induction : it consists essentially in the fact that if any good 
 conducting body is introduced into the electric field surrounding any system 
 of charged bodies it will in general be found to be electrified, certain parts of 
 its surface however exhibiting positive electrification and the other parts 
 negative electrification. This phenomenon arises from the action of the 
 electric forces in the field in pulling the free electrons in the metal about, 
 causing them to concentrate in certain parts of the metal, which will thus 
 be negatively electrified, whilst the parts from which they have been driven 
 will be positively electrified. Such a process will go on until the force due 
 to the original field in the interior of the metal is balanced by the force in 
 the new induced field which arises from the distribution of charges thus 
 brought about, which of course, as explained above, can only exist on the 
 surface of the metal. The question then naturally arises : can we determine 
 the distribution of charge thus induced on any conductor if the original 
 inducing distribution is specified and also the mechanical force of attraction 
 of the conductor which results from it? 
 
 Now whatever distribution is induced on the conductor the potential 
 </> of the new total field must be a regular function satisfying the following 
 conditions : 
 
 (i) V 2 < - 
 
 at all points of space where there is none of the inducing charge with a finite 
 volume density p, at other points it satisfies the equation 
 
 V 2 </> + 477/3-0. 
 
 We shall temporarily assume that the inducing charge can be completely 
 specified by the distribution of volume density p. 
 
 (ii) </> must be constant on the surface and throughout the interior of the 
 conductor and it must be the same as the potential of the inducing field at 
 a great distance from the conductor where the effect of the induced charge is 
 negligibly small. 
 
 (iii) There is also the further condition that the total charge on the 
 conductor is unaltered by the induced charge and is (generally) zero. 
 
 * Jour. f. Math. LXX. (1869), p. 105. Cf. also E. B. Christoffel, Ann. di. mat. (2), i. (1867); 
 iv. (1870). 
 
111-113] The theory of electrostatic induction 111 
 
 The solution of a problem of this kind thus turns on a determination of 
 a regular solution of the differential equation 
 
 V 2 </> + 477/9 = 
 
 which shall satisfy the remaining conditions of the problem. It is obvious 
 that it is only in certain cases where the inducing field is expressible in 
 analytical form and the shape of the conductor is of a geometrically simple 
 type that the solution can be effected. 
 
 113. As an example of the method we may consider the comparatively 
 simple case of a spherical conductor in a part of the field far removed from 
 all the inducing charges (which are therefore at infinity, as we say), where 
 the electric force intensity is practically constant in both magnitude and 
 direction over a region large enough to contain the sphere. We shall refer 
 the field to rectangular axes with the origin at the centre of the sphere and 
 the x-axis parallel to the direction of the lines of force in its neighbour- 
 hood. Thus near the sphere the potential of the given field can be written 
 
 in the very approximate form 
 
 <=</>- Ex, 
 
 <f> being the potential at the origin and 
 
 E = - 
 
 \dxJ 
 
 the force there. 
 
 This is the inducing potential. When the sphere is introduced certain 
 charges will be induced on its surface and their field will be superposed on 
 the field just specified. Let <f> 1 be the potential of these induced charges so 
 that the potential of the total field is 
 
 $ = <f>j + ^>o ~~ E X - 
 This must satisfy the following conditions : 
 
 (i) V 2 (/> = everywhere in the field at a finite distance from the origin. 
 
 (ii) (f> = <f> Ex at a comparatively great distance from the sphere 
 where the effect of the local disturbance produced by the introduction of the 
 sphere is negligible. This of course assumes that the form <f> Ex represents 
 the original field at such distances : if it does not the proper value must be 
 inserted ; in any case the argument is not appreciably modified. 
 
 (iii) <f> must be constant over the surface and throughout the interior of 
 the sphere ; and finally 
 
 (iv) the total charge on the sphere is zero so that 
 
 the integral being taken over the surface of the sphere. 
 
112 The characteristic properties of the electric field [CH. n 
 
 114. Let us try a solution of these conditions with 
 
 Ax B 
 
 </>i = -3- + - 
 r 3 r 
 
 outside the sphere and <j = Ex 
 
 inside. These forms are specially chosen to satisfy the first two conditions : 
 
 they satisfy the third if 
 
 4 -:*- o, 
 
 a 3 
 i.e. if A = a*E. 
 
 We have finally to put down the condition that the total charge on the sphere 
 is zero ; it is of course all on the surface and its density is determined by 
 
 J_ W = ty 
 4w dn 4:TT dr 
 
 _1_ /B 3Ex\ 
 * " 
 
 + 
 
 where we have used the condition that A = a 3 E. Thus the condition for no 
 charge requires that 5 = 0. We thus satisfy all the conditions in the field 
 with the potential function 
 
 </> = < -z(l-^ 3 ) 
 
 outside the sphere and <j> = < 
 
 inside, these two values agreeing as they must do at the surface. We have 
 therefore completely determined the circumstances for this special case of 
 the general problem. The density of the charge induced on the sphere is 
 given as above by 
 
 4:770, 
 
 being positive on one side and negative on the other. 
 
 Before proceeding to a discussion of the mechanical relations of this field 
 we shall illustrate by a diagram the type of disturbance here obtained. This 
 is done in the figure below where the lines of force in the part of the field 
 near the sphere are drawn. 
 
 This diagram is easily constructed if it is noticed that the field induced 
 outside the sphere which is superposed on the original uniform field, viz. the 
 field of the potential 
 
 is precisely the field of an electric doublet at the centre of the sphere with 
 its axis along the direction of the field and of amount Ea 3 . 
 
 The method followed in this simple case is typical of the more general 
 one to be followed in any case of the present type. The given field involves 
 
114, 115] The theory of electrostatic induction 113 
 
 a harmonic of the first order and so we try for the additional field harmonics 
 
 00 
 
 of the same type : outside the sphere we must use 3 instead of x because the 
 
 latter is regular at infinity, whereas inside the sphere the latter, type is the 
 correct one to try since the other one is not regular at the 'origin. 
 
 Fig. 29 
 
 115. Now let us examine the mechanical relations of the field thus 
 determined. The total work required from external systems to establish 
 the sphere in its position in the field is represented by the integral 
 
 taken over its surface ; it is of course presumed that the inducing system is 
 either rigid or at least too far away to be affected by the introduction of the 
 sphere. Inserting the values for cf> and a we get 
 
 w = 
 
 -- 
 
 4:770, 
 
 Thus no energy at all is required from external systems. The energy for 
 the mechanical work actually performed will thus all come from the store 
 of internal energy in the sphere, part of which is set free by the separation of 
 the charges on the conductor. The amount of energy thus set free is 
 
 L. 
 
114 The characteristic properties of the electric field [CH. n 
 
 and thus the amount gained in mechanical work (raising weights, etc.) 
 during the introduction of the conductor is 
 
 a*W 
 
 IT ' 
 
 We may conclude that the force tending to move the sphere is 
 
 so that it will tend to move into regions of stronger force. 
 
 116. We can verify this result directly but it requires a closer in- 
 vestigation taking into account the next approximation of the original 
 field. Owing however to the intrinsic interest of the problem it seems 
 worth while indicating the analysis. The inducing field must now be taken 
 in the more general form* 
 
 The field after the introduction of the sphere must satisfy the usual funda- 
 mental conditions besides reducing to the constant value (</> ) at the surface 
 of the sphere and agreeing with the above value at a great distance. Bearing 
 in mind the remark made above we see that we may take for the new field 
 outside the sphere 
 
 /, 
 
 1 1 - 
 
 the inside potential being still (f> . The density of the charge on the sphere 
 will be similarly given by 
 
 The force on the sphere will be identical with the force exerted by the external 
 system on the system of charges induced on it and its ^-component will be 
 f f/3^\ /8 2 </A (&4>\ 
 
 F - = J " t(i), + * (w\ + y (sU + z 
 
 which to the second order of approximation reduces to 
 
 * / ' 
 
 * It is not now necessary to assume that the x-axis is parallel to the direction of the field 
 at the point. 
 
 f The suffix denotes the values of the functions at the origin. 
 
115-117] The theory of electrostatic induction 115 
 
 provided that ' I-f + hr$ + hr = > 
 
 J- \ /~\npt i \ /-J2/" / \ /Ho'*5 / 
 
 \\jJu /Q \ t/ */ / 
 
 which is necessarily satisfied. Thus 
 * = 2" dx ]\dx) 
 
 /3 
 
 or generally F = -^ grad E 2 , 
 
 A 
 
 as above. 
 
 The theory thus completely accounts in a general way for the whole 
 series of phenomena associated with the electrification of a conductor by 
 induction. 
 
 117. As a second example of these principles we may consider the 
 slightly more general case of a conducting ellipsoid (axes a, 6, c) intro- 
 duced into a field which is practically uniform in the neighbourhood of 
 the ellipsoid. Again choosing axes at the centre of the ellipsoid, with how- 
 ever the principal directions now along the principal axes of the ellipsoid, 
 the field in the neighbourhood of the origin may be expanded in the 
 approximate form 
 
 4>=(j> Q -E x x- E y y -E g z. 
 
 We shall first consider the case where the inducing field is parallel to the 
 axis of x so that the potential is 
 
 & = fa - E x x. 
 
 When the ellipsoid is introduced charges will be induced on it and the potential 
 function </> of the new field will be a regular function satisfying the fundamental 
 
 equation 
 
 V 2 </> = 
 
 at all points of space, reducing to a constant on the surface of the ellipsoid 
 and agreeing with the original field at a great distance. The appropriate 
 type of solution could be obtained directly but we have indirectly obtained 
 it in the first section where the fields of certain ellipsoidal distributions 
 were directly examined. Kemembering the results there given we are 
 induced to try solutions of the type 
 
 <t> = (> Ex Lx 
 
 outside the ellipsoid and <f> = </> 
 
 inside : in the first of these expressions A is the positive root of the equation 
 
 2 ,Z ?2 
 
 4- ^ - 4- - 1 
 
 9 i J ' ll9 i J ^ 2 i f "* 
 
 a* + t b* + ^ c* 5 + ^ 
 
 82 
 
116 The characteristic properties of the electric field [OH. n 
 
 This form of (/> satisfies all the conditions except constancy of (/> on the surface 
 of the ellipsoid, and it satisfies this if 
 
 dt 
 
 _ 
 
 o (a 2 + V(a 2 + t) (6 2 "+ t) (c 2 + t) 
 We use as above 
 
 A = i_ a 3 bc 
 
 __ 
 
 o (a z + t) V(a z + t) (6 2 + t) (c 2 + t) ' 
 
 a 3 bcE x 
 so tnat L/ x . . 
 
 We are thus enabled to satisfy all conditions with the outside potential in 
 the form 
 
 a 3 bcE x x r 00 dt 
 
 2A 
 and the inside one in the form j _ A 
 
 118. The density at the point (x, y, z) on the surface is given by 
 
 1 O(j) 1 fpx 9<^ py d<f) pz 9<^\ 
 
 I i - 1_ I 
 
 T-'TT (j'Yt i TT \ Ct (jIT h (Jtl f* fj? / 
 
 where p is the central perpendicular on the tangent plane at the point, 
 
 fpx py pz\ 
 
 U 2 ' 6 2 ' c 2 / 
 
 being then the direction cosines of the normal to the surface ; but remembering 
 the rule for the differentiation of a definite integral function with respect to 
 a parameter occurring in one of its limits we see that this is 
 
 a s bcE x x / 2p \ 
 h A I <rr~ ) 
 
 4.TT ) A \ n&hf t 
 T:7T . --/I xCc/^OO/ 
 
 "l^Z' 
 and adf 
 
 taken over the surface of the ellipsoid vanishes, as it should do, there being no 
 charge on it. 
 
 In the more general case we may easily see that the potential in the field 
 outside the ellipsoid is 
 
 </> - E x x - E v y - E z z 
 
 abc r | a*E x x b*E y y J?EzZ_ \ dt 
 
 " 2 J, \A (a 2 + ) + B (6 2 + t) + C(c* + t)j V(a 2 + t) (6 2 + (c^+J) ' 
 whilst the density on the surface is 
 
 p fxE x yE v zE z \ 
 ~ TT\A B ~C- 
 
117-120] The theory of images 117 
 
 119. The energy function of the mechanical forces on the ellipsoid can 
 be calculated as before either directly or as the equivalent of the energy 
 required to establish the separation of charges on the ellipsoid : it turns out 
 to be 
 
 / 75 /r 2 /? 2 u 2 E 2 Z 2 E 2 \ 
 
 TTT I A' / *-* X V "^ "tf "^ Z \ JJ? 
 
 w = r- + -- + -7H #. 
 
 . 87rV 4 5 C 
 
 the integral being taken over the surface of the conductor, and this is equal to 
 
 abc/E x * E y * E z * 
 ' T\A " B " C 
 
 The mechanical forces tending to drag the ellipsoid into the field are obtained 
 as the gradients of this function. In addition to this there are forces tendijig 
 to rotate the ellipsoid. To obtain some idea of these we can specialise our 
 field slightly and put 
 
 E x = E cos 0, E y = E sin 0, E z = ; 
 
 abcE* /cos 2 sin 2 
 we have then 
 
 B 
 
 and the couple tending to increase the angle 6 is 
 
 dW abcE 2 . n fl I 
 
 Now if a > b, then A < B, so that the couple is negative, in other words the 
 tendency is to set the long axis of the ellipsoid along the field. 
 
 There are other general methods of attacking problems of the type 
 just examined; but as they depend essentially on some rather important 
 analytical results deduced from Green's theorem we shall find it more con- 
 venient to discuss these first. 
 
 120. Green's equivalent stratum general theory of images *. According 
 to the general formula of Green applied to space external to a given surface 
 /, the value of <f> satisfying the usual continuity conditions at any point P 
 outside the surface is given by 
 
 . dv f /a<i d<f> 2 \df' 
 = - J VV 7 - l f (I,' - |?) i 
 
 where the first integral extends to all space outside the surface /, the second 
 and third to all surfaces of discontinuity outside / and the fourth to / itself. 
 
 * Cf. Kelvin, Cambridge and Dublin Math. Jour. 1848, 1849, 1850. Reprint, 55 et seq. 
 also 208 et seq. See also Thomson and Tait, Treatise on Natural Philosophy, u. sees. 499-518. 
 
118 The characteristic properties of the electric field [CH. n 
 
 Now suppose that the surf ace /is so charged that at each point of it there 
 is a surface density a and a double sheet distribution T, where 
 
 L - 
 brdn' ^477' 
 
 the potential of this distribution is 
 
 . JL 
 
 4-77 
 
 We now interpret <f> P , in the above formula, in the language of electric 
 theory as a potential function of an electric field arising from a charge distribu- 
 tion obtained from it in the usual manner. Now superpose on this electrical 
 field the electrical field of the imaginary distribution on the surface / defined 
 above. The total potential is now 
 
 , r dv ' /%>! 8<M df r 8 /i\ 
 
 V'cb I -~ - -= -j- I [91 <z> 2 ) 51 I aj 
 
 J r .f\dn dn J r J /> T ' dn\rj 
 
 That is <f> P ' is due solely to the charge distribution external to the surface 
 /. The imaginary distribution therefore completely cancels the effect of the 
 charge distribution inside / at all points external to that surface. This 
 distribution with its sign changed thus completely represents the external 
 system at all external points. 
 
 This is Green's equivalent stratum. Any electrical system can be replaced 
 by a distribution similar to that above, on any surface completely surrounding 
 it, as long as its action at external points is under review. 
 
 121. If in the above example the surface/ had .been an equi-potential 
 of the whole original field the double sheet distribution would not be 
 required, since as <f> is constant over the surface the potential due to this part 
 of the distribution at external points is 
 
 The distribution on the surface / would then be the same as if the surface 
 were conducting, because the distribution 
 
 !_a^ 
 
 4?r dn 
 
 on it is in agreement with the fact that it is an equi-potential surface. 
 We know for example that the potential of the point charge q at is 
 
 2 
 
 r' 
 and the equi-potential surfaces are concentric spheres : if we take the one 
 
 whose radius is ( -jr- 1 its potential is </> , and thus the external potential of this 
 
 Wv 
 
120-122] The theory of images 119 
 
 sphere charged to potential </> is the same as that of the point charge q at 
 its centre and the charge distribution on it is uniform : this agrees with what 
 we have found above. 
 
 Another example is provided by the case of three point charges a-^ Q at 
 O l5 a 2 </> at 2 and - - </> a * 0, where however divides the line O t 2 * 
 
 C 
 
 in the ratio of the squares of % and 2 an ^ c 2 = OjOa 2 = a^ + 2 2 - I n 
 case the equi-potential surface in the field on which </> = <f> is the outer 
 portion of two intersecting orthogonal spheres of radii a l5 2 , centres at 1 
 and 2 respectively, and is the mid-point of their common chord. Thus 
 
 Fig. 30 
 
 these three point charges effectively represent the external electric field of 
 the free equilibrium distribution of charge on the conductor which makes its 
 potential </>; the amount of the charge is equal to the sum of the point 
 charges inside and is therefore 
 
 The analysis for this case, which is quite straightforward, will be obvious 
 when we have treated the more general problem now to be discussed. 
 
 122. In a case like this last it is often desirable to know the actual 
 distribution of the charge on the conductor and the relative amounts on 
 the different parts of the surface. This of course is easily obtained as soon as 
 the potential function is specified completely, as is for instance always the 
 case when the image system can be reduced to a series of point charges 
 finite in number. We can however find the total charges on different 
 portions of the surface by an application of Gauss' theorem to non-closed 
 
 * Maxwell, Treatise, i. 166. This author also solves the general case of spheres cutting at 
 any angle ir/n. 
 
1 20 The characteristic properties of the electric field [OH. n 
 
 surfaces. For example, in the case of the conductor formed by the outer 
 parts of two orthogonal spheres the total induction through the sphere of 
 radius a is 2?(l + cos0), 
 
 where 6 is the angle of the cone subtended by the image q at the circle of 
 intersection of the spheres : in this case this is 
 
 a, a, , 
 - - - - <t>o 
 
 and thus the charge on this sphere is 
 
 _ ^o L a i2 , 
 
 477 2 T 1 
 
 The case where a 2 is small compare v d with t enables us to approximate to 
 the effect of a small knob on an otherwise perfectly spherical surface. 
 
 If a v is very large the solution is of the type appropriate to a plane conductor 
 with a hemispherical boss on it. 
 
 123. Now consider the problem in another form. Suppose we have 
 any two electrical systems A and B for which combined we can determine 
 the equi -potential surfaces by calculation. Suppose that <b = </> is one of 
 these surfaces which divides the system A from the system B, the system A 
 being inside the surface and B outside it. 
 
 Now consider the surface c/> = </ as a conductor charged to potential < n 
 and under the influence of the electrical system B outside it; what is the 
 potential function and distribution of charge. The function has to take the 
 value (f> = (f> on the given conductor and <f> = at infinity and has also t'o 
 include the discontinuities of the system B in the legion between. Now 
 the original potential function <{> of the combined systems is a function which 
 outside (f> = <J>Q satisfies all these conditions and must therefore be the required 
 solution of the problem. The surface density on the conductor <^> is 
 
 i a< 
 
 a = A 3 
 
 4?T on 
 at any point. 
 
 Thus the distribution on the conductor just obtained has at all external 
 points the same potential as the old system A and consequently also the 
 same force intensity. Also the total quantity of electricity on the conductor 
 is the same in amount as the total of the system A. 
 
 For internal points the charge on the surface and the B system produce 
 a constant potential </> and therefore the force is zero : thus the charge and 
 the system B produce the same force inside but not the same potential unless 
 in the special case </> = 0. 
 
 When </> is not zero we may however regard the charge on the conductor 
 <f> = </> as consisting of two charges superposed, viz. (i) a charge which would 
 
122-124] 
 
 The theory of images 
 
 121 
 
 be exactly equal and opposite to B in its. internal effect and (2) a charge 
 which would produce a constant potential </ on the conductor in a field by 
 itself. 
 
 In a generalised sense the distribution in the system A may be regarded 
 as the image of the distribution B in the conductor (^ = </> when it is at a 
 potential <^> . 
 
 Notice that all electrical images are virtual ; the system A must be com- 
 pletely enclosed by the surface <j> = </> . 
 
 124. If the system A consists of the single point charge q' at A' and 
 the system B of the point charge q (> q') at A, then the equi-potentials of 
 the combined system are determined by 
 
 Fig. 31 
 
 r and r' denoting the distances AP, A'P from A, A' respectively to any 
 point in the field. The surface </> = is the sphere on BE' as diameter, where 
 
 AB _ AW _ 
 BA'~A'B'~ q'' 
 
 a 2 
 If the centre of this sphere is and OA =/, then OA' = -^ , where 2a = 
 
 and therefore 
 
 We thus see that if a sphere of radius a is kept at zero potential under the 
 influence of the point charge q at A, the charge on the sphere acts at all 
 
 external points just like the charge 
 
 aq 
 
 7 
 
 at the inverse point A' would do. 
 
122 The characteristic properties of the electric field [CH. n 
 
 Under these circumstances the normal component of force outside the 
 sphere must be the same as that due to the two charges and we can therefore 
 calculate the density of the induced charge on the sphere; for at any point 
 Ponit 
 
 a (normal force inwards) 
 
 (X. 
 
 qa 
 
 C S ^ + fTA'P 2 C S 
 
 <f> l} <f> 2 being the angles marked in the figure. Thus 
 
 q [ 1 ., a ( a 2 
 
 - * (/cos * - a) + '* a ~ 
 
 a 3 / a 2 
 
 The density varies inversely as the cube of the distance from the external 
 
 point. We know that the total charge on the sphere is - - and so we 
 
 ^ 
 conclude that a distribution of density -3 on any sphere of radius a, where 
 
 r is the distance from a fixed external point distant / from the centre, has a 
 total mass or quantity 
 
 and that it acts on all external points as if it were condensed at the inverse 
 
 point; but it acts on all internal points as if , ' were condensed at the 
 
 / 2 - 2 
 external point. 
 
 125. In considering the more general case, with the same notation, in 
 which the system A consists of - -^ at A' and a<f> at 0, the general potential 
 function with the same system B is 
 
 q aq a<f> 
 (t> -r~fr' + ~It> 
 
 R denoting the distance from to any point in the field. Now the surface 
 
 <A = 0o 
 
 is the sphere R a : and this shows that the image of the charge on the 
 sphere still under the influence of the point charge q at A, but now at a 
 
 potential </> , consists of the point charges - ~ at A' and </> at 0. 
 
124-127] The theory of images 123 
 
 The total charge on the sphere in this case is the sum of the images and 
 is therefore 
 
 and thus if Q had been given instead of ^> , we could determine </> from this 
 relation as 
 
 The density on the sphere is determined as before and is given by 
 
 AP 3 
 
 In the particular case of the first example, where q= q', the sphere 
 degenerates into the infinite plane which bisects perpendicularly the line A A'. 
 
 126. The mechanical forces tending to move the conductor can also be 
 simply determined from the image system, for it ;is equal and opposite to 
 the force of reaction on the external inducing system and this depends only 
 on the field in its immediate neighbourhood ; so that it is independent of 
 whether it arises from the actual distribution on the conductor or the 
 internal image distribution which effectively represents it at all external 
 points. 
 
 Thus, for example, there is a force tending to pull the charged sphere 
 of the above example towards the point charge q of amount 
 
 . aq 
 
 L/ 2 / 2 (/ 2 - a2 ) 2 J ' 
 if q, Q are of the same sign it is positive when the charge is near the sphere 
 but negative when it is at a great distance away. 
 
 127. Returning now to the case of the conductor formed by the larger 
 portions of two orthogonal spheres we see at once the reason for the par- 
 ticular choice of the point charges : the point charge - <j> Q is the image 
 
 
 
 of the charge at either centre in the other sphere. Let us now consider 
 the case of the conductor uninsulated and under the influence of a point 
 charge q at an external point P. Now we know by geometry that if P x is 
 the inverse point of P in the first sphere 0, and P 2 is the inverse point of P 
 in the second sphere and if X P 2 and 2 P l intersect in P 3 , then P 3 is the 
 inverse of P 2 in the first sphere and of Pj in the second sphere and also that 
 
124 The characteristic properties of the electric field [CH. n 
 
 Thus if O^P =f 1 and 2 P =/ 2 and we put charges at P 1? P 2 an d ^3 respec- 
 tively of amounts 
 
 a t q a. 2 q 
 
 " ~ 
 
 the potential over the surface of the conductor is constantly zero. The 
 denominator in this last fraction is the common value of 1 P . 2 P]_ and 
 2 P.0 1 P 2 . 
 
 Fig. 32 
 
 If we want the more general case when the conductor is insulated and 
 charged we must insert the appropriate images at O l5 and 2 to give the 
 total charge right. 
 
 The force on the conductor can also be obtained as the resultant of the 
 forces on the images inside it. 
 
 128. Finally let us briefly consider the simple case of the charges 
 induced on two infinite plane conductors at right angles to one another 
 and under the influence of a point charge </"in the angle between them. The 
 system of images is obvious, they will be q, + q, q respectively at the 
 corners of a rectangle symmetrical round the angle and with one corner at 
 the position of q. Thus if r x , r 2 , r 3 , r 4 denote the respective distances from 
 the four corner charges the external potential of the charge distribution 
 induced on the planes is 
 
 _1 + 1_^ 5 
 r 2 r 3 r 4 ' 
 whereas the internal potential is 
 
127, 128] 
 
 The theory of images 
 
 125 
 
 The law of distribution of charge on the conductor can now readily be obtained 
 as above. The total charge on the horizontal conductor turns out to be' 
 
 tan" 1 ^, where a and 6 are respectively the distances of q from the 
 
 6 
 vertical and horizontal planes. 
 
 Fig 33 
 
CHAPTER III 
 
 THE ELECTRICAL AND MECHANICAL RELATIONS OF A SYSTEM 
 
 OF CONDUCTORS 
 
 129. On the relations between the potentials and charges of a system of 
 conductors. The discussion of the two former chapters indicates as the 
 main theoretical problem in electrostatic theory, the determination of the 
 distribution of charge on a given system of conductors under given 
 conditions and the deduction finally of the mechanical relations of these 
 conductors. All electrical experiments of a static nature are made with 
 various forms of conductors under different conditions and a knowledge of 
 their mutual relations when charged is therefore essential to a correct inter- 
 pretation of the results of such experiments. 
 
 Of course if we can determine the potential function of the field our 
 problem is completely solved; but this is the difficulty. Whatever this 
 function it must be continuous and regular at infinity and satisfy 'the equation 
 
 V 2 < = 
 
 at each point of space. In addition to this it must be constant throughout 
 each conductor. The question of the existence of such a solution is not one 
 we can enter into now, and we shall content ourselves by saying that from 
 the physical point of view there is a certain amount of evidence in favour of 
 at least one solution. 
 
 If we assume the existence of a solution it is easy to prove mathematically 
 that if the charges or the potentials of all the conductors are specified the 
 whole circumstances of the field are uniquely determinable*. 
 
 130. (i) In the first case when the charges are given the potential (/> 
 of the external field has to satisfy the Laplacian equation and the usual 
 regularity conditions, and it must in addition be constant over each con- 
 ductor and such that 
 
 1 
 
 taken over the surface of the rth conductor is equal to Q r , the charge on that 
 conductor. 
 
 * Cf. Maxwell, Treatise, I. ch. in. 
 
129-131] The uniqueness theorem 127 
 
 Suppose that there are two solutions <f> and (f> + ^ of these conditions : 
 then <f> 1} the difference of these two solutions, must satisfy all the conditions 
 as before except that 
 
 - I ^ df 
 4*T) fr dn ajr 
 
 is zero for each conductor. 
 
 If therefore we consider the integral 
 
 J 
 taken throughout the whole of space, it is equal to 
 
 the sum denoting the sum of integrals relating to the surfaces of the separate 
 conductors : the last term refers to the infinite boundary. This latter term 
 tends to vanish, </>! is regular at infinity ; and also since (f> is constant over 
 the surface of each conductor 
 
 for each of them. The original volume integral is therefore zero, but it con- 
 sists of essentially positive elements, which must therefore separably be zero 
 
 or 
 
 d<fti = <tyi = <i = Q 
 
 dx dy dz 
 
 at all points of space : this means that <f> 1 is constant, and being regular at 
 infinity, it must be zero. There is only one solution of the conditions. 
 
 (ii) A similar proof applies to this second case : there is no need to detail 
 it out in full. 
 
 131. We can now establish the important principle of superposition. 
 
 If for a system of conductors the potentials are < 15 2 , </> when the 
 charges are Q : , Q 2 , ... ^ w and^> 1 ',^> 2 ', ...(f> n f when the charges are Qi', Qz Qn ', 
 then when the charges are Q l + $/, Q 2 + Q 2 ', Q n + Qn th e potentials will 
 
 be (/>! + (f>i, 4>2 + < / ) 2 / ) <f>n+ 4>n 
 
 Let (f>, (f>' be the potentials of the electrostatic fields in the first two cases 
 and <!> the potential when the charges are superposed. 
 
 Then besides the usual conditions of regularity the first two functions 
 satisfy 
 
 (i) V 2 </> = 0, V 2 </>' = at all points. 
 
 (ii) <f) = (f) 1; </> 2 , ... <f> n , <f>' = <f>i, <f>2> fin on the respective conductors. 
 
 (iii) 5-- df r = - 4:TrQ r , ^ df r = 4wQ r ' for each conductor. 
 
 J / on . / on 
 
128 The electrical relations of the conductors [CH. in 
 
 The function <f> satisfies, besides the regularity conditions, also 
 
 V 2 <f> - 0, 
 and it is constant over each conductor and such that 
 
 for each of them. It is completely determined by these conditions ; but an 
 obvious solution of them all is 
 
 $> = <]> +<', 
 
 and since there can be but one solution this is the solution. But then the 
 potentials of the various conductors are 
 
 01 +</>/> 02 + 02'> " 0n+0n'- 
 
 Thus statical electric fields are superposable. 
 
 132. Now suppose that the result of placing unit charge on the first 
 conductor and leaving the others uncharged is to produce potentials 
 
 "11? i2> "in 
 
 on the n conductors respectively ; then the result of placing Q 1 on this same 
 conductor and leaving the others uncharged is to produce potentials 
 
 bnQi, b ln Q l . 
 
 Similarly if placing unit charge on the second conductor and leaving the 
 others uncharged gives potentials 
 
 0-213 b 2n , 
 
 then placing Q 2 on this conductor and leaving the others uncharged gives 
 potentials , n , n 
 
 2lV2lJ 2nV2- 
 
 In the same way we can calculate the result of placing Q 3 on the third 
 conductor, Q 4 on the fourth and so on. 
 
 If we now superpose the solutions thus obtained, we find that the effect 
 of simultaneous charges Q l} Q 2 , ... Q n is to give potentials l5 2 , ...</> where 
 
 01 = &llQl + &1202 + -+&l0n, 
 
 \ 02= b zl Qi + b 22 Q 2 + ... +b 2n Q n , 
 
 03 = 031$1 + &32$2 + " + b 3n Q n , 
 
 These equations give the potentials of each of the conductors as linear 
 functions of the charges on all of them. The coefficients 6 H , 6 22 > b lz , ..., 
 called the coefficients of induction do not depend on either the potentials or 
 charges, being purely geometrical quantities, which depend on the size, shape 
 and relative position of the conductors. 
 
131-133] 
 
 TJte coefficients of induction 
 
 129 
 
 By linear solutions of these equations regarded as equations in the charges 
 
 we obtain 
 
 Q l = c ll (f) l + c 12 <j> 2 + ... -I- c ln <f> n , 
 
 Qz = C 2l<f>l + C 22</>2 + ... + C 2n <j) n , 
 
 The coefficients c rs , the coefficients of capacity for the conductors are given 
 by the system of equations 
 
 Cn Cio 
 
 12 
 
 "12 5 b 13 , ... b ln 
 
 
 
 "21? "315 "nl 
 
 6 22 , 6 23 , ... b 2n 
 
 
 "235 6335 ... O w3 
 
 h 
 
 
 77 7 
 
 "325 
 
 
 245 345 4 
 
 6 31 , ... 
 
 133. There are many important relations which exist between these 
 coefficients in all cases, and which are in fact direct consequences of certain 
 general properties of the field surrounding them. 
 
 Many of these can be derived directly by a simple application of the general 
 reciprocal theorem due to Gauss which was given in the first chapter. 
 This theorem takes a simple form when the only charges in the field are those 
 on the conductors, which may be put in the words : 
 
 If two electrical systems are simply one system of conductors charged in 
 two ways with systems of charges Q l , Q 2 , . . . Q n in the first case and 
 Qi> Qz> Qn in the second and if the potentials of the conductors in the 
 two cases are <f>i, <f> 2 , </> n an( i ^i'j ^2'* ^n respectively, then 
 
 This follows directly from the general theorem, for although the charges on 
 each conductor are spread out over its surface, the positions it occupies are 
 positions of constant potential in the other distribution. 
 
 Now take the system of conductors to be that discussed above and suppose 
 the first distribution of charge is represented by zero charge on all the con- 
 ductors except the rih which has a unit of charge ; the potentials are then 
 respectively b h h h h 
 
 u r\ 5 . "r2 3 u rr u rs 5 u rn 
 
 similarly if the second distribution is unit charge on the sth conductor and 
 zero on all the others the potentials are 
 
 "si 5 o s2 , ... b sr , ... b ss , ... b sn , 
 and the application of the theorem at once shows that 
 
 "rs == "srs 
 
 which is probably the most important property of these coefficients : it will 
 be shown in the next paragraph that it is an immediate consequence of the 
 principle of energy as applied in these matters. 
 
 L. 9 
 
130 The electrical relations of the conductors [OH. in 
 
 In a similar manner we can deduce that 
 
 CTS C sr) 
 
 or this may be inferred as an algebraical consequence of the equality of b rs 
 and b sr . 
 
 134. Next suppose the rib. conductor carries a charge Q r but that all the 
 others are uncharged ; the potentials are then 
 
 kriQr, b r2 Q r , ... b rr Q r , ... b rn Q r . 
 
 The corresponding charge Q r must be at infinity where the potential is 
 zero. Thus the rth conductor is the only one on which the charge is entirely 
 positive, and infinity (or to be precise, the earth) is the only one with an 
 entirely negative charge : thus Q r b rr and are respectively the greatest and 
 least of the potentials in the field ; we must therefore infer that all the 
 coefficients b rr , b rs , ... are positive, but that 
 
 b rr >b rs . (s= 1, 2, ...n) 
 
 We may prove in a similar way that c rr is positive, but that c rl , c r2 , ... are all 
 negative and such that 
 
 CTI + c. r2 + ... c rr + ... c rn > 0. 
 
 135. The particular case in which one of the conductors is enclosed in 
 another one is of such great importance practically that it is worth detailed 
 treatment here as an example exhibiting 
 
 certain properties of the coefficients. 
 We shall suppose the second conductor 
 encloses the first. If Q = the second 
 conductor becomes a closed conductor 
 with no charge inside, so that the poten- 
 tial in its interior is constant or </>j = </> 2 . 
 Putting 0, = the relation <, = 
 gives 
 
 (b iz - & 22 ) Q z + (bi 
 and this must be true for all values of Q 2 , Q 3 , ..., so that we must have 
 
 Oi2 22 > #13 = t>23, .... 
 
 Now suppose that the second conductor is earthed so that </> 2 = 0. Then 
 if Q t = it follows that </> x = also. Hence from the equation 
 
 $1 = C ll^l + C 12^2 + C ln<f>n, 
 
 we obtain in this special case that 
 
 a relation which must be satisfied whatever the values of </> 3 , ^> 4 , ... may be. 
 This means that 
 
 ^ia yid V. 
 
133-136] Special cases 131 
 
 The second system of equations therefore reduce to, still with d> 2 0, 
 #1 = Cn<i, 
 
 Q-2 = C l2<f>l C 23</>3 + C 24^4 + > 
 
 These equations show that the relations between the charges and potentials 
 outside the second conductor are quite independent of the electrical conditions 
 which obtain inside this conductor. So also the conditions inside the second 
 conductor are not affected by those outside this conductor. These results 
 become obvious when we consider that no lines of force can cross the second 
 conductor either from inside or from out and that there is no way except by 
 crossing this conductor for a line of force to pass from the first conductor to 
 any other one outside the second. 
 
 An electric-system which is completely surrounded by a conductor at 
 zero potential is said to be electrically screened from all systems outside 
 the conductor; for charges outside this screen cannot affect the screened 
 system. This principle of screening is often used in electrostatic instruments 
 to shield the instrument from action by extraneous fields. 
 
 136. These properties may be illustrated by the calculation of the 
 coefficients in a special case. 
 
 We consider the simple case of two concentric spherical conductors, the 
 inner one of radius a^ and the outer one of radius a z . The equations con- 
 necting the potentials and charges are 
 
 <t>i = b nQi + b 2l Q 2 , 
 <f>2 = bi 2 Qi + b 22 Q 2 . 
 
 A unit charge placed on the outer sphere raises both to the potential , 
 
 Q/n 
 
 so that on putting $1 = 0, Q 2 = 1 we must have ^ = <f>. 2 = - . Thus 
 
 " If the outer conductor is uncharged and the inner has unit charge the 
 field of force is that investigated above in 87. Hence 
 
 bu=, b 12 =~~, 
 
 1*1 **2 
 
 results which verify that 6 12 = 6 21 , 
 
 and the relation peculiar to electric screening 
 
 "12 "22- 
 
 92 
 
132 The electrical relations of the conductors [CH. in 
 
 We have then fa = ^ + ^, 
 
 <f>2 = -^ + ~- 
 Solving for Q l and Q 2 in terms of (f> l and </> 2 we obtain 
 
 aft * * ft /y 
 
 I C*j u<2 MI 
 
 , t , t n 2 
 
 Lvo t*i Cvo Cvi 
 
 4 JL A JL 
 
 so that 
 
 C ll ~ ~ r~> C 12 = C 21 ~5 C 22 = ~ ^~- 
 
 t* 2 ttj tt 2 Cij 1*2 **1 
 
 These verify the general properties of the coefficients of capacity given 
 above. 
 
 137. The case when the spheres are eccentric or external to one another 
 is much more difficult to analyse. Approximate results can however be 
 obtained by using certain results obtained in the section of the last chapter 
 dealing with the general theory of images. We first suppose the spheres 
 carry charges Q^ and Q 2 . 
 
 Fig. 35 
 
 If the spheres are at a great distance c apart and of radii a x and a 2 their 
 charges will be uniformly distributed, the field of the one not disturbing the 
 distribution on the other. This is the first approximation. The next 
 approximation is obtained by taking account of the influence of one charge 
 on one sphere on the distribution on the other. To a first approximation 
 the charge Q l on the sphere a t will influence the second sphere just as if it 
 were collected at its centre : the distribution on the second sphere is then 
 
 such that the outside field is the same as that of the images Q z + - at 
 
 G 
 
 the centre and Q 1 at the inverse point 7 2 of Oj in the sphere 2 : this 
 
 C 
 
136, 137] The problem of two spheres 133 
 
 a 
 
 2 
 
 is at a distance -- from 2 towards O l . The distribution of charge on the 
 
 first sphere will to the same order of approximation give a field outside 
 it just like images 
 
 r\ a l 
 
 at its centre and - (^ 2 
 
 
 
 a 2 
 at the inverse point / x of 2 in the sphere, i.e. at a distance - from O l . 
 
 This gives the second approximation : to correct it and obtain the third 
 approximation we calculate the full effect of the two images in Q l in the 
 second sphere, i.e. we insert their images just as before in order to keep the 
 potential of the sphere constant : this leads to a second image point /j' and 
 1 2 f in each sphere at distances respectively 
 
 2 
 
 2 
 
 from the centres ; the charges there being 
 
 The charges at the centre are now 
 
 c* a 2 
 
 ft ft 
 
 2 
 
 and Qz + Qi^-Q 
 
 respectively : the other images are now 
 
 and 
 
 By calculating and compensating the effect of the new images we can 
 proceed to the next approximation. 
 
 The potential of each sphere will then be equal to the sum of the images 
 which fall at the centre of the sphere divided by its radius and will be therefore 
 
 respectively 
 
 / 1 
 
 + " 
 
 2 2 
 
 a l c 2 2 
 
 ;2 \a 2 c 2 - a x 2 
 so that the coefficients of potential of the spheres are 
 
 b - _ 2 
 11 c 2 - 2 
 
 "21 ~ ~ ~ 
 
134 The mechanical relations of the conductors [OH. in 
 
 It is more usual however to assume that the potentials are (f> l and <f> 2 , given 
 quantities, and to calculate the charges. This is easily done in the same way : 
 to a first approximation the charges on the spheres are a^j, a z <f> 2 respectively : 
 on inserting the images of these charges we get the next approximation just 
 as above. In this way it is found that the coefficients of induction and 
 capacity are 
 
 (c 2 - 2 2 ) 2 - 
 
 Cvi ~ : 21 = c (c 2 - af - 
 
 * n n 2 ft 3 
 
 2 1 1 2 
 ^22 == ^2 "T 
 
 ~ i ~9 9 ~T~ / 9 9\9 99* 
 
 s*& si _ ( f n \" n */^ 
 
 ** 
 as tar as -=*. 
 
 c 3 
 
 138. The energy and ponderomotive forces in the field of a system of 
 conductors. We have next to consider the mechanical relations of the 
 systems of conductors whose .electrical conditions have just been examined. 
 It is ultimately by the mechanical forces which these conductors exert on 
 one another that we are able to examine and measure their state of 
 electrification, and in fact it was only in this way that such a state was 
 discovered at all. A precise formulation of the connection between the 
 electrical and mechanical conditions of the conductors is therefore essential 
 to a complete theory, in as far as it indicates the ultimate means of testing 
 that theory. 
 
 We consider the case of n conductors in the field carrying charges Q l , 
 Q 2 , ... Q n and at potentials </> l5 < 2 , </> Now the work required to bring 
 up small increments of charge SQ 15 SQ 2 , ... BQ n to the respective conductors 
 from a remote distance is 
 
 This follows directly from the work definition of the potential </> at any point 
 of the field. 
 
 The work required to increase the charges on the conductors by the 
 specified amounts is thus ^ = Y;^ g^ ? 
 
 which is the fundamental differential form of the characteristic equation of 
 energy for the system. If it is integrable, i.e. if a potential energy function 
 W exists, then we deduce at once that 
 
 dW 
 
 dQ ^ ^ ^ = 
 
 * The first determination of the problem of two spheres was given by Poisson, Mdm. de Vinst. 
 xii. 1, p. 1 (1811), 2, p. 163 (1811). The present solution is due to Kelvin, Phil. Mag. 1853 
 (Reprint, p. 86). In Maxwell's Treatise (ed. 1892), J. J. Thomson discusses a solution by Kirch- 
 hoff and gives further references. 
 
137-139] The energy 135 
 
 but the differentia] form contains all these n partial equations in one 
 expression. Thus if we know the energy we also know the potentials. These 
 n equations involve the following relations^ 
 
 and conversely, if all these relations are true then SPF is a complete differen- 
 tial, the function W exists and then the partial equations are also true. 
 The truth of these reciprocal relations is thus the analytical criterion for 
 the existence of the potential of the system and then the rest is a direct 
 expression for the energy principle. 
 
 As a matter of fact we know that in the case under consideration the 
 potentials (f> 1} </> 2 , ...</> are linear functions of the charges Q l , Q 2 , ... Q n . 
 Therefore W is a quadratic function of these quantities and by Euler's theorem 
 
 dW 
 
 so that W = p;</v&*. 
 
 'If we multiply each charge by half its potential and add up over the whole 
 system we get the general value for the potential energy of the system 
 relative to the configuration in which all the conductors are at zero potential. 
 
 139. Having thus determined the potential energy of the system we 
 can at once proceed to an examination of the mechanical forces between 
 the conductors, the mere existence of which is involved in the idea of the 
 energy of the system. To obtain these forces we need only give the con- 
 ductors small virtual displacements and include the virtual work in these 
 displacements in the general expression of the work done on the system 
 during a general virtual change in its configuration. 
 
 If the positions of the various conductors are determined by the generalised 
 coordinates l} 6 2 , ... d n , in the usual Lagrangian sense and if the force 
 components corresponding to the coordinates are l5 @ 2 , ... @ m then the 
 work done on the system during a general virtual displacement is 
 
 m 
 
 -S0.80.. 
 
 s=l 
 The work done in bringing up small additional charges to the conductors is 
 
 i<f> r 8Q r , 
 
 r=l 
 
 if the conductors are fixed ; if however in addition the conductors receive 
 a small virtual displacement we must add the work done against the forces 
 
 * Kelvin, Glasgow Phil. Soc. Proc. m. (1853), Math, and Phys. Papers, i. 61. Helmholtz, 
 Vber die Erhaltung der Kraft (1847). 
 
136 The mechanical relations of the conductors [CH. in 
 
 acting in these displacements. The general formula for the work done on 
 the system during the most general virtual change in its configuration is 
 
 n m 
 
 8W='Z <f> r 8Q r - 2 S S0 S , 
 r=l s=l 
 
 and the principle of the conservation of energy asserts that this 8W must be 
 an exact differential, or that the function W exists ; otherwise we should have 
 perpetual motion. 
 
 Thus when W is obtained as a function of the Q's we have as before 
 
 dW 
 
 2Q-=<t>r, 
 
 dW 
 
 and also = S , (s = 1, 2, ... m) 
 
 dV s 
 
 from which a whole series of reciprocal relations of the type 
 
 d& _ a. 
 
 de s dQ r 
 
 can be deduced. The principle of the conservation of energy leads to all 
 these relations; in fact the experimental test of the conservative system is 
 that all these reciprocal relations hold. If they are true then we know that 
 the energy in the system is conserved, i.e. none of it is frittered away into* 
 heat. The real property which indicates conservation of the energy is of 
 course the reversibility of changes in the system. If all the operations for 
 changing the system from one configuration to another can be reversed the 
 system is conservative. The above relations are merely the analytical form 
 of this property. 
 
 140. The relations so far discussed depend on an expression for W in 
 terms of the charges on the conductors : the form of results is therefore that 
 suitable for the discussion of the relations in the system when the con- 
 ductors are insulated and their charges are constant. We wjtint however to 
 be able to extend the results to a system in which some or all of the 
 bodies are maintained at constant potential, perhaps by connecting them to 
 batteries. The most suitable form of W is then its expression in terms of 
 the c's. 
 
 What now are the relations when W is expressed in terms of the </>'s? 
 To obtain them we proceed by a universal method, viz. to get an expression 
 for STf involving 80 l5 S(/> 2 , ... instead of SQ l , &Q 2 , The procedure is to 
 
 write fa&Qi 
 
 in another form ; it is in fact 
 
 *(fc$)-rOl&fc. 
 
 so that we have 
 
 S (W - S fa Q r ) = - S Qrfyr - S 0.80,, 
 r=l r=l s=l 
 
 and this is the required transformation. 
 
139-141] The ponderomotive forces 137 
 
 The usual argument shows that 
 
 is an exact differential and so we can deduce another set of reciprocal relations 
 
 of type 
 
 dQ r _ d(f> r /r = 1, 2, ... n\ 
 
 307 = 30,' U= 1, 2, ... n) 
 
 This set of reciprocal relations must of course be equivalent to the first set 
 because they both are the analytical test of the principle of the conservation 
 of energy. It is in fact an easy example in the calculus to show that the one 
 set can actually be deduced from the other. Either set is in itself a complete 
 expression for the conservation of the energy. 
 
 141. In the general case the new function W introduced above is aot 
 the energy ; it is an entirely new function, but in the present case It so 
 happens since 2 JF = ZQ r <f>r, 
 
 that W'= -W, 
 
 and so we have 8W S$ r S</> r + 20 S S$ S . 
 
 If then we use the two symbols WQ, W^> to represent the energy, when this 
 is expressed as a function of the Q's and </>'s respectively we have the two 
 following expressions for 8W Q and STF $, which are the characteristic equations 
 representing the principle of energy 
 
 The first is equivalent to a series of relations of the types 
 
 (i) = &, ('=1, 2, ...n) 
 
 (ii) = -S s , (s=l, 2,...ro) 
 
 which are to be compared with the equivalent series for the second 
 
 (i) 
 
 (ii) - = + B s . (-!, 2, ...m) 
 
 00, 
 
 The striking contrast is in the last (ii) in each set. When the bodies are 
 insulated the forces exerted by the system tend to diminish the potential 
 energy; the equation of energy being 
 
 W Q = -S0.80., 
 
 the charges being constant. This is of course right as the work of these 
 forces has to come out of the potential energy of the system. 
 
138 The mechanical relations of the conductors [CH. in 
 
 On the other hand when the conductors are maintained at constant 
 potential (by connection with batteries) they exert forces tending to increase 
 their energy ^ = S0 S S0 S . 
 
 The forces are of course still the same as in the previous case, but the regions 
 into which the conductors tend to move are now regions of greater electrical 
 energy. 
 
 The batteries have now to supply not only the increased electrical energy 
 SW 4, but also the equal amount of energy for the work done by the mechanical 
 forces P s Sp s . Energy flows out of the batteries and half of it goes in 
 mechanical work and the other half in increasing the intrinsic electrical 
 energy. 
 
 Thus everything is summed up in the doctrine of energy, as long as we 
 confine ourselves to static systems. 
 
 142. In the case of a mixed system in which some of the conductors 
 are maintained at constant potential and others are insulated the relations 
 can be obtained by a partial transformation of 8 W Q , of the form 
 
 h h n m 
 
 8(TF-sa< r ) = -sas</> r + -s ^ r sg p -s0.s0 a , 
 
 r-l r=l r=h+l s=l 
 
 and the right-hand side is again an exact differential of some function W. 
 But now this function is something quite new and different from W. 
 
 An interesting point to notice is the analogy between the analysis here 
 exemplified and that which occurs in the examination of the Hamiltonian 
 transformation of the equations of dynamics. 
 
 143. Of course if the full details of the charge distribution on the con- 
 ductors are known we need not resort to these general methods of treat- 
 ment, as the forces and energy themselves can be immediately deduced 
 by simple integrations when the results of 67, chapter I are taken 
 into account. Let us consider for example the simple case of the forces 
 on a single conductor in the field. The force inside the conductor is 
 zero and outside it is normal to the surface so that the average force on 
 each element of charge on the surface is a normal one of amount ^E n per 
 unit charge at the point. There is thus a resultant force on each element 
 of a charged conductor which is normal to the surface at the place and of 
 amount 11? _ 
 
 2 \ M -'n u 
 
 per unit area ; a is the density of the charge distribution at the point on the 
 surface. This force is an outward pull. If the conductor is surrounded by 
 air then 
 
 E 
 
141-145] The condensing system 139 
 
 so that the normal force per unit area on its surface amounts to 
 
 E 2 
 
 I - 2 2 
 
 at any point. 
 
 We can thus evaluate the force on the conductor by mere integration of 
 the forces on each element as thus specified. 
 
 This result was first given by Coulomb. 
 
 144. On electrical condensers. The most important applications of the 
 foregoing principles are to the construction of instruments for the exact 
 measurement of the electrical conditions of conductors, or in fact of any 
 other body : such measurements have not only a theoretical interest in that 
 they provide us with the ultimate basis for our theory, but they are of 
 great practical importance in the technical applications which have been 
 made of the principles of this subject. 
 
 The method is to work out the theory as exactly as possible for some 
 simple type of conductor system and then to construct these conditions in 
 an actual case so that the essential consequences of the theory may be tested 
 as to their validity. In all simple systems of conductors the circumstances 
 are always more or less complicated by the presence in the surrounding field 
 of other conducting bodies, and the protection afforded by partial screening 
 is not always theoretically satisfactory. Means have therefore to be adopted 
 for restricting the fields under examination, and for such purposes the con- 
 densing arrangement of conductors is most effective. In this arrangement 
 the field of the charges on the conductors is always more or less confined 
 within certain simple and easily defined spaces, so that the circumstances in 
 it are susceptible of exact mathematical specification. 
 
 The theory of the action of a condenser may be treated as an example of 
 the foregoing general theory; but we can with certain advantages discuss it 
 from a more elementary standpoint. The subject is best approached by the 
 consideration of a simple example of the ordinary type of electrostatic field 
 surrounding conductors as discussed above. 
 
 145. Let us consider the electric field surrounding two charged con- 
 ductors so placed as to have a large part of both surfaces very close together 
 and as nearly parallel as possible (see diagram). We shall first consider that 
 the conductors have equal and opposite charges so that all the lines of force 
 from one conductor go to the other and they cut each conductor normally. 
 
 Thus at a point between the near surfaces the lines of force are practically 
 straight portions of the common normal to the two surfaces. The tubes of 
 force are therefore practically cylindrical and have the same cross-section all 
 the way across. The force is therefore uniform between the surfaces and 
 
Special systems of conductors 
 
 [CH. Ill 
 
 the surface densities on the opposed faces are equal but opposite. Thus if 
 E is the intensity of force in any tube whose length is t and if a is the density 
 of the charge at each end of this tube (+ a on one end and a on the other) 
 then 
 
 and since E is uniform all across the difference of potential <j> l (f> 2 between 
 the two conductors is given by 
 
 and thus 
 
 01 02 
 
 a = - , 
 
 Fie. 36 
 
 and is positive on one side and negative on the other. The density of the 
 distribution on the surfaces is thus inversely as the distance across. If the 
 surfaces are very close together and the difference of potential is finite, 
 a must be very large. 
 
 Now consider any other line of force not between the near surfaces. Such 
 a line is much longer than those between the surfaces but the total fall in 
 potential is just the same, the rate of fall, i.e. the force intensity along such 
 lines must therefore be very small and therefore also the charges at each end 
 on the conductor are also small. Thus nearly the whole of the charge is 
 concentrated on the near opposing faces of the conductors ; the two equal 
 
145, 146] 
 
 The condensing system 
 
 141 
 
 charges as it were bind one another together there and prevent spreading out. 
 At the edges and outer parts of the conductors the field and charges are 
 uncertain and difficult to estimate, but in ordinary cases when the opposing 
 areas are big we can neglect the field and charges beyond that in the space 
 between the conductors. 
 
 The total charge on either face is 
 
 and this is also practically the charge on the conductors : thus 
 Q = 
 
 Thus Q is proportional to (f> l </> 2 ', the constant of proportionality being 
 
 477.' t 
 
 If the surfaces are very close together, or at least if C is large, and there is 
 a finite difference of potential, the charges are equal and opposite and also 
 very large. The arrangement is therefore called a condenser; it accumulates 
 a large charge for an ordinary difference of potential. The constant C is 
 called the capacity of the condenser; so that 
 
 The capacity is equal to the charge when the difference of potential is unity. 
 
 146. The commonest form of condenser in actual use is known as the 
 Leyden jar and is illustrated diagrammatically in 
 the figure. It consists of a vessel made of thin 
 glass; the inside and outside surfaces of this 
 vessel are coated with tin foil. An electrode is 
 connected to the inside of the jar in order that 
 electrical connection can easily be made with it. 
 If A is the area of each coat of tin foil, t the 
 thickness of the glass, i.e. the distances between 
 the surfaces of the tin foil, then, neglecting the 
 effect of the glass as a dielectric medium the 
 capacity is approximately 
 
 A^ 
 
 4:7Tt' 
 
 Fig. 37 
 
 For purposes of more exact measurement however it is usual to adopt forms 
 of conductors more readily susceptible of mathematical treatment and in 
 such cases the simpler types indicated below are adopted. 
 
 Green, Essay etc., pp. 43-45. The second approximation is also obtained by this author. 
 
142 
 
 Special systems of conductors 
 
 [en. in 
 
 +cr 
 
 -er 
 
 147. The parallel plate condenser. We are given two parallel conducting 
 plates with equal and opposite surface densities a. The 
 distance between the two plates is so small compared with 
 their dimensions that we may regard them as very large and 
 neglect any irregularities arising from the edges. 
 
 From the symmetry of the arrangement we may assume 
 that the lines of force are normal to both planes and go 
 straight across between them. The tubes of force are cylindrical 
 and thus the electric force E = kira is uniform all across. From 
 the previous general considerations we may conclude that the 
 field exists only between the plates; in reality there is a field 
 in the surrounding space and also charges on the backs of the 
 plates, but when the capacity of the arrangement is large these 
 are all negligible in comparison with the field and charges 
 between the plates. 
 
 If the plates are at a distance t apart and the potentials are 
 (j> and (f) 2 
 
 Et = </>! </> 2 , 
 
 so that (f)^ <f> 2 = 4t7rcft. 
 
 Thus if A is the area of a plate the charge on it is 
 
 Fig. 38 
 
 The capacity of this arrangement is therefore 
 
 ^Cil V 
 
 ~\ r /7 '/* 
 
 or j - per unit area ; in agreement with our general formula 7 I 
 vn 47TJ Z 
 
 148. TAe spherical condenser. The two spherical radii are a and 6 > a. 
 The sphere a carries the charge + Q and the sphere b the charge Q. The 
 whole space is divided into three parts. 
 
 (i) Inside the inner sphere a there is no electric intensity, it is internal 
 to both spherical distributions E^ = 0. 
 
 (ii) In the space between the spheres the electric intensity is due to the 
 inner sphere alone and is 
 
 (iii) In the space outside the two spheres the field is due to both spheres 
 
 Q Q 
 
 E, = - _ o. 
 
 3 r i r ?. 
 
147-150] The cylindrical condenser 143 
 
 The electrical field is therefore entirely confined to the space between the 
 spheres and then 
 
 ~ dr = r 2 ' 
 
 ~ & Q 
 and thus <f>i <pz = 
 
 where fa and (f> z are the potentials of the spheres. The capacity is 
 
 /~"i _ 
 
 If the sphere b becomes infinitely large then 
 
 ab a 
 
 b a 1 _ a 
 ~b 
 
 so that the capacity of the sphere a alone is equal to a, a result which we 
 might have deduced from previous results. 
 
 149. The cylindrical condenser. By proceeding in a manner exactly 
 analogous to the above we may find the capacity per unit length of a condenser 
 formed of infinite coaxial cylinders of radii a and b. If the charges are 
 Q and Q per unit length the field between the cylinders is radial and at 
 
 a distance r from the axis is ; thus integration for the potential gives 
 
 so that the capacity per unit length is 
 
 1 
 
 150. These results are the particular cases of a more general theorem. 
 
 If we know the equi-potentials in the field of any freely charged conductor 
 C, then we can determine the capacity of any condenser formed by conducting 
 surfaces in the shape and relative position of any two equi-potentials in this 
 field say A and B. 
 
 Suppose </> is the potential function of the conductor C with a charge 
 Q and let the values of </> on A and B be denoted by (f> A and </># . 
 
 Now consider the problem of the determination of the potential for the 
 two conductors A and B charged in any manner with quantities Q A and Q B . 
 We have to find a regular potential function </>' which will be constant on 
 A and B and give the right charges. 
 
144 
 
 Special systems of conductors 
 
 [CH. Ill 
 
 Now supposing the surface B encloses the surface A ; then a possible 
 form -for the potential of this new distribution is obviously 
 
 a 2 (f> 
 
 the suffices 1, 2, 3 referring respectively to the three regions the first inside 
 A, the second between A and B and the third outside B. 
 
 These potentials must of course be continuous at the surfaces, requiring 
 
 that 
 
 Fig. 39 
 
 and they must give the charges right, requiring that 
 
 a dcf> 
 
 but remembering that A and B are surfaces surrounding C in the first case 
 we see that each of the surface integrals on the right of these equations is 
 equal to ^Q. Thus 
 
 QA = a 2 Q, 
 
 Qz = (3 - a z) Q, 
 
150, 151] Condenser combinations 145 
 
 and there are therefore enough equations to determine a l , a 2 , a 3 and 6; 
 solving them we find 
 
 Q 
 
 Q 
 
 Q 
 
 In the particular case when Q A = Q B = Q these potentials are 
 
 , A B, 
 Q 
 
 and the field is confined to the space between the conductors. The system 
 then acts as a condenser whose capacity is 
 
 Numerous results can be deduced as special cases of this general formula. 
 
 151. Combinations of condensers. It often happens that the capacities 
 of the separate condensers at our disposal are not sufficiently large for the 
 purposes in hand, or they may be too large to produce the required potential 
 with a given charge. It is however possible to connect a number of such 
 condensers so as to secure these advantages. We consider n condensers in 
 space, so distant from one another that they do not mutually influence each 
 other. 
 
 (i) Parallel condensers : we connect the positive plates of all the con- 
 densers together and all the negative ones as well. The total charge in the 
 system 
 
 and (f) l and c/> 2 are the same for all the positive and negative plates respectively, 
 so that if we write Q = C (fa - fa), 
 
 C=SC r> 
 
 r-l 
 
 the capacity of this arrangement is the sum of the capacities of the separate 
 condensers. 
 
 (ii) Condensers in series : we connect the negative plate of each condenser 
 with the positive plate of the next succeeding condenser and charge the first 
 plate of the first condenser to potential ^ and the second plate of the last 
 to potential ^ 2 ; the potential of the positive plate of the (r + l)th condenser 
 is then equal to that of the negative plate of the rih or 
 
 fa"=fa*^. 
 
 L- 10 
 
146 Special systems of conductors [CH. in 
 
 Moreover the charges on all of the plates are equal to Q, so that 
 
 and consequently 
 
 </>i <^2 = Q^ T = 
 
 The reciprocal of the capacity of the combination is obtained by adding the 
 reciprocals of the separate capacities. 
 
 Thus if we want to get a higher potential difference than that usually at 
 our disposal we need only charge a battery of condensers in parallel and then 
 rearrange them in series. In the first place, if the condensers are equal in 
 all respects n Q = n c^ 
 
 where < is the directly accessible potential difference and C the capacity of 
 a single condenser of the battery, and then afterwards 
 
 152. On electrometers and their use. Having now secured certain 
 simple conductor systems whose behaviour can be accurately specified, we 
 can compare the actual mechanical behaviour of these systems with that 
 which might be expected on theoretical grounds and thus secure a decisive 
 test for the theory. Having then convinced ourselves of its general validity 
 we may reverse the argument and regard the mechanical relations of the 
 system as determining its electrical conditions, and then the arrangement is 
 called an electrometer. The principal types of such instruments, practically 
 all due to Lord Kelvin, will now be briefly described. 
 
 If two parallel metal plates of area A at potentials F x and F 2 form 
 a condenser whose capacity is 
 
 if we disregard the complications at the edges as being too small to affect 
 matters. 
 
 The energy of this combination is 
 
 where a is the surface density of the charge on the plates ; but 
 
 toTa=-=* t 
 so that W -- 
 
 Srr t 
 
 * Kelvin, Proc. R. S- (1867), Reprint, 352, also 401-426 and 427-9. 
 
151-153] % 
 
 The cylindrical electrometer 
 
 147 
 
 The force tending to increase t is 
 
 = ^LJL = _ 27rAa 2 = - 
 
 01 O7T V 
 
 dt dt STT \ t 
 
 so that there is really an attraction per unit area of amount lira*. 
 
 This is the idea of Kelvin's absolute electrometer (trap door)*. The one 
 plate of the condenser is hung on a balance and the force of attraction towards 
 the other plate measured by compensating it by a weight. To eliminate the 
 irregularities at the edges of the plates which are not taken account of in 
 the theory Thomson extends the plates by surrounding them by so-called 
 guard-rings. In this way the field between the effective parts of the plates 
 is made to conform to that theoretically described. 
 
 153. The cylindrical electrometer^. Consider now another similar case: 
 two coaxal cylinders one of which (the outer) is fixed vertically 
 and the other is suspended inside it but partly projecting at the 
 upper end. If both cylinders are charged the. inner cylinder 
 will be sucked down into the other ; with what force ? 
 
 In the middle the charge distribution is uniform and 
 even; it is only near the ends that the unknown irregularity 
 in the distribution occurs. The unknown distributions are 
 however not pertinent because they do not alter to any 
 appreciable extent when the inner cylinder drops down a short 
 distance ; provided of course that there is a good length of 
 either cylinder projecting at each end. The unknown distri- 
 butions remain the same whatever the depth of immersion and 
 they therefore do not matter. The energy W of the arrangement 
 ma,y therefore be calculated in the form 
 
 W = W r + C, 
 
 where W r is the energy calculated as if the distribution to the 
 actual depth z of immersion were uniform and there were 
 nothing else and C is the constant correction to be added for 
 the irregularities at the ends. If the cylinders are circular 
 (radii a, b) and the difference of potential is </> Fig 49 
 
 W = 
 
 the capacity per unit length being 
 
 * W. Thomson, Reprint of Papers on Electricity and Magnetism, 358, 360, Phil. Mag. (4), 
 vni. (1854), p. 42. 
 
 t W. Thomson, Reprint etc. p. 38. Cf. also Maxwell, Treatise, i. 129. Bichat and Blondlot, 
 Jour, de Physique (2), v. (1886), p. 325. 
 
 102 
 
148 Special systems of conductors 
 
 There is therefore a force 
 
 [CH. Ill 
 
 drawing the inner cylinder down ; a knowledge of which in a particular case 
 would enable a determination of <j> to be made. 
 
 The importance of this type of electrometer is that its capacity can be 
 varied at will by simply lowering or raising the cylinders. 
 
 154. The quadrant electrometer*. This is probably the most suitable in- 
 strument for determining relative potential 
 differences. It consists of a flat cylindrical 
 metal case divided into four equal quadrants 
 by perpendicular central sections. The, two 
 opposite quadrants (diagonally) are in each 
 case metallically connected and are at 
 potentials </>! and </> 2 : a flat paddle shaped 
 needle with its plane parallel to the top 
 and bottom of the box and which is capable 
 of turning about the vertical axes of sym- 
 metry of the box, is at a potential </> 3 . If 
 ^ = < 2 = ^ the paddle lies symmetrically 
 between the quadrants. The whole appa- 
 ratus is enclosed in a metal case at zero 
 potential. The charges on the three portions 
 are 
 
 Q l = c n ^ 
 
 </> 2 
 
 12 <> 2 
 
 3 </> 3 , 
 
 2 < / > 
 
 Fig. 41 
 
 or since c rr > and c rs < we can write this in the form 
 
 (/2 == Q-21 \r*2 *rl) ~^~ ^23 (02 *r3/ ' ' 
 u/o = Ct 3 i (0a T*!/ ~l~ ^32 vr3 T*2) 
 
 where all the a's are positive and a rs = a sr . 
 The energy is 
 
 " ~ 2" { a !2 (r*! T*2) 2 ~t~ a !3 (r*l T'S) ~^~ a 23 ( < T2 03/ "^" ( 
 
 The couple on the needle is 
 
 ey 
 
 = 36 ' 
 
 W. Thomson, Reprint etc. 345. 
 
153-155] The quadrant electrometer 149 
 
 where 6 is the angle turned through from the position of rest. Now the 
 construction of the needle enables us to deduce the dependence of the a's on 
 6. The broad form of the paddle and the smallness of the sections between 
 the quadrants show that very few, if any, lines of force get away from the 
 edges of the paddle to the case, so that in an infinitely small displacement 
 dd the number of lines of force which run from the paddle to the second pair 
 of quadrants increases by a quantity proportional to dd. It thus follows by 
 symmetry that* 
 
 Ottoq 60,10 , 
 
 s? _ ld Z. QQV 
 
 dd ' 80 ' 
 
 whilst all the other constants are independent of 6, because the number of 
 lines of force which go from the quadrants or needle to the case or from one 
 quadrant to the other, do not vary during the small displacement of the 
 paddle. 
 
 Thus = -<> 2 -<>-<> 2 
 
 155. Different arrangements enable us to determine various potential 
 differences. The couple is always independent of the position of the needl^ 
 provided k is constant and is determined by balancing it against the torsion 
 couple of its suspension. If it is not too large the angular deflection of 
 the needle is proportional to the couple which balances it in the final 
 position of equilibrium of the needle. If | < 2 <f> 3 > (/> x <f> 3 \ the couple is 
 positive and the needle is thus always drawn towards the quadrants whose 
 potential differs most from its own. 
 
 If the needle is connected to one pair of quadrants, i.e. if (f>^ = </> 3 , then 
 
 is proportional to the square of the potential difference (</> 2 </> 3 ). This 
 arrangement is adopted for the accurate measurement of large differences of 
 potential. 
 
 If on the other hand (<f) l </> 3 ) is always very large compared with (f>^ <f> 2 , 
 then we can write approximately 
 
 0r^(&--6M*iHJi)> 
 
 and thus if ((/> x </> 3 ) is maintained constant G is proportional to <^ x (j> 2 . 
 This furnishes a very convenient arrangement for comparing small differences 
 of potential and for this purpose the paddle is very highly charged, so that 
 (f> 3 is large, so that we may even write 
 
 G = k (</>! - <f> 2 ) (f> 3 . 
 
 * The positive sign for k is chosen to correspond to the positive direction of quadrantal order 
 1-2 for increasing 0. 
 
150 Special systems of conductors [CH. in 
 
 If as is often the case the second pair of quadrants is earthed then </> 2 = 
 and then , G= _ &^ 3 . 
 
 156. This instrument may also be used for measuring a charge of 
 electricity and the arrangement adopted is similar to that last described for 
 measuring small potential differences. The paddle is highly charged whilst 
 both pairs of quadrants are originally earthed : a quantity Q Q of electricity 
 will then be induced symmetrically ; and if (f> 3 is the potential of the paddle 
 
 Qo = c<f> 3 , 
 
 where c is the common value of c 13 and c 32 when the paddle is in the 
 symmetrical position. If now one pair of quadrants is insulated and given 
 a charge Q the paddle will be deflected and in the final position we shall 
 
 ^^ 01 = 0+00=^1 + ^3, 
 
 so that Q = c ll (f) l + (c 13 c)<f> 3 . 
 
 The couple on the paddle is still very approximately given by 
 
 G = ^!^ 3 , 
 
 and if the final deflection of the paddle from the symmetrical position is 
 through the angle 6 
 
 and c 13 = c kdO = c kd, 
 
 if k and K are independent of 6 as is usually the case with sufficient approxima- 
 tion. We have then 
 
 /dig + k*<f> 3 \ 
 = I -- j-j ] 
 
 \ K() 3 
 
 and is proportional to 6 if (f> 3 is kept constant. 
 
 . It is interesting to notice that when the potential of the needle is increased 
 beyond a certain point the deflection of the needle due to a given charge 
 Q on the quadrants diminishes as the potential of the needle increases, hence 
 to obtain the greatest sensitiveness when measuring electrical charges we 
 must be careful not to charge the needle too highly. 
 
 157. Now let us consider the practical application of this instrument to 
 determine the potential of any conductor which exists in the field of a given 
 system of charges. By connecting this conductor by a long thin wire to one 
 pair of quadrants of the electrometer and the other pair of quadrants to earth, 
 the difference in the potential between the two quadrants then measures the 
 potential of the conductor under investigation relative to the earth*. It is 
 
 * Assuming that the capacity of the quadrants is sufficiently small compared with that of 
 the conductor, unless this latter is maintained at a constant potential. 
 
155-157] Practical determination of capacity 151 
 
 assumed that the wire is so thin that its presence does not affect the conditions 
 of the system and that the electrometer is so far removed from the system so 
 as not to affect it or be affected by it. 
 
 The charges on the conductors could 'be determined by transferring them 
 to one pair of quadrants and determining them in the manner described 
 above, but this is in general not very practical and it is usual to determine 
 what is known as the capacity of the conductor whence the charge can be 
 inferred from its potential. This is easily done provided three condensers 
 are available and the capacity of one of them can be varied. The three 
 condensers may be of any of the simple theoretical types illustrated above. 
 
 We need only discuss the problem of determining the capacity of a con- 
 denser consisting of two conductors : the capacity of a single conductor is 
 in reality the capacity of the condenser formed by it and the earth. Let 
 therefore A, B be the plates of the condenser whose capacity is required: 
 
 Fig. 42 
 
 C, D; E, F ; G, H the plates of three condensers whose capacities are 
 known. Connect plates BC together and to one pair of quadrants of an 
 electrometer* ; also connect F and G together and to the other quadrants. 
 Connect DE together and to one pole of an apparatus for producing a 
 potential difference, and connect AH together and to the other pole of 
 this apparatus. In general this will cause a deflection of the electrometer ; 
 if there is a deflection then we must alter the capacity of the condenser whose 
 capacity is variable until the vanishing of this deflection shows that the 
 plates BC, FG are at the same potential. When this is the case a simple 
 relation exists between the capacities. 
 
 Let 6j , 6 2 , 6 3 , 6 4 be the capacities of the respective condensers. Let </> be 
 the potential of AH, </>' that of BC and therefore also of FG when there 
 is no deflection in the galvanometer, (f> that of D and E. 
 
 The charge on B is then 6j (<$>' <), 
 
 * It is again assumed that the capacity of the quadrants is very small ; otherwise it has to be 
 eliminated or allowed for by additional measurements. In actual practice different methods are 
 employed depending on the theory of electric current flow. Cf. Pidduck, A Treatise on Electricity 
 (Cambridge, 1916), p. 143. 
 
152 Special systems of conductors [OH. in 
 
 whilst that on C is 6 2 (</>' <A ) ; 
 
 these two must be equal and opposite so that 
 
 Again, the charges on F and G must be similarly equal and opposite and 
 they are respectively 
 
 so that we may conclude that 
 
 6i6 3 = 6 2 6 4 , 
 
 6,6, 
 
 and thus 
 
 / 2 i/ 4 
 
 and is known as soon as 6 2 , 6 4 and 6 3 are known. 
 
 Thus if we have standard condensers whose capacities are known, we can 
 measure the capacity of other conductors and condensers. 
 
 158. On the Proofs of the Inverse Square Law. In the experiments 
 described in the first chapter it was found that on inserting an electrified 
 body into the closed metal vessel and then bringing it into contact with the 
 vessel the electrification entirely left the body and appeared outside the 
 vessel ; and on the withdrawal of the body there were no signs of electrifica- 
 tion either on it or on the interior of the vessel. 
 
 This fact was found to follow as a general principle from Gauss' theorem 
 on normal induction which depends essentially on the validity of the inverse 
 square law for the action between the electric charge elements. Cavendish 
 reversed the argument* and by a direct experimental determination of the 
 limits to the probable truth of the fact that the force in the interior of a 
 charged conductor is zero, he was able to conclude that the law of force 
 between electrical charges can only differ very slightly from the inverse 
 square law. Cavendish's work although carried out some time before 
 Coulomb's determination of the law of force however remained unpublished 
 until Maxwell saw it. 
 
 Maxwell improved slightly on the apparatus used by Cavendish and 
 repeated the experiments and his measurements provide us with the most 
 exact experimental verification of law which has ever been given. 
 
 In this experiment two concentric spherical conductors of which the inner 
 one is charged are put into connection and the residual charge of the inner 
 one is measured. 
 
 159. The first problem to solve is that of finding the potential at any 
 point due to a uniform spherical shell, the repulsion between two units of 
 electricity being any given function of the distance! 
 
 * Ct. Maxwell, Treatise, i. pp. 80-86. 
 

 157-160] The proofs of the inverse squares law 153 
 
 Let </> (r) be the repulsion between two unit^ at a distance r, and let / (r ) 
 be such that 
 
 Let the radius of the shell be a, and its surface density cr, which by symmetry 
 is uniform over the whole surface ; then if Q denote the whole charge of the 
 
 shel l Q = 47ra 2 CT. 
 
 Now let b denote the distance of the given point from the centre of the shell 
 and r its distance from any point of the shell. If we refer to spherical polar 
 coordinates with centre of shell as pole and the radius b as axis then 
 
 r 2 = a a + 52 _ ^ab cos 0. 
 The charge on the element of shell is 
 
 o-a- 2 sin 6ddd(f>, 
 
 and the potential due to this element at the given point is 
 
 ' 
 
 ..-! - aa z smd 
 
 and since rdr = ab sin Odd this is 
 
 (W*V ' ( r ) drd(f>, 
 
 which has to be integrated over the whole surface of the sphere : the result 
 is that 
 
 where r { is the greatest value of r, which is always (a + b) and r 2 is the least 
 value, which is (b a) when the given point is outside the shell and (a 6) 
 when it is inside. 
 
 The potential at an external point is 
 
 ^ = 
 and at a point on the shell 
 
 and inside < - (f ( + 6) -/( - b)}. 
 
 160. Now let us determine the potentials of the shells in Cavendish's 
 experiment on the supposition that the one has a charge Q and the other 
 a charge Q 2 ', the potentials </>! and </> 2 are given by 
 
154 Special systems of conductors [CH. in 
 
 In the Cavendish experiment the two spheres are in connection and so 
 
 </>i = </>2 = <f> 
 so that the charge on the inner sphere 
 
 6/(2a)-a{/(a + 6)-/(g-6)} 
 
 * 
 
 (2a)~/(2&) - {/(a + 6) -/(a - 6)' 
 
 The hemispheres forming the outer sphere are then removed to a distance 
 and discharged and the potential of the inner shell is determined ; it would be 
 
 If now we assume with Cavendish that the law of force is some inverse power 
 of the distance, not differing much from the law of inverse squares, we can 
 
 put ^ ^ = r5 - 2j 
 
 and then / (r) = ^ r q+l , 
 
 and if we suppose q small we can expand this in the form 
 
 (q log r) 2 
 logr + ^ 
 
 so that to a first approximation in q 
 
 4a 2 a , a + b 
 <^T 2 " b log a- b 
 from which q may be determined if Q 1 is measured. Experiment will thus 
 determine an upper limit to q and it has in fact been shown that q must 
 certainly be less than 10~ 5 . This is probably the best experimental test of 
 the law that there is in existence. 
 
 161. Laplace's proof of the inverse square law*. A second proof was 
 given by Laplace and is based on the two definite experimental facts that 
 there is no electric force inside a conductor in electrostatic equilibrium and 
 that no free charge resides in the interior of the conductor, it is all on the 
 surface. It is quite easy to show that these two results require that the 
 law of action should be the inverse square law. 
 
 We prove that no function of the distance except the inverse square 
 satisfies the condition that a uniform spherical distribution of surface charge 
 exerts no force on a particle within it. 
 
 If in the previous example we suppose that Q z is always zero we can apply 
 Laplace's method to determine /(r). For then we have 
 
 bf (2a) -af(a + b} + af (a - b) = 0, 
 
 for all values of b : differentiating twice with respect to b we get 
 
 /"(a +6) =/"(a- 6), 
 
 * Mec. celeste, i. p. 163. 
 
160, 161] The proofs of the inverse squares law 155 
 
 which is only satisfied for all values of b < a if 
 
 /" (r) = c constant, 
 i.e. /' (r) = c a r + c, . 
 
 Hence 
 
 Thus </> (r) = -| 
 
 2' 
 
 We may observe however that though the assumption of Cavendish, that 
 the force varies as some power of the distance, may appear less general than 
 that of Laplace, who supposes it to be any function of the distance, it is 
 the only one consistent with the fact that similar surfaces can be electrified 
 so as to have similar electrical properties. 
 
CHAPTER IV 
 
 THE FARADAY-MAXWELL THEORY OF ELECTROSTATIC ACTION 
 
 162. Action at a distance or transmission through a medium*? The fact 
 that certain bodies after being rubbed appear to attract other bodies at a 
 distance from them was known to the ancients. In modern times a great 
 variety of other phenomena have been observed and found to be related to 
 these phenomena of attraction at a distance. The first really definite notion 
 of this interaction of two bodies at a distance apart however first arose in 
 Newton's law of gravitational attraction which states that one body attracts 
 another according to a simple law, an essential point being that any inter- 
 mediate matter does not affect the action. This law of gravitation proved so 
 successful in astronomy that it was made the pattern for the solution of more 
 abstract problems in physics and attempts were made to model the whole 
 of natural philosophy on this one principle, by expressing all kinds of material 
 interaction in terms of laws of direct mechanical attraction across space. 
 Of course if material systems are constituted of discrete atoms, separated from 
 each other by many times the diameter of any one of them, this simple plan 
 of exhibiting their interactions in terms of direct forces between them would 
 probably be exact enough to apply to a wide range of questions, provided we 
 could be certain that the law^s of force depended only on the positions and 
 not also on the motions of the atoms f. 
 
 Then when it was discovered that the laws of electric and magnetic 
 actions were of the same type as that of gravitation it was only too natural 
 to suppose that they were direct distance actions, without any reference to 
 any medium which may occupy the space between. 
 
 This doctrine of action at a distance which thus expresses the view that 
 our knowledge in such cases may be completely represented by means of 
 laws of action at a distance expressible in terms of the positions (and possible 
 motions) of the interacting bodies without taking any heed of the intervening 
 space, was specially favoured by the French and German scientific schools 
 
 * Cf. the article "Aether" by Sir J. Larmor in the Encyclopaedia Britannica (1911). Also 
 the book Aether and Matter by the same author. 
 
 f The most successful of these applications has been in the theory of capillary action 
 elaborated by Laplace, though even here it appears that the definite result* attainable by 
 the hypothesis of mutual atomic attractions really reposed on much wider and less specia 
 principles those, namely, connected with the modern doctrine of energy. 
 
162, 163] Distance or medium action? 157 
 
 and in W. Weber's hands electric theory was built up on it to a most 
 wonderful perfection*. The doctrine was however strongly repudiated by 
 Newton himself and hardly ever became influential in the English school of 
 abstract physicists. 
 
 The modern view of these things, according to which the hypothesis of 
 direct transmission of physical influences expresses only part of the facts, is 
 that all space is filled with physical activity and that while an influence is 
 passing across from a body A, to another body B, there is some dynamical 
 process in action in the intervening region, though it appears to the senses 
 to be mere empty space. The problem is of course whether we can represent 
 the facts more simply by supposing the intervening space to be occupied by 
 a medium which transmits physical actions, after the manner that a continuous 
 material medium, solid or liquid, transmits mechanical disturbances. The 
 object of the following pages is to answer this question in the affirmative 
 following closely along the lines laid down by Faraday in his wonderful 
 experimental researches and followed by Maxwell in his mathematical dis- 
 cussion of the results of these researches from which he evolved a definite 
 justification I of the fundamental notion. 
 
 163. We are thus at the outset confronted with these two opposite 
 notions, pure action at a distance and transmission through and by the 
 'aether' of space. The two notions are to be contrasted but are in reality 
 not so much opposed to each other as at first sight appears. Both theories 
 can and have been logically developed and up to a certain stage give a very 
 good account of the phenomena. There is however one question which can 
 decide between the two theories. Are the effects of gravitational and 
 electrical attractions propagated in time? If time is taken to transmit a 
 condition there must be something to transmit it. If the change appears 
 to be instantaneously set up then it is either actually instantaneous or is 
 too fast to be observed ; in either case we can neglect the medium as it is of 
 no use talking of a medium which acts too quickly to be followed. 
 
 In electrical theory this question has been definitely settled. The 
 machinery underlying electric and magnetic attractions is the same as that 
 in light phenomena. Oersted was the firs to state this but before his time 
 it was an obvious guess. Ampere treated the question very fully but no one 
 constructed a theory of it till Maxwell put Faraday's ideas into mathematics 
 and thereby arrived at a definite formulation of the connection. Experi- 
 mental confirmation of Maxwell's theory was provided twenty years after its 
 
 * An historical account of the developments of electrical theory on this basis is given with 
 full references by Reiff and Sommerfeld in Ency. der math. Wissench. v. 12, pp. 1-62. 
 
 f The "justification" is not absolute. In fact in the view of the more modern "reletavists" 
 the existence of a material aether is denied, although the mathematical form of Maxwell's theory 
 is accepted in its entirety. 
 
158 The Faraday-Maxwell theory of electrostatic action [CH. iv 
 
 publication by Hertz, who then succeeded in producing and examining the 
 long electric waves, whose existence was an essential consequence of Maxwell's 
 fundamental hypothesis on the electrodynamic properties of the aether. 
 
 As far as gravitational theory is concerned the fundamental question is 
 however still an open one. No velocity of gravitational action has yet been 
 detected experimentally. 
 
 The great success achieved by Maxwell's theory has now made it very 
 improbable that electric and magnetic forces (and probably also gravitational 
 ones) acting in distances large compared with molecular dimensions are pure 
 distance actions. The question of molecular forces is still an open one. 
 
 164. The theory of electric actions developed in the previous chapters 
 is essentially a distance action theory, inasmuch as it depends merely on the 
 concept of the electric charge with a definite law of action between point 
 charges. No reference whatever is made to the medium between the charges 
 and in fact our theory is true only if one single homogeneous medium 
 pervades the whole of space. Such is in general not the case and the 
 theory thus needs generalisation on the lines suggested by Faraday and 
 worked out mathematically by Maxwell. 
 
 165. Faraday's* ideas on the nature of the electric action between 
 charged bodies. Faraday firmly believed that the action between two 
 charged bodies was transmitted through and by the medium between them. 
 To test his ideas experimentally he tried to alter the medium by inter- 
 posing between the charges different dielectric substances. The procedure 
 actually adopted to obtain the exact effect was to use two equal spherical 
 condensers, one with an air dielectric and the other with some other 
 substance, such as sulphur. By connecting the two inner spheres of the 
 condensers together and the outer ones to earth any charge is divided 
 between them. If the condensers are of exactly the same size the charge 
 should be divided equally if the presence of the sulphur makes no difference. 
 Faraday found however that they were not equally divided ; but were in a 
 definite ratio e : 1 ; the presence of the sulphur increases the capacity of the 
 one condenser e times. This constant e was found to be typical of the 
 substance used in the second conclenser and is therefore called the specific 
 inductive capacity (s. i. c.) of the substance or simply its dielectric con- 
 stant f. 
 
 In attempting to find some cause or theory of this new effect Faraday 
 was induced to a closer investigation of the electric field in the cases when 
 the otherwise simple circumstances are complicated by the presence of some 
 
 * Cf. Experimental Researches, especially i. 1231, 1613-16; in. 3070-3299. 
 t Dielectric constants were independently determined by Cavendish in 1773. Cf. Maxwell, 
 Treatise, I. p. 54. 
 
163-166] The mathematical formulation of the theory 159 
 
 dielectric material, his method being to trace out experimentally the lines of 
 force and to form them into tubes of force in the manner indicated in the 
 previous chapter. He intuitively got thejdea that the amount of electricity 
 at one end of the tube was the same as that at the other and he then 
 succeeded in verifying it experimentally even in the case when dielectric 
 media of great complexity were present in the field. Moreover he dis- 
 covered* that the other important property of tubes of force, viz. that the 
 product of the force intensity at any point in a tube by the cross-section 
 at that point is constant all along, still held in the general case. 
 
 Such obviously fundamental results naturally led Faraday to think that 
 there must be some physical cause for this equality and he put it down to 
 some physical action in the tube. His idea was that the tube was full of 
 an incompressible fluid and the behaviour is just as if the liquid were pushed 
 along the tube, whatever excess is produced at one end has an equivalent 
 diminution at the other, both being equal to the amount crossing any section 
 of the tube during the establishment of the displacement. 
 
 This notion of Faraday's was the ruin of his recognition by contemporary 
 mathematicians. The theory of simple attractions gave satisfactory proofs 
 and explained everything known at the time and no new idea seemed 
 necessary. Faraday however went deeper and tried to explain the mechanism 
 of the attractions but he could only offer suggestions which appeared rather 
 crude against the completeness of the attraction theory. 
 
 The essential point in this idea of Faraday's is that an electric field arises 
 through an electric displacement or induction along the curved lines of force 
 resulting in an accumulation of positive charge at one end and negative at 
 the other. The term 'electric displacement' thus introduced is however not 
 to be taken too literally in this sense. The idea that survives is that it is 
 some vector of the same nature as the displacement of a fluid which is related 
 in the usual manner to the lines of force which experiment maps out. 
 
 166. In the mathematical formulation of this scheme it is first necessary 
 to define the electric force independently of the idea of simple attraction. 
 We define the electric force intensity E at a point in the field so that ESg- is 
 the force on a small charge S<? placed there. Before however we put the charge 
 there we must make room for it, we must remove the matter inside a small 
 cavity surrounding it. The question as to whether this removal affects the 
 force at the point is reserved for future consideration and for the present we 
 shall assume that the force intensity thus defined is a mathematically definite 
 vector quantity. 
 
 The force E as thus defined is still the gradient of a potential function </>. 
 A simple justification of this statement could be based, in the manner pre- 
 viously indicated, on the perpetual motion idea, the argument being that 
 
160 The Faraday-Maxwell theory of electrostatic action [CH. iv 
 
 under steady conditions carrying a charge round a closed circuit ought to 
 involve no work on the whole, assuming that the conditions are the same at 
 the end as at the beginning. We shall therefore assume the existence of 
 a potential function <j> at each point of the field. 
 
 We must next define the electric displacement or something akin to an 
 electric displacement. The intensity of the displacement D, in any direction 
 at a point is such that the total amount displaced across any small area 
 df s perpendicular to the direction during the establishment of the field is 
 
 D s ^/ s . 
 
 Suppose now we consider that the field 
 is established by slowly increasing all 
 charges proportionally : the configuration 
 of the field at each instant as regards the 
 lines of force and displacement will then be 
 similar to the final one. Now consider a 
 tube formed by the lines of displacement 
 drawn through any small area df, which is 
 perpendicular to its mean axis, and let 
 df $ be any adjacent slanting cross-section 
 making an angle 6 with df. This tube is 
 a tube of displacement at each instant 
 
 during the establishment of the field and thus the totality of displacement 
 across any section of it is the same all along : that is 
 
 but df & cos 6 = df and therefore 
 
 D s = D cos 6, 
 
 and thus the quantity D so defined is an ordinary vector, its direction being 
 at each point along the line of displacement through that point. 
 
 It is easily verified that the final result of any more complicated method 
 of establishing the field would be expressible by the same quantity D. 
 
 167. Thus in our electric field we have at each point two kinds of 
 vectors : 
 
 (i) the force intensity E which is the gradient of a potential function 
 and 
 
 (ii) the flux intensity P. 
 
 Now our theory implies that E is the cause of D ; the displacement at 
 a point is conditioned by the force intensity there. The simplest possible 
 law of causality we can have is that in which there is a simple proportionality, 
 
166-168] Gauss's theorem 161 
 
 so that if we double the cause we double the effect. In this case the com- 
 ponents (D x , D y , Dj of the displacement would be linear functions of the 
 components (E^, E v , E z ) of the electric force, i.e. 
 
 477-D,,, = euEj, + e 12 E,, + e 13 E 2 , 
 
 e 33 E 2 *, 
 or in symbolic vector form ^-Q _ t \ jj. 
 
 For isotropic media, i.e. for those having the same properties in all direc- 
 tions, this would be more simply expressed by the vector relation 
 
 D = -.- E. 
 
 477 
 
 This is the simplest possible form of the theory. Subsequent develop- 
 ments will show that it is very approximately the correct one. Let us now 
 follow it to some of its more important consequences, confining ourselves 
 however to the case of isotropic media. 
 
 168. Draw out all the lines of displacement in the field and form 
 them into tubes. As we have assumed the total displacement across any 
 cross-section of a tube is constant along the tube. Moreover the displace- 
 ment at one end of a tube where it abuts on a charge is equal to that along 
 the tube and on the simplest assumption is measured by the charge there. 
 If we are dealing only with conductors carrying charges and if a tube ends 
 at a place where the surface density is a and the cross-section there is df^ , 
 then if df is the cross-section at any other part of the tube where the flux 
 density is D Ddf=adf 1 . 
 
 On this idea the surface density, or charge at the ends of a displacement 
 tube, is merely the terminal aspect of the displacement in the tubes. The 
 way the displacement reveals itself is by piling up surface density. 
 
 In this theory the ordinary Gaussian surface integral theorem has a 
 distinct physical significance. Our notion of the displacement means that 
 if we take a closed surface of any kind in the field and integrate the normal 
 displacement over it, then the total thus obtained must be equal to the total 
 charge inside. Thus if D n is the normal component of the displacement at 
 the position of the element df of this surface 
 
 where Q is the total charge inside the surface /. This means that if the 
 charge inside is a volume charge of finite density p then 
 
 ( D n df=Q= [ pdv, 
 Jf Jf 
 
 * The 47r is introduced in conforming to the usual notation. 
 
 11 
 
162 The Faraday-Maxwell theory of electrostatic action [CH. iv 
 
 which by Green's lemma reduces to 
 
 f f 
 
 divDdv = pdv, 
 
 Jf Jf 
 
 and as this is true for any volume /we must have 
 
 div D = p 
 
 at each point of space : if p = 0, as is usually the case in all electrostatic 
 
 problems, then 
 
 div D = 0, 
 
 and as much is displaced out of any region as flows into it, i.e. the displace- 
 ment is like the flux of an incompressible fluid, as assumed by Faraday and 
 Maxwell ; D then satisfies the conditions for a stream vector. 
 
 169. We have already seen that the electric force intensity at each 
 point of the field is a vector whose components are 
 
 (^x > "y ? "z) ~ 
 
 moreover our assumptions imply that the displacement D is conditioned by 
 the electric force E and the relation adopted was 
 
 (e)E (e) 
 
 hence we have the characteristic equation of the field in the form 
 
 div ((e) grad</>) = 4wp, 
 
 which is the equation that replaces Poisson's equation for our theory. It is 
 the characteristic equation of the potential function on the new generalised 
 method of procedure invented by Faraday. 
 
 If the dielectric medium is isotropic this equation reduces to 
 8 / d(f>\ d ( d<f>\ 8 
 
 dz V dzj 
 and if it is homogeneous as well it becomes 
 
 V72 J. ^Tr/a 
 
 v*-- , 
 
 a modified form of Poisson's equation. This result stated in words implies 
 that the result of having the dielectric throughout the region is that the 
 same distribution of potential requires a distribution of charge e times 
 larger than in free space. Thus the constant e of homogeneous isotropic 
 medium which has been here introduced merely as the physical constant in 
 the relation between the electric force and displacement, is identical with 
 Faraday's dielectric constant e. 
 
168-171] The boundary conditions 163 
 
 170. As in the previous method of analysis of the electric field it is 
 essential that we consider the alterations in the above analytical procedure 
 necessitated by singularities in the distribution of charge, so that we may 
 know how to deal with them when they turn up in any applications. The 
 most important case is that in which p is infinite along a surface so that 
 there is a surface distribution of density a; we shall confine our attention 
 to this example. 
 
 In the first place it is obvious that discontinuities can arise only in the 
 neighbourhood of the surface distribution. Moreover (f> must be continuous 
 as we cross the surface as otherwise its gradient across would be infinite. 
 If we now apply our generalised Gauss' theorem to a small flat cylinder 
 enclosing a part of the surface of area df we can conclude at once that the 
 normal component of the displacement across the surface is discontinuous 
 by an amount + a, i.e. if D n is the component of D normal to the surface 
 on which the surface density is situated and if also suffices 1 and 2 denote 
 the values of the functions at near points on the same normal one at each 
 side of the surface 
 
 D B1 - D W2 = a, 
 
 but, as above, in the case of isotropic media 
 
 n ^iMi 
 
 4rr dn ' 
 
 n _ 2 302 
 
 TT dn ' 
 
 where we have also included for generality the possibility of different values 
 of e for the substance on the two sides of the surface. Thus 
 
 5i 30i _ 2 302 _ 
 4-7T dn 47T dn 
 
 If the surface is one of discontinuity in the isotropic dielectric medium, 
 without any charge, a = and thus 
 
 30! 30 2 _ 
 
 l dn ' ^ dn = ' 
 
 and these equations give the form of the boundary conditions to which is 
 subject in the present form of the theory. 
 
 171. It must now be noticed that the theory here given agrees perfectly 
 with that deduced from the old attraction ideas for a vacuum. We have 
 only to put e = 1 to get all the results of our previous theory. We have 
 however gone deeper into the matter; instead of talking of mere attractions 
 we have attempted to see what is going on and have generalised the theory 
 to include the properties of the medium conveying the action. Instead of 
 a simple theory of attractions we have now a theory of flux stimulated by 
 
 112 
 
164 The Faraday-Maxwell theory of electrostatic action [CH. iv 
 
 electric force. The exposition is of course merely of the nature of an 
 explanation ; no proof can be given that it is the correct view of the affair. 
 We have merely invented a consistent scheme whose continued existence 
 depends only on the test of its reality. 
 
 172. On electric displacement. The present scheme of course turns on 
 the electric displacement. What sort of thing is this electric displacement ? 
 Why is it different when a dielectric substance is present ? The first question 
 presents one of the fundamental difficulties of the theory, whose solution has 
 not yet been effected. The second question was answered by Kelvin by 
 comparing the behaviour of dielectrics under electric force with the behaviour 
 of magnetisable bodies under magnetic force. The magnetic behaviour of 
 iron had just been worked out by Poisson* and Kelvin f simply transferred 
 the theory to dielectric media and it fitted splendidly. In this theory electric 
 excitement is always accompanied by separation ; the old fashioned method 
 is that there are practically infinite amounts of positive and negative electricity 
 in each body and the electric force pulls on them and of course in opposite 
 directions and thus tends to separate them. In a conductor the two are 
 pulled right apart and as the supply is practically unlimited the separation 
 will go on while the electric force exists. If on the other hand the body is 
 an insulator the extent of the separation is limited by certain elastic resilience 
 forces tending to hold the charge elements to the particles of matter. The 
 result of the small separation which can be produced is to render each molecule 
 (or perhaps molecular groups) of the substance bi-polar, like a little magnet. 
 A fuller explanation will be given when we return to the subject of polarised 
 media in a subsequent chapter. For the present we merely regard the electric 
 force as actually doing two things : 
 
 (i) moving electric charges, 
 
 (ii) producing electric displacement in a dielectric. 
 
 It is part of a physical theory to explain how it is capable of acting in 
 these two ways, why it is that one really involves the other. 
 
 There are two* further remarks respecting the electric displacement which 
 it might be convenient to mention before concluding. Firstly notice that the 
 direction of the vector D in isotropic media is in the positive direction of the 
 lines of force and thus since the lines of force go outwards from a positive 
 charge the displacement is also away from the charge. This rule of signs 
 is opposite to that which one might expect and thus makes it clear that the 
 'displacement' is not a displacement of electricity in the usual sense of 
 those words. 
 
 * Memoires de VAcademie, 5 ( 1826), pp. 247, 488 ; 6 ( 1827), p. 441. Cf. also Maxwell, Treatise, 
 n. 385. 
 
 t Camb. and Dublin Math. Journ. (1845). Reprint, 447. The physical idea of polarisation 
 in dielectrics was introduced by Faraday, Experimental Researches, 1295 (1837). 
 
171-174] Electrostatic problems with dielectrics 165 
 
 Secondly it is important to emphasise that the vector D represents a 
 totality of displacement. If the rate of change of the displacement at any 
 time t be denoted by a vector R, then the amount displaced across any surface 
 element df during the small time dt is"R n dtdf, and during the finite interval 
 since the initial instant t required in setting up the field the total displace- 
 ment across the surface element is 
 
 D n df=dff t R n dt, 
 
 ' to 
 
 and this is the vector D of our theory. 
 
 173. The inverse electrostatic problem with dielectrics. The considera- 
 tions of this chapter have slightly modified the conditions governing the 
 problem of the determination of the circumstances in any electrical system 
 under given conditions. 
 
 The potential function is now a regular function </> satisfying the following 
 conditions : 
 
 (i) div (e grad <f>) = ^np at any point in the dielectric which for the 
 present purposes is assumed to be isotropic ; in a vacuum or practically 
 speaking in air we shall have still 
 
 = - 47775 ; 
 
 (ii) at any surface of discontinuity in the field (excluding double sheet 
 distributions which we need not here consider) the normal component of the 
 induction is discontinuous by the amount 47r<r so that 
 
 8</>! d(f> 2 
 
 e l -= A e 2 -~ + 477-cr = 0, 
 dn on 
 
 but the potential and tangential force must be continuous. 
 
 The problems of any practical interest mainly concern the estimation of 
 the modification of the field of given charges by the introduction into their 
 field of one or more dielectric masses. We must attempt to obtain a solution 
 of the above conditions subject to the restrictions imposed by the conditions 
 implied in the special data for the problem. The essential features of such 
 problems are illustrated by the following simple cases. 
 
 174. (a) To find the effect of 'introducing a uniform dielectric sphere into 
 a uniform electric field. 
 
 The potential of the original undisturbed field can be written in the form 
 
 </> = Ex, 
 
 if the a;-axis of coordinates is parallel to the uniform direction of the field 
 near the sphere. The potential </> of the final field must agree with this 
 field at remote distances, where the disturbance due to the sphere is inappreci- 
 able : it must also satisfy y2J, _ Q 
 
166 The Faraday -Maxwell theory of electrostatic action [CH. iv 
 
 both inside and outside the sphere (e being now constant), and be continuous 
 at the boundary of the sphere, i.e. between the two dielectric media. These 
 conditions determine </> completely and following the hints suggested by the 
 analogous problem of the conducting sphere already examined in detail, we 
 choose the origin of coordinates to coincide with the centre of the sphere, 
 whose radius is a, and then try solutions for </> in the two regions 
 
 (i) ( f) i = Ax 
 
 inside the sphere and 
 
 Bx 
 
 outside. These satisfy all but the surface conditions; they also satisfy 
 these if 
 
 a 3 
 
 and eA = E ; 
 
 a 3 
 
 the first of these equations expresses the continuity of the potential and the 
 second the continuity of normal induction (e-^-J. We have therefore 
 
 3E 
 
 ~<TT~2' 
 
 and thus the problem is completely solved by 
 
 The force inside the sphere is uniform but diminished in the ratio 3 : e + 2. 
 
 175. (b) The effect of introducing a homogeneous dielectric ellipsoid 
 in a uniform field. 
 
 If we take first the case when the original field is parallel to an axis of 
 the ellipsoid, its potential in the undisturbed state is 
 
 <f> = - Ex. 
 
 Now, noticing that the various forms obtained for the potentials of charge 
 distributions on ellipsoids in the first chapter must be solutions of the potential 
 equation, we are induced to try solutions in this case in the form 
 
 (i) for outside space 
 
 T r dt 
 
 d> = <6 Ex + Lx I . , 
 
 J A (a 2 + t) V(a z + t) (b 2 + t) (c 2 + t) 
 
174, 175] Induction in an ellipsoid 167 
 
 (ii) for inside space J, _ A _|_ JJx 
 
 and we must then try and find L, L' to satisfy the continuity conditions at 
 the boundary. 
 
 Continuity of potential requires 
 
 E L , L r * 
 
 J o (a 2 + t) V(a 2 + t) (b z + t) (c 2 + t) 
 Continuity of induction requires 
 
 r f 00 dt 2L 
 
 - E + L I .. - - - = *L', 
 
 .' o (a 2 + t) V(a 2 + t) (6 2 + t) (c 2 + t) a*bc 
 
 whence using, as in the first chapter, 
 
 A = ia 3 6c . . , 
 
 J o (a 2 + \/(a 2 + t) (6 2 + (c 2 + t) 
 
 a 3 bc . j, 
 
 we have ~- (e - 1) & 
 
 L' = - 
 
 so that the correct forms of the potential for the disturbed field are : 
 outside the ellipsoid 
 
 a 3 6c ( e - 1) Ex r dt 
 
 2 i,+ 4(c.- 
 and inside 
 
 We see at once that we can generalise this result. If the original field were 
 given by 4 = 4 - tf e x - E y y - E z z, 
 
 then the outside potential would be 
 
 abc . 
 <j>=<t>v-E x x- E y y - E z z + - (e - 1 
 
 f 
 
 1 + 5(6-1)" 1 + 0(6-1) ~ A J' 
 
 whilst the inside one is 
 
 9 9o ~~ i _i_ j /,. . _ i\ ~" i _i_ D / i \ ~ i i n (^ _ i\ ' 
 
 , . r * 
 
 wherein A\ = I 
 
 J x (a 
 
 2 + V(a 2 + t) (b z + t) (c 2 + t)' 
 and similarly for 5* and C A . 
 
168 The Faraday -Maxwell theory of electrostatic action [OH. iv 
 
 176. (c) The effect of introducing an infinite slab of dielectric with one 
 plane face into the field of a single point charge. 
 
 Let the point charge be q at A^ and at a distance c from the slab : take 
 AZ on the normal A t N to the plane face of the slab so that A 1 A Z = 2c. 
 
 Now try a solution of the conditions for the potential function 
 
 A 2 P' 
 
 <f>2 = 
 
 (f> 1} (f> 2 being the potentials in the respective regions. These forms satisfy 
 everything but the conditions of continuity at the boundary. 
 
 Fig. 44 
 
 Continuity of potential gives 
 
 Continuity of normal displacement gives 
 
 whence with 
 
 we satisfy all the conditions for </> x and </> 2 . Thus if we use r and r 2 for 
 A^P and A 2 P we have 
 
 $1 = r~? ~j 
 
 ' nr c \ IT 
 
 ' i t n^ j. / n 
 
176, 177] Condensers with material dielectrics 
 
 169 
 
 177. The effect of a dielectric substance on the capacity of a con- 
 denser. We have seen how Faraday investigated the modifying action 
 of dielectric media in electric fields by comparing condensers in which the 
 air space is either wholly or partially occupied by such substances. In 
 order to interpret the results of these experiments we must examine on our 
 theoretical basis the details of the circumstances in some practical cases, 
 where the air space of a condenser is occupied as in Faraday's experiments 
 by some dielectric substance. 
 
 Fiz. 45 
 
 (a) In the parallel plate condenser of the previous chapter we introduce 
 a parallel slab of dielectric of constant e and thickness t 2 . The densities of 
 the charges on the plates are still a and a and by symmetry the lines of 
 force go straight across and the tubes are cylindrical, and so the electric 
 displacement along them is constant. At the positive plate the electric 
 force is 
 
 E = 47TCT, 
 
 the dielectric being air. In the dielectric slab the force is constant across it 
 
170 The Faraday -Maxwell theory of electrostatic action [en. iv 
 
 owing to its homogeneity and from the equality of the normal displacement 
 at its surface on both sides we must have 
 
 E 
 
 E' = - 
 e' 
 
 and then again in the air near the other plate the force is again 
 
 E = bra. 
 The difference of potential between the plates is easily obtained, 
 
 (pi 02 = I Jit (Lt 
 
 J 
 
 /! being the thickness of the first air slab. The total charge on a plate is 
 
 AE 
 V Ao' ~; , 
 
 47T 
 
 and thus the capacity of the arrangement as a condenser is 
 
 A 
 
 We might describe t 2 as the 'reduced thickness' of the dielectric slab, the 
 thickness of an equivalent layer of air. 
 
 This result can be generalised for several slabs ; the potential is 
 
 [tit, fl\ 
 t-^-tz+l . 
 J ii e J 
 
 178. (6) There is a more general theorem* of which the above is a 
 particular case. 
 
 Supposing C is any conductor for which the equi-potentials are known 
 when it is freely charged : if A and B are any two of these equi-potentials 
 we can determine the capacity of a condenser formed by conducting surfaces 
 A and B between which there is a dielectric stratified so that e is constant 
 along the equi-potential surfaces of C. 
 
 Supposing (f> is the potential function of C with a charge Q when alone in 
 the field, and let us assume that the potentials of the conductors are <$> A 
 and (/>#. 
 
 When the dielectric is absent the capacity of the condenser formed by 
 the surfaces <f>^ and <f>B is 
 
 c _ Q 
 
 <f>A <j>B 
 * This theorem is virtually due to Kelvin, Reprint.. 45. 
 
177-179] Condensers with stratified dielectrics 171 
 
 Now suppose that the dielectric substance is inserted so that at any point 
 init 
 
 then the potential in the new field is obviously of the form 
 
 * 
 
 = a 
 
 and the constants a, b can be determined to make this satisfy all conditions. 
 The charge on the conductor C is still Q and thus 
 
 Thus a = c , 
 
 c denoting the value of e at the surface of C which we can assume is unity. 
 The difference of potential between the surfaces A and B is now 
 
 4>A $B = I t 
 
 / ** 
 
 and thus the new capacity of these two surfaces with the dielectric between 
 them is 
 
 If the dielectric were uniform this would give 
 
 C" = eC, 
 which is Faraday's result. 
 
 179. As a particular case of the general theorem we may also quote the 
 results for a spherical condenser (radii of surfaces a, b). He e we have 
 
 r' 
 
 and thus in the new case 
 
 n dr , A 
 = - aQ j + 6. 
 
 o er 2 
 
 Thus (f)A <f>B = at 
 
 and the capacity is C" = , . 
 
172 The Faraday -Maxwell theory of electrostatic action [en. iv 
 
 In the particular case when e = e x for all values b < r < c and e = e, for all 
 values c < r < a the capacity is 
 
 1 
 
 C' = 
 
 c dr_ a 
 
 180. On the refraction of the lines of force in an electrical field on 
 passing across a discontinuous dielectric surface. It is interesting to notice 
 how the lines of force are suddenly bent on crossing an interface between 
 two dielectrics. The conditions of transition tell us at once what the law 
 of refraction is. 
 
 If the forces are E v , E 2 on either side of the interface and their directions 
 make angles i 1} i 2 with the normal at the point of incidence, then we have : 
 
 (i) the tangential component of the force is continuous, if there is no 
 double sheet distribution, thus 
 
 E! sin *! = E 2 sin i z ; 
 
 (ii) the electric displacement across the surface is continuous as there 
 is no surface charge, so that 
 
 e l E l cos ij = 62-^2 cos i z . 
 
 These are the two conditions to be satisfied; from them we deduce at 
 once 
 
 tan *\ tan i z 
 
 ei e 2 
 
 The lines of force are refracted according to a law of tangents (Maxwell). 
 In entering a denser medium (e bigger) the line is bent away from the normal. 
 
 This can be generalised by considering a double sheet distribution to be 
 present on the surface as well. 
 
 181. The distribution of the electrostatic energy. In the previous 
 chapters we found that the total energy in the electrostatic field was ex- 
 pressible in terms of the charge distribution in such a way that the amount 
 of energy required to increase the volume density of charge at any point by 
 Sp and the surface density by Scr was given in the form 
 
 = (A$pdv+ t 
 
 J Jf 
 
 the first integral being taken throughout the whole of the field and the second 
 over all those surfaces where there is a surface density. 
 
179-182] The distribution of energy in the field 173 
 
 Although the previous discussions have entirely ignored the dielectric 
 medium and its effect on the electrical conditions of the system the result 
 just quoted is perfectly correct whatever the complexity of this dielectric 
 medium; this results from the general definition of the potential function 
 <j) which we have given above. 
 
 Now the simple hypothesis of action through a medium regards the electric 
 charges merely as manifestations of a varied condition in the whole of the 
 medium throughout the field. In such a theory therefore the energy of the 
 electrical system is distributed throughout the space of the field so that each 
 element of volume furnishes a part to the total amount, which part depends 
 solely on the electrical conditions existing in the element. Now on the 
 Faraday-Maxwell form of the theory the essential specification of the con- 
 ditions at any point of the field is involved in the vector D, the electric 
 displacement, so that the alteration of the energy produced in the system 
 by a small arbitrary charge in its specification should be expressible in terms 
 of the alteration of the conditions of the medium, viz. by SD, at each point- 
 But in the form of the theory adopted, this increase of displacement SD at 
 each point of the field is effected by the agency of the electric force E, which 
 produces it and is alone effective in altering it, and if the analogy with 
 material phenomena implied in our choice of names is valid the work done 
 by the force intensity E in producing an additional virtual displacement 8D 
 throughout the small volume element dv would be 
 
 (E . SD) dv, 
 
 so that the total increase in the energy of the system should be expressible 
 by the integral 
 
 f(E . SD) dv 
 
 taken throughout the whole field. 
 
 182. Owing however to the great indefiniteness in our knowledge of the 
 true nature of the vector D, this deduction of an expression for the energy 
 cannot be regarded as anything more than a mere analogy. It can how- 
 ever be shown that the result obtained on this analogy agrees exactly in 
 the total amount with the former estimate. In fact we have at each point 
 of space div J) = P} 
 
 and at points on surfaces on which there is a charge density CT 
 
 D, - D, = a, 
 
 in n 
 
 and thus the variations SD are connected with the variations 8p and Scr by 
 the conditions 
 
 div SD = Sp 
 at each point of space and 
 
 SD ln - SD 2n = Scr 
 
174 The Faraday -Maxwell theory of electrostatic action [on. iv 
 
 at each surface distribution. Thus the first estimate of W is equivalent to 
 
 </> div SDdv + <f> (8D lB - SD 2 J df, 
 
 and a simple transformation of the first integral by integration by parts 
 shows that this is equal to 
 
 8W = - [(SD. V)<f>dv 
 = [(E . 3D) dv*. 
 
 Thus if the change in the total energy of the whole system is distributed 
 throughout the field with a density (E . SD) at each point the total amount 
 is consistent with our other theories. For the above mentioned reason we 
 have however no really definite theoretical basis for regarding this distribution 
 of the energy as the actual one. Maxwell assumed that it wasf and there are 
 certainly many points in favour of his view. It is the simplest distribution 
 which suits the case, and for this reason alone it might be regarded as the 
 correct one. More we cannot say except that future developments confirm 
 the assumption. 
 
 183. Now let us examine the equation of energy in this latter form a 
 little more closely : 
 
 SJF = f(E . SD) dv. 
 
 Now 8W is the work supplied by external agency to the system during the 
 arbitrary change of its configuration specified by SD (a virtual displacement 
 in the ordinary mechanical sense), and the principle of the conservation of 
 energy asserts that it is an exact differential, or. in technical terms, a function 
 W exists; otherwise we should have perpetual motion. The argument, as 
 we had it before, is that if we suppose the system to be taken through a definite 
 series of changes from a general configuration which we shall denote by 1 to 
 any other configuration which we shall denote by 2, then the work done on 
 the system during the change, which is expressed by 
 
 f 2 f 2 f 
 
 8W = (E . SD) dv, 
 
 J i ' iJ 
 
 must be independent of the actual series of changes through which the system 
 is taken from the one configuration to the other; provided of course that 
 each series of changes is reversible, so that any work gained to the external 
 agency in traversing it one way would be turned into an equal loss in going 
 
 * To make this argument quite rigorous it is necessary to include a bounding surface in 
 the field at a great distance from the origin and to examine the integral over it which results 
 in the integration by parts. In any real case however the field will be always regular at infinity 
 and this surface integral tends to vanish. 
 
 f Treatise, i. p. 167. 
 
182-184] The law of induction 175 
 
 the other way. If this were not so we could pass from configuration 1 to 
 configuration 2 through one series of changes and back again through another 
 series reversed, and could thus gain work on the whole in the complete cycle. 
 The system being finally in precisely the same state as before we should have 
 gained work from nothing and by repeating the process as often as we like 
 we could get an indefinite amount of energy out of nothing, a fact which is 
 highly improbable if not actually impossible. Thus under such conditions 
 
 2 SJF 
 i 
 
 depends only on the initial and final configurations 1 and 2 and not at all 
 on the series of changes through which the system is taken from one to the 
 other. This can only be true when 8PF is an exact differential when it is 
 
 i 
 We thus see that under ordinary conditions 
 
 8W 
 
 is an exact differential. Owing to the arbitrariness of SD this means that 
 (E . SD) must be a complete differential of some function of D or rather of its 
 components. 
 
 184. Writing this out in full we have, denoting the function by w, 
 
 E X SD X + E,8D V + E Z SD, - 8w, 
 
 so that w is a function of D x , D y , D. whose partial differential coefficients 
 with respect to these variables are respectively equal to the corresponding 
 components of E. But by hypothesis the various components of D are 
 functions of the various components of E and thus by substitution we can 
 interpret the function w as a function of E^, E^, E z and this is the more 
 fundamental form since E is the independent variable in the physical theory. 
 But having made this transformation we see at once that 
 
 (DSE) = 3 {w - (ED)} - 8w', 
 
 say; thus (DSE) is also a complete differential of a function w' which is a 
 function of the variables E x , E v , E z ; its partial differential coefficients will 
 be similarly equal to the corresponding components of D. Moreover this 
 function w' must remain unaltered if the direction of the electric force E is 
 simply reversed*, so that it can only involve even powers of the components 
 (E,,., "Ey, E ) and for weak fields would be practically quadratic. If we imply 
 this restriction to small fields then we can write the function w' in the form 
 
 ~ (e n E x 2 + e 2 E, 2 + sa E,i + 2e 12 E a E, 7 + 2 e23 E,E, f 2e 31 E,E x ), 
 * Provided there is no permanent displacement, as is generally assumed to be the case. 
 
176 The Faraday -Maxwell theory of electrostatic action [CH. iv 
 
 and thence we deduce that 
 
 4-7rD a; = e n E x 
 
 e 31 E x + e 32 E y + e 33 E, , 
 
 in agreement with the results of our previous speculations. We^see here 
 however that the doctrine of energy requires that 
 
 e !2 = 2l) e !3 = e 31? e 23 = 32? 
 
 a fact not involved in the former arguments. 
 In this case the energy in the field is 
 
 W - (*nE* 2 + .- + *i&& + ..-) dv = \ (ED)*. 
 
 For isotropic media these relations simplify very considerably, for then 
 
 4rrD = eE, 
 
 so that W = ^- leWdv\, 
 
 and this is the real form of Maxwell's result. The energy of an electrical 
 system may in a restricted sense be considered as distributed throughout the 
 
 eE 2 
 field with a volume density - - at each point. 
 
 O7T 
 
 185. A simple application of Green's analysis will show that in the 
 general case of a linear relation between E and D the total energy of the 
 system is given also by 
 
 W ^ I p(f)dv + \ I o(f>df, 
 
 J J / 
 
 which is the form suitable for applications based on the distance action theory. 
 In fact in such a case 
 
 W = /(ED) dv 
 
 and this transforms by the analytical theorem to 
 
 W= - 
 
 
 (D m - DJ df, 
 
 the first integral being taken throughout the whole of the field and the second 
 over those surfaces where D is discontinuous as regards its normal component. 
 
 * Maxwell, Treatise, i. p. 147. 
 
 f Kelvin, Math, and Phys Papers, i. 61. 
 
184-186] The transmission of force 177 
 
 The integral over the infinite boundary as usual tends to zero and is neglected 
 altogether. But $ lv jj _ _ 
 
 and D m - D no = - a, 
 
 so that W = + \ pd>dv + | aJxlf, 
 
 Jf 
 
 which is identical with the form obtained in the more restricted theory given 
 in the first chapter. The present analysis indicates the necessary restrictions- 
 which must be placed on the method of argument used on the former 
 occasion*. 
 
 186. On the transmission of force through the dielectric medium. We 
 
 must next turn to another fundamental problem in the present mode of 
 formulation of electric theory : this concerns the explanation of the pondero- 
 motive forces between charged bodies whose existence is implied in the 
 determination of the potential function just found and which can be derived 
 from this function, suitably expressed, in the manner elaborated in detail in 
 the first chapter. 
 
 The transmission hypothesis underlying the Faraday-Maxwell theory of 
 electric action here under discussion regards all electrical phenomena merely 
 as the result of a certain state of affairs established in the surrounding dielectric 
 field. It would therefore be necessary on such a theory to regard the pondero- 
 motive forces resulting from the attractions and repulsions between the 
 charges as the terminal aspects of some state of stress in the medium between. 
 We must now enquire as to a possible representation of the manner in which 
 these forces are transmitted across the space between the bodies. The 
 problem reduces itself to finding the state of stress in the elastic medium 
 which agrees with the known boundary values. If we knew the nature of 
 the elasticity of the medium we could solve this problem completely; but 
 this is just Avhat we do not know in the case of the aether. We can only 
 guess and of course there can be any number of guesses. Faraday divined 
 a very simple scheme which has high claims on account of its simplicity, 
 but as" we shall see later, it is not general enough for our purposes. 
 
 After experimentally investigating the nature of the electric fields around 
 conductors Faraday came to the conclusion that the forces between them 
 could be accounted for by a pull along the tubes of force, i.e. as if they were 
 tending to contract like stretched elastic bands. This would obviously 
 account for the attraction, but with it alone the elements of the transmitting 
 medium could not be in equilibrium. He then saw somehow that in addition 
 there must be an equal pressure in all directions perpendicular to the tubes: 
 
 * It must however be emphasised that the restrictions are of theoretical interest only. Tho 
 simple linear relation satisfies all the requirements of actual fact. 
 
 L. 12 
 
178 The Faraday -Maxwell theory of electrostatic action [OH. iv 
 
 the tube tends not only to contract itself along its length but also to expand 
 against a normal pressure all round. Under such a stress system the medium 
 would be in equilibrium as regards its own parts but would transmit the force 
 from one body to another. Although Faraday invented this scheme he was 
 unable to prove its reality : it was Maxwell who formulated it mathematically 
 and put it in a very precise form. 
 
 187. As we are here going rather deeply into this question it might be 
 as well to indicate how the stress in any medium is analysed*. Consider 
 the medium in the neighbourhood of any point separated over a small 
 interface there. Forces would then be required to be applied at each of the 
 exposed surfaces to hold the medium in equilibrium : the same forces would 
 be required for each of the two interfaces since they represent the action 
 and reaction which were exerted between the parts of the medium on the 
 two sides of the slit before they were separated. We shall consider these 
 forces measured as so much per unit area. If for the slit made of area 8/ 
 we know that the force required to hold the medium on either side of the 
 interface in equilibrium is TS/, then the vector T defines the stress for this 
 particular direction of df. If we knew so much for every direction of the slit 
 we should have a complete knowledge of the state of the stress at the point. 
 It can however be shown that it is quite sufficient to know it for three 
 directions only. 
 
 If in fact we consider the equilibrium of a portion of the medium included 
 in any closed surface / we see that it is in equilibrium under the actions of 
 (i) the rest of the substance across the surface /, and (ii) the external bodily 
 forces applied to the material in the tetrahedron such as the force of gravity. 
 If we denote these latter forces generally as a force F per unit volume at any 
 point then the linear equations of equilibrium of the mass are vectorially 
 
 Vdv + JW/=0, 
 
 the latter integral being taken over the surface / enclosing the volume over 
 which the former integral is extended. The angular equations are similarly 
 of type 
 
 f(F,y - P,z) dv + I (T,y - T,z) df = Of, 
 j . / 
 
 the regions of integration being the same. 
 
 From the form of these two equations we can deduce a result of great 
 importance. Let the volume of integration be very small in all its dimensions, 
 
 * The general theory is discussed by Love, Mathematical Theory of Elasticity (2nd Ed. 1906, 
 Cambridge). 
 
 f The analysis is of course referred to a convenient system of rectangular axes which 
 also form the base for the reduction of the forces. 
 
186-188] The specification of a stress 179 
 
 and let I s denote this volume. If we divide both terms in each of these 
 equations by I 2 and then pass to the limit by diminishing I indefinitely we 
 find that 
 
 U-1 
 1=0 
 
 and three equations of the type 
 
 which show that the tractions on the elements of area of the surface of any 
 portion of a body which is very small in all its dimensions are ultimately, to 
 a first approximation, in equilibrium among themselves. 
 
 188. Now let us suppose that the stress components per unit area for 
 directions of the small interface perpendicular to the axes of coordinates at 
 any point to be respectively 
 
 C\} T T T 
 
 \ i f -*- xx) *- xy ) -*- 
 
 xz> 
 
 l\\\ T T T 
 
 \ Li ) - L yx) - L yyi -J-yz) 
 \m) -*-zx> -*-zy> -*-zz> 
 
 then the force on any small area dydz perpendicular to the x-axis has com- 
 ponents 
 
 T xx dydz, T xv dydz, T xz dydz. 
 
 Now consider a very small cube of the material with its edges parallel 
 to the coordinate axes. To a first approximation the resultant attractions 
 exerted upon the forces perpendicular to the x-axis are 
 
 T A T A T A 
 
 - 1 xx *-> * xy 1 -*) J -xz L ^ 
 
 for the face on the positive side and 
 
 _TA - T A - 7 1 A 
 
 a 4 *! ***> - t xz'- x 
 
 for the other face, A being the area of a face. Similar expressions hold for 
 the other faces. The value of the integral 
 
 for the cube can be taken to be simply 
 
 Ik (T yz - T zv ), 
 
 where I is the length of any edge ; but the cube being to a first approximation 
 in equilibrium under these forces we must have 
 
 7 1 T 
 
 * yz * zy ' U J 
 
 and therefore T yz = T zy , 
 
 similarly T xz - T zx , T yz = T zy , 
 
 so that the nine quantities just given reduce to six. 
 
 122 
 
180 The Faraday -Maxwell theory of electrostatic action [CH. iv 
 
 189. We next consider the equilibrium of a portion of the medium 
 included in a small tetrahedron of which three edges are parallel to the 
 axes of coordinates and of lengths (dx, dy, dz) : let us suppose that the 
 fourth face is of area df and is normal to. the direction whose cosines are 
 (I, m, n). Since the surface tractions over the four faces of this tetrahedron 
 form to a first approximation a self-equilibrating system of forces we must 
 have for the component action on the fourth face of the tetrahedron parallel 
 to the ce-axis 
 
 T M- T a. T ~ 
 
 *- x^J ~ } * xx cy ' * yx c> * o 
 
 (A & L ) 
 
 - (IT XX + mT vx + nT zx ) df, 
 
 because it must balance the forces in the same direction on the other faces. 
 Thus the components of the stress across df are 
 
 and therefore the stress at any point in the medium is completely specified 
 by the six quantities T xx , T yy , T zz , T yz = T zy , T zx * T xz , T xy = T yx . 
 
 The stress across this fourth face will be normal to the face if these three 
 quantities be proportional to I, m, n ; that is, if 
 
 IT XX + mT xy + nT xz _ lT yx + mT m + nT vz IT ZX + mT za + nT zz 
 
 I m n 
 
 These equations we know determine three directions at right angles, viz. the 
 axes of the ellipsoid 
 
 T xx x* + T yy y* + T zz z* + (T xy + T yx ) xy + (T xz + T zx ) xz + (T zy + T vz ) yz = l. 
 
 These directions are the directions of principal stress at the point and were 
 we to refer the ellipsoid to these axes it would become 
 
 where T 1 , T 2 , T s are the principal stresses. 
 
 190. If we examine the matter a little more closely we shall find that 
 the tractions over the surface of any element of the medium do not exactly 
 balance among themselves as assumed above. There is a small outstanding 
 part which must balance the bodily force. To find this part we have only 
 to consider the statical equilibrium of the medium enclosed in any surface/:, 
 the forces acting on it are the bodily forces F per unit volume at any point 
 and the tractions over the surface/ Resolving along the a>axis the condition 
 for equilibrium is 
 
 - (IT XX + mT xy + nT xz ) df, 
 
189-191] The determination of a stress 181 
 
 (I, m, n) now denoting the direction cosines of the normal to the surface 
 element df. A transformation by Green's lemma and the usual line of argu- 
 ment shows that 
 
 3T dT xy dT 
 " 
 
 dx " dy 
 
 F_ 0-*- xy i V-*- yy . 
 11 ~^ i o 
 
 F 
 z 
 
 dx dy dz ' 
 dT dT 7 , 
 
 dx dy dz 
 
 It is the small differences or space gradients of the stress components that 
 balance the extraneous forces. 
 
 Reversing the argument we see that if the applied forcive F can be deter- 
 mined in such a manner as will enable .us to express its x-component in the 
 form 
 
 dx 
 
 and obtain similar forms for the other components, then we may say that 
 the bodily forces on that part of the system enclosed in any surface / can 
 be represented as the result of an elastic stress traction over the surface / : in 
 fact 
 
 = J(IX X + mX v + nX s ) df, 
 
 and thus we can at once write 
 
 T- Y T - Y T - Y 
 
 xx **** *- xy **K 5 * xz * 5 
 
 as providing a sufficient solution of the problem. 
 
 191. Returning now to the consideration of the stresses in the dielectric 
 medium of an electrical system, we know that the x-component of the bodily 
 forcive acting on the part of the system inside / is 
 
 fv x dv = I P E x dv + i [ a (E^ + E M ) df, 
 j j . f 
 
 the first integral being taken throughout the volume and the latter over all 
 surface charges. But p = div D and a D ni D w ., so that 
 
 div Dcfo + i (D ni - D 
 But we can easily verify that 
 
 E x div D = - i ( - 2^^ + (B, D)) - (BJD.) - (E A 
 
182 The Faraday- Maxivell theory of electrostatic action [CH. iv 
 
 so that 
 
 <Z/' 
 
 I (- 2E,D X + (ED)) - m (2E X DJ - n (2E X D 2 
 - i J {I (- 2E X D* + (ED)) - m2E JD, - n2BD v } <*/ 
 
 where in the integrals/' are now included also all surfaces inside /over which 
 the dielectric medium is discontinuous. 
 The surface integral over/' is 
 
 (D M1 - D 
 
 2E x D n + I (E, D) 
 
 na - i 
 
 which reduces to 
 
 Tl\ /T? T\\ 1 j_ /T3 1 T7 1 \ fT\ _1_ Tl \~\ rtf' 
 
 , D) 1 (E, U) 2 ) + (Bft *J (U n2 + U n JJ Of 
 
 If 
 Hence we have in all 
 
 f 
 
 {I (- 2E,D X + (E, D)) - m (2E X DJ - n (2E,D 2 )} df 
 + tfw* - E,J (D n2 - D B1 ) + I ((ED), - (BD) 8 )} df 
 
 This is the most general result for which no further reduction is possible. 
 We must therefore conclude that in the most general possible cases we are 
 unable to reduce the forces on the system inside any surface to a representation 
 by means of a system of tractions over that surface and they cannot therefore 
 be regarded as the result of an applied stress system throughout the medium 
 inside/. The outstanding terms representing a bodily forcive whose -com- 
 ponent is 
 
 I nD-x E -> dv 
 
 2 J ( dx 005 ] 
 
 and the surface forces on the various, surfaces of discontinuity shew that 
 a stress system which would otherwise be determined by the first surface 
 
 integral above in the final expression for ]1? x dv would not in general be self- 
 balanced when applied to the system ; it could not of itself keep the elements 
 of volume of the medium of the dielectric in equilibrium when acting alone. 
 
 * I, m, n are the direction cosines of the normal to the surface / at the position of the 
 element df. 
 
191-193] Special cases of Maxwell's stress 183 
 
 192. The failure to obtain the desired result in the most general case, 
 although of theoretical importance, in reality affects the usefulness of the 
 underlying ideas but little, because in the cases of practical importance 
 most of the outstanding terms vanish and the attempt to obtain the re- 
 presentative stress is entirely successful. 
 
 Electrostatic problems are in general concerned mainly with charged 
 conductors, no other charge distribution being generally possible. In this 
 case whatever the complexity of the dielectric medium the resultant electric 
 force at any point of a conductor is always normal to the surface outside and 
 zero inside and thus the part of the integral 
 
 [(E,, - EJ (D,,, - D ni ) + I {(ED), - (ED) 2 }] df> 
 
 corresponding to the surfaces of the conductors vanishes, for the integrand 
 in such parts is 
 
 - E Xl D ni + 1 (ED), = - m n p ni + m n p ni _ o, 
 
 so that the outstanding terms arise solely from discontinuities in the dielectric 
 medium ; if there are no sudden discontinuities in the dielectric medium (and 
 any such could be considered as a rapid but continuous transition) these 
 terms are completely represented by the volume integral 
 
 ^SEx ' 9D\ , 
 D - - E -.- dv, 
 ox) \ ox) 
 
 which in the case of isotropic media reduces to 
 
 This would mean that the stress system, determined by the first surface 
 integral over / and more fully specified below, when applied to the medium 
 between the conductors would leave an outstanding pull on each element of 
 volume which would have to be balanced somehow. This requires an 
 additional finite force per unit volume and if the change in the character 
 of the medium as specified by e is very rapid this force is very large. No 
 such force has however ever been contemplated. 
 
 This integral however also vanishes for homogeneous media, however 
 anisotropic, provided only that the linear relation between D and E holds ; 
 this is easily verified by the substitution of the usual linear functions for the 
 components of D, a reference to the principal crystalline axes having previously 
 been made. 
 
 193. The conclusion is therefore that for homogeneous dielectric media 
 the mechanical force acting on the part of an ordinary electrical system 
 inside any surface / drawn in the field can be represented in the main by a 
 system of interfacial tractions over the surface. These surface tractions 
 
184 The Faraday -Maxwell theory of electrostatic action [CH. TV 
 
 themselves are the representatives of a stress system specified at any point 
 by nine components * 
 
 T xx = \ (EJD, - E^ - E,D Z ), 
 
 T n = I (- EJD, + E y D y - EJU 
 
 T zz - | (- E X D. - E.D, + By),), 
 
 2V = E^, T zl/ = E Z D V , 
 
 T ZX = EJ> X , T XZ = EJ) Z . 
 But in the general case of crystalline medium, for which 
 
 V x D V .D Z ^E X E, : E 2 , 
 
 this stress is not self-conjugate. It does not therefore represent a stress 
 system of the ordinary mechanical type which is always self -conjugate, as 
 we proved before. The difference between the cross-terms in the matrix 
 show in fact that there is a torque on the element of volume of amount 
 
 [EDj dv, 
 and therefore the medium cannot be in equilibrium. 
 
 We thus see that, except in the very special case of isotropic homogeneous 
 media, it is impossible to obtain an ordinary elastic stress system to represent 
 the mechanical actions between electrified conductors. The specification of 
 the necessary stress system even in homogeneous media is more general than 
 is possible in simple ordinary elastic solids and necessitates a modification of 
 our views on the constitution of dielectrics. As we shall see in the next 
 chapter the specification of the stress in media with electric or magnetic 
 polarised molecules is much more general and involves a solution of most of 
 the difficulties here encountered. 
 
 194. In homogeneous isotropic media for which the ordinary law of 
 induction 
 
 D e - E 
 
 ~ 477 
 
 holds, the actions between electrified conductors may be completely repre- 
 sented as the result of an imposed stress system whose six components are 
 represented in the matrix 
 
 6 
 47T 
 
 2B.B,, 
 
 2BA, -E/ + E/-E, 2 , 2E,E, 
 
 2EE 2EE _E 2 E 2 -fE 2 
 
 ""^H - J *-'y*- i zi *-*x *-*y T **z 
 
 which is Maxwell's self-conjugate stress system f; the cross terms are sym- 
 metrical. The stress quadric for this system is obviously 
 
 * Heaviside, Phil. Trans. 183A (1892), p. 423, or Electromagnetic Theory, i. p. 84. 
 
 f Gf. Treatise, i. p. 159. The only case dealt with by Maxwell is that in which e = l. 
 
193-195] Alternative stresses 185 
 
 2 (E^z 2 + ... + 2E x E a xy + ...)- (E. 2 + E, 2 + E, 2 ) (x* + y z + 2 2 ) = ^, 
 
 or 2 (EL x x + E v y + E,z) 2 - (E x 2 + B, 2 + E, 2 ) (x 2 + y 2 + z 2 ) = 
 This quadric is one of revolution and its axis is in the direction 
 
 E* : E tf : E 2 , 
 i.e. along the lines of force. Referred to the principal axes its equation is 
 
 2E 2 z 2 - E 2 (x 2 + y* + 2 2 ) = - 7T , 
 
 
 so that the principal stress system consists of the three components 
 
 EZ _ e ?L 2 _ C J!L 2 _ ^ 
 
 47T 'STT' 8^' 877' 
 
 and in this form we see that it consists of a uniform hydrostatic pressure 
 
 eE 2 eE 2 
 
 - combined with a tension along the lines of force ' , and thus we can 
 
 07T 4:77 
 
 identify it with the system given by Faraday. 
 
 The important conclusion is however that unless the dielectric medium is 
 uniform and isotropic this Maxwellian stress system could not of itself keep 
 the elements of volume of the dielectric medium in equilibrium when acting 
 alone. This fact renders the system quite impossible for any real case 
 because the only realisable homogeneous medium is the unattainable empty 
 space. 
 
 195. The Maxwellian stress system is not the only one obtainable under 
 the specified conditions. Special emphasis has however been placed on it 
 on account of its particular disadvantages. It is however possible to obtain 
 two different stress specifications by a slight transformation of the outstanding 
 terms obtained in the general reduction given in 191. The surface terms 
 arising on account of the discontinuities are susceptible of the same treatment 
 as before but it is possible to reduce the outstanding volume integral in the 
 expression for F^ to the form 
 
 which in the simplest type of media for which 
 
 4?rD = eE, 
 reduces to 
 
186 The Faraday-Maxwell theory of electrostatic action [CH. iv 
 In this case the stress remaining is specified by the components 
 
 T = E D - E 2 
 
 '*. *>x"x g^' 
 
 E D - E 2 
 
 ~~ z z 877- ' 
 
 * * -jy , yx 
 
 T w - E.D,, T = B.D,. 
 
 T tt = B,D B , T XZ =E X D Z , 
 
 and differs from Maxwell's system only as regards its uniform pressure 
 constituent which is now ~- W instead of | (ED). 
 
 O77 
 
 This particular type of stress is analogous to Maxwell's magnetic stress* 
 and will be more fully dealt with in the next chapter. When however the 
 dielectric medium is regarded, "as at present, as a mere transmitter of the 
 forces between the electric charges, this stress is just as impossible as the 
 former one, as it also requires the existence of body forces on the dielectric 
 medium to maintain it in equilibrium. The type of force required is however 
 very different to that discussed above and admits of a simple physical 
 explanation. 
 
 196^ Yet another type of stress system can be obtained by reducing the 
 outstanding terms in the general reduction to 
 
 In this case the stress is specified in the same way as above but with leading 
 terms of the type T - F n 
 
 * -* xx I - l x' J x 
 
 This system is of course quite an impossible one as it would require a force 
 on the elements of the dielectric medium to maintain it in equilibrium, even 
 if that medium were free aether. It is however of interest because it is pre- 
 cisely the type of stress which should have been derived from the general 
 theory of electrostatic dielectric forces originated by Kortewegf, formulated 
 in general terms by von HelmholtzJ and further developed by Lorberg, 
 Kirchhoff, Hertz || and others. This general theory, based on the method 
 of energy, is usually made to lead to the first type of stress discussed above, 
 
 * Treatise, n. p. 278. 
 
 f Wied. Ann. 9(1880). 
 
 J Wied. Ann. 13 (1882); Abhandlungen, i. p. 798. 
 
 Wied. Ann. 24, 25 (1885k- Abhandlungen, Nachtrag. p. 91. 
 
 || Wied. Ann. 41 (1890). 
 
195, 196] Alter native stresses 187 
 
 but the analytical argument by which this is obtained has been criticised 
 by Larmor*, who shows that, properly interpreted, it can only lead to this 
 third type of stress, and must therefore involve some radical fallacy, as in 
 fact is obvious on physical grounds. This criticism appears however to have 
 been entirely overlooked and Helmholtz's procedure is still tacitly accepted 
 and reproduced by all recent writers on the subject f.' 
 
 * Phil. Tram. A. 190(1897), p. 280. Cf. also Livens, Phil. Mag. xxxn. (1916), p. 162. 
 
 f Cf. Cohn, Das electromagnetische Feld, p. 87 (Leipzig, 1900); Abraham, Die Theorie der 
 Elektrizitdt. I. p. 434 (2nd Ed. Leipzig, 1907); Jeans, Electricity and Magnetism, p. 172 (1st Ed. 
 Cambridge, 1908) ; also the articles by Lorentz, Pockels and Gans in the Encyclopddie der 
 mathematischen Wissenschaften, Bd. v. 
 
CHAPTER V 
 
 THE THEORY OF POLARISED MEDIA 
 
 197. Introduction. We have already seen how the simple laws of action 
 of electric charges are very considerably modified by the introduction into 
 the field of certain substances which we called dielectrics. We have also 
 seen how, up to a certain point, the general properties of the electric fields 
 when such bodies are present can be mathematically expressed in terms of 
 definite quantities definable in terms of the usual functions of the theory, 
 the fundamental assumption being that the properties of electrostatic fields 
 of any dielectric complexity whatever are expressible in terms of the assumed 
 general properties of a continuous dielectric medium occupying the field, the 
 continuity being the essence of the affair. We saw however that this simple 
 theory led us into difficulties when we attempted to investigate the mode of 
 action of the medium, the conclusion being that in general the non-homo- 
 geneous dielectric medium cannot transmit the actions between electric 
 charges by means of a simple self-balanced stress system which leaves the 
 elements of that medium in equilibrium. 
 
 We are therefore induced to start again on a new path and analyse more 
 intimately and from a different standpoint the origin and cause of the 
 modification of electric actions by the interposition of dielectric substances. 
 
 If there be brought near to a charged body A, a rod composed of some 
 dielectric or conducting material, the usual phenomena of electric induction 
 are observed : the ends of the rod near to and remote from the charged 
 body behave just as if they carried respectively charges of the opposite and 
 of the same kind as A. If the rod is made of conducting material it can be 
 charged permanently in this way : on cutting the rod at any point between 
 its two ends and removing it from the neighbourhood of A the separated 
 fragments are found to retain the charges which they appeared to carry under 
 the influence of the charge on A. But if the rod is made of insulating material 
 the separated fragments will be without charge at whatever point the rod 
 be cut. The old-fashioned explanation of this fact is that the neutral rod 
 is supposed to contain in each of its elements or particles equal quantities 
 (comparatively large) of electricity of opposite sign which normally counter- 
 balance one another. When the body is brought into the neighbourhood of 
 the charged body A the electricity on that body attracts that one of the two 
 charges in each element of the neutral body which is of the opposite sign and 
 
197, 198] The physical basis of the theory 189 
 
 repels the other, the result being a separation of the charges. In a conductor 
 the charges are quite free to move about as they like and the displacement 
 may be of finite extent. In a dielectric on the other hand the charges appear 
 to be bound to the molecules of the body by restraining forces of some kind*, 
 quasi-elastic forces we may say, and the displacement is thereby limited to 
 very small molecular dimensions, the charges each settling down into an 
 equilibrium position where the electric forces of the field balance these quasi- 
 electric fofces. On such a view all electrification is merely electric separa- 
 tion, i.e. separation of positive and negative charges; this separation may 
 be of finite amount in conductors, but in dielectrics it is confined within the 
 molecule itself. 
 
 This theory has been completely substantiated by the discovery of the 
 atomic structure of electricity and the consequent developments of experi- 
 mental science in the elucidation of the 'electron theory' to which this 
 discovery gave rise. According to this theory every atom contains as an 
 essential element of its constitution a certain number of electrons more or 
 less tightly bound in it, in addition to the necessary positive charge to make 
 it neutral. The application of an electric field will then as above displace 
 the negative electrons relative to the positive ones and thus render each 
 atom bi-polar in 1 the above sense. 
 
 198. The theory of polarised dielectric media is a molecular one; the 
 polarisation is supposed to belong to the individual molecules of the dielectric 
 substances or perhaps to molecular groups ; the essence of the affair is that 
 the physical element (the smallest thing we are concerned with) is a bi-polar 
 one, i.e. it has two equal and opposite poles of electric charge. Each molecule 
 of a substance contains one or more elements of electrical charge of each 
 sign, which are originally practically coincident ; the application of an electric 
 field would then tend to pull them about until some elastic resilience (supposed 
 for the present to be proportional to the displacement) balances the electric 
 pull. 
 
 There is another theory of polarised media which assumes that the 
 separate molecules are permanently polarised (i.e. the charges never coincide) 
 but are usually arranged in all sorts of directions. It is owing to the for- 
 tuitous distributions of the directions that the total statistical effect is null in 
 the normal condition. The application of an electric field would then turn 
 each molecule round, all towards a definite direction, against elastic resilience 
 and their separate fields would no longer cancel. 
 
 Either of these theories would do for the present purposes but we prefer 
 the first for reasons which will afterwards appear. The statistical view of 
 the two is the same, and this is, after all, all that we are concerned with at 
 
 * Mossotti assumes that the molecules are like small conductors insulated from one another. 
 
 Cf. Sur les forces qui re'yissent la constitution intime des corps (Turin, 1836). 
 
190 The theory of polarised media [CH. v 
 
 present. There are certain facts which suggest that there is, at least in some 
 cases, a certain justification for the second view but these will be dealt with 
 separately. 
 
 199. Mathematical formulation of the scheme*. Having thus formed 
 a definite idea of the polarisation in the molecule we must now examine the 
 field of a polarised medium regarded as a whole. The mathematical analysis 
 to be here presented was started by Poisson in its application to the theory 
 of magnetisation, but his notions were very elementary principles constructed 
 on the distance action theory. Poisson's theory was transformed into a more 
 physical theory by Kelvin, who added nothing to Poisson's results but 
 explained his mathematical formulae with physical ideas. When Faraday 
 discovered the corresponding phenomenon in dielectrics, Kelvin was at once 
 able to explain it as the analogue of magnetisation in iron. We shall present 
 the analysis in its application to dielectrics first and then transfer the results 
 to the treatment of magnetism later on. 
 
 'P 
 
 Fig. 46 
 
 We start by analysing the electric field of a polarised dielectric mass. 
 The element of the analysis is the simple bi-pole consisting of two point 
 charges (+ q, q) placed at a small distance apart. The law of inverse 
 squares is assumed for each constituent. If the pole + q is at A and q at 
 B, then the potential due to this element at any point P in the field is 
 
 A- J-.. JL 
 V~PA PB' 
 
 Let be the mid-point of AB and 6 be the angle AGP ; also put A = 8s, 
 OP = r, then since Ss is very small we have 
 
 PA = r -g- cos 6, PB = r + cos 
 
 *j ^j 
 
 so that (f> = q ' 
 
 qBs cos # 
 = 2 ^ practically. 
 
 * The mode of presentation here adopted is due to Larmor. Cf. Aether and Matter, A pp. A, 
 *'On the principles of the theory of magnetic and electric polarisation." 
 
198-200] Polar constitution of dielectrics 191 
 
 In dealing with elements of this nature we do not as a rule know either q or 
 8s, only the product q$s ; but this is all we are concerned with in investigation 
 of the field at distances large compared with the dimensions of the doublet; 
 we introduce a single symbol m for it and we call it the moment of the element : 
 the line AB is called the axis of the bi-pole. 
 
 If we put the bi-pole at the origin of rectangular coordinates and use 
 (A, n, v)Je for the direction cosines of its axis, then the potential of the element 
 at a point at a distance r in a direction (I, m, n) is 
 
 m /JN 
 
 = ~2 (/ A + m/z + nv) 
 
 N I m n 
 
 = mX.- 2 + mp. . -t + mv . - 2 , 
 
 but the first term is the potential of the x-component of the bi-pole. The 
 potential of the element is the sum of those of its components. Thus if in 
 the specification of a bi-pole by its moment we also imply a knowledge of the 
 direction of its axis, we see that the moment can be regarded as a vector 
 quantity, i.e. a directed quantity which is resolvable by the parallelogram 
 law. We may therefore adopt our vector notation and use e for the moment 
 of a bi-pole. 
 
 200. The molecule or element of a dielectric medium may consist of a 
 whole system of simple doublets of the kind here examined. The molecule 
 may contain a whole lot of positive and negative elements and if we group 
 them in pairs (positive and negative) we have a system of bi-polar elements. 
 We could find the resultant of the moments of all these separate elements and 
 we should then define this as the moment of the molecule. If we are con- 
 sidering their effects at ordinary distances the different positions of the 
 centres of all these bi-poles in the molecule will not concern us; for all 
 practical purposes we can regard the centres as coincident on account of the 
 extreme smallness of the molecule. It thus appears that we can treat each 
 molecule, however complicated it may be, just as if it were a simple bi-pole 
 with a definite moment, obtained perhaps by considering the positive 
 charges as practically equivalent to a positive charge at its mean centre 
 and similarly with the negative. The essential point is thus that we can 
 treat the molecule as a simple element and we need not for the present 
 trouble ourselves with details of how it is built up. 
 
 Now suppose we have a whole lot of these bi-polar molecules forming 
 a finite mass. We must then treat the thing as a whole and take as the 
 element of our analysis a volume element Sv. The aggregate of the moments 
 of all the bi-poles in this element is again obtained by combining them all 
 vectorially without regard to their different positions in the element. The 
 resulting moment must be ' proportional to Sv if that volume is small*; 
 
 * ' Physically small.' 
 
192 The theory of polarised media [CH. v 
 
 suppose it is PSt', where of course P is a vector. This quantity P expresses 
 the way in which the body is polarised ; it is the polarisation per unit volume 
 at the position, and in the language of physics is called the intensity of the 
 polarisation. This vector expresses all that we can know or recognise 
 experimentally about the polarisation of the body. 
 
 If the bi-polar elements or molecules are distributed anyhow in all different 
 directions, P = ; but if there is any degree of convergence of their axes to 
 a Definite direction, then P has a definite value different from zero. 
 
 201. Having now denned this quantity P which completely specifies the 
 electric state of the polarised body, we can by its means determine the 
 electric field in the neighbourhood of the body, without at present stopping 
 to consider the actual method by which this polarisation is produced. 
 
 Each volume element 8i\ of the body is like a little bi-pole of moment 
 PjSvj and thus its potential at the point P is evidently 
 
 Tj denoting the unit vector along the direction of the radius r 1 from the 
 position of the element Svj to P. The potential of the whole body at any 
 point P is therefore 
 
 wherein S denotes a sum taken over all the elements &v of the body. This 
 may also be written in the form of an integral 
 
 which is the same as 
 
 the vectorial operator V involving differentials with respect to the coordinates 
 of P. 
 
 The intensity of force at the point P in the field can now be written down 
 in an analogous manner. It appears as a vector the negative gradient of 
 the potential < E = - grad</>. 
 
 One proviso, and an important one, has been missed out of the above state- 
 ment : the point P must be well outside the dielectric substance. 
 
 202. The integral definitions here given necessarily involve some sort 
 of continuity in the distribution of the polarisation intensity, and rather 
 more than is actually the case in a real medium constituted of discrete 
 molecules. To give them a definite sense in a strict mathematical theory 
 
200--203] The specification of the field 193 
 
 we can however replace as in the first chapter the real medium by a per- 
 fectly continuous distribution of polarisation with the proper intensity P at 
 each point. This hypothetical distribution is effectively equivalent to the 
 real one at all points of space which are not too near it. 
 
 We may next enquire as to how close to the distribution does this repre- 
 sentation of the force and potential by integrals remain valid. In this 
 connection it must first be noticed that since we have assumed the nucleus of 
 the bi-pole which forms the basis of this theory to be entirely confined within 
 a molecule or atom it may reasonably be supposed that the law of its action 
 as defined above remains valid up to within a physically small differential 
 distance from the molecule, which is a length defined so as to include a large 
 number of molecular diameters ; in other words the effective combination 
 of positive and negative elements of charge in the molecule into doublets 
 in the manner specified above is valid up to within this small distance from 
 the molecule. But the substitution of an effectively continuous charge 
 distribution for the distribution of both positive and negative elements thus 
 combined is valid to the same extent, so that we may conclude that the 
 hypothetical distribution of polarisation effectively replaces the actual 
 discrete one as regards its field up to within a physically small differential 
 distance from the polarised body, this being the distance at which the actual 
 distribution of the charge in any small volume element ceases to be irrelevant. 
 
 203. The next process in the development of the mathematical theory is 
 to specify the electric field at points in the interior of the polarised medium. 
 Waiving for the present any other difficulties involved in the extension, let 
 us assume that the hypothetical continuous distribution of polarisation 
 effectively replaces the old one at all points of the field however near to the 
 medium it may be and let us examine the force and potential at the internal 
 point. We might jump to the conclusion that in this case the force and 
 potential are correctly represented by the integrals as given above for 
 external points; but any attempt to use these definitions for internal points 
 for which the corresponding integrands both become infinite must be preceded 
 by a justification of their convergence at such points. If the integrals 
 representing them are convergent at internal points, then the force and 
 potential so defined will have definite meanings at such points. 
 
 From a physical point of view we see that the contribution to the values 
 of these functions at any point from adjacent parts of the medium involves 
 a large factor in the integral and we^ want to know whether their aggregate 
 is comparable with that from the rest of the body. If this is the case the 
 integrals are at best semi-convergent and the definitions are almost useless, 
 because we do not know anything about the local configuration of the elements ; 
 it may be anything for all we know. 
 
 L. 13 
 
194 The theory of polarised media [CH. v 
 
 But we have already seen that the integral expressing the potential is 
 absolutely convergent, so that on the present basis, the local contribution 
 due to the continuous distribution of polarity near the point under investiga- 
 tion is negligible : strictly speaking the effects of these adjacent parts involves 
 the dimensions of their volume linearly and thus in the aggregate their 
 effect is negligible compared with that of the rest of the body. The physical 
 way of stating this is, as we had it before, that the scooping out of a 
 vanishingly small cavity round P makes no difference to the integral; the 
 shape of the cavity does not matter so long as it is indefinitely small. 
 
 It is however otherwise with the integral for the force at P. The 
 ^-component of this force is in fact represented by the integral 
 
 '1 
 
 which, if we use (x 1 , y 1} %) as the coordinates of the element di\ and (x, y, z) 
 as those of P, can be written in the form 
 
 P 
 
 II * -y <-. * *r\ l I* d/l I , _ % \ 
 
 dv\ -| + 3P,, r^ + r^ (y - yi* v + z- ZjP g ) , 
 
 />f o *'/*> /v5 ^J j L y ' i *> \ ' 
 
 which is precisely of the type which was stated in the introduction to be 
 semi-convergent. The local parts of the polarisation even in the hypothetical 
 continuous medium thus have an effective influence on the value of the force. 
 Thus our definitions of the force, at least, in the internal field by means of 
 the effectively continuous distribution of polarisation breaks down ! This 
 does not however seriously disturb our formulation of the theory because 
 we can proceed as in the first chapter to modify the definitions to make them 
 more consistent with a physical theory. 
 
 204. The necessary modification will come better if we first obtain 
 Poisson's* transformation of the potential integral given above. We shall 
 assume that the point P at which the field is investigated is well outside 
 the dielectric substance, so that there is no doubt about an application 
 of the above definitions and the consequent effectiveness of the continuous 
 distribution of polarisation. If P is actually inside the medium we just 
 remove a part of the medium inside a physically small cavity round P, 
 so that it is still in free space. We can then adopt without any further 
 hesitation the above definitions for the force and potential at P. The 
 potential is in fact 
 
 but since V - - V x , 
 
 * L.c. p. 164. 
 
203-205] Poissoris transformation of the potential 195 
 
 where V 1 denotes the same vector differential operator as V but taken with 
 respect to the coordinates (x^ , y-^ , zj of dv instead of (x, y, z) as above, this 
 is also 
 
 the integrals in each case being taken throughout the volume of the polarised 
 medium, excluding the part removed from the small cavity about P if it is 
 made, a simple transformation by Green's theorem shows that 
 
 ^-^ d 
 1 
 
 where the surface integral is extended over the surface of the body (including 
 the walls of the cavity if P is inside) and the volume integral over the volume 
 of the body. 
 
 This means that the potential of this polarised body is the same as the 
 gravitational potential of the mass distribution specified as : 
 
 (i) a volume density p = div P 
 
 throughout the body excluding the cavity if made ; although, as a matter of 
 fact, the cavity may be filled in with the continuation of this distribution 
 throughout its volume without making any appreciable difference to the 
 integral for cf> which is convergent at the point. 
 (ii) a surface density CT _ p * 
 
 over the surface of the body and cavity. 
 
 All this applies only to a point outside the substance of the polarised 
 medium. If the point is right outside the medium the values of the force 
 and potential due to such a mass distribution are quite definite and are in 
 fact identical with those already given on the more direct definition; this 
 distribution of attracting charges or masses effectively replaces the distribution 
 of polarisation as regards its action at all external points. It is however in 
 the analysis of the field at internal points that this mode of treatment helps us. 
 
 205. If the point P is inside the polarised medium we can draw round 
 it a small surface whose linear dimensions are physically small. The dis- 
 tribution of polarisation in the medium outside this surface can then as 
 regards its action at P be effectively replaced by tHe continuous distribution 
 of attracting masses just described. We thus see that the total field at P can 
 be separated into distinct components. Firstly the volume distribution p 
 
 * At an interface between two different dielectric media there is a surface distribution of 
 density 
 
 132 
 
196 The theory of polarised media [CH. v 
 
 outside the surface gives a definite force and potential at P, no matter what 
 size or shape the cavity may be, provided only that it is physically very 
 small. Similar remarks apply to the surface distribution a on the outer 
 surface of the body. But the distribution a on the walls of and the dis- 
 tribution of molecules inside this cavity give a potential and force at P which 
 depend entirely on the shape and size of the cavity even if it is very small 
 and will in general be comparable with the other parts. This latter com- 
 ponent of the field is however a purely local part depending entirely on the 
 molecular configuration round the point and the conditions of polarisation 
 existing in them : as we do not know the local molecular configuration, 
 which may be changing rapidly, we cannot know what this local part 
 really amounts to; but we have succeeded in separating it from the main 
 part of the action due to the rest of the body. 
 
 We now adopt the arbitrary course of simply neglecting this local 
 molecular part of the field, so that we can confine our discussions entirely 
 to that definite part of the force which is due to the medium as a \vhole, 
 i.e. the molar part. This is merely following a usual method in physics and 
 involves but a simple extension of the ideas underlying the Young-Poisson 
 principle of the mutual compensation of molecular forcives employed in their 
 theory of capillary actions* . It requires that such local forcives shall set up 
 a purely local physical disturbance of the molecular configuration in the 
 material, until it is thereby balanced. Another example of this principle is 
 provided in the ordinary theory of elasticity where in addition to the local 
 strain forces in an elastic medium there are the comparatively very powerful 
 cohesive forces, which are however presumed to form an equilibrating system 
 and not to affect the phenomena as a whole. It is fortunate that we can in 
 this way eliminate the influence of the neighbouring elements. 
 
 We can therefore define the electric field inside the body as that field 
 when the effect of the local part is omitted and this definition will apply quite 
 consistently to outside points as well. We have been able to separate the 
 local part from the total, and the field of force of our subsequent discussion 
 is that due to the rest. On this definition the integrals expressing the force 
 and potential are always convergent and apply to inside as well as outside 
 points as they are then just like the corresponding functions of our former 
 theory involving only ordinary volume and surface distributions of charge. 
 The electric force is then always the gradient of the potential. 
 
 206. Having thus fitted up these consistent definitions of the functions 
 involved let us see how they work in the theory. The electric force vector E 
 
 * Cf. Larmor, Aether and Matter, App. A; Young, "On the Cohesion of Fluid?," Phil. Trans. 
 (1805); Poisson, Nouvelle Theorie de V Action Capillaire (Paris, 1831); Rayleigh, "On the theory 
 of surface forces," Phil. Mag. 1883, 1890, 1892; especially 1892 (1), pp. 209-220; Van der Waals, 
 Essay on the continuity of the liquid and gaseous states. 
 
205, 206] General relations of the field 197 
 
 is still the gradient of the potential (/>. Also since </> is due to the specified 
 volume and surface distribution 
 
 -- + 4-n- divP ................................. (i), 
 
 at each point of the field and at the boundary of the dielectric in air 
 
 &_ i = _ 4 = - 
 
 but E = grad (f> = V(f>, 
 
 so that 47T div P = V 2 < = - div V< = - div E, 
 
 or div (E + 4-nP) = 0. 
 
 The vector (E + 47rP) is therefore a streaming vector, it satisfies the usual 
 equation of continuity of incompressible fluid flow. We call it the vector of 
 electric displacement * and denote it by 4irD ; the factor 4rr is introduced for 
 a reason which will subsequently appear. This electric displacement is the 
 important vector of the theory : its importance lies in the fact that the flux 
 or displacement through any surface only depends on its boundary so that 
 we can take the flux as estimated as so much through a circuit. For we 
 
 have 
 
 div D = 0, 
 
 so that if we take any closed surface / in the field we get 
 
 f(div D) dv = 
 
 taken throughout the region bounded by /. But by Green's theorem this 
 consists of 
 
 together with the surface integrals arising from discontinuities when we pass 
 into the polarised medium ; these are the integrals of 
 
 - (D ni - DJ 
 over the parts of the surfaces concerned which are inside /, or 
 
 which is zero : we thus have 
 
 (D n df=0. 
 
 Now suppose we have an unclosed surface / abutting on the closed curve 
 s, then we can take another surface /' with the same curve s as boundary 
 and the two surfaces / and /' together form a closed surface. Thus if n 
 
 * Maxwell, Treatise, I. p. 64. The vector D is the displacement proper. 
 
198 The theory of polarised media [OH. v 
 
 denote the normal to the element of either surface in a, definite sense through 
 the circuit, the above equation gives 
 
 [ I at 1 
 
 f f 
 
 and the result is as stated. 
 
 The real significance of these results does not however appear until we 
 discuss the subject of electromagnetism, it is hidden by the more general 
 circumstances under which the present theory has to be developed. 
 
 207. The general problem. We have so far merely discussed the fields of 
 polarised media without any reference to the way the polarisation is created. 
 As a rule however we cannot have dielectrics polarised .unless they exist in 
 an external electric field, i.e. in an imposed field due to an extraneous 
 electrical system. The introduction of the dielectric into the field results in 
 each element of it being turned into a little bi-pole in the manner above 
 indicated and the total field of all these bi-poles has alone been under investi- 
 gation, although as a matter of fact it merely represents the addition to the 
 original field brought about by the introduction of the dielectric substance 
 into it. For the general case therefore we must superpose on the field above 
 investigated that original field which existed before the dielectrics were 
 introduced and which we shall for the present suppose to be due to certain 
 volume and surface densities p and o- . The above discussion of the conver- 
 gence of the force and potential integrals is not hereby affected, since the 
 additional parts of these functions due to such a distribution are already 
 known to have definite values at all points of space. The electric force is 
 therefore still the gradient of a potential function. We now use </>! for the 
 potential of the field above investigated, </> for the potential of the original 
 field, and </> for that of the total field ; a similar suffix-notation is also adopted 
 for the other quantities involved. We have now 
 
 and E = - grad^> = V<f>. 
 
 Thus V 2 </> = V 2 </> + V 2 </. 15 
 
 and V 2 </> = 47rp , V 2 ^ = 4:7rp l = 4?r div P, 
 
 so that we now have ^iv (E + 47rP) = 47r/> . 
 
 The induction or displacement vector is no longer a stream or solenoidal as 
 we say ; if we again use 4?rD to represent it we have 
 
 208. At a boundary surface of the dielectric medium which also 
 carries a surface charge of density <T O we have 
 
 nj dn 2 
 
 4:7Ta 1 47TCT0, 
 
206-209] The law of induction 199 
 
 or - E Bl + E w , 
 
 or - B ni + (E W2 
 
 In free space, i.e. on side 1 of general surface of discontinuity, 
 
 47rD = E ; 
 in the dielectric medium however 
 
 47rD = E + 47rP, 
 so that the above surface condition can be written as 
 
 Dm-D na = <7 - 
 
 The normal induction is discontinuous across the surface charge by an amount 
 cr ; if there is no surface charge 
 
 D ni = D ws . 
 
 Similar conditions are also found to hold at charged or uncharged surfaces 
 of discontinuity in the dielectric medium itself, i.e. surfaces separating not 
 the dielectric medium from a vacuum, but one medium from a second different 
 one. 
 
 We have thus a complete specification of the field in terms of the electric 
 force E, the electric induction or displacement D and the intensity of polarisa- 
 tion P where we know that at each point of the field 
 
 4-n-D = E + 4?rP 
 
 in the vector sense. These three vectors give us the distribution of force 
 induction and polarisation ; but any two of them are sufficient as the third 
 is determined when the other two are known. Although the vector P is 
 perhaps the more fundamental physical one we shall regard the first two 
 vectors as the independent variables ; they turn out to be the more significant 
 ones of the theory. 
 
 209. The law of induction. We have now examined the electrical field 
 under the conditions of the dielectric being present and having induced in it 
 a polarisation of intensity P at each place. But how do we know what 
 polarisation will be induced in a given dielectric substance and of what use 
 is the above analysis ? We evidently want an additional physical principle 
 to complete the scheme. 
 
 The electric force at any point in the dielectric medium is E (neglecting 
 the local part) and the electric displacement induced is D, and as we have 
 pictured the affair to ourselves the polarisation and therefore the displacement 
 is conditioned by the electric field. Thus if there is to be any law about the 
 matter at all one of these quantities is a function of the other. The simplest 
 possible relation we could have is a simple proportionality so that if we double 
 the cause we double the effect : this is moreover the only workable relation 
 mathematically and so our procedure is to work out the results on it and see 
 
200 The theory of polarised media [CH. v 
 
 how they are experimentally justified. Expressed mathematically a relation 
 of this kind means that the components of the displacement are linear functions 
 of the components of the electric force 
 
 e 12 E y + e 13 E z , 
 
 31X 82 , 33 z , 
 or expressed shortly by a vector equation 
 
 47rD = (e) E. 
 For homogeneous media this relation would assume the simpler form 
 
 4?rD = eE. 
 We might of course assume more generally that 
 
 4-rrD - eE + e^ 2 + ..., 
 
 but we presume that if e is small the other terms beyond the first are negligible 
 and we find that it fits the facts. In any case the simpler form is right for 
 very small fields and anything more complicated is mathematically unwork- 
 able. 
 
 It might be thought that it would be better to take the polarisation as 
 proportional to the total electric force including the local part. The local 
 influences have however been regarded as equal and opposite actions and 
 reactions occurring in and between the molecules concerned and cannot add 
 anything to the total result in any definite direction. The presumption is 
 that these local effects are erratic and cannot influence a vector effect at the 
 place. 
 
 210. We can now complete our formulation of the scheme : we have 
 in the vector sense 
 as our physical relation ; but 
 
 -E 
 
 // \ 
 
 and thus 
 
 and P the electric polarisation intensity is thus also proportional to E. 
 
 Also on this theory 
 
 47rD = ( e ) E = - (e) grad <f>, 
 
 and since div D = p, 
 
 we nave div {(e) grad <f>} = 47r/3 , 
 
 which is the characteristic equation of the theory; the potential function 
 (f> must always satisfy this equation when there are dielectrics about. 
 Notice that for isotropic media this becomes 
 
 ^ 
 
 ^ I C 7^ 
 
 CX 
 
209-212] The energy relations of dielectrics 201 
 
 The boundary conditions to apply at a surface of discontinuity in the 
 medium on which there is also a surface charge of density CT O are obtained 
 in the usual way ; they are A = A ^ 
 
 , . e, , , <> 2 
 <"> 8? - ^ & 
 or if CT O = simply 
 
 (f \ 8< ^i _ (e \ ?^?* 
 (6l) dn (2 > dn 
 
 211. We are thus enabled to express everything in terms of the poten- 
 tial function of the field which must satisfy the above characteristic equations 
 and relations at each point of the field. We notice that these relations are 
 identical in form with those obtained from the simple Faraday-Maxwell 
 theory in the previous chapter, the constant e being the same as that there 
 adopted : the electrical* conditions in the field are therefore the same on 
 either theory, as of course they must be ; the only difference lies in the 
 physical basis for the mode of action of the dielectric medium in modifying 
 the electric field by which it is surrounded. The advantage of the present 
 theory lies however in another direction. When examining the energy and 
 ponderomotive forces in the electric field containing dielectrics on the 
 Maxwellian theory we got into difficulties. We found that an ordinary 
 medium with the characteristic elasticity of solid bodies could not support the 
 supposed state of strain between electrified bodies and be in equilibrium in 
 its elements, the application of an extraneous couple of amount proportional 
 to its surface being in general necessary to keep it balanced. If however we 
 imagine an ordinary elastic medium full of elementary bi-poles of the type 
 now under investigation with orientations distributed according to some law 
 or even at random, and in internal equilibrium either in its own or an external 
 field, then on rotational distortion a couple will be required to hold each 
 element in equilibrium : the present mode of formulation of the theory will 
 thus in all probability help us out of our former difficulties in this connection. 
 
 212. The energy relations of dielectric media. We can now analyse 
 on the bases of the foregoing principles the distribution of energy and 
 transmission of force in an electrical field throughout which dielectric media 
 of the kind specified are distributed. The problem is however much more 
 complex now, involving as it does all the difficulties and refinements of 
 the statistical method in the mechanics of molecularly constituted media. 
 
 As an introduction it will be perhaps best to give a short sketch of the 
 .general principles underlying the simple dynamical problem of the motion of a 
 body or system of bodies composed of an enumerable aggregate of molecules, 
 
 * These conditions are perhaps not so happily expressed as they might have been. In the 
 general case of anistropic media the expression of the normal component of the induction vector 
 in terms of the potential gradients requires great care. 
 
202 The theory of polarised media [OH. v 
 
 each of which may be treated, for the purposes of theoretical dynamics, as 
 a single mass particle. In such a problem we have to deal with the large 
 number of mass-particles which we may typify by the mass m r at the point 
 whose coordinates at time t referred to any convenient rectangular axes are 
 (x r , y r , z r ). Each of these masses will in the general case be subject to the 
 action of forces which may be of different types ; firstly there is the definitely 
 controllable force exerted directly by external systems, which we may take 
 to be equivalent to a resultant force r on the mass m r ; these are the so-called 
 external forces : there is next the direct resultant forces of type F rr ' exerted 
 on this same mass m r due solely to and conditioned by the presence of the 
 other mass particle m r > in the system; these are called the internal forces. 
 Lastly there are the forces of constraint exerted indirectly as a consequence 
 of some condition restricting the motion of the system, such as for example 
 that certain points of it are forced to move along prescribed paths : these 
 forces are typified by their resultant F/ on the mass m r and are called the 
 reaction forces on the system. The equations of motion of the mass m r are 
 then of type 
 
 2 denoting a sum taken over all the mass points excluding the one m r whose 
 
 r' 
 
 ds 
 velocity is represented by the vector -p. There is one vector equation of 
 
 this type for each particle in the system ; they can however all be summed 
 up into one equation, viz. the general differential equation of virtual work 
 in the system : if 8s r denotes vectorially an arbitrary virtual displacement of 
 the mass m r , this equation is 
 
 Z,m r ~* 3s r = S (F r Ss r ) + 22 (F r/ Ss r ) + S (F/3s r ). 
 
 The terms on the right represent respectively the total amounts of the work 
 done during the displacement by the external forces, the internal forces and 
 the reactions from the constraints'. If the displacement system typified by 
 8s r is consistent with the implied geometrical restraints in the system, the 
 total work of the reaction forces vanishes and the equation simplifies to 
 
 ^ Ss> r ) - S (F r Ss r ) + SS (F rr ^s r ). 
 
 (11 J r r r' 
 
 If the geometrical restrictions on the motion of the system are such that in 
 the actual motion the reaction forces do no work, a still simpler form can be 
 deduced from this equation ; for a possible displacement system in the last 
 equation is that actually taken by the system during the next succeeding 
 small interval of time so that 
 
 ds 
 
212, 213] The energy relations of dielectrics 203 
 
 and the equation becomes with these values of Ss r 
 
 //72e .] \ 
 
 We now use 8E {} = HF r Ss r 
 
 r 
 
 for the work done on the system by the external forces ; 
 
 m t = 22 (F r/ Ss r ) 
 
 r r' 
 
 for the work done by the internal forces which might also with the reverse 
 sign be counted as the increase of the mutual potential energy of the internal 
 configuration of the system : and finally 
 
 /Y/Q \ 2 
 71 IV u *r 
 
 1 - m - 
 
 for the total kinetic energy of the system. The equation of virtual work in 
 the special form to be adopted is thus 
 
 ST = 8E + Ei 
 
 which merely expresses that the increase in the kinetic energy of the system 
 is effected partly at the expense of the external systems and partly at the 
 expense of the store of potential energy in the medium. 
 
 213. But for purposes of the dynamical phenomena of material bodies, 
 which we can only test by observation and experiment on matter in bulk, a 
 complete atomic analysis of the kind here involved would (even if possible) 
 be useless* ; for we are unable to take direct cognisance of a single molecule 
 of matter. The development of the theory which is to be in line with 
 experience must instead concern itself with an effective differential element 
 of volume, containing a crowd of molecules numerous enough to be ex- 
 pressible continuously as regards their average relations, as a volume density 
 of matter. 
 
 It will therefore be necessary for us to interpret the above equation in 
 terms of average quantities for the medium as a whole. As regards the 
 kinetic energy of the motion we can interpret it as the kinetic energy of the 
 average drifts or orientation of the molecules ; but this always leaves out of 
 account an average residue of energy concerned with the motion of the 
 
 o o/ 
 
 molecules devoid of any recognised regularity (which is superimposed on the 
 regular motion sorted out) and of which we know nothing except its quantity: 
 this part of the kinetic energy is the thermal part and is a function only of 
 the temperature of the medium. It is the only part that exists if there is 
 no visible motion of the medium as a whole. 
 
 Again the actual interactions between the molecules represented by forces 
 of the type F rr - are also necessarily presented to us divided into various 
 
 * Cf. Larmor, Aether and Matter, p. 87. 
 
204 The theory of polarised media [on. v 
 
 groups, which would be the subject of perception by different senses, so that 
 their -virtual work consists of several distinct parts. There is a part involving 
 the interaction, with any molecule under consideration, of other molecules 
 of the system at finite distances, which integrates into an energy function 
 of the applied mechanical forces exerted between the various elements of 
 the system. Of the remainder of this work, which arises from the mutual 
 actions of neighbouring molecules, a regular or organised part can be 
 separated out which represents the energy of elastic stress and is a function 
 of the deformation of the volume treated as a whole : this stress arising from 
 the immediate surroundings in part compensates for the element of mass 
 under consideration, the applied mechanical forces aforesaid. The remaining 
 usually wholly irregular part may be considered as compensated on the spot 
 by other such forces of different origin that are not at present under review*. 
 If therefore we use T for the kinetic energy of the organised motion of 
 the medium, T f for that of the irregular heat motion, W t the potential 
 function of the mechanical forces between the different elements of the 
 system and W s the energy of elastic stress, the virtual work equation can be 
 written in the form 
 
 if we limit ourselves to the statical case with the dielectric media as a whole 
 at rest, then T = and the equation is simply 
 
 8r = 8# - 8IF, - SJF,, 
 
 and 8E Q 8W if which represents the work done by all the directly observable 
 mechanical forces on the elements of the media, may be treated as a single 
 quantity 8W , which is the potential function of these forces 
 
 82\ - STF - 8W t , 
 an equation of fundamental importance in thermodynamics. 
 
 214. In the present theory of polarised media the problem is however 
 rather more complex than that just discussed. The molecules of a dielectric 
 medium are not simple mass particles, but each involves in its essential 
 constitution a more minute system of electrically charged particles (the 
 electrons), all more or less lightly bound to it by some quasi-elastic resilience. 
 The application of an electric field would thus be effective in producing 
 a strained condition within the molecules themselves, of a more fundamental 
 type than that already discussed. The average stress in this intra-molecular 
 deformation would be superimposed on that discussed above, so that 8W S 
 now consists of two parts the first corresponding to the ordinary elastic stress 
 which we shall still denote by &W S and the second to the internal quasi- 
 elastic stress in the molecules, which we can denote by STF/f. 
 
 * Cf. Larmor, Phil. Tram. A, 190 (1897), 48, 49. 
 
 f It is fairly obvious that since the two strains here involved are absolutely independent 
 their energies would be additive. 
 
213-216] The energy relations of dielectrics 205 
 
 The kinetic energy T t would also now contain a part due to the relative 
 motions of the electrons in the molecules, we may call this part T/ using 
 T { still for the thermal energy ordinarily associated with the irregular heat 
 motion of the molecules*. 
 
 The general equation of virtual work is thus 
 
 8T< + 8T/ r 8 ^o - %W S - 8W,'. 
 
 A simplification is now introduced into the theory by the fact that observation 
 tends to confirm the view that processes directly concerned with the internal 
 constitution of a molecule are, with few exceptions, almost entirely inde- 
 pendent of the temperature of the substance in which the molecule exists: 
 in other words the terms 8T; and BW S and a part perhaps of STF of the 
 general equation above depending on and affecting the temperature balance 
 naturally among themselves leaving the general equation of virtual work of 
 the electrical system and its mechanical reactions in the form 
 
 8W ' being the part of the work of the mechanical forces directly concerned 
 with the electrical attractions, it would not for instance include the part of 
 SJF , if such existed, corresponding to the external pressure balancing the 
 fluid pressure at the boundary of a gaseous dielectric which depends 
 essentially on the temperature, although of course it would in the main be 
 represented by the work of applied boundary pressures superimposed on these 
 thermal ones if they exist. This simplification is however not always possible 
 and then the more general equation with the undashed letters must be 
 retained. In any case the argument is the same. 
 
 215. Now let us apply these considerations to the case of a mass of some 
 dielectric substance in an electric field. The applied forces on the mass will 
 then be mainly of two kinds; the mechanical forces applied generally as 
 pressures on the outer boundaries of the media and the electrical forces 
 exerted on the charges rigidly connected therewith. We shall denote the 
 work in these two parts by $W and 8W e respectively and then in the most 
 general case our equation of virtual work assumes the form 
 
 where we use 8E { generally to denote the change of the total internal energy 
 in the medium of both elastic and motional types 
 
 8ff*--8r, + 3fP,. 
 
 216. But 8W e is easily calculated : in fact the energy required to 
 establish any constituent bi-polar element at any point in the medium can 
 be regarded as mathematically equivalent to the work required to separate 
 
 * The part T^ would be the kinetic energy of motion of each molecule with its mass 
 collected at its centre of gravity : T/ would then be that of the motion relative to the centre 
 of gravity: these two parts are additive in the ordinary way. 
 
206 The theory of polarised media [OH. v 
 
 the positive pole + q from coincidence with the negative pole q. If the 
 intensity of force at the point is E, supposed uniform in the neighbourhood 
 of the element, and e represents the vector moment of the doublet finally 
 established, this work is easily seen to be equal to 
 
 (Ee). 
 
 Thus summing for all the doublets in the element of volume 8v we have the 
 total work done in polarising the element equal to 
 
 ZW e = 2 (Ee). 
 
 In the present theory of polarised media we saw that the force E at any 
 point internal to the medium consists of a definite molar and an irregular 
 molecular part which we succeeded in separating by means of the ideal 
 volume and surface densities of Poisson; the method consisting essentially 
 in computing the force by combining opposed poles of neighbouring elements, 
 instead of taking the single polarised element as the unit. It appears that 
 the adjacent poles nearly compensate each other except as regards a simple 
 volume density whose attraction has no local or molecular part and a surface 
 density partly at the outer surface and partly at the surface of the cavity 
 which contains the point under consideration. The effect of the latter 
 surface density depending as it does wholly on the immediate surroundings 
 is the molecular or cohesive part of the average forcive. It is the irregular 
 part of the forcive on the contained element of the dielectric which arises 
 from the excitation of neighbouring molecules and is expressed in terms of 
 them alone. It is not transmitted by a material stress ; but forms a balance 
 on the spot with cognate internal molecular forcives of other types. Thus 
 in seeking for the mechanical relations for the dielectric as a whole we shall 
 be justified in neglecting this local part of the total force and its associated 
 energy. We can thus use E as the electric force as defined above, omitting 
 the local part ; and it is then clear that E will be practically constant through- 
 out the small volume element Sv and thus 
 
 8W C = (E, Se), 
 
 but Se = P8v, 
 
 and thus the work done in polarising the element 8v to intensity P is 
 
 BW e = (EP) 8v, 
 and for the whole medium the work done is 
 
 This is the energy required by the system as a whole on account of the 
 polarisation induced in it. 
 
 217. As explained above this energy is to be regarded as consisting 
 partly in the mechanical potential energy of the polarisations of the elements 
 
216-218] The energy relations of dielectrics 207 
 
 of volume and partly in mechanical work done against internal quasi-elastic 
 forces preventing displacement of the elementary charges ultimately con- 
 stituting its polarisation. To effect a separation of the two parts thus involved 
 we proceed by the method, usual in such problems, of varying the configuration 
 of the system generally and calculating the coefficients of each part of the 
 variation in the general expression for the virtual work thus obtained. 
 
 An infinitesimal displacement of the volume 8v from a place where the 
 field is E to where it is E + 8E involves a total change in W e equal to 
 
 SW e - [(ESP) + (PSE)] Sv 
 
 and then it is at once obvious that the second part with its sign reversed, 
 viz - - (PSE) 8v, 
 
 is the virtual work 8W of the mechanical forces performed during the 
 shifting of the element, for it is the part of $W e which remains when the 
 polarisation of the element is held rigid during the displacement so that no 
 work is done in the internal degrees of freedom corresponding to the displace- 
 ments involved in it. 
 
 The other part of the total energy &W e expended during the displacement 
 of the volume element Sv corresponds to 8E t and thus 
 
 BE, = (ESP) Sv, 
 
 or integrated throughout the system for the total. This represents work 
 done against the quasi-reactions to the setting up of the polarisation. It 
 has nothing whatever to do with the mechanical forces on the dielectric as 
 a whole, but is stored up in the polarisation of the medium as internal energy 
 of intra-molecular strain*. 
 
 218. We may now draw some important conclusions respecting the 
 relations between P and E, which follow directly from a simple application 
 of the energy principle with the forms for the energy above determined. 
 Confining ourselves to the element Sv we see that the work supplied by it 
 to outside systems, which it drives, in traversing any path is 
 
 - IBW = dv [(PSE), 
 
 the integrals being taken along the path. If P is a function of E so that the 
 operation is reversible this work must vanish for any closed cycle, otherwise 
 energy would inevitably be created, either in the direct path or else in the 
 reversed one, for the complete system of w T hich dv is an element. The 
 negation of perpetual motion therefore requires in this case that 
 
 (PSE) = <ty 
 is a complete differential of some function of E. Moreover this function </> 
 
 * The argument here employed is given implicitly by Larmor, Phil. Trans. A, 190 (1897); 
 and in more detail for the cognate magnetic problem in Proc. R. S. 71 (1903), pp. 236-239. 
 
208 The theory of polarised media [OH. v 
 
 can only involve even powers of (E x , E^, E 2 ) and for weak fields is practically 
 quadratic. Its coefficients are then the six electric constants for general 
 anisotropic media, no rotational quality in the polarisation being thus 
 allowable by the doctrine of energy; 
 
 ^ = i [^u'E* 2 + 6 22 'E, 2 + e 33 'E 2 s + 2 ei2 'EJE, + 2e 23 'E,E 2 + 2e 31 'E,E,], 
 and then P x = ej/E^ + e ia 'E v + e 13 'E 2 , 
 
 PI/ = 2i'E x + eaa'E^ + e 23 'E 2 , 
 
 PS = si'E x + e 32 'E 2/ + e 33 'E 2 . 
 
 This is the most general linear relation between the polarisation and polarising 
 force which is true for crystalline bodies. We can simplify by taking the 
 proper coordinates for cf> so that it becomes 
 
 </> = i (e/E, 2 + 62 ' E / + e 3 'E 2 2 ), 
 
 and this determines the principal electro-crystalline axis of the substance : 
 the law of polarisation is now given by 
 
 (P* , P v , P z ) = (fi'E^,, 2 'E y , e 3 'E 2 ), 
 which for isotropic media becomes simply 
 
 P = e'E, 
 results agreeing with those deduced from our previous considerations. 
 
 219. If we assume this simple linear law of polarisation, viz. 
 
 P = e'E*, 
 
 then the part of the energy expended by the medium, as it is displaced, in 
 mechanical work against the electrical attractions is 
 
 (dv 
 
 which is \dve'W - $ f(e')E 2 ^, 
 
 whilst the internal molecular stored energy is increased by 
 
 fife 
 
 o 
 Thus the two parts are now equal and distributed throughout the medium 
 
 with a density 
 
 e'E 2 
 
 2 ' 
 
 or at least may be considered as so distributed. 
 
 The whole of this analysis of course depends on the reversibility of the 
 operations. As long as the polarisation is slowly effected against the re- 
 silience of reversible internal elastic forces the whole of the second part of the 
 
 * Notice that e' * ^- when e is the constant of our former theories. 
 
218-221] The energy in the field 209 
 
 work done is stored up in the medium as potential energy, but any want of 
 reversibility involves degradation of some of it into internal molecular energy 
 of another type, while if the field were instantaneously annihilated the particles 
 would swing back and vibrate so that ultimately all of it would be lost. This 
 is the principle underlying the hysteretic loss of energy which is of such great 
 importance in the correlative subject of magnetism; we need not however 
 dwell on it here as it is of but little importance in the present connection. 
 When viscous and other hysteretic effects are practically absent the changes 
 of polarisation keep step exactly with those of the polarising field and P is 
 a function of E, so that the whole of the average organised energy involved 
 in the polarisation is mechanically available. 
 
 220. The complete expression for the energy in any electrostatic field. 
 
 So far we have tacitly assumed the existence of a rigid inducing electric 
 field, in the discussion of its effects on polarising a body moving about in 
 it. But in a complete analysis of any problem it is necessary for us to take 
 all coexistent systems into account, since none of them are in reality rigid 
 and the motion of one usually affects the distribution in the others. Thus 
 in order to have a full account of the field it is necessary to specify it 
 more closely. The simplest method would be to specify the distribution of 
 free charge throughout the field ; the other quantities being then defined by 
 the relations established in the previous paragraphs. In this way results can 
 be obtained equivalent, in the simpler cases, to those deduced in the 
 previous chapter, but a more general and more fundamental interpretation 
 can be attributed to them than was there possible. 
 
 We shall therefore suppose the electric field to arise from a continuous 
 volume distribution of charge throughout the field of density p at any point : 
 surface charges, if they exist, may be regarded as limiting cases of such a 
 distribution and do not therefore need special treatment. 
 
 221. The total mechanical work done in establishing the field may 
 now be calculated by building up the charge distribution p gradually in the 
 presence of the dielectric media, the induced polarity simultaneously taking 
 the appropriate value at each stage of the process. By the general definition 
 of potential, the work done in bringing up small increments of charge 8p at 
 each point of the field in which the potential is (j> is 
 
 but generally p = div D, 
 
 so that the small variation of D at any point of the field induced by the 
 above change in p is defined by 
 
 8/j = div SD, 
 r, 14 
 
210 The theory of polarised media [CH. v 
 
 whence SJF = ( div SDdv, 
 
 and by Green's lemma this is 
 
 - (SD. V)<f>dv 
 
 the latter integral being taken over an indefinitely extended surface bounding 
 the field. For a finite charge system this integral vanishes and thus 
 
 = - (8D. 
 but E = - V</>, 
 
 so that 8W = f(E . SD) dv, 
 
 ' i 
 
 as in the previous theory of Maxwell-Faraday. But now D is a composite 
 function 
 
 E 
 
 =l +f > 
 
 so that SD = ~ SE + SP, 
 
 *7T 
 
 and thus 8W = ^ f (ESE) dv + f(ESP) dv, 
 
 and therefore the total work done in establishing the field can be calculated 
 in the form 
 
 W = I STF = -i- Idv ( E (ESE) + dv P (E3P) 
 
 .'0 4-7T .! J o . . 
 
 = j- [wdv + !dv (ESP). 
 wr .' .' . o 
 
 The first term in this expression represents the electrical potential energy 
 stored up in the electrical field on account of the distribution of electricity 
 involved in the charges and polarisations. It may be regarded, as in the 
 ordinary theory, as the potential function of the mechanical forces equivalent 
 to the electrical attractions between the various charged and polarised 
 elements of matter in the field. The second part represents the energy 
 stored up in the dielectric media as a consequence of the strained condition 
 involved in its polarisation. 
 
 222. For a linear isotropic law of polarisation we have 
 
 P - e'E, 
 and thus the total energy is 
 
 W = J- [E 2 <fo + i [e'Wdv 
 
 07T J J 
 
 = J- E 2 (1 + 47re') dv 
 
 O7T 
 
 = ~ 
 
 OTT 
 
221-223] The constituent parts of the energy 211 
 
 and as before we may regard it as distributed continuously throughout the 
 field with a density at any point equal to 
 
 -E 
 
 877 
 
 the part - E 2 
 
 O7T 
 
 belonging to the aether and the rest to the matter. This is the usual result 
 with which our present theory therefore agrees. It is however shown to be 
 restricted to the case when the polarisation is induced without hysteresis and 
 follows a linear law of induction. 
 
 223. Of the total energy associated with the electrical conditions in the 
 aether, viz. 
 
 ."'*' 
 
 the part - tdv t (PSE) 
 
 J J o l \ 
 
 corresponds to the part of the mechanical energy stored as electrical potential 
 energy of the polarisations : it is concerned mainly with the electrical 
 attractions, or their equivalent mechanical forces, exerted by the field on 
 the polarised media as a whole, of which it may be considered as the potential 
 
 function. 
 
 j 
 The remainder 
 
 i r r r E r r /E \ r C E 
 
 - -\Wdv + \dv (P8E) = \dv\ L + P, 8E)=U; (D8E) 
 
 07T ' J J J J V*T / J Jo 
 
 is concerned with the attractions exerted by the field on the elements of the 
 free charge distribution, being balanced by the material elastic or other 
 purely mechanical reactions necessary to maintain them in their specified 
 distribution. This part may be transformed to 
 
 dv I (DV) <, 
 .' o 
 
 and by Green's lemma, with the assumption of finite fields, this is easily 
 
 shown to be 
 
 r r* r r* f" /* 
 
 \dv divDSc/. = \dv pS<f> = dv P E s 8s, 
 
 J J o . .' o J . o 
 
 in which form it is at once recognised as the electrical potential energy stored 
 up in the system of the free electric charges, or as the potential function of 
 the reacting mechanical forces on the material masses in the field arising on 
 account of their charges. 
 
 It is of course assumed in this argument that the free electric charges are 
 in reality absolutely free to move about in the matter without restraint 
 
 142 
 
212 The theory of polarised media [CH. v 
 
 except as far as they may be prevented by the direct action of mechanical 
 forces of rigid restraint at the boundaries of conductors or in the interior of 
 the dielectrics ; so that they are completely effective in converting electrical 
 attractions into mechanical bodily forcives by reaction. 
 
 224. It is important to notice that although the total work done on the 
 system in building it up as aforesaid is equal to the sum of the electrical 
 potential energy of the charges and of the polarisations together with the 
 internal stored energy in the medium or in symbols 
 
 {dv [ P (ESD) = dv | E (D8E) + (- dv f .(PSE)! + [dv f P (ESP), 
 
 J J . .' \ . .' / J J 
 
 yet it is only in the case when the law of induction is linear when 
 
 (P3E)= f P (ESP), 
 J o 
 
 that the total work done on the system is equal to the electrical potential 
 energy of the charges in their specified distribution. 
 
 The fact that any want of reversibility in the elastic forces resisting the 
 establishment of the polarisation is completely accounted for by a corre- 
 sponding irreversibility in the purely electrical energy of the polarisation 
 induced so that (ESP) + (P8E) 
 
 is a perfect differential under all circumstances verifies that the forces 
 resisting the displacement of the free electric charges are reversible so that 
 
 (BSD) + (D3E) 
 is a complete differential. 
 
 225. The mechanical relations of the polarised medium*. Having 
 obtained in the previous paragraphs a definite value for the potential 
 energy of the mechanical forces acting on the dielectric medium as a whole, 
 we may at once proceed to investigate the nature of the forces of which it is 
 the energy function. 
 
 To deduce the forces from their potential energy function we vary this 
 function with regard to the geometrically possible displacements of the 
 medium as a whole. In any such displacement however the polarisation 
 must be kept constant, for it is determined wholly by the internal degrees 
 of freedom of the medium. We see at once by giving the medium a small 
 linear displacement that the force acting on it is the vector 
 
 f 
 
 - grad W = grad (P . E) dv, 
 
 the differentials however not operating on P. 
 
 * Cf. Larmor, Phil. Trans. A, 190 (1897), p. 248. 
 
223-226] The mechanical relations of dielectrics 213 
 
 This is the same as 
 
 JV (P . E) dv, 
 
 the operator V not affecting P. 
 
 This determines the linear components of the force; there may also be 
 a torque. To obtain its components give the body a small vectorial rotation 
 8a> : in this displacement the element dv goes into a position where E has 
 the value E + [E . 8a)] and so the variation of the energy is 
 
 SW = [ (P . [E . Sco]) dv = - f(8o> . [E . P]) dv, 
 and thus the couple is 
 
 226. There is however also a simple alternative method of deducing 
 these results from the ideas involved in the theory of the polarisation in the 
 medium. The mechanical force acting on a single doublet of moment e at 
 a point in the field where the electric force intensity is E is represented by 
 the vector / e y\ g 
 
 Thus by simple addition over all the doublets in the volume element dv of 
 the polarised body, we obtain that the linear force on the element of the 
 
 medium is equal to S (e . V) E = (Se . V) E 
 
 = (P . V) Edv 
 or (P . V) E per unit volume. 
 
 In this calculation the value taken for E excludes the local part of the 
 total forcive which acts on any pole at the place. In such cases when we 
 are dealing with a summation throughout the element of volume, the local 
 actions in each charge element, which really arise from the other elements 
 in the volume, must all be cancelled by complementary reactions, so that in 
 the aggregate such terms will not occur. 
 
 This expression for the linear component of the mechanical force per 
 unit volume on the medium is slightly different to that obtained from the 
 energy, the difference being actually 
 
 (P . curl E), 
 which is however zero if cur j jj _ Q 
 
 i.e. if the electrical forces have a potential which is true in the present 
 statical theory: the difference should however be noticed for future reference. 
 
 The elementary theory also shows that the simple bi-pole above mentioned 
 is subject to a torque of amount 
 
 [e.E] 
 
214 The theory of polarised media [CH. v 
 
 and thus we obtain for the torque on the element of volume of the polarised 
 medium [P . E] dv, 
 
 or in all a vector [P . E] per unit volume. This is the same as the previous 
 result. 
 
 If the medium carries a volume charge of density p there is to be added to 
 the linear constituent of the force on it the vector 
 
 pE = E div D. 
 
 227. In pursuance of our general plan we can now show that this bodily 
 forcive can be represented by means of an applied stress system identical 
 with one of the types found necessary in the previous chapter. The general 
 method is to write down the bodily forcive on any element of the dielectric 
 and then try and exhibit them as the result of an elastic stress acting over 
 the surface of the element.To accomplish this we must as usual try and put 
 the x-component of the bfl B^rcive F per unit volume into the form 
 
 ^B^^ccc , V-*- xv _|_ V *- xz 
 
 dx dy 3z ' 
 because then J? x dv taken throughout any volume can be expressed as a 
 
 surface integral over the surface of that volume. Now in the genera] case 
 
 "C 1 ClT? ^f ^^^ ^^^ ^51^ 
 
 _ OEA X _ OJlij, _ OCi z CU X _ CLf y _ OL) Z 
 
 Now remembering that E is the gradient of a potential and using the sub- 
 stitution 
 
 E 
 
 we find that 
 
 w + * i - 1 a < E * 2 + E - 2 + E ^ + ** div D 
 
 . , 
 
 dx oy 
 
 thus f..|(H.I> < -|) + |(B.D,) + |(.D.). 
 
 We have thus succeeded in our aim. We .can write the other components 
 
 in the same way and thus we see that this linear part of the bodily mechanical 
 
 forcive on any element of the medium can be expressed by means of surface 
 
 * It is for the present always to be noticed that V does not affect P. 
 
226-229] 
 
 Stresses in polarised dielectrics 
 
 215 
 
 tractions over its outer boundary. The complete specification of the stress 
 system defining these tractions is given in the matrix form 
 
 E,D a 
 E,D, 
 
 E*D, 
 EjA- 
 E,D, 
 
 E 2 
 
 E D 
 
 *-'*' 
 
 E 2 
 
 228. We notice again that in the general case this stress system is 
 not self -conjugate, the diagonal or cross terms are not equal : this means 
 that there is a bodily torque which is equal per unit volume to the 
 differences in the cross terms ; for instance the ^-component is 
 
 T T = ED ED, 
 
 This is precisely the same torque which was determined from the elementary 
 principles, so that this stress specification includes everything. It is moreover 
 identical with the second type of stress examined in the previous chapter 
 for the general dielectric medium with no discontinuities. But now there is 
 no contradiction of principle : our material dielectric is now polarised and 
 will thus be more than a mere medium of transmission as regards the 
 mechanical force. In the Faraday-Maxwell theory the function of the 
 dielectric, essentially continuous, is merely to transmit the forces without 
 adding anything to them, but on the present theory the elements of the 
 dielectric medium are subject to a definite additional type of strain the 
 reactions to which are sufficient to account for the more general form of 
 stress found necessary to transmit the electrical actions. 
 
 229. To obtain a simpler expression* of this stress let us choose con- 
 venient axes. Choose the (x, ij) plane as the plane of D and E and the 
 cc-axis as the internal bisector of the angle 26 between these two vectors. 
 The z-axis is chosen to form the usual right-handed system. Now 
 
 (E x , B,, E,) = (E cos B, -E tin d, 0), 
 and thus the matrix is now 
 E* 
 
 ED cos 2 6 
 
 8V 
 
 ED sin 6 cos 6, 
 
 - ED sin 6 cos 6, 
 
 0, 
 which can be dissected into parts. 
 
 STT' 
 
 o, 
 
 
 
 
 
 E* 
 
 ' STT 
 
 * Cf. Maxwell, Treatise, n. p. 280. * 
 
216 
 
 The theory of polarised media 
 
 [CH. V 
 
 (i) The terms ^-- make a uniform hydrostatic pressure -- throughout 
 
 O7T 077 
 
 the medium. 
 
 (ii) The two terms in the diagonal give a torque per unit volume ED sin 26 
 and the remaining terms represent 
 
 (iii) a tension along the z-axis ED cos 2 6, 
 
 (iv) a pressure (-) along the ?/-axis ED sin 2 6. 
 
 Fig. 47 
 
 In a diagram they are 
 
 EDSin I 
 
 ED Sin 2 
 
 EDCOS 2 
 
 This is the general result and is fairly complex but if we take the medium 
 to be isotropic considerable simplification results. In this case 9 = and 
 the bodily torque vanishes. Moreover the other parts reduce to a pull or 
 
229, 230] The force on a surface of discontinuity 217 
 
 tension along the lines of force or induction equal to ED and a hydrostatic 
 pressure all round equal to 5 E 2 . This is identical with Maxwell's stress 
 
 _ O7T 
 
 system, when the dielectric medium is free aether, which is the only case 
 with which he deals. 
 
 230. The reduction of the bodily forcive on dielectric media to a repre- 
 sentation by means of an imposed stress system, which we have just dis- 
 cussed, is valid only in so far as the medium is without discontinuities. 
 When there are in the electric field interfaces of transition between different 
 media, there will also exist surface tractions on them which may be 
 evaluated either as the result of the Maxwellian tractions towards the two 
 sides of the interface or by considering an actual, somewhat abrupt, interface 
 to be the limit of a rapid continuous variation of the properties of the medium 
 which takes place across a layer of finite thickness. Let then the total 
 displacement D with its circuital characteristic where there is no free charge 
 be made up of the dielectric material polarisation P and the displacement 
 
 E 
 
 proper . We have then 
 
 div E = 477 (p + />'), 
 
 wherein p' is the Poisson ideal volume density corresponding to the polarisa- 
 tion, and p is the volume density of free charge, surface distributions being 
 now, by hypothesis, non-existent. Also 
 
 divP = -p'. 
 
 The mechanical forcive acting on the dielectric is per unit volume a force 
 F and couple G where 
 
 F = (P . V) E + pE, 
 
 G = [PE]. 
 
 e 
 
 The linear force acting on the whole transition layer is the value of I "Fdv 
 
 integrated through it. This integral is finite although the volume of integra- 
 tion is small, on account of the large values of the differential coefficients 
 which occur in the expression of VE. To evaluate it we endeavour by 
 integration by parts to reduce the magnitude of the quantity that remains 
 under the sign of volume integration, so that in the limit we may be able 
 to neglect that part : thus we obtain 
 
 Ydv = E . D n df + (p + p') 
 
 Now by the definition of the electric force E it is the force due to a volume 
 distribution of density p + p' and to extraneous causes ; so that in the limit 
 
218 The theory of polarised media [CH. v 
 
 when the transitional layer is indefinitely thin, we have, by Coulomb's 
 principle. 
 
 [ (p + p')Edv = $ f (<T + <y') (E x + E 2 ) df 
 
 ' f (E x - E 2 ) (Ei + E 2 ) df, 
 
 \ J.*j A*i / \ ' if \J 
 
 E l5 E 2 being the vectors of electric force on the two sides of the layer and 
 E x , E 2 the normal components of these forces, all measured towards the 
 side 2, while a and a' are the surface densities constituted in the limit by 
 the aggregates of p and p' respectively taken throughout the layer. Hence 
 in the limit 
 
 P n Edf 
 
 J- I (Ei - E 2 ) (Ei + E 2 ) <Z/. 
 
 Q_ I V I* ^n/ v J <!/ ^ 
 
 Thus the electric traction on the interface of transition may be represented 
 by a pull towards each side, along the direction of the resultant force E ; this 
 pull is. on the side 2, of intensity 
 
 P 2n E 2 + =- (E ln - E 2 J E 2 , 
 
 or what is the same thing 
 
 P 2n E 2 - 1 (P 2?? - P ln - a) E 2 = \ (a + P ln + P 2n ) E 2 , 
 in the direction of E 2 ; on the face 1 the pull is 
 
 i (a - P! - P 2 ) E! , 
 
 4 v x n ^n' " 
 
 now in the direction of Ej. As the tangential component of the electric 
 force E is under all circumstances continuous across the interface the total 
 traction on both sides is along the normal and equivalent to 
 
 i (P lw + P 2n ) (E 2n - E ln ), 
 
 together with tractions |E 2 er, ^Ejcr acting on the true charge a, all the 
 quantities being now measured positive in any the same direction. In the 
 case where there is no surface charge this simply reduces to a normal pull 
 towards the side 2 of amount 27rP 2n 2 + 27rP lw 2 . 
 
 When the interface is between a dielectric 1 and a conductor 2, the traction 
 is only towards the side 1 and is equal to | (P ln + a)E 1; or ^Dj E 1? per 
 unit area, along the normal, which is now the direction of the resultant 
 force. 
 
 231. All this is quite independent of the law of the connection between 
 the polarisation and the electric force in the material medium. Thus under 
 the most general circumstances as regards electric field, the forcive on the 
 material due to its electric excitation consists of the interfacial tractions thus 
 specified together with a force F and a torque G per unit volume given by 
 the above formula, viz. p _ (PV)E 
 
 G - [PE]. 
 
230, 231] The stresses in fluid media 219 
 
 The assumptions underlying this analysis that the transitions are gradual, 
 will be sufficiently satisfied even if the intermediate layer is only one or two 
 molecules in thickness, for as these molecules are arranged slightly in and 
 out, and not in exact rows along the interface, their polarity can still be 
 averaged into a continuous density, as above. The aggregate tractions over 
 a thin layer of transition thus do not depend sensibly on the nature of the 
 transition, but only the circumstances on the two sides of the layer. 
 
 In the case of a fluid medium, the bodily part of the forcive produces 
 and is compensated by a fluid pressure 
 
 (PcZE), 
 
 where P, being polarisation induced by the electric force E, is for a fluid in 
 the same direction as E and a function of its magnitude. This pressure will 
 be transmitted statically in the fluid medium to the interfaces (i.e. a reacting 
 
 pressure | (PdE) exerted by the interface will keep the medium in internal 
 
 equilibrium) ; combining it there with the surface tractions proper, it appears 
 that the material equilibrium of fluid media is secured as regards forces of 
 electric origin, if extraneous force is provided to compensate a total normal 
 traction towards each side of each interface, of intensity 
 
 In the case usually treated, in which a linear law of induction is assumed, so 
 that the relation between P and E is 
 
 477 
 
 the mechanical result of the electric excitation of the fluid medium is easily 
 shown to be the same as if each interface were pulled towards each side by 
 
 eE 2 
 a Faraday-Maxwell stress made up of a pull -^ - along the lines of force and 
 
 O7T 
 
 an equal pressure in all directions at right angles. But this imposed 
 geometrical self-equilibrating stress system would not be an adequate repre- 
 sentation of the mechanical forcive in a solid medium ; for then the bodily 
 forcive, instead of being wholly transmitted is in part balanced on the spot 
 by reactions depending on the solidity of the material. The forcive acting 
 on isotropic material may however in every case, whether the induction 
 follows a linear law or not, be expressible as an extraneous or imposed system, 
 
 made up of a bodily hydrostatic pressure I (PdE) (which in the case of a fluid 
 only relieves the ordinary fluid pressure and so diminishes the compression) 
 
220 The theory of polarised media [CH. v 
 
 together with normal tractions on the interfaces between dielectric media, 
 of intensity 27rP n 2 PdE acting towards each side, and tractions 
 
 iD B E-J(P<ZE) 
 on the surfaces of conductors acting towards the dielectric. 
 
 232. As a single example of these general principles let us consider the 
 energy and ponderomotive forces of a homogeneous dielectric ellipsoid in 
 an otherwise uniform field. 
 
 We have already solved this problem as far as obtaining the field is 
 concerned. We then showed that the field inside the ellipsoid has, with the 
 usual notation, a potential 
 
 E x x E v y E z z 
 
 ~ 1 + A (e - 1) 1 + B ( e - 1) 1 + C ( - 1) ' 
 
 and thus the force intensity at any point inside the ellipsoid has components 
 
 / E x E y E z \ 
 
 and the components of the polarisation intensity are therefore 
 1 / E v EV E, 
 
 ~' c + 
 
 ' e- 1 
 
 We thus conclude that the portion of the total energy of the field which is 
 associated with the polarisation of the material of the ellipsoid and which 
 serves as an energy function of the bodily forces on that ellipsoid is merely 
 
 !(PE) dv = i^- 1 f&dv, 
 
 OTT 
 
 the integral being taken throughout the volume of the ellipsoid, E denoting 
 the force intensity of the total field. 
 This is equal to 
 
 - 1 
 
 STT |_ (1 + A (e - 1)) 2 J 3 
 in the present case when the ellipsoid is homogeneous. 
 
 The mechanical force can either be obtained from the energy function 
 or by the elementary methods. Adopting the former we see that it is a 
 wrench of intensity F and in which the couple is G where 
 
 , |V^ 2 , 6-1 
 
 F = grad dv, e -- - , 
 
 B denoting a generalised angular component. 
 
231-233] On electric displacement 221 
 
 We thus see that if the field changes very slightly in the space occupied 
 by the ellipsoid or if -- is uniform throughout the ellipsoid 
 
 000 
 
 F r= 7rabc r grad E*, 
 
 so that it will be attracted into places of stronger force (not necessarily higher 
 potential). 
 
 To obtain some idea of the nature of the couple let us specialise our 
 assumptions slightly. If the ellipsoid is fixed to rotate about its c-axis and 
 the field is in a direction perpendicular to this axis, then 
 
 E x = E cos 6, E y = E sin 0, E z = 0, 
 
 determining the angle between the plane of the c-axis and E and the plane 
 of the c- and a-axes, then 
 
 E 2 (c-I)[ cos 2 sin 2 "1 
 
 W = ' h ~ r> 
 
 so that 
 
 G = 
 
 .{1 + B ( e - 1 (I + A [e -T)p rS 
 If a > b then A < B < 1 and the quantity in the square bracket is negative, 
 
 thus G e is negative when < -~. We thus see that the ellipsoid always tends 
 to turn its longest axis into the direction of the field. 
 
 233. On electric displacement*. Since the vectors of the present theory 
 satisfy exactly the same conditions as those of the original Maxwell-Faraday 
 theory they must ultimately represent the same quantities. But the theory 
 just developed is based on elementary physical notions regarding the behaviour 
 of the dielectric medium when introduced into an electric field. The present 
 ideas must therefore help us to explain the former theory and by means of 
 it we should be able to obtain some insight into the nature of 'electric dis- 
 placement.' This is best accomplished by considering a particular problem, 
 viz. that of a parallel plate condenser with large surfaces with equal positive 
 and negative charges ; a plane slab of some dielectric substance (constant e) 
 is inserted parallel to the plates. We treat the air spaces as a vacuum 
 because its density is so slight. The solution of this problem is easy and if 
 the surface densities of charge on the plates of the condenser are a the 
 
 electric force is kira all across the air spaces and is in the dielectric. (The 
 
 lines of force are straight across by symmetry and at the surface of the 
 dielectric eE 2 = E x . ) 
 
 * Cf. Larmor, Aether and Matter; Lorentz, Versuch einer Theorie der electrischen und optischen- 
 Erscheinungen in bewegten Korpern. 
 
222 
 
 The theory of polarised media 
 
 [CH. V 
 
 Now on the present theory of the matter the dielectric substance is 
 polarised : the molecules have a positive and negative pole and owing to 
 the presence of the field the axes have a convergence towards a definite 
 direction, viz. straight across between the plates, so that their moments no 
 longer cancel. The intensity of polarisation of the medium is thus at each 
 point directed straight across between the plates. 
 
 If now we consider a small rectangular volume element of the substance 
 parallel to the lines of force straight across 
 we see that the polarisation of all the 
 molecules in it is equivalent to a small 
 polar distribution in the volume, which is 
 just the same as if it had a positive charge 
 of density + a t on one end and a negative 
 charge of density a t on the other. All 
 the little molecular moments can be 
 summed up into a uniform polarisation : 
 the irregular molecular distribution is 
 smoothed out into a uniform average. At 
 least this is an effective representation of 
 the matter. It does not mean that we 
 assert that there is an actual charge on 
 each end of the little element but that the 
 aggregate of the polarisation in the element 
 can be replaced by these charges when 
 investigating its action at external points. 
 The essential thing for this purpose is the 
 electric moment of the element and any 
 distribution giving the right moment is an 
 effectively correct one. 
 
 Now by combining all these small rectangular elements so polarised into 
 the finite piece of dielectric we see that there will be an uncompensated part* 
 of the surface density (which is not necessarily the same for each element) 
 where one rectangular block abuts on the next one and at an end at the 
 boundary of the dielectric itself there remains the complete surface polarity. 
 This amounts to what we have called the ideal electric distribution of Poisson ; 
 the outstanding parts throughout the medium correspond to the volume 
 density and the complete polarity remaining at the surface of the medium 
 corresponds to the ideal surface density. Regarded in this way it is obvious 
 that this theoretical distribution and the actual one will not give the same 
 field in the immediate neighbourhood of an element of the substance. The 
 ideal distribution has been smoothed out from the other and it is only at 
 a distance that it is effectively equivalent to the actual polarity. 
 
 * In the particular case examined this uncompensated charge is zero. 
 
 Fig. 49 
 
233, 234] On electric displacement 223 
 
 In the example under immediate consideration the field in the dielectric 
 is uniform and so the intensity of polarisation will also be uniform throughout 
 the medium : thus the charges on the ends of adjacent small elements will 
 be the same and thus when put together there will be no uncompensated 
 polarity : we shall merely have a surface density of ideal electric charge 
 - a' on one face of the dielectric and + a' on the other. In the old-fashioned 
 way of describing these things a (the charge density on the plates of the 
 condenser) might be called the free charge and a' the bound charge (as it 
 cannot be moved) ; a' is only the end aspect of the polarisation in the medium 
 which has a counterpart at the other side of the medium and they cancel across. 
 
 On our theory a' is equal to the normal component of the polarisation 
 at the surface and this is 
 
 e- I, 
 
 _ E, , 
 
 but since E, = = we have 
 
 
 
 So that a a -. 
 
 
 
 234. Let us now examine another point. The polarisation of the 
 element can be expressed by saying that an electric displacement in the 
 element from one end to the other has taken place. Initially the positive 
 and negative charges effectively coincide and cancel but on the application 
 of an electric field they are separated and the electric mqment can be con- 
 sidered to arise from the electric displacement of one charge relative to the 
 other. We can at least theoretically imagine it to be like this. There is 
 thus an actual movement of electric charge. Essentially the movement 
 consists in the molecules being really strained round a bit, but when we 
 aggregate these up for the small rectangular volume element as before, the 
 effect is the same as if the positive charge were moved from one face of the 
 volume element to the opposite one. If this is the case how can we measure 
 the displacement ? The proper measure is the product 
 of each charge element by the distance through 
 which it is moved and the total sum of the quan- 
 tities so obtained because if we moved the same 
 charge in each case through half the length it ought to give half the 
 measure of the displacement. Thus the total electric displacement in our 
 small rectangular volume element of end area Se and length 8/1 is 
 
 cr'Se X 81 = a'dr, 
 
 where ST is the volume of the element ; but this is the moment produced in 
 the element. Thus an effective measure of the displacement in the volume 
 element is the intensity of polarisation multiplied into the volume. 
 
224 
 
 The theory of polarised media 
 
 [CH. V 
 
 Thus for the slab of dielectric in the example considered the result of the 
 total electric displacements in the medium is merely to displace a surface 
 charge a' from one side of the slab to the other straight across. There is 
 a true displacement of electricity equal to a displacement right across. The 
 flux of electricity measured in this way is a true electric displacement. 
 
 But on the Faraday-Maxwell theory of electric action the electric dis- 
 
 eE 
 placement in any small volume dv is taken to be -. which is equal to 
 
 (1 + 47re') E E 
 
 5 ' dv = dv + Pdv, 
 
 4:7T 4:77 
 
 so that in addition to the true electric displacement represented by the term 
 Pdv as in the present theory there is something else which is quite a new 
 thing altogether. If we call the true electric flux or induction a displacement 
 
 E 
 
 the term - dv represents quite an extraneous thing altogether. This part 
 
 still exists in empty space when there are no dielectrics present so that it 
 cannot possibly be ascribed to an electric displacement. Maxwell's flux 
 vector is therefore not all electric displacement as part of it remains when 
 there is no electricity present at that point. The part Pdv is as we have 
 
 E 
 
 already seen a true electric displacement and the other part -r- dv we call the 
 
 aethereal displacement. This latter part has the same properties as the former. 
 Thus if we want to retain the analogy between the simple displacement 
 theory of Maxwell and the polarisation theory just developed we must 
 introduce this new, type of displacement so that the total electric displacement 
 of Maxwell includes the true electric displacement of the present theory and 
 the aethereal displacement. 
 
 235. The real significance of the matter is however best exhibited in 
 another manner. Consider again the ex- 
 ample of the condenser with the dielectric 
 slab ; what happens when it is charged ? As 
 far as we are at present concerned the con- 
 denser may be charged by transferring a 
 positive charge + Q round a wire connecting 
 the two plates thereby leaving, in defect, a 
 charge Q on the one plate and creating 
 an excess of charge + Q on the other plate. 
 While this is being accomplished a displace- 
 ment is taking place in the dielectric (the 
 polarisation is being gradually set up) and 
 a charge Q' is displaced across the medium 
 from one side to the other. This is all the 
 electric motion that takes place; an actual 
 
 Fig. 50 
 
234-236] On electric displacement 225 
 
 charge Q moves round the wire and. a change of polarisation in the dielectric 
 corresponds to a motion of a charge Q' across the dielectric. 
 
 But since Q = Aa and Q' = Aa', where A is the equal area of the parallel 
 faces of condenser plates and dielectric (the charge is assumed uniformly 
 distributed) we have that 
 
 so that the displacement of charge across the dielectric is not equal to that 
 round the wire : and there is nothing at all in the free spaces between. 
 
 If we stop here we shall have a very complicated state of affairs. It was 
 merely in order to avoid all these complications in electrodynamic theory 
 that Maxwell made the assumption that the displacement which takes place 
 during any electric change is always circuital; that is always takes place 
 in complete circuits. If we, with Maxwell, make the postulate that electricity 
 always flows in complete cycles we shall find electrodynamic theory much 
 easier to handle. This will appear later. 
 
 In the example above, Maxwell would therefore postulate a hypothetical 
 total displacement equal to Q in the air and - in the dielectric ; this being 
 
 all that is required to complete the flow of the quantity Q all round. 
 
 E 
 
 Estimated per unit volume this would mean adding a displacement -7'- at 
 
 4r7T 
 
 each point of space between the condenser plates. (It is assumed that E = 
 everywhere except between the plates.) 
 
 236. This is easily seen to be the general result. If E is the force 
 intensity at any point of an electric field Maxwell's theory adds a displace- 
 
 E 
 
 ment equal to j at that point to any true electric displacement that may 
 
 occur there. If we do this then the flux of displacement is always in closed 
 cycles. This additional displacement is not true electric displacement at 
 all, as it exists at points in a vacuum; it is an aethereal displacement 
 possessing all the electrodynamic properties of true electric flux. 
 
 A dynamical theory of electromagnetic actions should give a reason for 
 this action in the aether, for the existence of this aethereal displacement 
 which has the same properties as a flow of electricity but is not itself a flow 
 of electricity. The hypothesis is however experimentally correct and it 
 simplifies the theory immensely and there we shall leave it for the present. 
 
 On this view of the matter the aether is to be regarded as the seat of 
 part of the energy associated with any electrostatic field, viz. that part 
 associated with the production of the aethereal displacement. On a previous 
 
 L. 15 
 
226 The theory of polarised media [CH. v 
 
 analogy this part may be taken as distributed throughout the field with 
 a density at any point equal to 
 
 a result which is verified by the fact that all the energy in the field is located 
 in the aether if no dielectric medium is present. 
 
 237. The relation of inductive capacity to density. One of the most 
 successful ways of testing a constitutional theory of the present type is to 
 formulate on the same basis the connection between the constitutional re- 
 lations involved in it and the physical or chemical constitution of the 
 medium. In the present case the whole constitutive character of the theory 
 is involved in the one constant introduced in it, viz. the specific inductive 
 capacity e. If therefore we can formulate a connection between this con- 
 stant and the constitution of the medium we shall have a definite means 
 of testing the general validity of our theory. It is quite easy to obtain a 
 relation between the constant e and the density of the medium in certain 
 simple cases and we shall find that it agrees very well with our experi- 
 mental knowledge on the same question. 
 
 Let the dielectric medium contain n molecules per unit volume, these 
 molecules being presumed to be merely concentrated when a change of density 
 of the medium occurs. Each of the molecules becomes polarised to a moment 
 p by the field of the electric force ; this field is made up of the extraneous 
 exciting field and that of the polarised molecules themselves; the latter 
 again consists of a part arising from the polarised medium as a whole and 
 a part involving only the immediate surroundings of the point considered ; 
 to obtain an estimate of these various parts let us consider again the method 
 of their separation. 
 
 238. The total electric force acting on a single molecule is derived from 
 the aggregate potential 
 
 This potential, when the point considered is inside the polarised medium, 
 involves the actual distribution of the surrounding molecules ; and thus the 
 force derived from it changes rapidly, at any instant of time, in the interstices 
 between the molecules. But when the point considered is outside the 
 polarised medium, or inside a cavity formed in it (whose dimensions are 
 large compared with molecular distances) the summation in the expression 
 for (f> may be replaced by continuous integration ; so that P denoting the 
 intensity of polarisation in the molecules of the dielectric medium 
 
236-238] Inductive capacity and density 227 
 
 and the force thus derived is perfectly regular and continuous. This expression 
 may be integrated by parts since, the origin being now outside the region of 
 the integral, no infinities of the function to be integrated occur in that region. 
 Thus 
 
 , . [P n ,, fdiv P , 
 <t> = df - - dv 
 j r J r 
 
 that is, the potential at points in free aether is due to Poisson's ideal volume 
 density p = div P and a surface density a = P n . When the point considered 
 is in an interior cavity, this surface density is extended over the surface of 
 the cavity as well as over the outer boundary. Now when it is borne in mind 
 that, at any rate in a fluid, the polar molecules are in rapid movement and 
 not in fixed positions which would imply a sort of crystalline structure, it 
 follows that the electric force on a molecule in the interior of the material 
 medium, with which we are concerned, is an average force involving the 
 average distribution of these molecules, and is therefore properly due to an 
 ideal continuous density like Poisson's, even as regards the elements of volume 
 which are very close up to the point considered. To compute the average 
 force which causes the polarisation of a given molecule we have thus to 
 consider that molecule as situated in the centre of a spherical cavity whose 
 radius is of the order of molecular dimensions ; and we have to take account 
 of the effect of a Poisson averaged continuous local polarisation surrounding 
 the molecule, whose intensity increases from nothing at a certain distance 
 from the centre up to the full amount P at the limit of the molecular range, 
 this intensity being practically uniform in direction and a function of the 
 distance only. 
 
 We therefore assume spherical stratification in the distribution of the 
 Poisson idea] volume density near the point under investigation. To estimate 
 the effect of an elementary shell in this stratification, the charge in it can be 
 reckoned as a surface density on it of intensity 
 
 SP cos 6, 
 
 8P denoting the small increment of P as we pass through the shell, and 6 the 
 polar angle between the direction of P and the normal at the point of the 
 shell. This shell thus contributes a force at the centre in the direction of 
 P equal to 
 
 f/> 2 3P t 
 
 1 U * *V> ' /I 7/1 7 / 
 
 I 2~ cos v sin Vdvd(f) 
 
 Thus on the whole the local part of the forcive is 
 
 4-7T C 4-7T 
 
 - _ sp _ ^ p 
 
 O I r ~ ~ ^~ * ' 
 
 J Jo <J 
 
 152 
 
228 The theory of polarised media [OH. v 
 
 The force polarising the molecules is therefore 
 
 E + y". 
 
 E denoting the total electric force. Now if the polarisation produced be 
 presumed to be proportional to the polarising force 
 
 and thus since P = 2p = wp, 
 
 we have P = we' (E + ~~ P) , 
 
 V 3 / 
 
 and by the definition of e we have 
 
 -1 
 
 rp, e 1 4?r , 
 
 Thus _ - _ - ', 
 
 from which we see that the function 
 
 e- 1 
 
 e + 2 
 
 must be proportional to the density of the medium. This is the usual Lorentz 
 formula* which has been satisfactorily verified in numerous cases. 
 
 239. On the mechanism of dielectric polarisation f. We have thus seen 
 that the only consistent view of the action of a dielectric in transmitting the 
 actions in electrostatic fields is the one based on the idea of the electric polari- 
 sation of the individual molecules of the matter in the field, each of which is 
 thus presumed to contain as an essential part of its constitution certain electric 
 charges more or less tightly bound in its interior, but which can be separated, 
 the positive from the negative; when an electric field is applied from without. 
 We have also mentioned the fact that in all probability these charges are 
 constituted, the negative of a certain number of more or less identical atomic 
 charges, the electrons, of extremely small mass and the positive in an as yet 
 uncertain manner. Although the explanations developed above have in 
 reality no reference to this very definite constitution of the involved charges 
 and the mechanism of its binding in the atoms, it seems certainly of theoretical 
 interest, if not of practical importance, to formulate the connection between 
 
 * This formula was determined for the optical case by R. Lorentz, Ann. Phys. Chem. (3), 11 
 (1880), p. 77; 20 (1883), p. 19: and independently by H. A. Lorentz, Ann. Phys. Chem. (3), 
 9 (1880), p. 642. The mode of deduction here given is due to Larmor, Phil. Trans. A, 190 (1897),. 
 p. 232. 
 
 t Lorentz, Arch. Norland (1892), p. 363. Cf. also Livens, Phil. Mag. xxiv. p. 285 (1912). 
 
238-241] The mechanism of the polarisation 229 
 
 the general theory and the special form of it treated as a branch of the general 
 electron theory of these things. Of course any such procedure naturally 
 brings us into close contact with the vexed question of the constitution of 
 the molecule which is occupying so much attention at present. We can 
 however obtain sufficient information in a tentative sort of way without 
 making any very special hypothesis. Of course the state of affairs exemplified 
 cannot in the least be said to depict the thing as it really exists. All we do 
 is to imagine some simple system that will coordinate and explain the majority 
 of the phenomena known to us. 
 
 240. We imagine therefore a body composed of innumerable molecules or 
 atoms, of particles, as Lorentz calls them, each molecule containing a certain 
 number of electrons and the necessary positive charge. Since there is as yet 
 no definite evidence as to the distribution of this positive charge, it is best 
 to adopt an hypothesis which renders most easy the task of deducing the 
 properties of the atom from its structure. It will be supposed therefore 
 that the 'positive electrification is distributed rigidly throughout the volume 
 of the atom and this condition is probably secured in any case to a sufficient 
 approximation on account of the comparatively large mass of the positive 
 constituents. The model atom as pictured by Kelvin and worked with 
 by J. J. Thomson*, was assumed to be spherical in form and the distri- 
 bution of the positive charge uniform, but we need not make these special 
 assumptions for our present theory, especially as it would seem to be incon- 
 sistent with the recent work on the question. 
 
 Now on the modern view of these things all electrical phenomena are 
 concerned with the, motions of this complicated electrical system, called an 
 atom. The main part of the motion is that of the negative electrons and it 
 is possible to discuss the subject in terms of these electrons. But of course 
 any method of procedure in which the motion of the individual electrons is 
 the object of our investigation is wholly useless when the distribution of the 
 atoms is highly irregular. We have thus as usual to express ourselves in 
 terms of statistical or averaged sums over all the electrons in the element 
 of volume. Statistically the effect of an electric force on a body is to polarise 
 the molecules, that is, to twist them round or alter them in some way so that 
 they have a definite polarity. We can express this polarity as an averaged 
 sum over all the electrons per unit volume. 
 
 241. The x-component of the moment of a simple bi-polar element is 
 
 (5JU ^~ C/J/j 
 
 (x 1 , y', z') being the coordinates of the positive pole, and (x, y, z) those of the 
 equal negative pole of the doublet referred to fixed rectangular axes. 
 
 * Phil. Mag. (5), XLIV. (1897), pp. 310. See also his books Electricity and Matter and <7or- 
 puscular Theory of Matter. 
 
230 The theory of polarised media [CH. v 
 
 If the element of volume is small enough we have simply to add up 
 vectorially the moments of the simple doublets contained in it to get the 
 resultant moment of the element, i.e. its polarisation. 
 The x-component of the intensity of polarisation is thus 
 
 Pa, = Se (x' - x), 
 
 where S is taken per unit volume over all the doublets. This is simply 
 "Lex, taken for all the electrons and the elements of positive charge per unit 
 volume, due regard being paid to sign. Thus 
 
 P x = Sex 
 
 is the x-component of the polarisation. Now let (x, y, z) be the mean centre 
 'of the positive charge distribution in the element of volume at any instant, 
 then, since the atom was originally neutral, this is also the mean position of 
 the negative charges before they are displaced relatively to the positive 
 charges. Thus Sex for the positive charges is 
 
 Se, 
 and for the negative charges it is 
 
 xLe + Sex, 
 
 where (x, y, z) is now the displacement of the negative electron e from its 
 neutral position of equilibrium. 
 
 But Se for the positive charges is equal but opposite in sign to Se for the 
 negative electrons, and thus p _ ^ ex 
 
 S now being taken only for the negative electrons, (x, y, z) denoting the 
 component displacements of that electron from its neutral position of 
 equilibrium. 
 
 We have thus expressed the polarisation in terms of the negative electrons 
 alone and it is in this form that we shall use it. 
 
 242. The position of an electron or better its displacement (x, y, z}, 
 and therefore the polarisation, depends on the forces acting on the electron. 
 These forces are in the present instance of two types only. 
 
 Firstly there is the electric force due to the electric field in the aether. 
 At first sight it might be thought that this is simply eE, where E is the electric 
 force intensity, and this is what is taken by most authors. On closer investiga- 
 tion it is however obvious that just as in the last paragraph a term must be 
 added on account of the polarisation in the medium. This local force may 
 generally be taken to be of the simple form* 
 
 * In isotropic media only. In the case where the medium possesses natural or induced 
 aeolotropic characteristics the local force would be different in different directions and not 
 necessarily parallel to the polarisation. Cf. Larmor, Phil. Trans. A, 190 (1897), p. 236. 
 
241, 242] The mechanism of the polarisation 231 
 
 and its direction is that of P ; a is some constant depending on the local 
 distribution of atoms or molecules and which therefore would constantly 
 vary with the temperature and pressure, but which in most cases is very 
 
 nearly equal to -^- . 
 
 o 
 
 Secondly there is the force of unknown origin and amount holding the 
 electron inside the atom. Before the external field is applied the internal 
 forces of this type will hold the electron in a certain position of stable 
 equilibrium, and if the field is not too strong the displacement from this 
 equilibrium position when it is applied will only be small*, so that the force 
 holding it back may be taken to be proportional to the displacement. We 
 can easily imagine in a general way that this quasi-elastic force has its origin 
 in the mutual electrical actions between the charges in the atoms. Thus 
 denoting by k a certain positive constant which depends on the properties 
 of the atom, and w r hich may be different for different electrons in the atom, 
 we may write for the components of this force 
 
 - Jc (x, y, z) 
 
 (x, y, z) being interpreted as before stated, for each electron as the components 
 in three definite directions of its displacement from the equilibrium position- 
 it occupied when the atom was undisturbed by any external actions. An 
 average isotropy is assumed for the intra-molecular forces f. 
 
 Now in a condition of equilibrium these two forces, the internal and external 
 forces, must balance so that for each electron we have three equations of the 
 
 ^P 6 -kx + e (E x + aP x ) = 0, 
 
 so that the displacement of the electron has an as-component equal to 
 
 - (E x + aP x ), 
 
 
 
 e 2 
 and therefore P^ = J^ex = -r (E,,. + aP^), . 
 
 K 
 
 the sum 2 being taken per unit volume over all the contained electrons, each 
 with their proper value of k. Thus 
 
 * The fields which can be created in actual practice are very feeble compared with those 
 that probably exist inside the atoms or molecules. 
 
 f The question of aeolotropic intra-molecular forces is considered by Voigt (Magneto u. 
 Electro -optik) as also is the case when the forces are not proportional to the displacement. The 
 necessity for this extension does not however appear to exist at present, since all the phenomena 
 which are explained by them can be more suitably accounted for on the assumption of aeolotropy 
 in the local forcive, which is entirely neglected by Voigt. 
 
23*2 The theory of polarised media [OH. v 
 
 Thus on, the present form of theory the dielectric constant of the medium is 
 e where 
 
 C-1+ 
 
 and it is thus interpreted completely in terms of the electronic constitution 
 of the medium. This formula was first obtained by Lorentz but owing to 
 the uncertainty in the value of k it is of very little practical use in the present 
 instance. 
 
 It might be thought that this view of the affair is necessarily restricted 
 in that it leads essentially to a linear relation between P and E, which although 
 apparently satisfactory from the practical standpoint, is nevertheless merely 
 an approximation to the real state of affairs; but it must be remembered 
 that our deduction is also essentially based on the linear relation between 
 the quasi-elastic force resisting displacement and the displacement of the 
 electron itself, and it is only so far as this assumption is justified that the 
 above result is valid. 
 
 243. Pyro- and piezo-electricity*. We have generally assumed in the 
 preceding discussions that the elements of all dielectric media are always 
 permanently neutral as regards their electrical effect on external systems so 
 long as they are not under the influence of an external polarising field. This 
 would imply that anything iD the nature of permanent polarity, which is such 
 an important feature in the correlative subject of magnetism, is non-existent, 
 or at least negligible, in the electrical case. Whether any such presumption 
 is really justifiable it is difficult to say but there are certain phenomena which 
 seem to suggest at least the possibility that it is not valid in every case. 
 
 Several substances like quartz and tourmaline which crystallise in 
 asymmetric forms always appear to be polarised immediately after their 
 temperature is changed, and in opposite directions according as it is raised 
 or lowered. The polarisation exhibits itself mainly as an apparent separation 
 of charge on the outer surface of the piece of the substance under investigation, 
 one part of which appears positively charged and the opposite negatively 
 charged. This is the phenomenon of pyro-electricity. If the substance is 
 maintained for any period at the new temperature the polarisation gradually 
 disappears and soon ceases to be observable at all. 
 
 244. Lord Kelvin explains f this phenomenon by assuming that the 
 elements of the crystal substance are permanently polarised to an extent, 
 
 * A complete account of these phenomena with all the associated experimental and theoretical 
 details can be found in Voigt, Lehrbuch der Kristallphysik (Leipzig, 1910). Cf. also the same 
 author's Kompendium der theoretischen Physik (Leipzig, 1896), Bd. n, Teil 4, 11-15, 20. 
 
 f NicoVs Cyclopedia of Physical Science, 1860. Math, and Phys. Papers, I. p. 315. 
 
242-246] Pyro- and piezo -electricity 233 
 
 however, depending on the temperature, and that they are arranged with 
 their electric axes in regular order in the crystalline media with which the 
 phenomenon is associated. If the material of the crystal and in particular 
 its outer surface are not perfectly non-conducting, the polarisation will 
 ultimately give rise to a* surface distribution of charge which neutralises the 
 effect of its electric field at all external parts of the field. If the polarisation 
 of the medium is altered by changing its temperature and the establishment 
 of the neutralising charge takes place slowly, it should be possible to detect 
 the polarisation before its external field is again neutralised. 
 
 This explanation of the phenomenon appears to be perfectly consistent 
 with all the characteristic properties of the effect and, in addition, with the 
 results of numerous experiments based mainly on the independent variability 
 of the polarisation and its neutralising charge which have been performed 
 with a view to testing it. It would thus appear to be highly probable that the 
 underlying assumption of permanent molecular polarity is largely justified. 
 
 245. There is an inverse effect associated with the phenomenon of pyro- 
 electricity, the existence of which was predicted by Lord Kelvin* but which 
 was not observed until quite recently. If there is a relation of dependence 
 between the polarisation of a medium and its temperature there must be a 
 path of transformation open between the kinetic energy of thermal agitation 
 of the molecules and the organised electric energy of their polarisations, and 
 if the transformation can be carried out in either direction (i.e. if the effect 
 is a reversible one) an alteration of the electric energy should produce a 
 corresponding change in the thermal energy. The electrical energy of the 
 polarisations may be altered by moving the body about in an electric field 
 and thus we conclude that any such movement will give rise to a temperature 
 variation in the substance. This electrocaloric effect has been observed by 
 Straubelf and LangeJ, who find that the quantitative relation established 
 for the phenomenon by Lord Kelvin by thermodynamic reasoning is satis- 
 factorily verified. 
 
 246. Another effect of an analogous nature and of even more widespread 
 character than the purely thermal effects just described has been observed 
 in a large number of substances . In these cases the observed polarisation 
 of the substance is produced not by changing the temperature but by the 
 application of pressure on opposite sides of the substance. This pressure 
 gives rise to an additional strain in the material the main effect of which is 
 that the constituents of the permanent polar elements take up new positions 
 in the substance and the old neutralising surface charge is no longer effective 
 in balancing their field at external points. 
 
 * Math, and Phys. Papers, I. p. 316, 1877. 
 
 t Gottinger Nachr. (1902), Heft. 2. 
 
 f Dissertation, Jena, 1905. 
 
 ,T. and P. Curie, Pans C. B. 91 (1880), pp. 294, 383. 
 
234 The theory of polarised media [CH. v 
 
 Associated with this so-called piezo-electric effect there is an inverse 
 phenomenon* in which an alteration in the pure elastic strain in the medium 
 is effected simply by changing the energy of the electric polarisations by 
 moving the substance about in an electric field of variable intensity. 
 
 247. Although these phenomena of pyro- and piezo- electricity seem to 
 require for their explanation the assumption of permanent polar elements in 
 the substance it is possible that the occurrence of such elements may be 
 a result merely of the mutual interaction of the molecules, for it is only observed 
 in crystalline substances in which the molecular structure is perfectly regular, 
 and in which therefore the local interaction between any molecule and its 
 neighbours would always be related in a definite manner to the crystal 
 structure. Such a view is supported by the fact that the polarity, which 
 in such cases is essentially a phenomenon of molecular grouping, depends 
 on the physical conditions as to temperature and strain in the medium, 
 which are just the conditions which are circumscribed by the mutual inter- 
 action of the molecules. 
 
 248. Associated with these reversible phenomena of pyro- and piezo- 
 electricity, which depend essentially on the presence of permanent polar 
 elements in the medium, there are irreversible phenomena arising from the in- 
 duced polarity f, when the extent of the induction in a given field is a function 
 of the temperature and strain conditions of the medium. These two new 
 effects can of course only be exhibited in their inverse aspects and appear as 
 a temperature and strain condition variation resulting from the polarisation 
 of the medium induced by an external field. The former of these effects has 
 never yet been detected and the latter is usually inseparably mixed up with 
 the strain produced by the mechanical forces proper on the medium resulting 
 from its polarisation (the effect of electrostrictiori), although arrangements can 
 be devised by which it can be observed J. 
 
 * Lippmann, Ann. chim. phys. (5), 24 (1881), p. 164; Jour, de phys. (1), 10 (1881), p. 391. 
 Cf. also Riecke, Gott. Nachr. (1893), pp. 3-13; Voigt, Gott. Nachr. (1894), Heft. 4. 
 
 f The thermal one was predicted by Lippmann, Ann. chim. phys. (5), 24 (1881), p. 171, and 
 the mechanical one by Larmor, Phil. Trans. 190 A (1897), 83. The magnetic aspect of these 
 phenomena is more important. Cf. below, p. 270. 
 
 % Bidwell, Phil. Trans. A (1888), p. 228; 'Proc. R. 8. 1894. 
 
CHAPTER VI 
 
 MAGNETO-STATICS 
 
 249. Introduction. We now turn to the consideration of another static 
 affair which occurs in electromagnetic theory : this is magnetism. The 
 elementary ideas of the theory are derived from the action of certain bodies, 
 called magnets, which if suspended so as to turn freely about a vertical axis 
 at any part of the earth's surface except the magnetic poles, will in general 
 tend to set themselves nra certain azimuth, and if disturbed from this position 
 will oscillate about it. Such bodies are the iron ore called lode-stone, and 
 pieces of steel which have been subjected to certain treatment. 
 
 It is found that the force which acts on the body tends to cause a certain 
 line in the body to become parallel to a certain direction in space. This 
 line we call the axis of the magnet. 
 
 Let us now suppose the axes of several magnets have been determined 
 and that the end of each which points north is marked. Then if one of these 
 magnets be freely suspended and another brought near to it, it is found that 
 two marked ends repel each other as do also two unmarked ends, but a 
 marked and an unmarked end attract. If the magnets are in the form of 
 long rods or wires uniformly magnetised along their length, it is found that 
 the greatest manifestations of force occur when the end of one magnet is 
 held near the other, and that the phenomena can be accounted for by supposing 
 that like ends of the magnets repel each other, that unlike ends attract each 
 other and the intermediate parts of the magnets have no sensible effect. 
 
 The ends of a long thin magnet are commonly called poles. In the case 
 of an indefinitely thin magnet uniformly magnetised in its length the 
 extremities act as centres of force and the rest of the magnet appears devoid 
 of magnetic action. In all actual magnets however the magnetisation 
 deviates from uniformity so that no single points can be taken as poles. 
 Coulomb however by using long thin rods magnetised with care succeeded in 
 establishing the law of force between two poles. 
 
 The force between two magnetic poles is in the straight line joining them 
 and numerically equal to the product of the strength of the poles divided by 
 the square of the distance between them*. 
 
 * Coulomb, M em. de VAcad. 1785, p. 603. Cf. also Biot, TraM de physique, t. iii. 
 
236 Magneto-statics [CH. vi 
 
 This law of course assumes that the strength of each pole is measured in 
 terms of a certain unit, the magnitude of which may be deduced from the 
 terms of the law. The unit pole is such that when placed at unit distance 
 in air from a similar pole, it repels it with a unit force. 
 
 The quantity called the strength of a pole may also be called a quantity 
 of magnetism (positive or negative), provided we attribute no properties to 
 it, other than those observed in magnets. The quantity of magnetism at 
 one pole of a magnet is always equal and opposite to that at the other so 
 that the total quantity in any magnet is zero. 
 
 Since the expression of the law of force between given quantities of 
 magnetism ha,s exactly the same mathematical form as the law of force in 
 electrostatic theory much of the mathematical treatment of magnetism must 
 be similar to that of electricity. We shall in fact transfer the results of our 
 previous analysis directly to the subject now before us. 
 
 250. If the middle of a long thin magnet be examined, it is found to 
 possess no magnetic properties, but if the magnet be broken at that point, 
 each of the pieces is found to have a magnetic pole at the place of fracture, 
 and this new pole is exactly equal and opposite to the other pole belonging 
 to the piece. It is impossible to procure, by any means, a magnet whose 
 poles are unequal. 
 
 If we break the long thin magnet into a number of short pieces we shall 
 obtain a series of short magnets, each of which has poles of nearly the same 
 strength as those of the original long magnet. 
 
 Let us now put all the pieces of the magnet together as at first. At 
 each point of junction there will be two poles exactly equal and of opposite 
 kinds, placed in contact, so that their action on any other pole will be null. 
 The magnet, thus rebuilt, has the same form as at first. 
 
 Since in this case we know the long magnet to be made up of little short 
 magnets, and since the phenomena are the same as in the unbroken magnet, 
 we may regard the magnet, even before being broken, as made up of small 
 particles, each of which has two equal and opposite poles. The same idea 
 is also found to be generally true for magnets of any shape. We are thus 
 induced to transfer to this case the analysis and methods of the previous 
 chapter on polarised media. We regard each particle of the body (molecule 
 or molecular group) as a little magnet possessing two magnetic poles of equal 
 strengths at a small distance apart. The magnetic field of force of such 
 a particle is deduced in a manner exactly analogous to that already employed. 
 The potential for instance at any point distant r from its centre in a direction 
 making an angle 6 with its axis is 
 
 . m cos . 
 9 = 3~ 
 
249-251] The magnetic field 237 
 
 m being the moment of the doublet. This may also be written in the 
 form 
 
 when we take cognisance of the vector sense of the moment m. 
 
 The magnetic force intensity is then deduced as before by differentiation 
 of the potential. 
 
 251. The field of a finite magnet is now obtained by regarding it as 
 composed of a large number of small magnetic particles of the above kind. 
 
 The condition of magnetisation at any point of the finite magnet is 
 completely specified by a vector I, called the intensity of magnetisation, 
 which is such that if Sv is any small element of volume of the substance at 
 any point then ISv is the resultant effective moment of all the little bi-polar 
 elements (magnets) in it. If the axes of these elementary magnets are 
 distributed anyhow in all different directions then 1 = 0, but if there is any 
 degree of convergence to a definite direction I has a finite value. 
 
 The discussion of the potential and force in the field of this finite magnet 
 then follows exactly the same lines as in the previous chapter. 
 
 The potential of the magnet at points external to the distribution of 
 polarisation is obtained by the addition of the potentials of each of its con- 
 stituent elements and is therefore 
 
 and the force is obtained from this potential in the ordinary way, i.e. as its. 
 negative gradient. The application of these expressions at internal points 
 however fails firstly on account of the uncertainty as to the law of action 
 of a doublet very close up to it and secondly, as regards the expression for 
 the force, on account of the non-absolute convergence of the integral. We 
 are then led to the introduction of Poisson's ideal magnetic matter* con- 
 sisting of a volume density 
 
 p = div I 
 
 at any part of the solid magnet together with a surface distribution of density 
 
 = In 
 
 on the surface of the magnet. This distribution effectively replaces the 
 distribution of bi-poles as far as the determination of the magnetic field outside 
 the magnetic matter is concerned, a determination which is valid up to 
 within a physically small differential distance from the matter. If however 
 the point at which we wish to investigate the field is inside the magnet we 
 must as usual put a physically small cavity round it and define the field there 
 
 * Mem. dc VAcad. 5 (1826), pp. 247, 488; 6 (1827), p. 441. Cf. also Maxwell, Treatise, n. 
 385. 
 
238 Magneto-statics [CH. vi 
 
 as the field inside this cavity due to the distribution beyond it which is 
 effectively represented as above by the distribution of ideal magnetic matter 
 and which therefore includes a part due to the distribution a = I n of magnetic 
 matter on the surface of the cavity : together with the field due to the matter 
 inside the cavity. This latter part of the field with the distribution on the 
 walls of the cavity are the parts, and the only parts, of the field at the internal 
 point which depend appreciably on the local molecular configuration, and 
 which therefore in any real case are quite unknown : we have however 
 separated them from the definite part of the field due to the magnetic body 
 as a whole which is specified as due to the distribution of magnetic matter of 
 density p throughout the magnet and a on its outer boundary, the field of 
 these latter involving no .appreciable local part. We then follow the usual 
 course in physical theories and ignore the local part of the field as being 
 ineffective as regards the conditions of the matter in bulk. 
 
 We thus define the magnetic field inside the body as that field when the 
 effect of the local parts is rejected. It is therefore completely defined as 
 due to the distribution of ideal magnetic matter throughout the volume of 
 the magnet and on its surface. The force and potential of the field due to 
 such a distribution are expressible by definitely convergent integrals at 
 internal as well as external points. Moreover on this definition the magnetic 
 force is the vector which is represented by the gradient of the potential 
 function. 
 
 252. If the direction of magnetisation 7 at each point of the body is 
 the same, say parallel to the cc-axis, then its potential at the point (, 77, ) is 
 
 8 /IN , f T 9 /r 
 
 <A = M o v 
 .'. ox\r. 
 
 and since 7 is not a function of (|, 77, ), but only of (x, y, z) this is 
 
 8 
 
 Hf - == '"Trf** 
 C?c 
 
 Thus if we know the gravitational potential at the point (, 77, ) of the 
 same body with a distribution of density p = 7 throughout its volume we 
 can at once deduce the potential of the field of the body magnetised at each 
 place parallel to the axis of x to intensity 7. 
 
 For example the gravitational potential of a sphere uniformly charged 
 to density p = 7 throughout its volume has a potential at external points 
 distant r from its centre 
 
 and thus if we consider the same sphere uniformly magnetised parallel to the 
 axis of x at each point to intensity / the potential of the magnetic field is 
 
251-254] The magnetic field 239 
 
 253. The mathematical relations of the magnetic field. We have now 
 a consistent scheme to which to apply our analysis and we proceed in this 
 exactly as before : we denote the magnetic force by H and the potential 
 by and thus H = - grad 0, 
 
 and the magnetic potential / satisfies the conditions that at each point of 
 
 space V 2 ^ = - 4rj/>, 
 
 and at a surface of discontinuity in the medium ^r is continuous but 
 
 _ 
 
 dn dn 
 
 p and a as before, in this chapter, denoting the volume and surface densities 
 of Poisson's ideal magnetic matter, so that 
 
 p = div I, 
 
 tf = I W5 
 whence the conditions for e/ can be interpreted in the form 
 
 V 2 = - div H = 4rr div I, 
 
 and H ln -H 2w = 47rl n . i 
 
 Thus div (H + 477!) - 0. 
 
 The vector B = H + 4wl, 
 
 is everywhere solenoidal : it satisfies the usual hydrodynamical equation of 
 continuity and is usually called the magnetic flux or induction vector : it 
 is of immense importance in the theory. The fact that it is solenoidal simply 
 means that the surface integral of the normal induction over any closed 
 surface whatever is zero. For if we take the integral 
 
 (div B) dv 
 
 throughout any region bounded by the surface / it must vanish for at each 
 point div B = ; but by Green's theorem it consists of 
 
 '/ 
 
 together with the surface integrals arising from discontinuities when we pass 
 into the magnetic matter : these are the integrals of 
 
 over the surfaces concerned or of 
 
 HW! ~ H M2 4:7Tl n , 
 
 which is zero : this establishes the result. 
 
 254. Thus if we plot the magnetic field in terms of the lines of magnetic 
 induction and form them into tubes, we see that the tubes of magnetic 
 induction are always closed and the product of the strength into the cross- 
 
240 Magneto-statics [on. vi 
 
 section is constant along each tube so that the normal induction over any 
 closed surface is zero. It follows that the value of the surface integral over 
 any part of a surface depends solely on the boundary curve and it must be 
 able to be expressed in terms of the position of the boundary. This is accom- 
 plished by means of Stokes' conversion of a surface integral into a line integral 
 and requires us to find a vector which is such that its components (A,,., A y , A z ) 
 satisfy 
 
 dy dz 
 and two similar equations : this is expressed in the form that 
 
 B = curl A, 
 and then for any 'surface / and the boundary curve 6- 
 
 B n df= I (Ads), 
 
 the former integral being taken over / and the latter round its boundary 
 curve. 
 
 The vector A is called the magnetic vector potential* and its significance 
 will subsequently appear. 
 
 255. We know that the potential of the particle placed along the axis 
 of z at the origin is 
 
 / _ mz 
 
 T = 73 ' 
 
 so that by tfefmition of the vector potential A 
 
 9A Z . 9Aj, _ difj _ 3mxz 
 
 H~~d* = ~ dx = r 5 ' 
 
 
 3myz 
 ~ 
 
 dA y dA x _ m 3wz 2 
 
 dx dy r 3 r 5 
 
 These equations are solved by 
 
 . my . _ mx . 
 
 A * = " ~ r 8 ' A V ~ ^ ' AZ = U ' 
 
 The result is that the vector potential in the case of a magnetic particle 
 is at right angles to the axis of the magnet and to the radius to the point and 
 
 is of magnitude - - ^ , where 6 is the angle between the axis of the magnet 
 
 and the radius to the point and its sense is that of positive rotation round 
 
 , , m sin 6 mr sin 6 , 
 
 tne axis 01 the magnet. Ihe expression - - = -- ^ shows that we 
 
 * Maxwell, Treatise, n. 405. 
 
254-256] 
 
 The vector potential 
 
 241 
 
 may break it up into components for mr sin 6 may be interpreted as twice 
 the area of the triangle formed by m and r and the vector is at right angles 
 to the area and its components are the projections. If we now use generally 
 
 Fig. 51 
 
 r to denote the radius vector from the centre of the magnet to any external 
 point in the field, m the vectorial moment of the particle and r x the unit 
 vector along r. then we see that 
 
 r 2 r 
 
 is the general expression for the vector potential at the point in the field. 
 
 256. The above formulae for the vector potential of a magnetic particle 
 may now be used to enable us to write down a vector potential for any finite 
 magnet. We have merely to replace m by Idv and then integrate over the 
 whole body. We get at once 
 
 a formula which certainly applies at points outside the magnetism. At 
 points inside the magnetism r can vanish in an element of the integral but 
 the integral is nevertheless quite convergent and A is thus representable 
 quantitatively, in a physical theory which neglects purely local actions, by 
 this integral at every point of the field. But as regards its differential 
 coefficients, by means of which B is derived from it, it is dependent on the 
 
 * Maxwell, Treatise, n. 405. 
 L. 16 
 
242 Magneto-statics [CH. vi 
 
 unknown distribution of the local polarity. This is of course quite in 
 keeping with the fact that the magnetic field inside the magnet, considered 
 as due to the aggregate of the molecular magnetic elements, by means of 
 which the vector potential is defined through the relation curl A = B, itself 
 involves this local contribution. But in the previous theory we were able 
 to discard the local part of the magnetic force, depending on the molecular 
 character of the distribution at the point, from which alone indefiniteness 
 arises. It may be surmised that we should in like manner discard from 
 the vector potential the purely local contribution which is the source of its 
 discontinuity. This may be effected as usual by the aid of integration by 
 parts. At a point inside the material medium the field may be then 
 separated into two parts ; the first due to the medium beyond a physically 
 small closed surface surrounding it and for this part the function A and its 
 first gradients can be represented analytically by integrals of the above type 
 which are entirely convergent and determinate since r cannot be less than 
 a finite lower limit ; the second part is due to the elements inside the surface 
 thus drawn. On integration by parts the contribution of the former to the 
 expression for A may be put in the form 
 
 curl I 
 
 - dv + 
 
 r 
 
 r 
 
 and is thus expressible as a volume integral together with an integral over 
 interfaces of transition of the magnetism, and also an integral over the surface 
 of the cavity : the volume integral is convergent and does not depend on 
 the form of the cavity, while the integral over the surface of the cavity is 
 finite and with the part due to the distribution inside the small surface drawn 
 is the sole representative of the influence of the local molecular configuration ; 
 in our present procedure it depends on the form of the cavity; in actual 
 practice it depends on the local molecular configuration. By the general 
 principle, the mechanically effective functions are the analytical integrals 
 obtained by excluding this undetermined local part. This leads to an expres- 
 sion for the total vector potential of the medium treated as continuous 
 
 r curl I, /"[nil],/-* 
 A = - dv + -d/*, 
 
 T T 
 
 the latter surface integral being taken over the transition boundary 'of the 
 magnetism. 
 
 257. As a final result we may quote the case of a uniform magnetic 
 shell of strength r on the surface /. This is obtained from the form for 
 the particle by replacing m by i^rd/and integrating over the surface of the 
 shell : this gives 
 
 * Larmor, Aether and Matter, p. 260. 
 
256-259] Permanent and induced magnetism 243 
 
 which on reconversion by Stokes's Theorem gives 
 
 ds 
 
 this integral being taken round the boundary of the shell. 
 
 258. So far we have been dealing with generalities and have considered 
 the actual distribution of magnetisation in a magnet as given explicitly 
 among the data of the problem. We have made no assumption as to 
 whether this magnetisation is permanent or temporary, except in those 
 parts of our reasoning in which we have supposed the magnet broken up 
 into small portions, or small portions removed from the magnet in such a 
 way as not to alter the magnetisation of any part. 
 
 We must now consider, as in the analogous dielectric problem, the 
 magnetisation of bodies with respect to the mode in which it may be produced 
 and changed. A bar of iron held parallel to the direction of the earth's 
 magnetic field is found to become magnetic, with its poles turned the opposite 
 way from those of the earth, or the same way as those of a compass needle 
 in stable equilibrium. 
 
 Any piece of soft iron placed in a magnetic field is found to exhibit 
 magnetic properties. If the iron is removed from the field, its magnetic 
 properties are greatly weakened or disappear altogether. On the other hand 
 a piece of hard iron or steel retains its magnetic properties acquired when 
 placed in a magnetic field. 
 
 259. If a magnet could be constructed so that the distribution of its 
 magnetisation is net altered by any magnetic force brought to act on it, 
 it might be called a permanently or rigidly magnetised body. There is no 
 known magnetic substance which perfectly fulfils this condition, but it is 
 nevertheless convenient for scientific purposes to make a distinction between 
 the permanent and temporary magnetisation, defining the permanent magne- 
 tisation as that which exists independently of the magnetic forces, and 
 the temporary magnetisation as that which depends on those forces. This 
 distinction is however not founded on a knowledge of the intimate nature 
 of the magnetisable substances : it is only the expression of a convenient 
 hypothesis*. 
 
 * Various forms of the mathematical theory of magnetism have recently been constructed 
 on the assumption of the existence of a distribution of permanent magnetic matter. In this 
 case the magnetic induction vector is no longer solenoidal, its divergence determining the density 
 of the magnetic matter. This procedure is adopted in order to secure a closer analogy with the 
 electric case and to remove certain discrepancies supposed to exist in the more usual form of the 
 theory : in reality it confuses the point at issue and only complicates a perfectly valid theory. 
 Cf. E. Cohn, Das dectromagnetische Feld, p. 510; R. dans, Ann. der Physik. 13 (1904), p. 634 and 
 Encyclopadie der math. Wissensch. Bd. v. Art. 15. 
 
 162 
 
244 Magneto-statics [CH. vi 
 
 Our theory thus divides itself into two distinct parts which although they 
 are involved and dependent on one another are conveniently treated separately, 
 we shall therefore begin by a short discussion of the relations between rigidly 
 magnetised substances and the field they create and follow it by the more 
 general case of induced magnetisation. 
 
 260. Permanent magnetism. The above analysis of course applies to 
 this case. If the magnetic field is due to the permanent magnets alone the 
 field of their polarised elements is all that there is. The potential iff of this 
 field is due to the distributions of ideal magnetic matter consisting of a 
 volume density p _ _ ^iv j 
 
 throughout the volume of the magnets with 
 
 ^0 n 
 
 over their surfaces. I is of course the vector defining the intensity of the 
 permanent magnetisation at each point of the body. The magnetic force H 
 is defined as the force due to the above distributions so that everywhere 
 
 H grad i/j, 
 
 and also V 2 = div H = - 47rp = 4?r div I, 
 
 so that if the induction is B = H + 47rl 
 
 we have div B = 0, 
 
 with the condition at the surface of the magnet that 
 
 H l n - H 2 n = 477l n , 
 
 or interpreted in terms of the induction 
 
 so that B is a solenoidal vector as in the general case. The vector potential 
 of the permanent magnetism A is thus derived so that 
 
 B = curl A. 
 
 These general results of the analysis of the field are useful only in as far as 
 they enable us to determine the really essential and accessible quantities of 
 such fields, viz. those defining the observable mechanical relations of the 
 magnets giving rise to it. Let us consider a few applications in this respect. 
 
 261. What is the work done in introducing a permanent magnet into a 
 magnetic field of specified intensity ? In other words, what is the mechanical 
 potential of the magnet when existing in a given position in any given field ? 
 
 In displacing a small elementary doublet of moment m through a small 
 distance ds the work done is, as before. 
 
 j 
 
 - m w ds > 
 
 H denoting the strength of the field at the position of the doublet. 
 
259-262] Permanent magnetism 245 
 
 If now we consider all the little magnets in the element dv of the magnet 
 in any position we can sum them up into a polarisation per unit volume, the 
 
 intensity I x being such that 
 
 Ij_dv = Sm; 
 
 and each of these elementary magnets receives the same small displacement 
 ds at a place where the total magnetic force is H (due partly to the original 
 field and partly to the magnet itself) and is practically the same for them all. 
 Thus the work done in the displacement of the small volume element of the 
 magnet is 
 
 or for the whole finite magnet the work done during the displacement ds of 
 each element is 
 
 If we make the usual assumptions as to the reversibility of the actions we can 
 conclude in the usual way that the potential energy of the magnet in the 
 field is 
 
 JP. 
 
 since Ij is constant for any displacement of the magnet which does not alter 
 its shape Or form. The linear component of the force in any direction acting 
 on the magnet in this position is 
 
 dw r/_ an\ , 
 
 F = - -3 - Ij 3- )dv 
 ds J \ os / 
 
 jH) dv. 
 
 262. But the magnetic field consists of two parts, one due to the original 
 field and the superposed part due to the magnet itself; denoting the inten- 
 sities in these separate parts by H 2 and H x respectively, so that 
 
 we have W = dA) dv + (IjB.*) dv, 
 
 and the first integral on the right is independent of the position of the magnet 
 in the field ; it is a constant for any displacement of the magnet which does 
 not alter its form or magnetisation. We may thus simply regard 
 
 as the potential of the given magnet in the external field H 2 . 
 
246 Magneto-statics [CH. vi 
 
 263. Now suppose this external field H 2 is the field due to a second 
 permanent magnet, the work of bringing the first magnet up from a great 
 distance to its relative position is, as we have seen, equal to 
 
 F 12 = I (IA) dv. 
 
 If however the second magnet had been placed in position first and the first 
 magnet brought up so as to obtain the same relative configuration, the same 
 work would obviously have been done, but this time it is expressed by 
 
 so that we must have 
 
 where the combined integral is taken over the whole of the space including 
 the two magnets. Since however 
 
 r r 
 
 I (liHj) dv and I (I 2 H 2 ) dv 
 
 are both constant for displacements of the two magnets we may take 
 
 t 
 
 W = 4 (li + I 2 , Hj + H 2 ) dv + constant 
 J 
 
 for the potential energy of these magnets relative to one another. If we use 
 now T T -i- T 
 
 JL Xj -\- X 2 
 
 to denote generally the polarisation in any element of the field (I I t in the 
 first magnet and I 2 in the second and is zero everywhere else), we have then 
 
 W = \ I (IH) dv + const., 
 
 the integral being extended throughout the whole field. 
 
 A convenient interpretation of this quantity in terms of the coordinates 
 of the respective magnetic masses would enable us to determine the whole 
 force system exerted by one magnet on the other. 
 
 264. The general properties of the fields of permanent magnets and 
 their mechanical reactions are illustrated by application to the simplest 
 case when the magnets are very small doublets ; we have first to examine 
 more closely the field of a single doublet and then we can easily determine 
 the mutual potential and reaction forces of two such doublets. 
 
 The field of a magnetic particle. The potential due to the small magnetic 
 particle AB of moment p at a point P distant r from its centre is 
 
 , [i cos 6 
 
263-265] 
 
 The field of a magnetic particle 
 
 The components of force in the magnetic field at P are 
 (i) along OP 
 
 0J/r 2/A COS 6 
 
 ~ = ~ 
 
 247 
 
 +m A 
 
 ~m B 
 
 Fig. 52 
 
 (ii) at right angles to OP 
 
 sin 6 
 
 rd6 r 3 
 
 Fig. 53 
 
 The resultant force makes with the radius vector r an angle 
 
 <f> = tan" 1 (| tan 6), 
 
 and the tangent to the line of force at P meets the axis of the magnet produced 
 at a distance equal to 
 
 r sin </> _ r r * 
 
 sin ((f> + 6} ~ cos B + 2 cos 6 ~ 3 cos ' 
 
 265. The mutual potential of two small magnetic particles*. The mutual 
 potential of the two particles is the potential of one in the field of the other. 
 
 * Tait, Quarterly Jour, of Math. Jan. 1860. Cf. his book on Quaternion*, 442, 443. 
 
248 
 
 Magneto-statics 
 
 [CH. VI 
 
 The two particles are /u, at AB and p at AB'. We calculate the potential 
 of the second magnet in the field of the first as the sum of the potentials due 
 to its constituent poles m' at A' and m' at B' '. It is therefore the limit of 
 
 (m f cos AOA' m'cosAOB') 
 
 = Lt 
 
 OA'* 
 
 m' 
 
 j 
 
 /* I QA^ l r S " T ( 
 
 m' 
 
 ~"T C086 
 
 = Lt ULT COS 
 
 - cos 
 
 ' J ~ m/ ( f ~ "5" cos ^') 
 
 COS 
 
 = *-*?- (cos e 3 cos cos 6'), 
 r 3 
 
 where e is the angle between the positive directions of the axes of the particles 
 and 6, 6' the angles these axes make with the radius joining the centres from 
 one definite magnet to the other. 
 
 Fig. 54 
 
 If now we use (Z 1} m l , %), (1 2 , m 2 , n z ) as the direction cosines of the axes 
 of these magnets and (X, JJL, v) as the direction cosines of the radius vector 
 the mutual potential can be written in the form 
 
 (1 2 X 
 
 ML 
 
 which we might evidently have obtained otherwise as 
 
 1 51 1 ( *s 5~ 
 
 L 3~ I \ tfi x 
 
 + 
 
 where 
 
 r 2 = x 2 + 2 + 
 
265-267] Two magnetic particles 249 
 
 This may be written in the form 
 ' 
 
 - 
 
 266. Determine the wrench exerted by one magnet on the other. This 
 can be determined from the above value of the mutual potential, for if this 
 
 dW 
 is W the force component parallel to the x-axis is -- - or 
 
 (I 2 x + m 2 y 
 
 A simplification is however introduced if we resolve the resultant force into 
 three directions parallel to the radius and the axes of the two magnets 
 respectively. If these components are I, II, III then the above expression is 
 
 I/! + IIZ 2 + III -, 
 and by comparison with the above we see that 
 
 11 4 
 
 ,j = 3^' /Ijx + m z y + n^z\ 
 r 4 \ r r r /' 
 
 r r r 
 
 m = q ""' 
 
 or returning to the original notation, 
 
 3m// cos 6' 
 1 = -. , 
 
 -|--|- VjLVfA/ OC/O 
 
 TTT 3u,u' COS e 15u.a' 
 III = _t^l_ - 
 
 cos cos 
 
 267. We must now determine the couple on the second magnet and to 
 do this we reduce the wrench on it to a force at its centre and the couple. 
 The force is as given above : to determine the couple we proceed in a slightly 
 different manner although it is possible to deduce it from the potential 
 energy. The couple acting on any small magnet in any field when the 
 centre is the base point is at once written down when the force at the centre 
 due to the field is known. If H be this force there is a pair of equal and 
 opposite parallel forces mH acting on the poles and the moment of the couple 
 
250 Magneto-statics [CH. vi 
 
 is jj,H sin (f> and it is in the plane of the force and the axis tending to set 
 the- axis along the force ; (f> is the angle between the axes and the direction 
 of the force. 
 
 [The potential energy of the small magnet in the field is 
 
 /, 9A dlfj dlff\ I -i 
 
 u, u ~ + m -- + n 7=?- = ^ti cos 0.J 
 
 V 80; oy dzj 
 
 We have already seen that the force at the centre of the second magnet 
 
 3u, cos 6 ,. 
 
 can be resolved into two components, one -j outwards along the radius 
 
 mH 
 
 and the other ^ parallel to the axis of the first magnet. It follows that 
 
 the couple on the second magnet in the plane of its axis and the radius 
 vector is towards the radius vector and of amount 
 
 cos 6 sin 6' 
 
 3 
 
 and the couple in the plane of the axis and a line parallel to the axis of the 
 first magnet through its centre is 
 
 //,/// sin e 
 -75 ' 
 
 268. There are two particular cases of these general results which are of 
 historical interest as they provided the means by which Gauss proved that 
 the law of magnetic attractions was that of the inverse square. 
 
 (i) (ii) 
 
 S D 
 
 D A B 
 
 A ty 
 
 Fig. 56 
 
 The positions of the magnets are clearly indicated in the diagrams, the 
 axes being in each case in one plane. 
 
267-270] Induced magnetism 251 
 
 In the first case the force at the centre of the second magnet is ^ and so 
 
 ' 9 ' 
 
 the couple on it is ^- ; in the second case the couple is -~- : in the one case 
 
 the couple is double what it is in the other and this is Gauss' result. 
 
 269. Induced magnetisation. We shall now investigate temporary mag- 
 netisation on the assumption that the magnetisation of any particle of the 
 substance may be dependent on the magnetic force acting on that particle. 
 This magnetic force may arise partly from external causes and also partly 
 from the magnetisation of the neighbouring particles. . 
 
 A body whose magnetisation is altered in virtue of the action of a magnetic 
 force is said to be magnetised by induction, and the alteration in magnetisation 
 is said to be induced by the magnetising force. 
 
 The problem before us is really the determination of the alteration of the 
 magnetic field due to the introduction of a piece of a magnetisable substance 
 (which may for generality also have permanent magnetism in it). The theory 
 states that this alteration is due to the fact that the substance becomes 
 magnetised, i.e. each little bit of it becomes a[jlittle magnet under the influence 
 of the field. If this is so then a knowledge of the intensity of magnetisation 
 induced would be sufficient to enable us to determine the whole of the 
 circumstances. But this is what we do not know. We can however make 
 a theory and see how it actually agrees with the facts. This is the procedure 
 adopted for dielectrics in the previous chapter and we shall follow the same 
 course of reasoning as there set out. 
 
 The presumption is that the magnetisation is conditioned by the magnetic 
 force and thus if there is to be any law about the matter at all we must have 
 the polarisation intensity I at any point in the medium as a function of the 
 total magnetic force H at that point 
 
 I=/(H). 
 
 If the theory is to be at all workable this relation must be a linear one, which 
 in isotropic media assumes the form 
 
 I = //H + I , 
 where I = if there is no permanent magnetisation. 
 
 This simple law is of course right if the field is small or when the substance 
 is but slightly magnetic. In other cases it is of no use but we can do no 
 better. 
 
 270. Now let us examine the magnetic field in which such a relation 
 holds. The mathematical formulation involves three vectors: (i) I, the 
 intensity of magnetisation, (ii) H, the magnetic force which is the gradient 
 of a potential tfj and (iii) B, the magnetic flux which is a stream vector ; and 
 
 then B = H + 47rl, 
 
252 Magneto-statics [CH. vi 
 
 so that only two of the variables are really independent in the general theory. 
 
 Now- we have I = I + /H, 
 
 so that B = H (1 + 47r/t') + 47rI , 
 
 where I represents the distribution of permanent magnetism. 
 
 The coefficient (1 + ^TT^') is called the coefficient of induction of the medium 
 and is usually denoted by /x, so that 
 
 B = fjR. + 47rI , 
 
 IJL is Kelvin's permeability* : if ^ is big the substance is very permeable to 
 magnetic force. 
 
 We can express everything in terms of the magnetic potential / : in the 
 most general case fa v jj = Q 
 
 and H = grad ?/, 
 
 so that div (pH + 47rI ) = 0, 
 
 or in other words div (JJL grad /r) = 4vr div I , 
 
 which is the characteristic equation for the magnetic potential in the theory. 
 To obtain a solution for the problem of the disturbance of the magnetic field 
 by the introduction of such magnetic media we have to find suitable integrals 
 of this equation in the various regions involved and fit them up at the 
 boundaries. 
 
 271. A knowledge of the conditions which hold at a surface of dis- 
 continuity in the medium is therefore essential to the theory. These are 
 easily obtained in the usual manner. In crossing any interface in the 
 medium the normal component of the induction is always continuous or 
 
 Bj -B 2 =0 
 
 L n 'n 
 
 suffices 1 and 2 denoting the different media on the two sides of the surface. 
 But B x = H, + 47T OI/H! + I 0l ), 
 
 and B 2 = H 2 + 477 (/u 2 'H 2 + I 02 )> 
 
 so that /*iH ln + 477l 0in = /*2H 8n 
 
 or in terms of the potential 
 
 If as is often the case one of the media is air then we can put /x 2 = 1, I , = 
 so that the condition is 
 
 " Theory of induced magnetism," Reprint of Articles oil Electricity, etc. p. 484. 
 
270-273] The general problem 253 
 
 or if there is no permanent magnetism at all 
 
 di/Tj 3j/r 2 
 
 dn dn ' 
 
 In addition it is obvious that the potential /r must be continuous across any 
 such interface and therefore also the tangential component of the magnetic 
 force is also continuous. 
 
 272. To solve the problem of the disturbance produced by and in a 
 given piece of magnetisable substance introduced into a magnetic field 
 we have to determine i/r, a continuous function, to satisfy the following 
 conditions : 
 
 1. In free space ^2,7, = Q 
 
 and at a great distance from the field / and the force components are zero. 
 
 2. In the magnetisable substances 
 
 and at the boundary, if ifj { and /r refer to internal and external potential 
 
 dtjfi = (tyo 
 
 " dn dn ' 
 
 where 8n is the element of the outward normal. 
 3. In the rigid magnets if any 
 
 V 8 = ^ d i v I, 
 and at their surface gradn . _ gradn ^ = 4^ 
 
 273. A sphere of soft iron* in a uniform magnetic field. Here iff is 
 regular except at infinity where it is like Hx and V 2 /r = everywhere. 
 r y \fj i = Ax inside, 
 
 i/r = Hx -\ 3- outside. 
 
 These satisfy all but the surface conditions. Continuity of potential at the 
 surface r = a gives 
 
 ,4 = -# + - 3 . 
 
 a 3 
 
 and continuity of induction gives 
 
 fj.A = -H -^, 
 
 so that A = - - H, B = '- -* #a 3 , 
 
 2 + n ft + 2 
 
 
 
 * Pnisson, Z.c. p. 164. Cf. also Somigliana, Re.nd. del r. Inst. Lomb. 2 (36), (1903); Boggio, 
 ibid. (2) (37), (1904), p. 123 and Nuovo Cimento (5), 11 (1906), p. 1. 
 
254 Magneto-statics [OH. vi 
 
 and the inside potential is 
 
 and the outside one 
 
 - 1 a 3 
 
 The strength of the field inside, jthe iron is 
 
 3H 
 
 2+7t' 
 
 and is considerably smaller"- than H if //, is very big. The intensity of 
 magnetisation on the other hand is 
 
 and since fj, = 1 + 47771,' this is 
 
 / = 
 
 and if jx' is large this is practically 
 
 3 77 
 4^^' 
 
 so that as the susceptibility of the substance increases the intensity of the 
 magnetisation induced in it by a given field approaches a limiting value beyond 
 which it cannot go. 
 
 274. The induction in the iron is 
 
 R _ fj _ 3 ^ JT 
 
 ~2 + ^ 
 
 but outside at a distance from the sphere 
 
 B = H. 
 
 Now consider what this means. Take a tube of magnetic induction whose 
 cross-section at a distance from the sphere is ds. Now we know that along 
 this tube 
 
 Bds 
 
 is constant so that the cross-section of this tube in the sphere is ds^ where 
 
 Hds = fjiHdSi, 
 
 ds 
 
 dSi = , 
 V- 
 
 so that if /x is very big the cross-section of the tube in the iron is very small 
 compared with that outside at a distance. This illustrates very vividly how 
 
273-275] Induction in the spherical shell 
 
 255 
 
 the tubes of magnetic induction are gathered up into the iron so that the 
 magnetic flux in the field converges into and is concentrated by the sphere. 
 This is of course precisely the important practical property. The function 
 of soft iron in dynamos is to collect the magnetic flux, but not the force. 
 The force in the iron is in fact very small if /* is big. 
 
 Notice again that as /u, (or 
 reaches the limit 3H. 
 
 Fig. 57 
 
 ) increases the magnetic induction also 
 
 275. The spherical shell of magnetic material in the uniform field* . Let 
 H be the strength of the field, then Hx is the part of the potential due 
 to the given field and it is evident that the expressions for the potential in 
 the three given regions must be of the form : 
 (i) inside the shell tff l = Ax, 
 
 (ii) in the material of shell 
 
 /r, = Bx 
 
 Cx 
 
 r S> 
 
 (iii) outside tfi 3 = Hx + 
 
 Continuity of potential gives 
 
 Continuity of induction gives 
 
 from which we obtain 
 A B C 
 
 Dx 
 
 2/* 
 
 a 3 (p - 1) 
 
 c 
 
 ...(1), 
 
 a z 
 D C 
 
 ...(2). 
 
 b 2 6 2 
 
 . (3) 
 
 a 3 A 
 2pC 2D 
 
 '\*'/> 
 
 (4) 
 
 6 3 cr 
 D -3# 
 
 V^/j 
 
 (2+ ft 
 
 * Poisson, I.e. p. 164. 
 
 (2/* + i) - ~ (p - iy 
 
256 Magneto-statics [CH. vi 
 
 i 
 
 The internal field has a potential 
 
 - SHxt , 
 
 (2 + ^(2^+1) --^Ou-1) 2 
 
 This is the interesting result. The field inside the shell is reduced in the 
 ratio 
 
 which is very small if /A is very, big however thin the shell may be. This 
 illustrates the problem of magnetic screening; in order to shield delicate 
 instruments from influence by dynamos or other external actions they are 
 enclosed in a shell of iron. 
 
 276. As a final example we may examine the induction in an ellipsoid* 
 of magnetic matter placed in a uniform field of force with one of its axes 
 parallel to the direction of the undisturbed field. The analysis is precisely 
 similar to that given for the analogous case in dielectric media and need only 
 be briefly indicated in this case. 
 
 The potential of the field in the undisturbed state is, except for a con- 
 stant, tfi= -Hx, 
 
 if the axes of coordinates are chosen along the -principal axes of the ellipsoid 
 with the origin at the centre. 
 
 We are then induced to try potentials 
 
 -00 7, 
 
 J/TO = Hx + Lx - 
 
 . A (a 2 + t) V(a 2 + t) (b z -j- t) (c 2 + t) 
 in outside space and ,/,. _ _ jj x 
 
 for the space inside the ellipsoid; and we must now try and find L, L' to 
 satisfy the continuity conditions at the boundary. 
 
 Continuity of potential requires 
 
 - rit 
 
 _ TT _ Tt _ T _ tt/t/ _ 
 
 . o (a 2 + t) V(a z + t)(b z + t)(c* + t)' 
 whilst continuity of induction requires 
 
 - o (a 2 + t) V(a 2 + t) (b 2 + t) (c 2 + t) a*bc 
 Whence using, as before 
 
 r dt 
 
 o (a 2 - 
 
 A = 
 
 * Neumann, Jour. /. Math. 37 (1848), p. 21; Vorlev&iigen uber die Theorie des Magnetismus 
 (Leipzig, 1881); GuUani, Nuovo dm. (3), 11 (1882). 
 
275-277] Neumann's theorem 257 
 
 a 3 bc u, 1 H 
 
 G T 
 
 we find that L ~ 
 
 A(p- l)> 
 
 so that the modified forms of the potential for the disturbed field are 
 
 . a*bc (a - 1) Hx f 00 dt 
 
 Ao = - Hx + - - , 
 
 2 1 + A (IJL - 1).' A (a 2 + t) V(a* + <) (6 2 + t) (c 2 + 
 
 - Hx 
 
 The strength of the field inside the ellipsoid is 
 
 H 
 
 i + A(p-iy 
 
 and is as a rule considerably smaller than H, at least if the ellipsoid is made 
 of iron for which /z is very big. The important new point is however that 
 the strength of this field depends on A, and if A is very small, it is in fact 
 identical with the applied field. This means that if the ellipsoid is very 
 long in the direction of the field, as compared w T ith its dimensions in the 
 perpendicular directions, the field in its interior is practically the same as 
 the applied field. 
 
 This fact is employed in the determination of the coefficient ju, ; for if we 
 know r the magnetisation induced in a piece of iron by a given field the ratio 
 of the two determines the constant // if the internal field of the magnetisation 
 is negligibly small, as would for instance be the case if the piece of metal is 
 very extended in one direction. For this reason long iron wires placed parallel 
 to the lines of the applied field are used in such determinations. 
 
 The question of the forces on the ellipsoid magnetised by induction is 
 identical with the corresponding question for dielectric media and the results 
 are analogous. We need not therefore stop to consider these questions 
 further*. 
 
 277. Neumann's theorem. There is an important reciprocal relation 
 established by Neumann which, although it really depends on principles to 
 be subsequently established, is worth quoting at this stage on account of 
 its importance in determining details of the polarisation induced in pieces of 
 magnetic metal in any field. If in a given inducing magnetic field H a piece 
 of iron has induced in it a polarity I and in another field HQ the polarity is 
 I then, if the polarisation follows a linear law, 
 
 both integrals being taken throughout the mass of the iron. 
 
 * The problem of two spheres has been examined by C. Neumann, Hydrodynamische Unter- 
 suchungen (Leipzig, 1883), p. 282; R. A. Herman, Quarterly Jour, for Math. 22 (1887), p. 204; 
 Boggio, Rend. del. r. Inst. Lomb. (2), 37 (1904), p. 405. 
 
 L 17 
 
258 Magneto-statics [CH. vi 
 
 Suppose the total field Avhen the iron is in position is in the first case 
 H + H' and in the second H + H '. 
 
 Now the law of induction gives us that the components of I are conjugate 
 linear functions of the components of (H + H') however aeolotropic the 
 medium may be. Similarly the components of I are linear functions of 
 (H + H ') with the same coefficients. It thus easily follows that 
 
 (I . H + H ') = (I . H + H'). 
 Thus the given result is true if 
 
 where of course H ' is the field due to the distribution of magnetisation 
 specified by I ', and H' that due to the I' : the left-hand side of this equation 
 is therefore the work required to bring a "body polarised to intensity I into 
 its position in the field H ', which is obviously equal to the work required 
 to bring the body polarised to intensity I into the field H. The result is 
 therefore established. 
 
 An important application of this theorem is obtained when we know that 
 one of the fields is uniform, H a say, for then we have 
 
 l(IoH) dv - (HO ;\l&v 
 
 If therefore ,we know the details of the distribution of polarisation when the 
 body is in the uniform field (i.e. I ) then we can find 
 
 Uv, 
 
 when the body is placed in any field. 
 
 We know for example the polarisation induced in an ellipsoid in a uniform 
 field. We could therefore write down the total moment of the same ellipsoid 
 in any field ; it depends upon the volume integral of the inducing force. 
 
 This is a short sketch of the general theory of induced magnetism con- 
 structed from the view which regards it as a particular example of the general 
 theory of polarised media. With the help of this theory we can investigate 
 the fundamental questions concerning the energy and ponderomotive forces 
 acting on such magnetic media. But before doing this a few r words may be 
 said as to the validity of the theory from the point of view of actual experience. 
 
 278. Paramagnetism, diamagnetism and ferromagnetism. The theory 
 of induced magnetism given above is based on the linear relation between the 
 magnetisation intensity I and the magnetic force H which in isotropic media 
 is of the form T _. ,/TT 
 
 * A* **> 
 
 or in terms of the permeability /u, 
 
277-279] 
 
 Ferro magnetism 
 
 259 
 
 If IJL > I we see that the magnetisation has the same sign as the magnetising 
 force but if p < 1 the signs are opposite. The important thing is now that 
 there are bodies of both classes : there are in fact many quite ordinary 
 substances for which yu, < 1 and in which therefore the direction of the 
 magnetisation induced is in the opposite direction to the inducing field. 
 Such substances are called diamagnetic substances to distinguish them from 
 the more regular substances in which p > 1 and which are called paramagnetic 
 substances. 
 
 In all diamagnetic substances the value of /// is extraordinarily small : 
 it is largest in bismuth where however it only amounts to 2'5 . 10~ 6 . In 
 paramagnetic substances it is always equally small except in substances of 
 the iron group. The above theory would therefore be completely effective 
 for all substances except perhaps those of the latter group, which are called 
 ferromagnetic substances. The typical examples 'of the ferromagnetic sub- 
 stances are iron, nickel and cobalt. 
 
 279. In the ferromagnetic substances, all of which are strongly magnetic, 
 the simple relation between the magnetisation and magnetic force on which 
 the above. theory has been built is by no means valid except perhaps when 
 the magnetic inducing force is very small in which case it is necessarily 
 sufficient. In such cases it is found best to represent the relation by a 
 graph which can be determined empirically for any particular mass of the 
 substance under consideration. The type of curve which is always obtained 
 is exhibited in the figure where the ordinates represent for a particular 
 specimen of iron the values of I, the magnetisation induced, the abscissae 
 the values of H, the magnetic force. 
 
 Kg. 58 
 
 For very small values of H the curve is straight indicating that the 
 permeability, which is still defined by the relation 
 
 i 
 
 = 1 
 
 H 
 
 172 
 
260 Magneto- statics [OH. vi 
 
 is independent of the magnetic force. When however the magnetic force 
 increases beyond about '02 absolute units the curve begins to rise rapidly 
 and the value of p is greater than it was before for small magnetic forces. 
 The curve rises rapidly for some time when the force is about five absolute 
 units : it then begins to get natter and there are indications that for very 
 great values of the magnetic force the curve becomes a straight line parallel 
 to the axis of the magnetic force. This means that the intensity of magnetisa- 
 tion does not further increase as the magnetic force increases. When thi& 
 is the case the iron or other magnetisable substance is said to be saturated. 
 For a specimen of soft iron examined by Ewing* saturation was practically 
 obtained when the magnetic force was about 2000 absolute units. For steel 
 the magnetic force required for saturation is very much greater than for soft 
 iron. 
 
 280. When a piece of iron or steel is magnetised in a strong magnetic field 
 it will retain a considerable proportion of its magnetisation even after the 
 applied field has been removed and the iron is no longer under the influence 
 of any applied magnetic force. This power of remaining magnetic after the 
 magnetic force has been removed, is called magnetic retentiveness ; permanent 
 magnets are a familiar instance of this property. This effect of the previous- 
 magnetic history of a substance on its behaviour when exposed to given 
 magnetic conditions has been studied in great detail by Prof. Ewing, who 
 has given to this property the name of hysteresis^. To illustrate this property 
 let us consider the curve above, Fig. 58, which represents the relation for 
 a sample of soft iron between the intensity of magnetisation (ordinates) and 
 the magnetising force (the abscissa) when the magnetic force increases from 
 zero up to ON, then diminishes from ON through zero to OM , and then 
 increases again to its original value. When the force is first applied we have 
 the state represented by the portion OP of the curve, which begins by being 
 straight, then increases more rapidly, bends round and finally reaches P, the 
 point corresponding to the greatest magnetic force applied to the iron. If 
 now the force is diminished it will be found that the magnetisation for a given 
 force is greater than it was when the magnet was initially under the action 
 of the same force, i.e. the magnet has retained some of its previous magnetisa- 
 tion, thus the curve PE, when the force is diminishing, will not correspond 
 to the curve OP but will be above it. OE is the magnetisation retained by 
 the magnet when free from magnetic force; in some cases this amounts to 
 more than 90 per cent, of the greatest magnetisation attained by the magnet. 
 When the magnetising force is reversed the magnet rapidly loses its magnetisa- 
 tion and the negative force represented by OK is sufficient to deprive it of 
 all the magnetisation. When the negative magnetic force is increased beyond 
 
 * His results are given in detail in his book Magnetic Indiiction in Iron and other metals. 
 t The first real hysteretic effect was examined by Kelvin, Phil. Trans. 170 (1879), p. 68. 
 Cf. also Warburg, Wied. Ann. 13 (1880), p. 141; Ewing, Proc. JR. S. 34 (1882), p. 29. 
 
279-282] Ferromagnetism 261 
 
 this value, the magnetisation is negative. After the magnetic force is again 
 reversed it requires a positive force equal to OL to deprive the iron of its 
 negative magnetisation. When the force is again increased to its original 
 value the relation between the force and induction is represented by the 
 portion LGP of the curve. If after attaining this value the force is again 
 diminished to ON and back again the corresponding curve is the curve PEK. 
 
 281. Various more or less successful attempts have been made to account 
 for this special complexity in the magnetic properties of iron and steel. The 
 matter is of the utmost importance from the technical point of view and has 
 consequently taken a prominent place in theoretical discussions but it is 
 only during the last few years that any substantial progress has been made 
 on the theoretical side. It has long been known that paramagnetisation is 
 mainly a constitutive phenomenon of the medium which exhibits it, and it 
 must therefore be very considerably affected by any accidental character in 
 the constitution of such medium. Now the structure of ordinary iron is 
 known to be very complex and extremely irregular, so that its magnetic 
 behaviour must be expected to be irregular to a corresponding extent. It 
 thus appears to be necessary to examine some other substance with a more 
 definite and controllable constitution and which exhibits the same para- 
 magnetic phenomena, before any attempt at a theoretical discussion can be 
 effectively made. The most suitable substances for this purpose are such 
 minerals as pyrrhotite, hematite or magnetite, which can easily be obtained 
 in regular crystalline form. 
 
 The crystals of pyrrhotite, which is a sulphide of iron, possess three 
 mutually perpendicular planes of magnetic symmetry and are much more 
 easily magnetised in a direction perpendicular to one of these planes than in 
 any other direction. Moreover the induction phenomena for this direction 
 are extraordinarily simple*. If the crystal is originally unmagnetised it 
 remains so until the magnetic force H reaches a definite critical value H t , 
 when the intensity of magnetisation suddenly assumes its saturation value 
 /,, which it retains perfectly constant until the magnetic force reaches the 
 value H^ At this point the magnetisation is immediately reversed to the 
 value I s , a value which is retained constant until the magnetic force again 
 passes through the value + H s when it is reversed. The hysteresis curve 
 showing the relation between / and H is thus a simple rectangle of area 
 4# s / s and the phenomenon is irreversible. 
 
 282. An obvious explanation of these phenomena at once suggests itself. 
 If we assume that the elements of the crystal are permanently magnetised 
 to a definite intensity, it would appear that the two positions of an element 
 in which its magnetic axis is parallel to the main magnetic axes of the crystal 
 
 * They were discovered by Weiss and are fully described in various papers in Jour, de Phys. 
 for 1905. 
 
262 Magneto-statics [CH. vi 
 
 are the only ones which are stable under the action of the internal constitutive 
 forces ; and further that either of these positions ceases to be a position of 
 stable equilibrium as soon as the opposing magnetic force reaches its critical 
 value. The equilibrium in either case is largely maintained by kinetic 
 action, so that, any configuration would be immediately destroyed as soon 
 as it ceases to be a stable one. 
 
 This explanation is still further supported by the fact that the magnetic 
 behaviour of the substance in any direction other than along its principal 
 axis is perfectly continuous and in complete accord with the theoretical 
 consequences of such a view. The fact that the magnetisation parallel to 
 the principal axis is much greater than that in any other direction suggests 
 that the forces holding the elements in position are extremely large. 
 
 Broadly speaking the other ferromagnetic crystalline minerals exhibit the 
 same general features as pyrrhotite ; there are important differences in detail 
 but they all possess different magnetic properties along the different axes of 
 symmetry and usually one axis of conspicuously easy magnetisation. 
 
 283. Whatever the ultimate explanation of these properties of the ferro- 
 magnetic crystals may be it is probable that they furnish an indication of the 
 direction in which we must look for an explanation of the behaviour of the 
 ferromagnetic metals. We can in fact fully account in a general way for the 
 more complex behaviour of such substances as iron on the assumption that 
 it is constituted of an irregular conglomeration of small crystals of the simpler 
 type. It is well known that an ordinary piece of iron is a complex aggregate 
 of minute crystals and when impurities are present, as is generally the case, 
 the crystals may vary considerably in size and composition, so that no 
 unnatural assumption is thereby involved. Moreover when the constitution 
 of the iron is known to be regular, as for instance when it is deposited electro- 
 lytically under the action of a strong magnetic field, its magnetic behaviour 
 is just like that for pyrrhotite, the hysteresis curve being rectangular. 
 
 284. The fact that the ferromagnetic quantity is essentially a con- 
 stitutional one is best illustrated by the behaviour of iron in particular, 
 which is the more important practical substance. At ordinary temperatures 
 the simple relation between polarisation and polarising force assumed in the 
 analytical developments is by no means true for the case of iron in a magnetic 
 field. At least the factor \j! in the relation 
 
 is a function of H. There are also in addition very considerable hysteretic 
 losses of energy, as our analysis would imply in any failing case. The reason 
 of all this is that the polarisation induced in the iron at ordinary temperatures 
 is a property of molecular groups and not of the single molecules. The 
 experimental test of this fact is obtained by breaking up the groups by heating 
 
282-285] The energy of induced magnetism 263 
 
 the metal. It has been long known* that iron at high temperatures is only 
 slightly magnetisable ; but the important discovery was made by Hopkinson f. 
 There is another property of iron which goes by the name of 'recalescence.' 
 If we heat a piece of iron to a bright heat and then let it cool gradually, it 
 first gradually darkens by loss of heat but then at a definite temperature 
 (about 800 C.) it suddenly brightens up again. This means that at this tem- 
 perature there is a sudden release of internal energy in the form of heat. 
 Hopkinson showed that the temperature at which this occurs is just the 
 temperature at which the magnetic property changes and the metal ceases 
 to be fen omagnetic. The release of internal energy must mean a change 
 in the state of aggregation of the molecules and this is also indicated, on 
 the present theory, by the change in the magnetic properties. A striking 
 and probably just analogy is drawn by Curie between the simple law of 
 magnetisation of a substance like iron at high temperatures and this sudden 
 change which it undergoes when the temperature is lowered beyond the 
 recalescence point and the simple law of expansion of a gaseous substance 
 at high temperature and the sudden change which it undergoes on lowering 
 the temperature beyond a critical point so that the mutual attractions of 
 the molecules come into play and produce the liquid state. 
 
 There is another but less marked critical temperature (1280 C.) on both 
 sides of which the substance behaves as an ordinary paramagnetic body, 
 but in the passage through this temperature the susceptibility suddenly 
 changes. This of course indicates that a further alteration of the molecular 
 configuration takes place at this temperature. 
 
 The occurrence of these properties is by no means confined to iron. 
 Similar features are presented by all the other known ferromagnetic substances 
 and also by some diamagnetic substances. In the case of tin a number of 
 transition points have been observed and the element is sometimes dia- 
 magnetic and sometimes paramagnetic according to its temperature. 
 
 285. The energy and mechanical relations of induced magnetism. We 
 
 can now turn to an examination of the fundamental mechanical relations of 
 induced magnetism. The analysis is however to a certain extent analogous 
 to the similar problem connected with polarised dielectric media discussed 
 in the previous chapter. We need therefore only to give a short resume of 
 the results as far as they are concerned with the present case, interpreting 
 them of course in terms of the vectors of the present theory. 
 
 The energy required to establish the polarisation I in the element dv 
 regarded as the mathematical equivalent of the work done in separating the 
 poles of each small doublet is 
 
 (IH) dv, 
 
 * Barrett, Phil. Mag. 44 (1873), p. 472. 
 t Phil. Trans. A, 180 (1890), p. 443. 
 
264 Magneto-statics [CH. vi 
 
 and this represents the total work required by the element as a whole on 
 account of the polarisation in it. It consists essentially of two distinct parts 
 of fundamentally different origins and we try and separate them as in the 
 previous chapter. 
 
 Before doing this however we must recapitulate some remarks made in 
 the last chapter respecting the energy of polarised media in general. It was 
 found that the general equation of virtual work for a polarised medium 
 existing in any field of .force (now magnetic) would be of the general form 
 
 ST. = m m + &W- STF , - 8TP ,', 
 
 where 8T t represents the variation in the thermal energy of the irregular 
 heat motion of the molecules; 8W m is the virtual work of the magnetic 
 forces of attraction due to the field; SW is the virtual work of the applied 
 mechanical forces ; 8W f is the increase of the store of ordinary elastic energy 
 and STF/ that of the intramolecular energy of quasi-elastic strain and motion. 
 However in the most important cases which come under review in the theory 
 of magnetism it is not possible to draw a close distinction between thermal 
 and magnetic processes so that this equation cannot now be simplified as 
 in the previous application in the last chapter. We can however write it 
 in the convenient form 
 
 - 8W m = SW - 8E t , 
 
 wherein SE^ represents the variation of the total internal energy of the 
 matter, both elastic and motional, molecular and intramolecular. 
 
 286. Pass* the magnetic body through a cycle by moving it around a 
 path in a permanent magnetic field H. An infinitesimal displacement of the 
 volume $v from a place where the field is H, to one where it is (H + 8H) does 
 mechanical work, arising from the magnetic attractions, of amount 
 
 (ISH) 8v. 
 
 The integral of this throughout the whole connected system gives the virtual 
 work for that displacement, from which the forces assisting it are derived 
 as usual. Confining attention to the element Sv the work supplied by it 
 from the field, to outside mechanical systems which it drives, in traversing 
 any path is thus $ v (iH), 
 
 the integral being taken along the path. If I is a function of H, that is if 
 the magnetism is in part thoroughly permanent and in part induced without 
 hysteresis, so that the operation is reversible, this work must vanish for a 
 complete Cycle ; otherwise energy would inevitably be created either in the 
 direct path or else in the reversed one of the complete system of which St- is 
 a part. Thus the negation of perpetual motion in that case demands that 
 
 (I8H) d<f>, 
 
 * The treatment here followed is given by Larmor implicitly in Phil. Trans. A, 190 (1897) and 
 in exteruo in Proc. R. 8. 71 (1903), pp. 236-239. 
 
285-288] The loss of energy by hysteresis 265 
 
 d(f> being the complete differential of a function <f> of (H^, Hj,, H 2 ) involving 
 only even powers. If the polarisation follows a linear law, as it must do if 
 the field is small, <f> is quadratic and in the simple case of isotropic media is 
 
 so that I = ^'H, 
 
 as we assumed before. 
 
 287. But if there is hysteresis, so that the cycle is not reversible 
 
 -Svf(ISH) 
 
 represents negative mechanical work done, or energy degraded in the cycle *. 
 But this is 
 
 - Sv (ISH) = - 
 
 (HI) 
 
 Sv + Sv 
 
 /(HSI), 
 
 and in the average of a large number of cycles | (IH) Sv = so that the 
 energy lost is also measured by 
 
 Sv (HSI). 
 It is also equal to 
 
 and the latter part vanishes in a cycle so that the loss of energy is equal to 
 
 -f<BSH), 
 
 taken throughout the cycle. This is the practical method of obtaining the 
 hysteresis loss of energy, the expression obtained being that of Warburgf and 
 EwingJ for the magneto-hysteretic waste of mechanical energy in driving 
 electric machines. 
 
 288. In addition to this energy concerned with the attractions, the 
 external field expends energy in polarising or orientating the individual 
 molecules against the internal forces of the medium, of aggregate amount in 
 the whole cycle 
 
 Sv [(HSI). 
 
 This part has nothing to do with the mechanical forces. It is stored up as 
 internal energy of a purely elastic or thermal character. If the polarisation 
 
 * It is important to notice that this expression represents the area of the hysteresis loop on 
 the magnetisation curve given in 278. 
 
 f Ann. Phys. Chem. (3), 13 (1881), p. 140. Cf. also Rapports Congres internal, de phys. 2 
 (Paris, 1900), p. 509, and Phys. Zeitschr. 2 (1901), p. 367. 
 
 % Proc. R. 8. 48 (1890), p. 342. 
 
266 Magneto statics [CH. vi 
 
 gradually breaks away some or all of this is lost in heat and the phenomenon 
 of hysteresis results. 
 
 In any case, whatever the hysteresis, the sum of this second part and the 
 first reversed is integrable independently of the path, giving 
 
 Zv ! (HI) , 
 
 namely, the change in the total energy in the element, thus vanishing for 
 a complete cycle which restores things to their original state, as it ought to 
 do. The former part of this total represents the average waste of direct 
 mechanical energy in moving the magnetic substance through the cycle and 
 accounts for the heat thus evolved which is represented by the second part. 
 
 When the relation between I and H is linear and isotropic the two parts 
 of the total energy, the mechanical and molecular parts are both equal 
 
 s f ,S(H 2 ) 
 (H8I) 
 
 and this is the usual- theoretical result. If there is no hysteretic loss of any 
 kind we may take 
 
 r 'TI2 
 I /* ** J 
 
 J 2 *' 
 
 where the integral is extended throughout the substance, as the potential 
 energy of the mechanical forces acting on the medium. 
 
 289. Now let us apply these results in a particular case. Suppose the 
 magnetic field arises from a distribution of rigid magnetic polarity of density 
 I at any point in the field. The total energy in the field can then be 
 calculated as the mechanical work done in building the rigid magnetism 
 up gradually in the presence of the magnetisable substances, the induced 
 magnetism simultaneously takes the appropriate value at any stage of this 
 process. Suppose that at any instant the force intensity in the field is H, 
 then the work done in bringing up an additional small increment of polarity 
 8I to each place in the field is clearly equal to 
 
 - | (H . 8I ) dv 
 integrated throughout the field. 
 
 290. If now we assume that there are no sudden discontinuities in the 
 magnetic distribution in the field (and any such might be treated as a con- 
 tinuous rapid transition) then we can prove that the integral 
 
t 
 
 288-291] The total energy of the magnetic field 267 
 
 when extended throughout the whole field is zero. If is the magnetic 
 potential it is in fact equal to 
 
 and by Green's lemma this transforms to 
 
 *jJB n df, 
 
 the latter integral being taken over an indefinitely extended surface bounding 
 the field. When the field is regular at infinity this latter integral vanishes 
 and, since div B = 
 
 everywhere, the former does also. Thus we have 
 
 (HB) dv = 0. 
 
 In a similar manner it is verified in particular that 
 
 f (BSH) dv = t (HSB) dv = 0, 
 
 if SB denotes the small variation in the value of B at any place consequent 
 on the above defined small variation on the distribution of magnetic polarity, 
 
 since div SB = div (B + SB) - div B = 0. 
 
 291. Now B = H + 47r(I + I), 
 
 where I denotes the intensity of the induced magnetic polarity corresponding 
 
 to the field of I ; thus 
 
 SB = 8H + 4rr (8I + 81), 
 so that 
 
 - (HSI ) = (H8I) + 1 {(HSH) - (HSB)} 
 
 ' ^ (HSI) + ~ (BSH) - ~ (B8B) + n (I + IQ, 81 + 8I ;. 
 Thus the work done in increasing the rigid polarity by 8I may be reckoned as 
 BW = - - f 8 {B 2 - 1677 2 (I + I ) 2 } dv + f (HSI) dv. 
 
 O7T J . / 
 
 The total work done in establishing the magnetic field is therefore 
 
 16rr 2 (I + I ) 2 } dv + f (HSI) dv. * ' 
 
 The second part of this total energy represents the internal elastic energy 
 stored up in the magnetic media on account of the magnetic polarity induced 
 in them. The first part therefore represents the true magnetic potential 
 
. 
 
 268 Magneto-statics [CH. vi 
 
 energy of the field and on a tentative theory we could regard it as distributed 
 throughout the field with a density 
 
 - ~ {B* - 167T 2 (I + I ) 2 } 
 
 at any place. 
 
 292. In the case of a linear law of induction and when the medium is 
 isotropic we have B = H (1-f brp') + 47rI 
 
 = ^H + 477-Ic, 
 
 so that 
 
 B 
 
 and I = Li'H = 
 
 477 ( fJl, 
 
 Thus the energy stored in the magnetic media on account of the polarisations 
 induced in them appears as the integral 
 
 1 {B - 47rI } 2 dv. 
 
 ) 87TJU 2 
 
 In this case also we have 
 
 1 
 
 so that the total energy of the aethereal field is the integral of 
 
 fay* 
 
 - 
 
 ! ** ' 
 
 The total energy associated with the magnetic field is 'therefore 
 
 _ 
 
 STT .' 
 
 The second or purely local part of this energy depending solely on I 
 would in most cases remain constant; but in any case it would be foreign 
 to a mechanical theory concerned with the system in bulk. Thus the 
 mechanically effective part of the energy associated with the system is 
 
 B 
 dv, 
 
 and it can therefore be regarded as distributed throughout the field with the 
 density 
 
 _B* 
 
 SrrfJi ' 
 
291-294] The total energy of the magnetic field 269 
 
 At points of the field where there is no rigid magnetism this is the same as 
 
 ~~877' 
 
 which is the expression usually given in the text books, but with the opposite 
 sign. The discrepancy of sign is due to inconsistencies in the usual formulation 
 of the subject, which gives a density for the static potential energy differing 
 in the general case from the density deduced above by the amount 
 
 -I(BH), 
 
 giving a zero total on the whole. The formulation adopted above is the only 
 one that appears to be consistent with the subsequent developments of the 
 theory in its dynamical aspects. 
 
 293. Of the true magnetic potential energy in the field 
 
 --^ [{B 2 -167r 2 (I + I ) 2 }^, 
 
 rrH 
 
 a part (ISH) dv 
 
 JJo 
 
 corresponds to the polarisation induced in the magnetic media : it is concerned 
 mainly with the magnetic attractions, or their equivalent mechanical forces, 
 exerted by the field on the polarised media as a whole, of which it is the 
 potential function. The remainder 
 
 - {B 2 - 1677 2 (I + I ) 2 } dv + I dv (^(ISH) 
 
 OTT . J . 
 
 corresponds to the rigid polarisations I and is a potential function of the 
 mechanical reactions on the permanent magnets giving rise to the field. 
 
 Since B = H + ITT (I + I ) 
 
 this pare may be written in the form 
 
 dv (HSB) -r dv (I SH), 
 J o Jo 
 
 which, since in every stage of the process 
 
 (HSB) dv = 0, 
 
 f 1 H 
 
 reduces to \ dv I (I 8H), . 
 
 Jo 
 
 in agreement with a former result when there are no magnetic substances 
 about. 
 
 294. The forces acting on a magnetically polarised medium can now be 
 obtained either from the energy expressions or by an analysis similar to that 
 given in the previous chapter. Excluding as there the part arising wholly 
 
270 Magneto-statics [CH. vi 
 
 from the interaction of neighbouring molecules, which is not transmitted by 
 material stress, but is compensated on the spot by molecular action due to 
 change of physical state induced by it, the magnetic force proper is made 
 up of a bodily force F and torque G where 
 
 F = (IV) H, 
 and* G - [IH]. 
 
 Under the usual circumstances these expressions are identical with the 
 ones given by Maxwell in his treatise. The remarkable property is there 
 established, and is the direct analogue of our result for polarised media, that 
 independently of the form of the relation between the magnetic induction 
 and magnetic force in the medium and whether there is permanent magnetism 
 or not, this bodily forcive can be formally represented in explicit terms as 
 equivalent to an imposed stress : viz. it is the same as would arise from 
 
 H 2 
 
 (i) a hydrostatic pressure ^ , (ii) a tension along the bisector of the angle 
 
 O77 
 
 6 between H and B, equal to HE cos 2 0/47T, (iii) a pressure along the bisector 
 of the supplementary angle, equal to EH sin 2 6/4v, together with an outstand- 
 ing bodily torque turning from B towards H equal to HE sin 20/4-Tr. When 
 B and H are in the same direction, the torque vanishes, and a pure stress 
 remains in the form of a tension (EH |# 2 )/47r along the lines of force and 
 a pressure H 2 /87T in all directions at right angles to them. There is of course 
 no warrant for taking this stress to be other than a mere geometrical represen- 
 tation of the bodily forcive. It is however a convenient one for some purposes. 
 Thus the traction acting on the layer of transition between two media, in 
 which H changes very rapidly, which might be directly deduced in the same 
 manner as the electric traction above may also be expressed directly as the 
 resultant of these Maxwellian tractions towards the two sides of the interface. 
 As there cannot be free magnetic surface density the traction on the interface 
 is represented, under the most general circumstances, whatever extraneous 
 magnetic field may there exist, by purely normal pull of intensity 2?7l re 2 
 towards each side. 
 
 295. On the thermal relations of the energy of magnetisation. We have 
 briefly explained in a previous paragraph the difference between paramagnetic 
 and diamagnetic substances, but we did not stop to think what the essential 
 difference between these two kinds of magnetisation might be. How is it 
 that both para- and diamagnetism exist in nature? The answer must 
 
 * A general form of the theory of magnetic stress based on the method of energy and analogous 
 to Helmholtz's theory for dielectrics has been given by Cohn (Das electromagnetische Feld, p. 510) 
 and further elaborated by Gans (Ann. der Phys. 13 (1904), p. 634, and Encydopddie der math. 
 Wissensch. v. 15), KolaSek (Ann. der Phys. 13 (1904),.p. 1) ; Sano (Phys. Zeiis. 3 (1902). p. 401). 
 This theory is however open to the same criticism as levelled by Larmor at Helmholtz's pro- 
 cedure. 
 
294-29(5] The thermal relations of magnetic energy 271 
 
 obviously be that there is an essential difference in the origin of the two 
 kinds and this assertion is fully confirmed by experiment. As a result of 
 an extensive investigation of the magnetic properties of matter the law has 
 recently been formulated by Curie* that in all feebly paramagnetic substances, 
 including gases, the coefficient of magnetisation varies inversely as the 
 absolute temperature, with a degree of accuracy which tends to perfection 
 at high temperatures : that in strongly magnetic substances such as iron 
 and nickel the same law is ultimately reached when the temperature is high : 
 while in diamagnetic substances the coefficient is usually nearly independent 
 of temperature and also of changes in the chemical state of the material. 
 The inference is made by Curie that this points to diamagnetism being an 
 affair of the internal constitution of the molecule, having only slight relation 
 to the bodily motions of the molecules on which temperature depends, in 
 accordance with the first views of polarisation of a medium. On the other 
 hand, paramagnetism is an affair of orientation of the molecules in space 
 without change of internal conformation, so that alteration of the mean 
 state of translational motion is involved in it and we should expect a tempera- 
 ture effect : this is the idea underlying the Weberian theory of magnetisation 
 (the second alternative hypothesis mentioned to explain the cause of 
 polarisation). This relation between paramagnetisation and temperature 
 proves to be so simple that it must be the expression of a theoretical principle. 
 The following considerations in fact derive it from Carnot's principle : the 
 argument is precise so long as the induced magnetisation is so slight that 
 the exciting magnetic force on the separate molecules is practically that of 
 the inducing field, but it loses exactness as soon as, owing to the diminution 
 of energy of agitation with falling temperature, the molecules begin to 
 exercise sensible magnetic control over each other, and thus introduce the 
 phenomena of grouping and consequent hysteresis that are associated with 
 the ferromagnetic state. 
 
 296. Consider f a mass of paramagnetic material, moved up from a 
 place where the intensity of the field is H to a place where it is H + SH. 
 The aggregate per unit volume of the total magnetic energies of its mole- 
 cules is thereby altered from (IH) to (I + 81, H + SH). The mechanical 
 work done by the mass in virtue of its attraction by the field is (I8H). Thus 
 there remains a loss in the total magnetic energy of the molecules, equal to 
 (H8I) ; this can only have passed into heat in the material ; for we can work 
 on the hypothesis that the field of force H is due to an absolutely permanent 
 
 * Ann. dc Chimie (1805). A considerable number of exceptions to the law have been found 
 particularly at low temperatures but these are attributed to changes in the molecular configura- 
 tion of the material. Cf. Kammerlingh Onnes and Perrier, Konink. Akad. Wetensch. Amsterdam 
 Proc. xiv. p. 115 (1911). 
 
 f The argument here is due originally to Larmor, Phil. Trans. A, 190 (1897), 71 and 72. 
 Cf. also Proc. R. 8. 71 (1903), p. 235. 
 
272 Magneto-statics [CH. vi 
 
 magnetic system, so that no energy is used up in producing magnetic dis- 
 placements in the inducing magnets. Now let us apply Carnot's principle to 
 a reversible cycle in which the material is moved up in the field at tempera- 
 ture 6 + Sd and moved back at temperature 6, with adiabatic transition 
 between these temperatures. Let h + dh be the thermal energy per unit 
 volume which it must receive from without at the higher temperature, and 
 h that which it must return at the lower, in order to perform the amount 
 of work 8W, equal to 
 
 ot> 
 in the cycle ; then by Carnot's principle 
 
 80 ~0' 
 
 but h = - (HSI) 
 
 91 , HSI 
 
 as above so that ^ 8H = pr- . 
 
 vv u 
 
 91 H 81 
 W = ~08H' 
 
 /H 
 
 so that I = / ( -0 
 
 / being some arbitrary function. When the magnetising force H is very 
 small or the temperature 6 very large this relation is approximately equi- 
 valent to 
 
 so that p = n ) 
 
 which is Curie's law. 
 
 # 
 
 297. Conversely, assuming Curie's law we can deduce that in para- 
 magnetic bodies magnetisation consists in orientation of the molecules 
 without sensible change in their internal energies. In an analytical form 
 the argument is as follows: 
 
 dh = (Mrfl) + Ndd, 
 
 and dE = dh + (Idl), 
 
 P 
 
 whence by the thermodynamic formula 
 
 Q /l\ 1 
 
 so that = _ f = -- 
 
 \fji ) /A 
 
 M Q d 
 
 = _ 
 
 I do \fji 
 
 by Curie's law ; hence dh = _ (H(fl) + 
 
296-298] The thermal relations of magnetic energy 273 
 
 so that at constant temperature 
 
 - h = J(HSI), 
 
 that is the heat that the material develops during magnetisation is the equi- 
 valent of the magnetic energy that is not used up in mechanical work. This 
 is precisely what we should expect if the material is a gas : for there is then no 
 internal work by which this energy could be used up, and the magnetisation 
 arises from the effort of the magnetic field to orientate the molecules which 
 are spinning about as the result of gaseous encounters. The law of Curie 
 thus indicates that the same is sensibly true for all paramagnetic media at 
 high temperatures : at lower temperatures they generally pass into the 
 ferromagnetic condition. It is the magnetisation so to speak of an ideal 
 perfect ferromagnet in which the controlling force that resists the orientating 
 action of the field is practically wholly derived from the magnetic interaction 
 of the neighbouring molecules, which for this purpose form elastic systems. 
 In ordinary paramagnetic substances this mutual magnetic control of the 
 molecules is insensible compared with the control due to other molecular 
 causes, and our conclusion is that these causes are such that the magnetic 
 energy expended in working against them is transformed into heat 'energy, 
 not into internal energy of any regular elastic type. 
 
 298. The argument for the simple case of a gaseous medium exhibiting 
 paramagnetic properties has been further extended by Langevin, who has 
 succeeded in calculating, on the basis of the kinetic theory, the actual form 
 of the functional relation between the intensity of magnetisation and 
 magnetising force for this ideal case. The problem considered is that in 
 which a simple gas, such as oxygen, the molecules of which are regarded as 
 rigidly magnetised to moment m, exists in a uniform field of magnetic force 
 of intensity H. In this case the organising potential energy of the molecules, 
 which is of a purely magnetic nature, the mutual interaction of the molecules 
 themselves being neglected, is balanced entirely by the disorganising effect 
 of the kinetic energy of thermal agitation. Now the potential energy of a 
 magnetic doublet m in a field of magnetic force of intensity H is 
 
 - mH cos x, 
 
 if x is tne angle between the direction of the force H and the axis of the 
 doublet. It follows then by the usual considerations of the kinetic theory 
 that the number dn of molecules per unit volume whose magnetic axes make 
 with the direction of the field an angle lying between x ai *d x + &%. is 
 
 dn = 2irAe Re s 
 
 wherein R is the usual constant of the kinetic theory, 6 is the absolute 
 L. 18 
 
274 Magneto-statics [CH. vi 
 
 temperature and A is a constant determined by the fact that the total number 
 of molecules per unit volume is 
 
 mH eos 
 
 . 
 
 h 
 
 (mH\ 
 
 ~ 
 
 Thus A -- 
 
 In this case the resultant intensity of magnetisation / is in the direction of 
 H, by symmetry, and is actually given by 
 
 r ^ 
 
 _ mHcosx 
 R6 
 
 e cos v sin 
 
 .0 
 
 
 sinh ( mH \] 
 
 COSh \R6j 
 
 Smtl u0; 
 
 mH 
 
 (mH\ 2 
 
 Rd 
 
 \R6J J 
 
 Substituting the value of A from above we find that 
 
 T , fmH\ Rd 
 
 I = nm\ coth - ) ~ 
 
 \R6J mH 
 
 Since m is determined solely by the structure of the molecules we see that 
 for a given density of the gas the intensity / of the magnetisation induced 
 is a function of H/6, in accordance with the conclusion drawn from the 
 thermodynamic reasoning Moreover for small values of the inducing force 
 the functional relation becomes a mere proportionality or 
 
 _ nm*H 
 ''~2R0 ' 
 so that the susceptibility is 
 
 and varies inversely as the absolute temperature in accordance with Curie's 
 law. 
 
 
 
 299. This theory can be applied to the calculation of the moment m of 
 the molecular magnet in the case of those substances easily obtainable in 
 the simple gaseous state. It has been extended by Weiss so as to apply to 
 a substance in dilute solution in a feebly magnetic liquid. In this way 
 determinations of m can be made for a large number of substances, and the 
 results obtained prove to be related to one another in a most remarkable 
 manner. It is in fact found that the average magnetic moment of any 
 
298-300] The theory of ferromagnetism 275 
 
 molecule is always equal, within the very narrow limits of experimental 
 error, to a certain small integral multiple of a certain universal constant. 
 The constant integers for the large number. of substances examined range 
 from 4 to 56 and extremely few cases are found which cannot be so expressed 
 with great accuracy. 
 
 To explain this regularity Weiss was led to the very natural hypothesis that 
 the paramagnetic properties of substances arise from the presence of an 
 ultimate unit, the so called magneton, in the atom of the substance. It is 
 apparently necessary that these elements should be capable of annihilating 
 each other temporarily, as the same substance may contain different numbers 
 of magnetons at different temperatures. 
 
 It has not yet. been found possible to justify this assumption of an ultimate 
 magnetic unit, and we shall not therefore make any essential use of it in our 
 future discussions. The importance of the idea however seems to warrant 
 the brief reference just given to it. 
 
 300. The theory of ferromagnetism. It has not yet been found possible I 
 to test directly the general relation deduced by Langevin connecting the I 
 magnetisation induced in a gas and the strength of the inducing field. I 
 The theoretical formula has however been applied with great boldness by 
 Weiss* to build up a theory of the behaviour of ferromagnetic substances 
 in general, and the success which has attended his investigations removes 
 a good deal of doubt as to the justification of his ideas. 
 
 According to Weiss the circumstances in ferromagnetic substances such 
 as the crystals described above differ from those in the simple case to which 
 we have just applied the kinetic theory, only in so far as the effects considered 
 are modified by the mutual magnetic influence of the molecules of the sub- 
 stance. In the case of a gas the mutual interaction of the neighbouring 
 molecules can always be neglected, but when the molecules are packed so 
 close together as they are in the case of a solid body, and when the intensity 
 of magnetisation becomes so large as is the case with iron, this local interaction 
 may greatly preponderate. It is of course quite impossible for us to know 
 very much about the local magnetic field f of force surrounding any molecule,- 
 even in such ideal substances as the simplest crystals appear to be ; but we 
 do know that under the most favourable circumstances it will vary rapidly 
 in distances quite inaccessible to our perceptions. A good deal of suggestive 
 information can however be obtained by taking, after Weiss, a physically 
 average view of the matter and assuming this local field at an average estimate 
 smoothed out from the highly irregular field that actually exists. This 
 
 * Jour, de Phys. vi. p. 661 (1907). 
 
 f Tt is interesting to notice that neither the theory nor the facts imply that this local field 
 is magnetic in nature, although it may be convenient so to regard it. Cf. Eucken. Die Theorie. 
 der Strahlung und der Quanten (Halle, 1914), p. 321 (Bericht Langevin). 
 
 182 
 
276 Magneto-statics [CH. vi 
 
 average local field arises entirely in the magnetisation induced in the medium 
 and its intensity will, on the simplest hypothesis, be proportional to the 
 intensity of this magnetisation, the constant of proportionality being however 
 related to direction in the medium, if this latter possesses aeolotropic charac- 
 teristics. We shall consider the case of isotropic media and denote the local 
 field strength by al. 
 
 301. Weiss now formulates the theory of ferromagnetism exactly on the 
 lines suggested by Langevin but with the force acting on the molecular 
 magnets now equal to jj _j_ a / 
 
 The value of 1 for a given value of H is then to be obtained from the formula 
 
 r 4.V \ m m , A* R o 
 
 I - mn coth (H + a/) - 
 
 The maximum value of the magnetisation or its saturation value is 
 
 I s = nm. 
 If we introduce this value in the above relation between 7 and H it becomes 
 
 r T r u \I 8 (H + al)} nRd 1 
 
 7 = / s \ coth 1- ^^ -} T . - T . 
 L ( nR6 \ I s (H + at)] 
 
 This is the relation on which Weiss builds his theory. 
 
 In the first case the phenomenon of permanent magnetisation is fully 
 accounted for. In this case H = so that the magnetisation is given by 
 
 r 7 T , /a//,\ nR6 ) 
 7 = 7 J coth - ^ -- Yj . 
 \nR9J all s J 
 
 Under ideal circumstances m and a are both independent of / and 6 and there 
 is then, in general, a solution of this equation for / which is different from 
 zero. Moreover this non-zero solution when it exists represents the stable 
 condition of the medium, because the magnetic potential energy in it is a 
 minimum. The zero solution corresponds to a maximum value of the 
 potential energy and is therefore in the general case unstable. 
 
 As the temperature rises the permanent magnetisation intensity gradually 
 "decreases and at a certain temperature 6 S it vanishes altogether. Beyond 
 this temperature the only possible real solution of the equation for 7 is 
 
 7 = 0, 
 
 so that the substance is then incapable of permanent magnetisation. The 
 temperature 6 S may therefore be interpreted as the temperature at which 
 the ferromagnetic quality disappears, i.e. the so-called critical temperature. 
 In the neighbourhood of this temperature the magnetisation is always small 
 so that using 
 
 _a77 s 
 X " nR6 ' 
 
300-303] The theory of ferromagnetism 277 
 
 we have approximately 
 
 1 dl d f 11 1 
 
 -jr -5- = -=- cotn x -- = 1 cosech 2 x + -= = 4. 
 
 <27 nR6 
 
 But j- = =r- , 
 
 ax alg 
 
 so that at the actual critical temperature 
 
 7 8 _ nR6 s 
 ^~ al g ' 
 
 al 2 
 * 
 
 302. If now we introduce this temperature into the relation determining 
 the intensity of permanent magnetisation it assumes the form 
 
 , 
 
 7 - 7. 
 
 so that the ratio I/I S is a function of 6/6 s and the function is the same for all 
 substances. 
 
 Thus if we express the intensity of permanent magnetisation in terms of 
 the maximum possible intensity I s and the absolute temperature 6 in terms 
 of the absolute critical temperature, we obtain a characteristic equation for 
 the intensity of permanent magnetisation at a given temperature, which is 
 identical for all ferromagnetic substances. This relation has been tested by 
 Weiss* in the case of magnetite and he finds that it is satisfied with great 
 accuracy except at very low temperatures ( 79 C.) and in the neighbourhood 
 of the critical temperature, where however the deviations are not large. 
 When it is observed that there are no disposable constants in the formula 
 this agreement between the observed facts and what at first sight appears 
 to be merely a provisory theory can only be regarded as remarkable. 
 
 303. The theory proves equally successful in explaining the observed 
 facts of the phenomenon of induced magnetisation. Of course, as we have 
 already noticed, any simple theory of the present type is hardly likely to be 
 directly applicable to the ferromagnetic metals, the magnetic behaviour of 
 which is complicated by various secondary causes. It is however found 
 that the simple phenomena accompanying the magnetisation of the various 
 crystalline ferromagnetic minerals when placed in a field parallel to their 
 principal magnetic axis fit in admirably with the theoretical conclusions to 
 be drawn from the theory. 
 
 The general relation between / and H is expressed by the equation 
 
 . \ ,IH + aI RnB 
 
 I=L\ coth 
 
 nR6 I s (H 
 
 * Jour, de Phys, vi. p. 665 (1907). 
 
278 Magneto-statics [CH. vi 
 
 This requires that in general / should be a continuous function of H when 
 6 is maintained constant, which is just the opposite to what was actually 
 found to be the case, the magnetisation being practically independent of the 
 magnetic field and equal to its saturation value at the given temperature. 
 This could be the case only when the internal local field of intensity al far 
 exceeds in actual magnitude any field that we can experimentally produce. 
 Now this actually appears to be the case, as we shall soon see ; so that under 
 all circumstances except when I is very small, that is in the neighbourhood 
 of the critical temperature, the intensity of magnetisation is determined by 
 
 . f , (all 
 I = I S coth 
 \ 
 
 -- 
 nR6 
 
 and is equal to the saturation intensity in all fields. 
 
 304. In the neighbourhood of the critical temperature, when / is small, 
 the previous argument is invalid. But then the general relation may be 
 simplified by approximation, and for all practical purposes it is equivalent to 
 
 or introducing the critical temperature 
 
 in which form it has been satisfactorily verified in numerous cases, even with 
 iron. This latter agreement is not surprising because in the neighbourhood 
 of the critical temperature the constitutional irregularities are becoming very 
 unstable. 
 
 In all cases where the last relation is experimentally verified, a determina- 
 tion of the constant a may be made, and hence an estimate of the strength 
 of the local magnetic fields. From measurements of this kind Weiss finds 
 that the intensity of the local field al in iron is 6-56 . 10 6 absolute units and 
 it is of a similar order of magnitude in other substances. This is very much 
 greater than the strongest magnetic field (5 . 10 4 units) which can be produced 
 in th laboratory, so that the peculiar result that the intensity of induced 
 magnetisation 7 S does not, under the simplest circumstances, appreciably 
 alter with the external field is satisfactorily accounted for. 
 
 It is not possible in the scope of the present work to include any further 
 account of many interesting developments of Weiss's theory ; but it is hoped 
 that the above discussion of its more elementary aspects will at least indicate 
 the great advance which this theory represents in the theoretical examination 
 of the behaviour of ferromagnetic substances. 
 
CHAPTER VII 
 
 ELECTRIC CURRENTS IN METALLIC CONDUCTORS 
 
 305. Introduction. When a conductor is introduced into an electric field 
 a separation of electricity takes place until the field in its interior is com- 
 pensated. Until this state is attained there is a flow of electricity, or a 
 current, as we say. To illustrate the matter more fully let us consider 
 a conductor somewhat in the form of that shown in the figure. The end 
 a is charged with a positive charge + q and the end 6 with a negative charge 
 - q. There is then an electric field partly inside and partly outside the 
 
 Fig. 59 
 
 conductor, the lines of force in which may cross the surface of the conductor. 
 Such a state of affairs if initially established is however not a possible equili- 
 brium one so that there follows immediately a separation of the charges at 
 each point of the conductor the total result of which is the final annulling 
 of the charges at a and 6. We shall now suppose that we can continually 
 renew the 'charges at a and b in such a way as will maintain a constant potential 
 difference between the two ends of the conductor. The manner in which 
 this is accomplished will be hereinafter discussed. 
 
 306. Now the force driving the charge is the electric force of the field and 
 so the initial charge flux must follow the lines of force. There will thus be 
 initially a displacement of electricity along a line of force such as that shown 
 in the figure by the dotted line acdb. But this displacement can only proceed 
 as far as c where the surface of the conductor is reached. There will thus be 
 
280 Electric currents in metallic conductors [CH. vn 
 
 an initial accumulation of positive charge on the surface of the conductor 
 and this excites a field in the interior of the conductor, whose normal com- 
 ponent at the surface is opposed to the normal component of the original 
 force which was directed along the line abed. This charge at c accumulates 
 until this normal component is actually compensated, i.e. until the original 
 line of force is altered into one running from a to b inside, and nearly parallel 
 to the surface of the conductor. 
 
 Thus the first part of the electric flow is concerned merely with charging 
 the surface of the conductor so that all the lines of force starting inside.it 
 from a remain inside it till b is reached. 
 
 In external space the normal component of the force originally along the 
 line (a, b, c, d) is not compensated by the field of the surface charge, since 
 they are both in the same direction. There is in fact in the external space 
 a complicated electrostatic field composed of the original field superposed on 
 that due to the surface charge. 
 
 It must however be said that the charges and force intensity in the field 
 thus brought into existence are both extremely small. The electric elements 
 in a conductor are so extremely mobile that it requires only a very small 
 electrostatic force to produce an appreciable current. 
 
 307. The current can now flow undisturbed from a to b in the interior 
 of the conductor along the new line of force and if the charges at a and 6 
 are continually supplied so as to maintain the constant potential difference 
 a condition of Stationary streaming is attained. The field in the whole space 
 then remains constant. Moreover the amount of electricity crossing any 
 section of the conductor per unit time must be the same as otherwise there 
 would be an accumulation of charge in the conductor and a slight accumu- 
 lation would create a back electromotive force which would tend to stop 
 the current. A very slight accumulation would produce a sufficient back 
 electromotive force to stop the current. 
 
 The process of starting the current thus requires a very slight accumulation 
 of charge on the conductor which is just enough to make the flow steady 
 or the current uniform and stationary. In this steady state the stream lines 
 of the electric flow are also the lines of force of the electric field inside the 
 conductor. 
 
 This idea of a current as a flow of electricity did not exist even for 
 a considerable time after the discovery of batteries. It was Ohm (1827) 
 who started the notion*. 
 
 308. Definition of an electric current. We must now define the electric 
 flow in such a manner as to render it susceptible of calculation. If we adopt 
 
 * Die galvanische Kette, mathematisch bearbeitet (Berlin, 1827). Translated in Taylor's 
 Scientific Memoirs. 
 
306-308] 
 
 The definition of a current 
 
 281 
 
 the general method we should specify the flux of electricity in any direction 
 at a point in the conductor by the amount Cds which crosses per unit time 
 a small surface ds placed perpendicular to the direction with its mean centre 
 at the point. We can easily show that this defines a vector quantity if 
 we choose rectangular axes with their origin at the point under consideration 
 and consider the flow in and out of the small tetrahedral volume OABC at 
 the origin of coordinates with edges 8x8y&z along the axes. The area ABC 
 has projections on the axial planes equal to 
 
 | (S^Sz, SzSx, SxSy) or (n lx , u ly , n lz ) 8s 
 where 8s is the area ABC and n x its direction vector. 
 
 Fig. 60 
 
 The equation of continuity of flow expresses that the aggregate flux out 
 of this volume is equal to minus the rate at which the total charge inside is 
 increasing. If p is the density of charge inside 
 
 Sv |^ = - (C^n^ + G y n ly + C z n lz ) 8s + C n Ss, 
 
 where G n is the flux component normal to 8s and C x , G v , C z those normal 
 to the axial planes. The volume 8v (ABCO) is infinitely small of the third 
 order and the surface Ss is infinitely small of the second order and thus 
 ultimately when the volume is very minute the left-hand side of this equation 
 
 is zero and thus 
 
 C n = 0^ + 0^+0^, 
 
 which proves that is a vector with components (C x , C v , C z ). 
 
282 Electric currents in metallic conductors [CH. vn 
 
 309. This is the general method of definition. In order however to 
 obtain a closer insight into the true nature of 
 the flux involved we must proceed in a slightly 
 different manner. Consider again the small 
 surface 8s and construct on it a small cylinder 
 of length 81 with its axis parallel to the re- 
 sultant motional velocity v 1 of the electric 
 charge at the point (the direction of the lines 
 of force at the point), this direction making 
 an angle a with the normal at 8s. If the 
 density of the positive electricity at the point 
 is P! the quantity of positive electricity in 
 this cylinder is p^lBs cos a and during a time 
 
 87 
 8t = all of this electricity flows out across Ss. Thus for this surface 
 
 cos a ^ 
 
 = v l p l cos aos, 
 
 j 
 ot 
 
 or Cj = p l v 1 cos a 
 
 measures the current of positive electricity in the direction normal to ds at 
 the point. 
 
 If there is at the same time a flux of negative electricity of density p z 
 we know that it takes place in the opposite direction to that of the positive 
 although perhaps with a different velocity v 2 . The current across ds normally 
 in the same direction due to the negative charge is therefore 
 
 C 2 = p 2 ( v 2 ) cos a = p z v 2 cos a 
 and the total current in this direction is 
 
 C = G! + C 2 = (pjVi + p 2 v z ) cos a, 
 
 from which again we see that C is the component of a vector (p^ + pz v z) 5 
 which we call the current density of the electric flow at the point. 
 
 310. Ohm's Law*. The force driving the charge and imparting to it the 
 motional velocity is the electric force : the positive elements of charge are 
 moving in the positive direction of the lines of force and the negative ones 
 in the opposite direction and the two motions having the same effect constitute 
 the current. This being the case there must be some relation between the 
 electric force and the current. Ohm tried to reason the relation out by con- 
 sidering the phenomena of currents as analogous to the conduction of heat 
 down a temperature gradient. There are two kinds of streaming motion 
 recognised in physics ; that of steady diffusion in which the velocity is pro- 
 portional to the driving force ; and that of free motion in which the change 
 
 * This law was anticipated by Cavendish in 1781. Cf. his Electrical Researches. 
 
309-311] Ohm's law 283 
 
 of velocity is proportional to the force. In diffusion the motion is so 
 modified by impeding frictional forces that a state of steady motion is 
 attained in which the velocity is proportional to the force. The conduction 
 of heat is the typical example of a process of steady diffusion. 
 
 If now we assume with Ohm that the electric flow in the conductor is 
 a process of steady diffusion under the action of the electric force we must 
 assume with him that both % and v 2 are proportional t6 the driving electric 
 force E. Thus if we now use C for the resultant electric current we have in 
 the vector sense C = /cE 
 
 where K is a physical constant, which is usually called the conductivity of 
 the substance at the point. 
 
 311. Now consider the case of a steady current flowing in a wire of 
 finite thickness. The surfaces of the wire form a tube of force for the internal 
 field and also a tube of flow for the current. The ends a, b of the wire are 
 
 Fig. 62 
 
 presumed to cut the lines of force of the internal field everywhere normally 
 and the'same assumption tacitly underlies the choice of any other cross-section 
 subsequently made. These sections will then be equi-potentials of the 
 internal field. 
 
 Now consider any cross-section of the wire of total area / and suppose it 
 resolved into small elements of area df. If C is the current density at a point 
 in the wire the total quantity of electricity crossing the section per unit 
 time is 
 
 J = | C B Z/; 
 
 f 
 
 1 is called the strength of the current in the wire, or simply the current. 
 
 Again since the normal to df is in the direction of the line of force in the 
 field at the place Q _ Q _ ^.g 
 
 so that J = 
 
 Now at any infinitely small distance from the section / draw another equi- 
 potential section/' and let Ss be the distance between corresponding points 
 of the sections, then 
 
 f kEBs ,, 
 
 "J/TT* 
 
284 Electric currents in metallic conductors [CH. vn 
 
 But ESs = 0(j> is a constant over the whole surface/, viz. the constant difference 
 of potential between the two surf aces / and /'. Thus 
 
 The quantity 
 
 is called the ' resistance ' of the portion of the conductor between the cross- 
 sections / and /' : thus at this point of the wire 
 
 8* 
 
 e/ = 57 . 
 
 OK 
 
 But J is a constant all along the wire and thus if <f> a and </> 6 are the potentials 
 at the ends a, b of the* wire A _ A _ jfc 
 
 f 6 
 where k I Sk is the resistance of the wire between the two ends. 
 
 This is Ohm's law in its original form. The procedure adopted by Ohm 
 was however rather different from that sketched above. He tried to extend 
 the mathematics just previously developed by Fourier for the conduction of 
 heat down a temperature gradient. In doing this he had of course to assume 
 something analogous to temperature and it did not require much to convince 
 him that the potential was 'the required quantity. The current in a wire 
 is proportional to the fall in potential from one en'd to another 
 
 This idea that currents go by diffusion was at first merely an hypothesis, but 
 on the modern theory of electrons it appears as the actual state of affairs. 
 
 312. There is an important hydrostatic analogy which enables us to. 
 picture the process more clearly. If liquid is forced through a tube blocked 
 by a number of small obstacles so that no eddies can be formed and if the 
 motion is a steady pushing through with the hydrostatic pressure as the 
 driving force the amount of the flow is 
 
 difference of pressures 
 resistance of channel 
 
 This is the more direct analogy with the electrical case. The term electric 
 resistance is coined on this basis. 
 
 The analogy goes even farther and enables us to talk of the driving force 
 in the electrical case as an electrostatic pressure. The modern theory of the 
 flow of electricity basing the current on the flow of electrons is in fact a direct 
 application of this analogy. A current consists largely of free electrons being 
 pushed through among the obstacles presented by the molecules of the matter. 
 
 * These considerations are of course confined to a steady system : it is only when a steady 
 state of flow has been attained that a potential exists (see Ch. ix). 
 
311-313] 
 
 The Volta potential difference 
 
 285 
 
 The notion that electric pressure is the same as potential dates back to 
 Volta's time. He knew that it was electrostatic pressure that pushed the 
 current and he made a condensing electroscope sufficiently sensitive enough 
 to show this. The distinction between free motion and diffusion was however 
 due to Ohm. 
 
 313. The Volta potential difference. We have so far assumed that we 
 are able to maintain a constant potential difference between two points on 
 the surface of a conductor. We can now discuss how this is attained. 
 The foundation of the method is Volta's discovery that when two conductors 
 or generally any two different substances, are in contact, there is a definite 
 potential difference between them*. 
 
 If we place two different conductors in contact, then an adjustment of 
 charge will take place so that the one conductor will acquire a definite negative 
 charge and the other a definite positive charge. The quantities of electricity 
 involved in the rearrangement depends on the different conditions such as 
 the form, size and relative position of the bodies, but with the same two 
 substances the potential difference thus set up between them has a definite 
 value as long as we always work at the same temperature. 
 
 If the substances are conductors as is usually the case then in the 
 equilibrium condition the potential of the electrostatic field which arises has 
 a constant value* at all points inside either conductor except very near the 
 surface where it is in contact with the other. The change from the potential 
 of the one conductor to that of the other thus takes place in the infinitely 
 thin contact surface so that we can speak of a sudden jump of the potential. 
 It therefore also follows that the origin of the action is situated in the immediate 
 neighbourhood of the surface of contact. Between the two substances at 
 the adjacent faces there is a certain 
 stress of chemical affinity which re- 
 sults in an inequality in the forces 
 exerted by each metal on the elements 
 of charge in the other. We could for 
 example explain the potential differ- 
 ence between zinc and copper by 
 imagining that there is a greater 
 attraction on the positive charges 
 exerted by the zinc than the copper, 
 and on the negative charges exerted by 
 the copper than the zinc. This electro- Fig. 63 
 
 chemical stress results in an electric displacement either by polarisation 
 
 * Ann. de Chim. 40, p. 225. Cf. also Gilbert's Annalen, 9, p. 380 (1801); 10, p. 425 (1802); 
 12, p. 498 (1803). 
 
286 Electric currents in metallic conductors [CH. vn 
 
 of the molecules (in non-conductors) or by actual finite separation of the 
 charge between them. The resultant of the electric displacement is that 
 one of the two contact surfaces appears, positively charged and the other 
 negatively, the double sheet thus created accounting for the jump of 
 potential. 
 
 314. When the substances are conductors the charges on each will dis- 
 tribute themselves over the surfaces of the separate conductors. The charges 
 on the conductors must be equal and opposite (their total must be zero) and 
 so will be practically all concentrated on the adjacent surfaces at the surface 
 of contact. The potentials of the conductors being ^ and </> 2 , the difference 
 01 ~~ 02 i g always the same for the same conductors under the given conditions ; 
 and is usually called their volta difference of potential. 
 
 When equilibrium has been established, which usually requires only -an 
 extremely short time, there is an electrostatic field surrounding the conductors 
 which obeys all the laws of electrostatics. In the interior of the surface of 
 contact between the adjacent surface charges the electric field is however 
 compensated by the contact forces of chemical affinity. Thus for example 
 in the Zn-Cu case mentioned the zinc attracts the positive charge from the 
 copper and the copper the negative charge from the zinc, the result being 
 that the zinc becomes positively charged and the copper negatively. But 
 each addition to the positive charge on the zinc repels the remaining positive 
 charge on the copper and so lessens the total attraction of the zinc on it. 
 The separation thus goes on till the attraction of the zinc for any further 
 positive charge becomes balanced by the repulsion of that charge by the 
 positive electricity on the zinc. 
 
 The electrostatic field of the conductors is practically that due to the 
 double sheet on the surface of separation, the small remaining charges on the 
 further parts of the surface having no effect. 
 
 315. Now consider several such conductors joined in a ring. There 
 would be a potential difference at each junction of two different conducting 
 surfaces caused by the creation of a double sheet as indicated above. There 
 would thus be a potential gradient at each junction in the circuit and thus the 
 system is ready for an electric current to flow. But no current can flow or 
 else the principle of the conservation of energy would be violated. The non- 
 existence of the current may be explained by the fact that the currents arising 
 from the single impressed electromotive forces at the separate junctions so 
 flow as to cancel one another out. 
 
 The energy principle is no longer violated and a current can result if only 
 we could supply energy from some external source at one of the junctions. 
 We shall presume the possibility of this supply, postponing the discussion 
 
313-316] The Volta potential difference 287 
 
 of the exact method in which it is applied. We should then have a current 
 in the circuit, its density at any point being determined by 
 
 C = *E, 
 
 E being the electric force intensity of the field in the interior of the conductor ; 
 but E is composed of parts E 1? E 2 , ...-so that 
 
 E = E, + E 2 + ..., 
 
 where E x , E 2 , ... are the components in the direction of the resultant E of 
 the several force intensities in the separate fields arising from the double 
 sheet distributions at the various junctions. 
 
 316. Adopting the notation of the previous paragraph and considering 
 for the present the current of strength dJ flowing through an elementary 
 tube of flow (or tube of force) in the circuit of cross-section df at any place 
 we have 
 
 dj 
 
 j, n . f K (E, 8s + E 2 Ss + ...)df 
 
 where dJ = Cdf = _ lg * s 
 
 j P, 
 
 8s 
 or by integration round the whole circuit, i.e. with respect to 8 
 
 / J f* 
 
 <W I -4>= I (Ei ds + E 2 ds + ...). 
 
 J Kdf J S V * 
 
 But I E! ds = <f>i c/> 2 = ( 12 
 
 is the volta potential difference of the two metals at the first junction of the 
 circuit reckoned in a definite sense round the circuit. In this integral the 
 element of the circuit between the double sheet concerned is of course not 
 included, although the current crosses through this sheet ; the argument for 
 this may be stated as follows. If there were equilibrium with no current 
 flowing the electric force intensity E in the contact sheet is exactly balanced 
 by the contact forces (of chemical affinity), say S, so that 
 
 S - E = 0. 
 
 If however the charges separated by the contact forces can break away and 
 flow off as an electric current there is no longer an-exact balance. But even 
 in this case the outstanding difference between S and E is small compared 
 with either of them, because sufficient charge accumulates in any case to 
 establish the volta potential difference. Thus since S and E are both large 
 of the first order the difference S E can be at most finite, say E', so that 
 
 S - E = E'. 
 
 Since now the value of K in the contact surfaces is certainly not large (it is 
 at most finite) the values of.E'c? and KE'd may be neglected in comparison 
 
288 Electric currents in metallic conductors [OH. vn 
 
 with finite quantities, d being the thickness of the sheet. Thus practically 
 the -only driving force for the current is that in the interior of the metals. 
 We have thus 
 
 7>=<^12 + ^23 + "> 
 KM/ 
 
 or j 
 
 where 
 
 represents the total resistance of the circuit. This is of course Ohm's law 
 for the circuit. 
 
 317. If no energy is supplied at any part of the circuit then J = and 
 thus </> 12 + </> 23 +...</> nl = 0, 
 
 which is Volta's law. This sum is however no longer zero when a current is 
 flowing or when energy is being added from outside at one of the junctions. 
 
 A direct consequence of this result is that the volta difference 'of potential 
 of two metals in contact is exactly the same as the potential difference oi 
 the same two metals connected through a whole series of other metals. No 
 direct electrometer reading will therefore ever detect the potential difference 
 here described, because the instrument merely records the difference between 
 the pieces of metal forming its quadrants (which are of course of the same 
 metal*). 
 
 318. If we integrate in the positive direction round only part of the 
 circuit, say that between the two sections a and jS we get 
 
 where <f> a , cf>p are the potentials at the sections a, [3 and X(/> r>r+1 refers to the 
 vplta potential difference for all the junctions occurring in the section of the 
 circuit between a and j8. Consequently we have also 
 
 r _ <a <ftg + 2<ftr,r+l 
 
 "" ~~ ~ ' 
 
 where & 0j/3 is the resistance of the part of the circuit concerned. 
 
 If the sections a, j8 are very near but on opposite sides of a given junction 
 the resistance k^ is very small and so 
 
 or 
 
 * By constructing electrometers with opposite quadrants of different metals it is possible to 
 determine the difference of potential between the metals (Kelvin). 
 
316-319] The voltaic element 289 
 
 so that the potential in the circuit jumps at each junction by the corresponding 
 volta difference of potential for that junction. If we put the sections a, j8 
 close together but so that practically the whole circuit is included between 
 them and then remove the small portion of the circuit not included J = 
 
 andthus *.-*- -s&,, +1 , 
 
 the potential difference between the ends of an open circuit is equal to the 
 electromotive force operative in the same circuit when closed. 
 
 319. The general practical method of obtaining an effective electro- 
 motive force of this kind in the circuit is obtained by the insertion in it of a 
 voltaic element or cell. The general principle involved may be illustrated 
 by the following particular example. If we put into a vessel which contains 
 dilute sulphuric acid a plate of copper and a plate of zinc at a small distance 
 apart, it will be found by connecting the ends of the metals projecting from 
 the acid to the quadrants of an electrometer that the pair of quadrants 
 connected with the copper is at a higher potential than the other. In this 
 experiment we have a series of conductors ; brass, copper, acid, zinc and 
 brass. The potential in the interior of each conductor has the same value 
 at each point but it jumps at each surface of contact and the observed 
 potential difference is the algebraic sum of the jjimps. If </>#, <j>c, <f>s> 
 <f>z, <f>B denote the potentials of the metals in order, and if also we use (f>BC 
 for (^B </>c then we have the observed difference equal to 
 
 4>BC + <f>CS + <j>SZ + <f>ZB 
 
 One or more of these terms may be negative and if the acid were replaced 
 by a metal the sum would be zero. It is only because we have an acid (or 
 fluid conductor) in the series that the expression has a definite positive value 
 different from zero. 
 
 The apparatus here described and many others of a like nature, which 
 are composed of rigid and fluid conductors and which possess the property 
 of creating a potential difference between two pieces of the same metal is 
 called a galvanic element or cell. They were first invented and used by Volta 
 and are called after him*. 
 
 With such an apparatus it is possible to produce a permanent steady 
 current by connecting the metal ends projecting from the liquid through 
 a simple metallic circuit. The chemical affinities of the elements of metal 
 and those of the acid produce an electric separation in the manner previously 
 described and the double sheet so produced makes the sudden jump of 
 potential in crossing from the metal to the liquid. Chemical attractions 
 prevent the separate induced charges combining across the liquid and so 
 they have to go round the metals closing the circuit. The double sheet thus 
 
 * Cf. the letter in Phil. Trans. (1800), p. 402. Also Gilbert's Annalen, 6, p. 340. 
 L. 19 
 
290 Electric currents in metallic conductors [CH. vn 
 
 dissipated by these charges going round is then continually renewed by 
 chemical action and this makes a permanent current. The energy supplied 
 at the junctions here, is the chemical energy of combination of the one sub- 
 stance with the other. The precise nature of the action involved will be 
 discussed in the next chapter. 
 
 320. On impressed forces in mechanics : an analogy*. The mechanical 
 distinction between the two cases here discussed, when the current can 
 exist and when it cannot, is easily recognised. The closed circuit of physically 
 and chemically homogeneous conductors in itself is a self-contained mechanical 
 system so that no current can flow in it unless there are external forces of 
 some kind acting. Thus if a current flows in such a circuit there must on 
 the whole be an impressed electromotive force equal to Jk (as this is the 
 electromotive force in the circuit) which arises from actions of a purely 
 external nature. 
 
 This conception of electromotive force corresponds exactly to the concep- 
 tion of external or impressed forces in the mechanics of ponderable bodies. 
 The idea is that of a force, not determined by the conditions which govern 
 the system under consideration, but which nevertheless acts on it in an 
 arbitrary manner and by means which are in no way in essential connection 
 with the system. For example in a system of elastic bodies, the elastic 
 stresses and internal forces are in perfect accord with one another and with 
 the internal deformations and motions ; but external forces may act on the 
 system in any arbitrary manner. Of course, these forces influence the distri- 
 bution of the internal stresses but only in the sense that they alter the 
 conditions under which the system exists. 
 
 321. There are two reasons why we introduce the idea of external forces 
 into ordinary mechanics. The first is that we can thereby limit our discus- 
 sions to the consideration of a particular system by itself. If in the above 
 example of elastic bodies the external forces are produced by weights, their 
 action is determined by mechanical laws; such forces would of course be 
 internal forces if gravity were also included in the mechanical system. 
 Another important example will be given later, it involves the electro- 
 motive forces which can be produced by the oscillations of a magnetic field 
 in the neighbourhood of a current circuit or to the motion of that circuit 
 through a magnetic field. Such forces are external as long as we prefer to 
 leave the magnetic field out of the calculation. They become internal 
 forces, whose action is determined by the ordinary laws of electrodynamics, 
 as soon as the magnetic field is considered part of the system. 
 
 The second reason for introducing external forces into mechanics is that 
 they often represent actions, which cannot be sufficiently well explained on 
 * Cf. Abraham, Theorie der Elektrizitdt, i. p. 199. 
 
319-322] The energy of an electric current 291 
 
 a purely mechanical basis. Examples are provided in connection with the 
 motions of magnets or electrically charged bodies. If mechanics attempted 
 to explain all the motions of natural bodies, it would tacitly ignore all those 
 motions which cannot be explained on a purely dynamical basis. In the 
 cases mentioned, for example, a complete dynamical description of the motions 
 is not possible until we have a mechanical explanation of the magnetic and 
 electric actions. Mechanics treats only of one side of natural phenomena, 
 it is useless when we are dealing with phenomena of a non-mechanical nature. 
 A modified usefulness is however attained by admitting ignorance of their 
 fundamental basis but representing the actions of these non-mechanical 
 processes by means of impressed forces. The present type of impressed 
 electromotive force in a circuit is of this nature. Such forces could not be 
 treated from the standpoint of pure electric theory, because we are in reality 
 involved in them in chemical and thermal phenomena, the laws in which 
 are known only in a few special cases. Thus if we limit ourselves to the 
 description of electrodynamic phenomena, we must in such cases resort to 
 the idea of the impressed electromotive force to explain the action of such 
 processes on those under direct review. 
 
 322. On the energy relations of an electric current*. We must now 
 enquire into the amount of work expended in driving the current. Consider 
 for this purpose any conductor in which a current J is flowing. Let ^ 
 and (f> 2 be the electrostatic potentials of the internal electrical field at two 
 sections of this conductor between which the resistance is & 12 . In a time St 
 an amount of electricity equal to JSt is transferred from the one section 
 to the other through the conductor (or at least this is the effective result of 
 the electrical flow during this time). This means that an amount of 
 electrical energy (^ _ ^ j^ 
 
 has been lost in this part of the conductor during the time St. Per unit 
 time this is j (^ _ ^ 
 
 What has become of this energy? The driving of the current is an affair 
 of diffusion, the electric force is pushing the electric charges along among 
 a large number of obstacles. The electric atoms get up a velocity, but 
 impart it by collision to the molecules of the matter, so that their own motion 
 becomes irregular. This is the essence of frictional resistance; the motion 
 of the electric elements becomes irregular through collision with the obstacles. 
 The wasted energy thus appears again as irregular motion of the electric 
 charges and also partly of irregular motion of the molecules of the matter, 
 that is it appears as heat in the conductor. Thus the heat developed per 
 .unit time in the portion of the conductor considered is 
 
 J (</! ^2)- 
 * Kelvin, Phil. Mag. Dec. 1851. 
 
 192 
 
292 Electric currents in metallic conductors [CH. vn 
 
 If we introduce Ohm's principle that 
 
 the heat developed appears as of amount 
 
 JZTf 
 J K 1Z , 
 
 per unit time. Thus in the whole circuit the total heat developed is 
 
 where Jc denotes the resistance of the circuit. This expression however not 
 only gives the total amount of heat but also its location. 
 
 323. If a voltaic cell is supplying the current the energy which appears 
 as heat in the circuit must also come from the cell. Moreover this energy is 
 available energy for if we had conductors of small resistance we could turn 
 it into work. (If the conductors are of big resistance the work is entirely 
 wasted in them.) This work is introduced into the circuit in the battery 
 at the places where the substances are decomposed (i.e. at the surfaces of 
 metals in liquid). The supply of available energy comes in from the liquid 
 and may be used to drive a machine somewhere if it does not waste. The 
 location of the energy supply is different from that of its emergence. Thus 
 an electric current is a means of transmitting power. 
 
 If a voltaic cell is the source of the current the total heat developed in 
 the circuit appears as the work required to raise the quantity of electricity 
 supplied by the current through any cross-section of the circuit through the 
 various potential jumps at the contact surfaces in the circuit. Because 
 (< x </ 2 ) for a whole circuit is simply 
 
 for that circuit. The work per unit time is thus 
 
 J . (2</v ;r+1 ), 
 
 or since S</> r>r+1 is the quantity measured as the electromotive force of the 
 cell, it is measured by the product of the current by the electromotive force 
 of the cell. 
 
 It is of course assumed that all the work done on the electric charges 
 in driving them forward is spent in increasing their velocity and is thus 
 dissipated by collision and appears as heat. This only applies when none of 
 the energy is turned into mechanical work as is usually the case in circuits 
 containing dynamos. 
 
 324. The above results, based on the idea of diffusion, were experi- 
 mentally tested and verified by Joule 75 years ago*. From his results he 
 formulated his law expressing that the total amount of heat developed as 
 expressed above is correct and also that the distribution given is correct. 
 
 * Phil. Mag. 19 (1841), p. 260. 
 
322-325] Joule's law of minimum dissipation 293 
 
 He was also able to formulate the principle for electric flow which in its 
 generalised form states that in all cases of diffusion the flow distributes itself 
 so as to give the least possible heat for a given current. In other words if 
 a given current is introduced into a network of conductors it distributes 
 itself among the conductors so that the energy wasted is least. 
 
 The generalised principle in the form that in any steady dynamical motion 
 of a given material system, when the forces are only frictional, the motion 
 is such as to make the waste of energy the least possible, was given and proved 
 by Lord Kelvin. The particular case here quoted is however usually called 
 Joule's law of minimum dissipation. 
 
 325. The proof of this law in the electrical case is easy. Let there be 
 
 M (/y% "1 \ 
 
 n electrodes joined by = wires and let there be given electromotive 
 
 a 
 
 forces in the wires and given conditions of supply and withdrawal of current 
 at the electrodes so that the currents in the wires are steady. 
 
 Let </> l5 (f> 2) ... </> be the potentials of the n points; Q 1) Q 2 , ... Q n be the 
 amounts of electricity supplied per unit time at these points so that in a 
 steady state & + &+... G = 0, 
 
 and let E rs be a possible internal electromotive force in the wire joining the 
 rth and sth points so that p _ 7? 
 
 ^rs ^sr > 
 
 and also let K rs denote the reciprocal of the resistance in this same wire so 
 that K rs = K^; we also use other symbols K :i , K 22 , ... having no physical 
 significance but which are such that 
 
 KU + .12 + ... + K ln = 0, 
 
 #21 +# 22 + ... + # an = 0, 
 
 etc. 
 
 In applying Ohm's law to each conductor and examining the flow at each 
 point we have 
 
 Ql= #12 C^ia+^l ^2) + #13 (#13 + </>! ^s) + + #ln(#ln+</>l ~ </>)> 
 
 Q z = # 21 (# 21 +</> 2 </>i) + + K 2n (E 2n + (f) z (f) n ), 
 
 These equations, usually ascribed to Kirchhoff*, can be written in the form 
 
 #11</1 + #12^2 + ' Kln<}>n = ~ Ql + #12^12 + Kln^m, 
 
 The sum of the left and right-hand sides of these equations is zero and they 
 therefore reduce to (n 1) independent ones. 
 
 * Kirchhoff. Ann. Phys. Ohem. 64 (1845), p. 512; 72 (1847), p. 497 ; Wheatstone, Phil. Trans. 
 2 (1843), p. 323; Ann. Phys. Chem. 62 (1844), p. 535. Cf. also Maxwell, Treatise, vol. i. 
 
294 Electric currents in metallic conductors [OH. vn 
 
 These are the conditions of flow, if the currents obey Ohm's law and we 
 have now to show that the heat developed in this case' is the least possible. 
 
 326. We multiply the above equations in order by fa, <f> 2 , ...<f> n and 
 then add the n equations together ; this gives us 
 
 n 
 
 K rs (fa (f> s ) z = Qrfa + ^K rs (fa </> s ) E rt , 
 
 whence also 2*Q r fa = E# rs (fa <f> s ) (fa ~ <t>s + E rs ), 
 
 but if J rs is the current in the conductor joining the rth and sth points we 
 
 have K rs (fa -fa + E rs ) - J rs , 
 
 and thus ^Qrfa = 2/r S (fa ~ </>)> 
 
 or also if we use k rs = l/K rs 
 
 either of which expresses the energy equation. The total heat lost in the 
 wires is equal to the total energy supplied at the junctions plus that drawn 
 from the cells in the circuits. 
 
 In these equations J rs is the actual current in the typical conductor 
 determined by Ohm's law. Suppose now that J rs + x rs be a modification 
 of this current which is compatible with the conditions of supply and output ; 
 so that 7. _i_ /*. _i_ x, = 0. 
 
 and then we have 
 
 Zk rs (J rs + x rs Y - 2k rs J rs 2 = I>k rs x rs 2 + 2Sa? r A*7 rs 
 
 = SAv.av." + 2Sav, (fa -fa), 
 
 if there are no internal electromotive forces in the circuits to complicate 
 matters. But then the last term on the right is 
 
 2S (x rs fa + x sr fa), 
 and is zero in virtue of the linear relations among the x's and thus since 
 
 2j& rs (J rs + X rs ) 2jK rs J rs , 
 
 is essentially positive in this case the heat developed in the actual state, 
 with no internal electromotive forces, is less than that in any other state. 
 The more general theorem when internal electromotive forces are included 
 will be discussed on a future occasion. 
 
 327. The general relations for a network of conductors. We have 
 already obtained the general equations for the distribution of steady con- 
 duction currents in a network of ^^ wires joining n electrodes. We 
 
 2i 
 
 can now indicate the general method in which they may be solved and also 
 the chief characteristics of the solution. 
 
325-328] The general equations for a network 295 
 
 It is required to find the currents in the wires when the following data are 
 specified : 
 
 (/> 15 </> 2 , ... (f> n the potentials at the n points, 
 
 Qi> Qz> Q n the supplies per unit time, 
 so that Qi+Qt+ ... Q = 0. 
 
 E TS the general type of internal potential in the wire joining the rth to 
 the sth conductor so that 
 
 E = E 
 
 *** -^ST) 
 
 and K rs the inverse' resistance in the same wire so that 
 
 K = K 
 
 "r " 
 
 We then introduced other symbols of type K rr so that 
 
 K rl -f K r2 + ... + K r (,D + K rr + . . . + K rn 0, 
 and then found that the general equations could be written in the form 
 
 ^11</>1 + Kl2<f>2 + Kln<f>n = ~ Ql + ^12^12 + ^13^13 + + ^In^ln, 
 
 The sum of the left and right-hand sides of these equations are zero and 
 so they are not independent; they reduce in fact to (n 1) equations which 
 determine the differences of potential, which is all we are concerned with. 
 On account of the relations between the K's these equations are equivalent 
 to the (n 1) equations 
 
 K rl <f) ln + K rZ (f>z n + ... K rtn -i<f> n - ltn = Q r + K rl E rl -\- ... r = 1, 2, 3, n 1, 
 where we have used (f> rn = (f> r </> 
 
 for the potential difference between the rth and nth electrode. 
 
 328. These equations can be easily solved for the </>'s and they lead for 
 example to an expression for <f> rn of the form 
 
 I _ / f\ i ~T7 El I ~I7 ~E> \ IT 
 
 where A = K 11} K 12 , K ls ... ^C 1(n -i 
 
 l w _ _! 2 K n -i n-l 
 
 and A., the minor of K rs in this determinant. 
 
 ~ 
 
296 Electric currents in metallic conductors [CH. vn 
 
 Having thus determined (f> rn we can easily determine the current in the 
 particular conductor joining the rth and nth electrodes for by Ohm's law this is 
 
 Krn (<f>rn + E rn ). 
 
 Suppose for example that the whole system of currents in the network is 
 produced by a current Q entering at the rth electrode and leaving at the 
 sih, there being no batteries in the network. Then all the E's are zero and 
 all the Q's as well except Q r and Q s , these being given by 
 
 Qr = ~Q.= Q. 
 
 We thus get that 
 
 - (A r/ - 
 
 similarly e/> s < n = - ^ (A r8 . - A ss <), 
 
 so that < rV = <f> r > n - fa n = - - ( A rr < + A SS ' - A s/ - A s 
 
 , 
 
 from the symmetry of which we see that the potential fall between the r'th 
 and ' th points in the network when unit current traverses the network from 
 the rth to the sth points is the same as the potential fall between the rth 
 and sth points when unit current traverses the network from the r'th to the 
 s'th points. 
 
 Many other relations of a like nature can immediately be deduced. 
 
 329. All these results follow however equally well without troubling 
 to solve the equations as above. In fact if the conditions in any second 
 distribution of currents are denoted by the same notation as above but with 
 dashed letters then we have a similar set of equations in these dashed 
 letters. Now multiply the first set of equations by c^' ', </> 2 ', ... and the 
 second set by < 1} <f> 2 , ... and sum each set : we evidently get the same 
 on the left-hand sides and therefore the right-hand sides must also be equal. 
 This leads to the relation 
 
 S,/,/ (- Q r + k rl E rl -+ ...) = Sc/v (- Q r f + K rl E rl ' + ...). 
 Let there now be no electromotive forces and let a current Q go in at the 
 rth point and emerge at the sth point and in the second state let the' current 
 Q' go in at the r'th point and come out at the s'th point. Now in the first 
 
 state Q r = Q, Q S --Q, 
 
 and in the second Q,, = Q', Q s - - - Q', 
 
 and the above theorem reduces to 
 
 (</>/ -<;) = '(</y-M 
 which is the same result as determined above. 
 
328-330] 
 
 Wheatstone's bridge 
 
 297 
 
 330. An important application of this theory is in the bridge arrange- 
 ment invented by Wheatstone* for comparing resistances. 
 
 The bridge is represented diagrammatically in the figure. The current 
 enters it at the point A and leaves it at the 
 point D ; these points being connected by 
 the lines ABD, ACD arranged in parallel. 
 The line ABD is composed of two con- 
 ductors (AB) and (BD) of resistances k 12 , 
 & 24 and the line (ACD) is composed of 
 two conductors of resistances k l3 , k u . 
 The points (B, D) are connected by a 
 wire of resistance k 2S . Thus with the notation as above, if <f> 1} (f> 2 , </>3, <f>4 
 are the respective potentials of the four points A, B, C and D and if the 
 rate of supply of current is Q units per second then 
 
 -^ii^i + Ki2<j>z + -Kis^s = Q> 
 
 ^21^1 + -^2202 + ^2303 + ^24^4 ~ ^J 
 
 ^3101 + KSZ^Z + ^33^3 + ^34^4 = 0> 
 
 ^42</>2 + -^4303 + ^44^4 = Q> 
 
 where the coefficients K, the reciprocal resistances, are subject to the condi- 
 
 Fi 
 
 tions 
 
 + 
 
 + 
 
 K 12 + K 1S = 0, 
 
 + -^23 + -^24 0, 
 
 _ + #33 + # 34 = 0, 
 
 These are the general equations from which the potentials </> l5 (f> 2 , </> 3 and 4 
 can be directly determined and thence also the currents in each of the wires. 
 
 The important case occurs when there is no current in the conductor (23) . 
 The condition for this is j, _ j, _ <j, sav 
 
 and then the second and third equation from each set give 
 
 or we must have 
 
 or since K ra = j this condition is equivalent to the condition that 
 
 K 
 
 4? _ ^24 
 
 C-to A/q-i 
 
 * Phil. Trans. 2 (1843), p. 309; Pogg. Ann. 62, p. 509. 
 
298 Electric currents in metallic conductors [CH. vn 
 
 Thus if we know the resistances & 13 , & 24 , & 34 we can conclude from the attain- 
 ment of the condition mentioned the resistance & 12 . In the simplest form 
 of the bridge, the line ACD is a single uniform wire and the position of the 
 point C can be varied by moving a ' sliding contact ' along the wire. The 
 ratio of the resistances k l3 : & 34 is in this case simply the ratio of the two 
 lengths AC : CD of the wire so that the ratio k lz : & 24 can be found by sliding 
 the contact C along the wire ACD until there is observed to be no current 
 in BC, and then reading the lengths AC and CD. 
 
 331. Stationary current streaming in three dimensions. As a further 
 example of the general principles of the present chapter we may now examine 
 the general problem of stationary currents in three dimensions. 
 
 The currents are presumed to be generated as a result of the application 
 of external electromotive forces, which will be generally taken to be distributed 
 throughout the field with an intensity at each point specified by E e . In 
 the process of establishing the current flow a slight accumulation of charge 
 will take place at certain parts of the conductor, these being necessary to 
 secure the subsequent uniform streaming. When the steady state is attained 
 the electrostatic field of these accumulated charges will be steady, and there- 
 fore possesses at each point a potential (/>. The additional electromotive force 
 is therefore E = -grad<. 
 
 The total force driving the steady current is E + E e and thus if the specific 
 conductivity of the medium is K, the current density in the steady streaming is 
 
 C = K (E + E e ). 
 
 We have another condition which states that in the steady state there 
 is no further accumulation at any point, and thus the integral 
 
 f 
 
 taken over any closed surface in the field must vanish : this means that 
 
 div C = 0, 
 
 or the current is a stream vector. 
 
 / 
 
 The driving of the current is a process of diffusion and its energy is thus 
 gradually being converted into heat energy ; the amount of heat developed 
 at any point in the conductor, is per unit volume 
 
 (C . E + E.) = K (E + E.) 8 , 
 and is supplied by the impressed electromotive forces. 
 
 The condition of steady streaming implies that (f> satisfies the character- 
 istic equation div (|C grad ^ = di v ^ ^ 
 
 isotropy of the medium being of course presumed. If K is constant this 
 reduces to 
 
330-333] Electric flow in three dimensions 299 
 
 It is therefore only in the case when div E e = that no current flow is 
 possible. This case occurs in every closed circuit composed of conductors 
 at the same temperature and which is such that no chemical changes whatever 
 result from an actual flow of current produced in any way. 
 
 332. The general problem in this subject is to obtain the appropriate 
 solutions of the general characteristic equation 
 
 div (K grad (/) = div E e , 
 
 in each different part of the field and then fit them up across the boundaries 
 between. This of course necessitates the specification of certain boundary 
 conditions which are always satisfied. These are obviously : 
 
 (i) The normal component of the current must be continuous across any 
 surface of discontinuity in the medium, otherwise there would be accumulation 
 of charge on the surface 
 
 or KI K 2 
 
 ^Wj 2 3w 2 
 
 (ii) The potential difference between near points one on each side of the 
 surface of discontinuity is equal to the contact difference of potential for the 
 
 two media in contact there A . _ JL 
 
 <Pi ~~ 92 9i2> 
 
 which, if not obvious, is easily deduced by an examination of the integral 
 
 E s ds = - lgra,d 
 
 taken along any path between two such points without going through the 
 interface. 
 
 The important point in this result is that it implies continuity of the 
 tangential electric force across the surface 
 
 
 We can easily prove in the usual manner that these conditions determine 
 (f> to an additive constant, provided of course the distribution of impressed 
 electromotive forces E e is specified. 
 
 333. If the tangential components of the impressed electromotive forces 
 vary steadily through a transition layer between two conducting metals 
 in the field or if they are zero then continuity of tangential electric force also 
 implies 
 
300 Electric currents in metallic conductors [OH. vn 
 
 and thus for the tangential components of the current intensities on either 
 side of the surface we have a relation 
 
 K l K 2 
 
 and the first condition gave us 
 
 C OTl = Cn 2 , 
 
 so that here we have a definite law for the refraction of the stream lines of 
 the current flow through the surface. If we denote by 04 and a 2 the angles 
 
 ( < 9 ) which C x and C 2 make with the normal to the interface at the point of 
 incidence under consideration, these equations show that 
 
 tan a x K I 
 tan a 2 K Z ' 
 
 and the 'incident' and 'refracted' currents lie in a plane with this normal*. 
 
 334. In the non-conducting parts of the field C = everywhere and the 
 above characteristic equations do not give us anything ; but we know that 
 in these parts div E is the density of the free charge at the points and if 
 we exclude the existence of such we have 
 
 V 2 c/> - - div E = 0, 
 at all points in the dielectric. 
 
 At a boundary between a metal and a dielectric (non-conducting) the 
 boundary conditions reduce to the simpler form 
 
 (i) C B = 0, 
 
 or ^ =E e , 
 
 \dn/i 
 
 wherein i refers to the internal field in the metal. 
 
 <f> io being a constant for the two surfaces adjutting. This implies 
 
 \dtJi \dtJ ' 
 o referring to the external field. 
 
 The conditions are still sufficient to determine the problem. 
 
 335. In the cases which actually occur in practice the impressed electro- 
 motive forces are applied outside the bodies, in which the current flow is to 
 be determined. The electric current is then supplied to these bodies through 
 electrodes. When the latter are composed of much better conducting 
 * Kirchhoff, Ann. Phys. Chem. 64 (1885), p. 497. 
 
333-335] Electric flow in three dimensions 301 
 
 material than the substances in which they are imbedded the problem is 
 determinate. If we put E e = everywhere and K = oo for the electrodes 
 then we must also have E = at all points in an electrode. This implies 
 that </> is constant on the electrodes. The problem in this case may be thus 
 stated as follows : 
 
 (a) To determine a regular function </> which inside the conducting 
 substances satisfies fa v ( K grad ^ = , 
 
 and in the dielectrics y2 ( /, __ Q 
 
 and which assumes given constant values (f>^, (f>g, cf>c, ... on the electrodes 
 A, B, C, ... and at other parts of the boundaries of the metals 
 
 ty o 
 5 = U. 
 
 on 
 
 If <f> is determined then E and C follow directly. 
 If the currents supplied through the electrodes are J A , J B , J c , ... so that 
 
 7 / f\ Jf J p Jf 
 
 J A I ^> n a J, JB = I'nqji 
 J A J B 
 
 then we obtain by multiplication of these equations by <$> A , cf> s , ... respectively 
 and adding 
 
 A J B 
 
 and this latter integral sum can be converted into the integra 1 
 
 taken over the volume of the conductors. This gives the energy equation. 
 Direct integration over the outer boundary of the conductor leads to the 
 continuity equation J A + J B + J Q + = 
 
 If only two electrodes are present, say A and B, then 
 
 JA = JB = J, 
 and thus the heat developed in the conductor is expressed by 
 
 H = J(<{> A -<f> B ), 
 or if we denote by k the effective resistance of the body denned by 
 
 <f>A <f>B = kJ, 
 then H - kJ z . 
 
 The general problem here defined reduces to the analogous electrostatic 
 one when the outer boundary of the conducting substance, where it does 
 not coincide with the charged surfaces, is chosen to be composed of lines of 
 force in the electrostatic field. Now consider the two following analogous 
 problems*. 
 
 * Of. E. Cohn, Das electromagnetische Feld. p. 155. 
 
302 Electric currents in metallic conductors [CH. vn 
 
 336. (1) Two conductors A and B are kept at a constant potential 
 difference <E>; the field of this arrangement in air is known and the dis- 
 tributions of charge on the conductors. Now suppose that a number of 
 complete tubes of force running between the two have the air dielectric 
 replaced by a homogeneous dielectric of constant e. The solution of the 
 new problem is obviously that 
 
 (a) The forms of the lines of force and equipotential surfaces are 
 unchanged. 
 
 (6) The potential and therefore also the force may be everywhere 
 unchanged provided that 
 
 (c) we make the density at the ends of the tubes of dielectric e times its 
 previous value. 
 
 We could interpret this result by saying that the parts of the conductors 
 where the dielectric tubes adjut on them contribute a part to the capacity 
 of the whole arrangement equal to 
 
 6 = S/]r# 
 <&Jdn J 
 
 when the dielectric is there and 
 
 when it is away; </> being the potential in the field in either case and the 
 integral is taken over the parts of the surface of either conductor covered with 
 dielectric. 
 
 337. (2) The two conductors A and B are now perfectly conducting 
 electrodes maintained at a potential difference <E> : the space previously 
 filled with dielectric is now filled with some homogeneous conducting sub- 
 stance of conductivity K. The solution of this problem obviously leads to 
 the same potential function </> and thus we deduce that the resistance of the 
 arrangement is 
 
 the integral being taken over the same part of the surface of the one conductor 
 as above. We have thus the relation between k and 6 
 
 Z > = & 
 
 1C* 
 
 338. A simple example of this arrangement is provided by the portions 
 of two very long concentric cylindrical electrodes included between two 
 planes through the axis making an angle a with each other. If the radii 
 of the cylinders are r : and r 2 the capacity of the part considered is 
 
 log- 2 
 g 
 
336-339] Electric flow in three dimensions 303 
 
 1 a/c 
 
 i ,1 
 and thus 
 
 , n 
 
 log- 
 
 & 
 
 Y = - . 
 
 K , Tn 
 
 Many other examples will at once suggest themselves. 
 
 339. A slight simplification of the general problem is obtained by the 
 assumption of very small spherical electrodes, which is quite a sufficient 
 approximation in many cases for the physical requirements. If these 
 electrodes are at a distance apart from one another large compared with 
 their radii they may be treated as points, so long as the investigation of 
 the field is not pushed too close up to them. We may in a case like this 
 formulate the general problem in the modified form. 
 
 (6) To determine the function (f> which satisfies the equation 
 
 div (o- grad (/>) = 0, 
 in the conductors and Y2J, = Q 
 
 in the dielectrics; with the exception of certain points a, 6, ... where it 
 
 A B 
 
 becomes infinite like - 
 
 r r 
 
 In this case 4:7rA, farB, ... are the current supplies at the electrodes and 
 thus 
 
 A + B + C+ ... = 0. 
 
 For the calculation of the heat developed as well as the resistances however 
 we must revert to finite dimensions for the electrodes : as an example we may 
 consider the simple case of an indefinitely extended conductor with one 
 plane face, in which there is a small electrode through which a current J is 
 supplied. The second electrode is presumed to be at infinity. This problem 
 is exactly analogous to an electrostatic problem already solved, and thus the 
 solution is obvious. The potential in the conductor is 
 
 where r l is the distance from the centre of the electrode and r z the distance 
 from its image in the boundary plane of the conductor. The resistance of 
 the conductor is thus 
 
 , _ 
 
 ~ 
 
 where h is the distance of the electrode from the surface and a its radius. 
 The potential in external space is 
 
304 Electric currents in metallic conductors [CH. vu 
 
 The image method can also be used in a similar way when both media are 
 conducting, with specific resistances K I and /c 2 . We then get 
 
 i Kn JL 
 
 <* J l 
 
 G)a ^= . 
 
 2-77 (K! + /c 2 ) r 
 and for the resistance 
 
 i i 
 a K, + 
 
 340. As a last example of these principles we will consider a single 
 example of a non-homogeneous conductor. An infinite homogeneous con- 
 ductor (KJ) is traversed by a uniform current of density C in the direction 
 of the x-axis. We are required to calculate the disturbance of this uniform 
 flow caused by the introduction of a sphere of conductivity /c 2 and radius a 
 with its centre at the origin of coordinates. 
 
 Here again the problem is exactly analogous to the problem of the 
 disturbance caused in a uniform field of electric force by the introduction of 
 a homogeneous dielectric sphere. We can therefore write the results down 
 at once. 
 
 The resulting field is obtained by adding to the original potential 
 
 - G 
 
 K i 
 
 an additional part which in the sphere is 
 
 K K C 
 
 and outside is 
 
 K K C a s x 
 
 ^ = ~~ if 4- 2*r ' iT ' ^r*~' 
 
 A. o \^ ^l\. 1 ** I * 
 
 r being radial distance from the centre of the sphere. 
 
 More general cases can be analysed by harmonic functions*. 
 
 341. The thermal relations of an electric current. We have mentioned 
 so far the voltaic cell as the only means of producing by chemical action the 
 necessary energy to drive a steady current in a linear conducting circuit. 
 Experience however has shown that heat is also very effective as a generating 
 agent for electric flow. In fact a current is at once observed in a circuit 
 consisting entirely of pieces of metal if two junctions between different metals 
 
 * Various cases have been examined by Helmholtz, Ann. Phys. Chem. 89 (1853), pp. 211, 253 
 (Wiss. Abhandl. p. 494); W. M. Hicks, Messenger of Math. 12 (1883), p. 183; R. Felici, Tortolini 
 Ann. 1854, p. 270; Kirchhoff, Berlin. Monatsber. (1882), p. 72 (Ges. Werke, p. 66); Greenhill, 
 Proc. Camb. Phil. Soc. (1879), p. 293. 
 
339-34*2] The thermal relations of a current 305 
 
 or even two points of the same metal are maintained at different tempera- 
 tures*. 
 
 The simplest case is that in which the circuit consists of two pieces of 
 different metal A and B joined at P l and P 2 into a complete circuit. 
 
 Any difference of temperature between P l and P 2 then causes a current 
 to flow in the circuit and the direction of the current is reversed by inverting 
 the temperature difference. If the junctions are at the same temperature 
 there is absolutely no sign of any electric motion whatever. 
 
 Fig. 65 
 
 It is possible to arrange all the metals in a series so that in a circuit of 
 the type described the current flows across the warmer junction in the 
 direction from the metal higher in the series to that which is lower down. 
 A few of the metals arranged in this order are : bismuth, platinum, lead, 
 copper, gold, silver,, zinc, antimony. 
 
 342. If the circuit just described is broken at a point Q and if the 
 junctions P x and P 2 are maintained at the different temperatures 6 1 and 6 2 
 the same cause which causes the current to flow in the closed circuit will 
 create a potential difference between the ends of the broken circuit even 
 if these ends themselves are at the same temperatures. This potential 
 difference which may be directly measured will then serve as a measure of 
 the electromotive force in the circuit. It appears that for small differences 
 of temperature the electromotive force is proportional to (6 1 6 2 ) but for 
 larger differences the simple proportionality is not even approximately 
 verified. 
 
 These thermoelectric currents of course obey Ohm's law as regards the 
 relation between the electromotive force and current and they also obey the 
 series laws relating to compound circuits similar to those formulated for the 
 volta potential difference. 
 
 * Seebeck, Gilbert's Ann. 73 (1823), pp. 115 and 430; Pogg. Ann. 6 (1823), pp. 1, 133, 253. 
 L. 20 
 
306 Electric currents in metallic conductors [OH. vn 
 
 343. In order to explain these phenomena we can suppose that the heat 
 motion in the junction between the metals A and B drives the electricity 
 towards the former metal (which we may take to be antimony, B is 
 bismuth), with a force which is a function of the temperature. Then as 
 long as the junctions P x and P 2 are kept at the same temperature the two 
 electromotive forces at these junctions (indicated by small arrows in Fig. 65) 
 are equal and opposite and therefore neutralise one another. On warming 
 or cooling one of the junctions relatively to the other this equilibrium is 
 disturbed and there is a resulting electromotive force in the circuit. 
 
 If in the case exhibited above the warming of the junction P causes 
 a current to flow in the direction of the larger arrow we should expect that 
 at this junction heat would be used up in order to provide energy for the 
 current flux. This is actually the case : Peltier* observed for instance that 
 if a current is driven by an applied electromotive force across a junction 
 between Bismuth and Antimony in the direction B A the junction is 
 always cooled : whereas if the current is driven in the opposite direction 
 the junction is warmed. This local development or absorption of heat is 
 of course a function of the temperature and if the complete circuit contains 
 two such junctions at the same temperature the development of heat at one 
 junction exactly balances the absorption at the other, whereas if the junctions 
 are at different temperatures so that more heat is absorbed than developed 
 the heat which disappears will appear elsewhere in the circuit either as 
 Joule's heat or as heat of chemical transformations or it may be used up 
 in other more effective ways as purely electrical energy. 
 
 344. Next let us take a simple circuit again consisting now of a piece 
 of iron and a piece of copper and steadily heat the one junction up. The 
 electromotive force in the circuit gradually increases, at first proportional 
 to the temperature, but subsequently more slowly" until the hot junction 
 reaches the temperature 280, beyond which point an increase of temperature 
 reduces the electromotive force, finally changing its sign after passing through 
 the zero value f. We might again explain this in terms of our local electro- 
 motive forces at the junctions which we may now call <^ x and </> 2 at the 
 respective junctions. The electromotive force in the circuit is 
 
 </>2i = <f>2 </>i- 
 
 Now suppose that at ordinary temperatures with 6^ > 6 2 then </>j < </> 2 so 
 that </> 21 is positive, and that (f> 1 decreases as the temperature ^ is increased 
 attaining however a minimum value zero at the temperature 6 l = 280 after 
 which it gradually increases again, then</> 21 would behave exactly as described, 
 so that so far the explanation is effective. 
 
 * Ann. de Ckim. et de Phys. 56 (1834), p. 371 ; Pogg. Ann. 43, p. 324. 
 t Gumming, Annals of Philosophy, 6 (1823), p. 427. 
 
343-346] The thermal relations of a current 307 
 
 The temperature 6 = 280 C. 
 
 is called the neutral temperature of the copper-iron combination of metals. 
 Similar neutral temperatures exist for all combinations of metals. Moreover 
 it is found that at the neutral temperature there is no Peltier effect, so that 
 if an electric current is passed across the junction at this temperature then 
 no development or absorption of heat takes place. This agrees generally 
 with our explanation for it is only when the local electromotive force at the 
 junction is zero (</> : = 0) that there is no development or absorption of heat. 
 
 345. Suppose now we have a circuit of the kind described in which 
 the first junction has a temperature equal to the neutral temperature, 
 i.e. </>! = 0. The total electromotive force in the circuit is then 
 
 + </>2- 
 
 The direction of the current will then be that in which the Peltier effect at 
 the upper temperature (0j) exists as a heat absorption and at the lower 
 temperature as a heat development. But in the present instance no heat 
 absorption does take place at the temperature 6 l . In spite of this however 
 electrical energy will be transformed into heat energy at the lower junction 
 and developed there and in addition there will be the usual development of 
 Joule's heat. We are therefore driven to the conclusion. that at some position 
 in the circuit other than at the junctions heat must be absorbed and trans- 
 formed into electrical energy. This implies that even in a single wire whose 
 ends are unequally heated an electromotive force may arise as the result of 
 an absorption of heat and if this phenomenon is reversible there must be an 
 additional development of heat in the circuit when a current passes along 
 an unequally heated conductor. This phenomenon was theoretically pre- 
 dicted by Kelvin* and he was soon enabled experimentally to justify the 
 prediction : it is therefore usually known as the Kelvin or Thomson thermo- 
 electric effect. 
 
 346. The effectiveness of the explanation here suggested for these 
 phenomena is supported and further substantiated by the application to the 
 transformations of energy of which they are the expression of the general 
 laws of thermodynamics governing all such transformations. There are 
 however difficulties of a fundamental nature involved in any such applica- 
 tion in the present case : it may in fact be argued that the whole thermo- 
 dynamic procedure may be invalid because it is applied to a case in which 
 degradation is continually going on, in the form of conduction of heat, along 
 the same circuit which conducts the current, and of amount depending on 
 the first power of the temperature differences : and it does not appear 
 that this fundamental objection to the procedure can be safely ignored, 
 considering that conductivity for heat is closely connected with conductivity 
 for electricity. It would of course be removed if the heat conduction 
 
 * Phil. Mag. [4], 11 (1856), pp. 214 and 281. 
 
 202 
 
308 
 
 Electric currents in metallic conductors [CH. vn 
 
 proceeds in entire independence of the electric current, except as regards the 
 transfer of the electric elements, the influence of which is reversible and is 
 taken into account in the Kelvin effect. The electric cycle can moreover 
 be completed in so short a time that the thermal transfer by ordinary con- 
 duction may possibly be neglected. Waiving these difficulties however the 
 argument of Lord Kelvin* may be put in the following formj. 
 
 Suppose that in the transfer of the amount 86 of electricity from a place 
 where the temperature is 6 to a place where it is 6 + 86 the amount a868Q 
 of heat is absorbed by the current and converted into electrical energy of 
 motion of the electricity; a is called the 'specific heat of electricity' for the 
 conductor and it may be either positive or negative. 
 
 347. Let us then ignoring the finite degradation by heat conduction 
 but realising that the electric flow may 
 be made so slow that the electric degra- 
 dation, proportional to the square of the 
 current, is negligible and that therefore 
 the operations are certainly electrically 
 reversible in Carnot s sense apply the 
 principle of energy and Carnot' s prin- 
 ciple to a circuit, formed of two metals 
 and including as a part of itself the 
 dielectric of a condenser having these 
 metals for its coatings, the tempera- 
 ture 6 varying from point to point 
 along the circuit. When the plates of 
 the condenser are moved closer together 
 without alteration of temperature its 
 charge increases, as the difference of 
 potential <f> between the plates remains 
 constant : so that there is an electric 
 flow round the circuit and there is at 
 the same time a gain of mechanical 
 
 work and of available energy each equal to %(f>8Q, or in all <f> per unit total 
 flow. Thus the plates of the condenser (Fig. 66) being at the same temperature 
 a , we have, by the energy principle and Carnot's principle, considering 
 unit electric flow round the circuit 
 
 "" (a - a') dd, 
 
 Fig. 66 
 
 * The thermodynamic reasoning was first developed by Clausius (Pogg. Ann. 90 (1853),. 
 p. 513), but in an incomplete form. 
 
 t Cf. Larmor, Aether and Matter, p. 306. 
 
346-348] The electron theory of conduction 309 
 
 where Hj is the Peltier effect at the temperature 6 of the junction of the two 
 metals, that is the amount of heat absorbed or set free on the passage of unit 
 current across the junction at this temperature ; and cr, a' are the specific 
 heats of electricity in them. Thus 
 
 d 
 
 and < = n 
 
 Hence for a temperature of the junction, everything can be expressed in 
 terms of the curve connecting the electromotive force (f> of the circuit with 
 6, by the simple relations 
 
 II d$ a -a' dty 
 
 ~e~~dd' e = " do*' 
 
 The Peltier effect appears in the expression for </>, in the form of an electro- 
 motive force at the junction, as surmised in the simple explanation offered 
 above. The chemical mutual attractions of the molecules across the interface 
 produce in fact a polar electric orientation of these molecules which gives 
 rise to an abrupt potential difference of contact equal to II, and each unit 
 of charge passing across the junction thus introduces an energy effect II which 
 involves absorption or evolution of heat at that place in the Peltier manner. 
 
 f 
 
 The other term in the potential, viz. I (a a') dd is thermodynamically 
 
 involved in a convection of heat by the current passing from a warmer to 
 a colder part of the wire : the exact mode in which this arises will appear 
 
 better in our next discussions on the mechanism of metallic conduction. 
 
 
 
 348. The mechanism of metallic conduction. Many attempts have been 
 made to develop into further detail the idea of the electric current as a process 
 of diffusion, but before the introduction of the electron theory these attempts 
 could hardly be described as very successful. The difficulty arises in the 
 absence of any sign of transport of matter or of any chemical change 
 accompanying the transport of electricity. Every current circuit must 
 consist of two or more portions composed of different materials which may 
 contain no element in common and we must suppose either that the particles 
 which carry the current can pass from one material to the other or that they 
 cannot. Either supposition lands us in enormous difficulties. If the particles 
 cannot cross the boundary between the different metallic conductors they 
 must remain piled up at these boundaries, even if they are identical with the 
 atoms of the material in which they move, and this piling up must alter the 
 distribution of the mass in the conductor, while, if they are composed of some 
 substance different from that of the rest of the material, some sort of chemical 
 separation should occur. But the most careful experiments on pure metals 
 
310 Electric currents in metallic conductors [OH. vn 
 
 and alloys have failed to show the slightest change in any properties of a 
 metallic conductor induced by the passage of a current through it. On the 
 other hand if we suppose that the particles can pass freely from one material 
 to another new difficulties arise. We know that the atoms and groups of 
 atoms of different elements differ markedly in their properties : and we could 
 certainly detect the presence in one substance of atoms derived from a foreign 
 element. Since the properties of the materials forming a non-uniform circuit 
 are unchanged by the passage of the current, if the electricity is conveyed at 
 all by diffusing particles, these particles must be of a nature common to all 
 elements, or at least to all elements forming metallic conductors. Previous 
 to the discovery of the electron however no such electrical elements were 
 known to exist. 
 
 The discovery of the electron and the consequent formulation of the 
 so-called ' electron theory ' removed in one stroke all the difficulties thus 
 inherent in the earlier theories. According to this theory* there are in 
 every metal a very large number of (negative) electrons freely movable in 
 the spaces between the atoms, and it is the diffusion of these electrons through 
 the metal under the action of the electric force in the external field that is 
 the essence of a conduction current. If there is no external field the velocities 
 of these free electrons will be distributed equally in all directions : there will 
 be no tendency for them to move in one direction rather than in any other ; but 
 if the metal is placed in an electric field the electrons are subject to a force 
 in a definite direction (that of the force in the field) in virtue of their charge, 
 and those moving in this one direction will have their velocities increased 
 whilst those moving in the opposite direction will have theirs decreased, 
 there will thus be a slight drift of the electrons in a definite direction and this . 
 constitutes a current of electricity : the velocity of drift is however kept 
 in check by the continued encounters of the electrons with the metal molecules 
 and with one another when additional forces come into play tending to 
 deflect the electrons from their forward motion : the essential conditions for 
 a diffusion flux are thus satisfied. 
 
 349. Problems relating to the motion of the innumerable number of 
 electrons in a piece of metal are best treated by the statistical method which 
 Maxwell introduced into the kinetic theory of gases, and which may be re- 
 presented in a simple geometrical form so long as we are concerned only with 
 the motion of translation of the electrons. Indeed it is clear that, if we 
 construct a diagram in which the velocity of each electron is represented 
 in direction and magnitude by a vector OP drawn from a fixed point 0, the 
 
 * Cf. the original papers by Riecke, Wied. Ann. 66 (1898), pp. 353, 545, 1199; Drude, Ann. 
 der Phys. (1900), 1, p. 566; 3, p. 369: Thomson, Rapports de Congres de Physique (Paris, 1900), 
 3, p. 318. The treatment here given follows that given by Lorentz, The Theory of Electrons. 
 p. 266, or Proc. Amsterdam Academy, 7 (1905), pp. 438, 585. 
 
348, 349] The electron theory of conduction 311 
 
 distribution of the ends of these vectors, the velocity points as we shall say, 
 will give us an image of the state of the motion of the electrons. 
 
 If the positions of the velocity points are referred to axes of coordinates 
 parallel to those that have been chosen in the metal itself, the coordinates 
 of a velocity point are equal to the components |, 77, of the velocity of the 
 corresponding electron. 
 
 Let dv be an element of volume in the diagram, situated at the point 
 (f, 77, ) so small that we may neglect the changes of (, 77, ) from one of its 
 points to another, but yet so large that it contains a great number of velocity 
 points. Then this number may be reckoned to be proportional to dv : 
 representing it by /( ^ ) dv 
 
 per unit volume of the metal, we may say that, from a statistical point of 
 view / determines completely the motion of the swarm of electrons. 
 
 It is clear that the integral 
 
 (/(, 77, 0fo, 
 
 extended over the whole space of the diagram, gives the total number of 
 electrons per unit of volume. In like manner 
 
 , Qdv, 
 
 represents the stream of electrons through a plane perpendicular to Ox : 
 i.e. the excess of the number passing through the plane towards the positive 
 side over the number of those which go in the opposite direction, both numbers 
 being referred to unit of area and unit of time. This is seen by considering 
 first a group of electrons having their velocity points in an element dv; 
 these may be regarded as moving with equal velocities, and those of them 
 which pass through an element df of area in the said direction between the 
 moments t and t + dt, have been situated at the beginning of this interval 
 in a certain cylinder having df for its base, and the height gdt. The number 
 of these particles is found if one multiplies the volume of the cylinder by 
 the number per unit volume. 
 
 Hence if means an integration over the part of the diagram on the 
 
 . + 
 
 positive side of the 77- plane, and I an integration over the part on the 
 opposite side, the number of the electrons which go to one side is 
 
 dfdtf f(g, 77, Qdv, 
 
 j .1. 
 
 and that of the particles going the opposite way 
 
 dfdtj_ -/( 77, Qdv. 
 
312 Electric currents in metallic conductors [CH. vn 
 
 The expression given above is the difference between these values divided 
 by dfdt. 
 
 If all the electrons have equal charges e, the excess of the charge that 
 is carried towards the positive side over that which is transported in the 
 opposite direction is given by 
 
 C = e Itfdv, 
 
 and it is easily seen that if we use 
 
 u* = 2 + r? 2 + 2 , 
 for the square of the absolute velocity of an electron, then 
 
 H = | ml gu z fdv, 
 
 is the expression for the difference between the amounts of energy that 
 are carried through the plane in the opposite direction. 
 
 The function / is to be determined by an equation that is to be regarded 
 as the fundamental equation of the theory, and which we now proceed to 
 establish on the assumption that the electrons are subject to a force giving 
 them an acceleration F equal for all the electrons in one of the groups 
 considered. 
 
 350. Let us fix our attention on the electrons lying, at the time t in an 
 element of volume dv of the metal and having their velocity points in the 
 element dv of the diagram. If there were no encounters of the electrons, 
 neither with other electrons nor with the metallic atoms, these electrons 
 would be found, at the time t + dt, in an element dv' equal to dv and lying 
 at the point (x + gdt, y + ydt, z + ,dt). At the same time their velocity 
 points would have been displaced to an element dv' equal to dv and situated 
 at the point (f + F x dt, rj + "F y dt, + F z dt). We should have therefore 
 
 /( + Y x dt, f] + V v dt, + Y g dt, x + $dt, y + ydt, z + dt, I + dt) dv'dv' 
 
 =/( "n, , s, y, 2, t)dvdv. 
 
 The encounters or impacts which take place during the interval of time 
 considered require us to modify this equation. The number of electrons 
 constituting at the time t + dt, the group specified by dv'dv', is no longer 
 equal to the number of those which at the time t, belonged to the group 
 dvdv, the latter number having to be diminished by the number of impacts 
 which the group of electrons under consideration undergoes during the time 
 dt and increased by the number of impacts by which an electron, originally 
 not belonging to the group, is made to enter it. Writing advdvdt and 
 bdvdvdt for these two numbers we have, after division by dvdv = dv'dv' 
 
 + Y x dt, r] + Y v dt, + F z dt, x + gdt, y + rjdt, z + dt, t -f dt) 
 
 ), , a, y, z, t) + (6 - a), 
 
349-351] The velocity distribution for the electrons 313 
 
 or what is practically the same thing 
 
 This is the general equation of which we have spoken. It remains to 
 calculate a and 6 or at least the difference (a 6) and here the difficulties 
 begin : we can however simplify the problem if we neglect the mutual 
 encounters of the electrons, considering only their impacts against the 
 metallic atoms, whose masses are so great that they may be regarded as 
 immovable : we shall further treat both the electrons and atoms as perfectly 
 elastic spheres so that the velocity of any electron at any two instants 
 separated by at least one collision are wholly independent of one another*. 
 
 351. Nowf in the absence of any extraneous forces a steady state of 
 perfectly chaotic motion will soon be established among the electrons and 
 one in which ^ ^ y> ^ t} s/o ( ^ ^ ^ ^ t} 
 
 and on the assumptions just made it seems reasonable to suppose that this 
 distribution will be similar to that which exists under similar circumstances 
 in gas theory and is specified by Maxwell's law so that we may take 
 
 where A = N A / 3 , 
 
 where N denotes the number of free electrons per unit volume and u m 2 the 
 mean square of their velocities. This is the perfectly chaotic distribution 
 of motions and any departure from it arises as a result, of the external forces 
 or condition gradients tending to organise the perfect irregularity which this 
 law specifies. 
 
 Now since the velocity of any electron after a collision is independent 
 of that before collision it follows that the distribution of velocities among 
 any set of electrons when taken each just after its next collision succeeding 
 the instant t will be wholly independent of the distribution at the time t and 
 will in general be different from it unless indeed this latter distribution is 
 that specified by Maxwell's law which is specially defined so as to remain 
 unaltered by collision. It follows therefore also that the distribution of 
 velocities among any set of electrons, each taken immediately after its next 
 collision after the instant t will in fact be precisely that specified by Maxwell's 
 law and is therefore the same independently of the state of motion that 
 may exist at the instant t : in other words the collisions completely obliterate 
 
 * It has been found possible to generalise the theory to the more probable case where the 
 interaction in collision is like that between centres of force. Cf. Bohr. "Studier over metallernes 
 elektrontheori" (Dissertation, Copenhagen, 1911); Kichardson, Phil. Mag. July, 1912; and 
 various papers by the author in the same magazine during 1915. 
 
 t The present form of the argument was "sketched by the author Phil. Mag. 30 (1915), pp. 
 112-124. Cf. also pp. 549-559. 
 
314 Electric currents in metallic conductors [CH. vn 
 
 any regularity which existed in the electronic motions before collision. 
 Thus" the number (bdvdt) of electrons which enter the specified group during 
 the small time dt is precisely the same as the number which would enter the 
 same group if Maxwell's law specified the distribution both before and after 
 the collision and it might be calculated on this basis. The number advdt 
 of electrons leaving the group in the same time would then be exactly the 
 same as bdt if there were no external forces or condition gradients to modify 
 the distribution established by the collisions. In the more general case it 
 is however at once obvious that the number 
 
 (a b) dvdt, 
 
 can be calculated as the number of electrons removed by collision during the 
 time dt from among the partial group of electrons contained in the specified 
 group at the instant t, which is the excess of the number in this group over 
 and above the number required by Maxwell's law : that is the number 
 
 (f-fo)dvdt. 
 
 We thus want to find the number of collisions which the electrons of this 
 group undergo in the small time dt. Each of the electrons in the group is 
 moving along a zig-zag path with the definite velocity u. Let us fix our 
 attention on one of these electrons and calculate the chance of its colliding 
 with an atom at rest in a unit of time. This chance is obviously equal to 
 the number of atoms in a cylinder of base TrR 2 and height u, R being the 
 sum of the radii of an atom and an electron ; it is therefore equal to 
 
 n being the number of atoms per cubic centimetre in the metal. 
 
 But in unit time the electron under consideration travels a distance 
 u, hence the chance of a collision of the electron with an atom per unit length 
 of its path is 
 
 n D 
 
 C = - - = mrR z , 
 u 
 
 and thus the mean free path of an electron is, as before, 
 
 C n-rrR*' 
 and is independent of u. 
 
 The chance of a collision in the time dt is thus 
 
 ~ , udt dt 
 Cudt = -j = , 
 
 '"m ' rn 
 
 where r m = is the mean time in a free path : thus the number of collisions 
 u 
 
 in the group of electrons specified during the time dt is 
 
 , n / \ 7 wl 
 
 and thus I, - a = - 
 
351, 352] The electric and thermal currents 315 
 
 352. It follows therefore that the equation for the function / in the 
 most general possible case of the present type, is 
 
 df df df f df df df 8/_/ / 
 
 F ^ + F ^ +Fz a| + ^ + r? 8^ + ^ + ^-^~^- 
 
 Now the difference between the functions / and / may be shown to 'be 
 extremely small in any real case, at least compared with/ itself and we may 
 therefore use / = / on the left-hand side of this equation so that we get 
 
 where of course / has the value quoted above : on inserting this we find that 
 
 or in vector notation, using u as the vector velocity of the electron 
 
 / = Ae~^ \l + 2 ? r TO (uP) - ?f (uV) A + T M (uV) <?J . 
 The density of the electric flux is then determined by its vectors 
 
 (C e , c,, c.) = 
 
 whilst the flux of kinetic energy is determined by the vector with components 
 
 r r [+<*> 
 (H x , H,, H a ) = 
 
 The integrals in each of these cases can be directly evaluated by the spherical 
 polar transformation 
 
 g = u cos 6, 17 = u sin 6 cos </>, = u sin 6 sin </>, 
 and it is in this way found that 
 
 & 4 ( 2gF - -j grad A J + 6 grad q , 
 
 whilst H = - U 6 ( 2^F - -j grad J ) + H grad g , 
 
 6 |_ \ ^i / 
 
 wherein we have used 
 
 but T = where Z TO is independent of u so that 
 u 
 
316 Electric currents in metallic conductors [CH. vn 
 
 and thus C = ?^= Ifaqg - -j grad A) + - grad q\ , 
 
 27rAml 
 
 354. Now let us consider the conduction of electricity in a homogeneous 
 bar of the metal which is kept at the same temperature in all its parts : let 
 this metal be acted on by an electric force E in the direction of its length 
 which we may take to be directed along the x-axis. The force acting on 
 each electron will then be eE so that 
 
 eE 
 
 F = F = F = 
 
 ** > * y *z w > 
 
 IH 
 
 and since the physical conditions of the metal are the same throughout its 
 mass the quantities A and q which depend on these will be constants so that 
 
 grad A = grad q = 0. 
 
 We have thus in this case a current of electricity denned by its flow per 
 unit area across a section of the bar 
 
 from which we may conclude that the conductivity of the metal under the 
 specified conditions is 
 
 This formula was first given by Lorentz. 
 
 354. In order however to exhibit all the beauties of this theory it is 
 necessary to consider the question of the conduction of heat in the metal. 
 A bar of metal whose ends are maintained at different temperatures may be 
 likened to a column of gas, placed for example in a vertical position and 
 having a higher temperature at its top than at its base. The process by 
 which the gas conducts heat consists in a kind of diffusion between the 
 upper part of the column, in which we find larger, and the lower one in 
 which we find smaller molecular velocities ; the amount of this diffusion and 
 the intensity of the flow of heat that results from it, depend on the mean 
 distance over which a molecule travels between two successive encounters. 
 Now in the present theory of metals it seems at least plausible to assume 
 that the conduction of heat goes on in a way that is exactly similar to that 
 just described, only the carriers by which the heat is transformed from the 
 hotter towards the colder parts of the body, are now the free electrons, and 
 the length of their free path is limited, not, as in the case of a gas by mutual 
 encounters, but by the impacts against the metallic atoms, which we have 
 supposed to remain at rest on account of their comparatively large mass. 
 
353-355] The thermal current 317 
 
 Of course, this idea seems to imply that some sort of thermodynamical equi- 
 librium exists between the metal molecules and the electrons, so that the 
 latter are partaking of a real heat motion. It is usual to assume that this 
 equilibrium is of the simple type that exists between different kinds of mole- 
 cules in a compound gas so that the mean kinetic energy of the electron 
 expressed in the above notation by 
 
 3m 
 
 imw TO 2== , 
 
 is determined by the simple law of the equality of mean energies. This 
 means that if R is the universal gas constant and 6 the absolute temperature 
 of the metal we may write 
 
 so that the relation 
 
 m 
 
 2B0' 
 
 determines the dependence of q on the temperature. 
 
 355. In the general case it is not possible to determine a definite value 
 for the quantity of energy of the irregular or chaotic part of the electronic 
 motions which is transferred during their average flux, because the energy 
 of each separate electron being in part kinetic energy and in part potential 
 energy relative to the metal molecules and the external field is known only 
 to an additive constant. In one special case however when the aggregate 
 flux of the electrons vanishes, so that there is no flow of electricity, will this 
 indefiniteness disappear and the flux of energy through the metal is com- 
 pletely determined by the vector H given above. As in this case the electric 
 current is zero we must have the condition 
 
 1 2 
 
 20F -. grad A + - grad q = 0, 
 A & fe 
 
 m 
 
 and thus - grad q. 
 
 $q* 
 
 We have however from above 
 
 2Rq* 
 grad q = - grad 6, 
 
 m 
 
 so that H = o~ir" grad 6, 
 
 and the conductivity for heat, as usually defined is therefore in this case 
 given by 
 
 y * 3q 2 e z ~ ' 
 
 and it depends only on the nature and physical conditions at the point in 
 the metal under consideration. 
 
318 Electric currents in metallic conductors [OH. vn 
 
 356. The ratio of the conductivities of heat and electricity is 
 
 y 
 
 Q > 
 
 K e z 
 
 and is therefore the same in all metals at the same temperature. This is the 
 well-known Wiedemann-Franz law which has been successfully verified in 
 numerous cases*: a few typical ones are exhibited below: 
 
 Aluminium ............ '706 . 1Q- 10 
 
 Copper ............... -738. 10- 10 
 
 Silver ...... ' ......... -760. 1C)- 10 
 
 Iron ............... -890. 10- 10 
 
 and seeing that the specific electrical resistances of these substances range 
 from '3 . 10~ 4 to 64 . 10~ 4 the agreement is remarkable. 
 
 The ratio of the two conductivities should also on the present theory vary 
 as the absolute temperature, and this is a law which had been empirically 
 formulated by Lorentz : the temperature coefficients of the ratio in the four 
 cases quoted are respectively 4-37, 3'95, 3'77, 4'32. 
 
 But not only are these general qualitative results satisfactorily verified 
 by our theory : the agreement is quantitative as well : in fact it is known from 
 gas theory that 
 
 - = 2-8 . 10- 7 , 
 
 where e is the electrostatic charge on the electron and thus 
 
 y - = 16 . 10- 14 . 0, 
 
 K 
 
 or taking an absolute temperature of 300 (i.e. 27 C.) this gives 
 
 y - = -48 . 10- 10 , 
 
 K 
 
 which is of the same order of magnitude as those quoted abovef . 
 
 This agreement between the theory and experiment, first noticed by 
 Drude, is one of the most beautiful results of this theory and points distinctly 
 to the conclusion that the assumption that both electricity and heat are 
 carried through the metal by the electrons, and that these electrons are in 
 a simple mechanical heat equilibrium with the metal molecules is completely 
 justified. 
 
 357. In the preceding paragraphs we have considered two special cases 
 of the transfer of heat and electricity in a homogeneous piece of metal. 
 We shall finally consider briefly the more general case which leads to an 
 
 * Full details of the experimental results to 1911 are given by K. Baedeker, Die elektrischen 
 Erschienungen in metallischer Leitern (Braunschweig. 1911). 
 
 t Still better agreement is obtained in the more general theory when the law of force between 
 the molecules and electrons is the inverse cube law. 
 
356-358] Some special cases of flow 319 
 
 explanation of the full circumstances in the thermoelectric phenomena dis- 
 cussed above. For the sake of generality we shall introduce the notion of 
 molecular forces of one kind or another exerted by the atoms of the metal 
 on the electrons and producing for each electron a resulting force along the 
 direction in which the metal is not homogeneous ; this is the main point of 
 the idea of Helmholtz's assumption that each substance has a specific affinity 
 for electricity which varies with the temperature: these forces will be 
 assumed to be such that the typical electron will on the average have a 
 potential energy e^ under the standard conditions relative to the metal 
 molecules surrounding it. Thus if we now assume that the impressed field 
 is derived from a potential cf> we shall have 
 
 F= - grad (<f> + /x), 
 so that we have the currents of electricity and heat 
 
 C = - K\ grad (<+/*) + -^ g rad A + grad 6 , 
 
 nr>/3 
 
 and H = - - C y grad 0. 
 
 From the first of these expressions we may deduce expressions for the rate 
 of fall of potential at each point and for the difference of potential between 
 the ends of the bar examined in the previous paragraph. It is however more 
 interesting to make the calculations for a more general case. We therefore 
 consider a circuit consisting of a thin curved wire, the dimensions of the 
 normal section at any point of which are small compared with the radius of 
 curvature of the curve of the wire at the point. We may then assume that 
 the nature and temperature of the metal are very nearly the same at all 
 points of a single cross section and we shall therefore only be concerned 
 with the component fluxes tangentially along the wire, the other component 
 being negligibly small. If we use s to denote a coordinate of distance along 
 the wire the equations for these fluxes are 
 
 [d .. . R0dA ZRddl 
 
 G s = - * hr (<A + /*) + ^r j~ + - 
 
 \_ds Ae ds e ds] 
 
 . r dd 
 
 and H s = .._C._ y -. 
 
 We now examine special cases of these equations. 
 
 358. (1) In an open circuit in which no current is flowing there is a 
 potential difference between the ends which may be regarded as a measure 
 of the electromotive force existing in the circuit when closed. In this case 
 
 we have 
 
 d$_ _dfj,_R6dA_2Rdd 
 ds ds Ae ds e ds' 
 
320 Electric currents in metallic conductors [OH. vn 
 
 so that on integration along the circuit from the point s^ to the point s 2 we 
 
 get _* 
 
 9i 92 (Pi Pz) e 
 
 If the temperature is uniform along the circuit this gives 
 
 - ^ L 
 e A 2 
 
 This equation shows at once that the potential difference between any two 
 points of the circuit will depend only on the nature of the metal at these two 
 points and will be zero if these metals are the same, for then ^ ju, 2 and 
 AI A 2 , these two quantities depending essentially on the character and 
 conditions of the metal and nothing else. The explanation of the volta 
 potential differences and the laws which it .obeys is now obvious : the 
 difference in the potential between the ends of a compound circuit may be 
 due either to a difference in the "affinity" potential //, of the electron in the 
 metals at the ends or to a difference in the concentration of the electrons at 
 these ends. 
 
 (2) We now consider an open circuit consisting of one kind of metal 
 only but in which the temperature is not uniform. In this case A and //, will 
 be functions of the temperature only so that the expression for d<j> will be an 
 exact differential with respect to 6 : again the potential difference between 
 two points of the circuit will only be dependent on the temperature of these 
 points and will be zero if these are the same : 
 
 ds ds Ae ds e ds ' 
 
 f 2 /dp RddA 2R\d6 J 
 
 i ~ 92 ~~ I ( ~JQ + ~ir ~JQ -- I T ds. 
 J j \d6 Ae dd e ) ds 
 
 359. (3) We finally consider the more general case of a non-homo- 
 geneous circuit in which the temperature varies. We shall confine ourselves 
 to the consideration of a circuit such as that described above in the text in 
 which three pieces of metal are joined up in a circuit; the first and third 
 pieces being however of the same material so that if they were joined up the 
 circuit would in reality consist of only two pieces of metal. The junctions 
 are P 1 and P 2 and we shall suppose that they are at temperatures 6 l and 2 
 and also that the temperatures at the two ends of the circuit are both 6 . 
 Still retaining the coordinates Sj and s 2 for these ends we have 
 
 , f 2 /dp , Rd dA 2R dd\ , 
 
 9i - 92 = [j+-r-j- + - --rid*. 
 / \ds Ae ds e ds) 
 
 Now 
 
 ds 
 
 , 
 
 ds 
 
358-360] The Peltier effect 321 
 
 and the integrated term vanishes because the metal and temperature at the 
 ends 1 and 2 are the same : thus 
 
 fdfji Rd dA 2R dd^ 
 
 f 2 fdu. R6 dA 2R d6\ 
 <-_ ir + , -j- ) 
 
 Jj \ds Ae ds e dsj 
 
 =-/:< 
 
 'dp 
 ,d8~ 
 
 e 
 
 2R\d6 , 
 
 -) j-M 
 
 e J ds 
 
 - .0 
 
 (1- 
 
 R, 
 
 -log A- 
 
 As 
 
 e / 
 
 re, 
 J i 
 
 (dp 
 \dd 
 
 R i 
 
 -log A 
 
 }dd 
 
 e ) 
 
 /* 
 
 + L, 
 
 /dp 
 \dd 
 
 *logA- 
 
 f }dO 
 e J 
 
 - /: 
 
 (dp 
 \dd 
 
 * e logA 
 
 2R\ d() 
 
 e ) .' 
 
 -/: 
 
 (|- 
 
 flog .4 
 
 2R\ j 
 f - - U0, 
 
 e / 
 
 and the first integral now refers to the one type of metal and the second to 
 the other so that the integrands are proper functions of the temperature. 
 Using suffices a, b to denote quantities referring to the different metals we 
 see that 
 
 The potential difference between the ends of the circuit depends therefore 
 merely on the temperatures at the junctions of the two different metals and 
 vanishes if these are equal. 
 
 360. Let us now examine the development of heat which takes place 
 in the same circuit when the current is allowed to flow round it. To do 
 this we shall assume that the conditions at each point of the circuit are 
 stationary, the temperature being suitably maintained constant by con- 
 duction (if the circuit consists of a thin wire -this may be done without 
 appreciably altering the conditions under which we are treating these 
 questions). 
 
 We now consider a small element of the circuit between the cross sections 
 at distances s and s + ds from the origin on the circuit and we find the 
 quantity of heat dH which must be extracted from such an element to maintain 
 its temperature constant. This quantity will of course be equal to the 
 difference between the amount of energy supplied to the electrons in the 
 element on account of the extraneous forces and the amount which is brought 
 into the element as a result of the electrons diffusing into it from other parts 
 of the circuit. The state of the flow being assumed to be stationary the same 
 number of electrons will flow into the element on one side as will flow out at 
 
 L. 21 
 
322 Electric currents in metallic conductors [CH. vn 
 
 the other and the amount of heat can be calculated from the expression for 
 H s which determines the amount of electronic energy, consisting in part of 
 kinetic energy and in part of potential energy, which is transferred through 
 unit area of a normal section of the circuit per unit time. If we take the 
 area of the cross section at s to be / then we shall have 
 
 or introducing the total current flow J determined by 
 
 ds ' 
 
 J 2 -r (du, Rd dA \ d ! r dd 
 
 we get da. = -*-+ J {-p + - -r~l r f/V T- 
 
 (_ JK \ds eA ds J ds \ ds 
 
 The first term in this expression, which is proportional to the square of the 
 current strength, indicates of course the Joule's heat developed in the element. 
 The last term, which is independent of the current, indicates the heat supply 
 to the element on account of true thermal conduction in the circuit. The 
 middle term is the important one : it denotes a development of heat in the 
 circuit 
 
 Rd dA dp 
 
 eA ds ds 
 
 which is proportional to the strength of the current and which therefore 
 changes sign when the direction of the current is reversed. It is this term 
 which contains the expression of the Peltier and Thomson effects, as we 
 see by examining two simple cases. 
 
 361. (a) We first consider a part of the circuit in which the tempera- 
 ture is constant and in which a transition from the metal a to the metal 6 
 takes place; this transition is assumed to be gradual and on integration 
 of the expression for dH' across it we find the corresponding amount 
 of heat developed per unit of time by the passage of the current across the 
 junction is equal to 
 
 ~R6 A A, 
 
 so that the Peltier effect as defined in the text is 
 
 (6) Now consider a portion of the circuit within which the metal is the 
 same but in which the temperature varies. We then find in a similar manner 
 that the amount of heat developed per unit of time in a small part of the 
 circuit in which the temperature changes uniformly from 6 to + dd will be 
 
 [_Ae dd dd 
 
 dd, 
 
360, 361] The Thomson effect 323 
 
 so that the Thomson effect as defined is 
 
 Re dA 
 ~3i 
 
 The expressions thus found for the quantities denoting the potential in a 
 circuit of the type under consideration and the Peltier and Thomson effects 
 are fully consistent with the relations between these quantities deduced by 
 Kelvin from thermodynamical considerations . and given above in the text. 
 
 This theory moreover effectively accounts for many features of the 
 phenomena which it is rather difficult to explain on any other basis and in 
 particular the phenomena associated with the Thomson effect, which on the 
 present basis becomes almost self-evident. 
 
 212 
 
CHAPTER VIII 
 
 ELECTRIC CURRENTS IN LIQUID AND GASEOUS CONDUCTORS 
 
 362. Electrolysis. So 'far we have been dealing with the relations of 
 electric flow without troubling much about the actual generation of that flow. 
 We have, it is true, been led to certain conditions which are essential to the 
 flow, but the exact way in which they are satisfied did not appear. We 
 shall now attempt an exposition of this other side of the subject and explain 
 how a current is actually generated. As a preliminary we must first explain 
 in detail some important facts connected with the flow of currents through 
 liquid conductors*. 
 
 Chemical compounds which can exist either as salts or acids, or have the 
 same general characteristics of these bodies, can conduct an electric current 
 when in the liquid form or in solution. In order to observe this it is merely 
 necessary to insert two conductors as electrodes into the liquid or solution 
 and connect them with the poles of a voltaic cell. A current will be found 
 to flow round the circuit and if this experiment is carefully examined it will 
 be found that 
 
 (i) No substance can conduct an electric current, without being resolved 
 into two constituents of which the one appears at the positive electrode 
 and the other at the negative. This resolution is called electrolysis : the- 
 substances which admit of such, electrolytes, and the two constituents, the ions. 
 
 (ii) The metal or the stuff which takes its place in the compounds 
 (Hydrogen in the acids) is always one of these constituents and it always 
 appears at the electrode where the current leaves the electrolyte (the negative 
 electrode or cathode). We call it the cation, the other part the anion. 
 
 (iii) Secondary actions may follow the deposition of the ions on the 
 electrodes, but they are not connected with the current. 
 
 If for example copper sulphate (CuS0 4 ) is in solution between copper 
 electrodes; when the current flows one electrode is covered with copper 
 but from the other a portion of the copper combines with the free S0 4 ion 
 again. The total quantity of dissolved copper sulphate thus remains the 
 same but the one electrode increases in weight and the other decreases by 
 the same amount. 
 
 * A complete account of the theory of conduction and the correlated phenomena in liquids 
 is given with references by Whetham, The Theory of Solutions (Cambridge, 1902). 
 
362-364] 
 
 The laws of electrolysis 
 
 325 
 
 B 
 
 Fig. 67 
 
 363. We can obtain a very simple picture of this phenomenon in the 
 following way. Suppose AB and CD are the 
 
 electrodes and suppose the current flowing from 
 left to right. We can now imagine that one con- 
 stituent P of the substance remains at rest but 
 that the other moves with the current. There 
 will thus appear a definite quantity of this latter 
 constituent in the free state at the electrode CD 
 and an equivalent amount of P left free at AB; 
 whilst each volume element in the middle of the 
 liquid contains just as at first equal amounts of 
 both constituents. 
 
 As far as the resolution of the substance is 
 concerned we might have imagined that the 
 constituent Q was at rest and the part P moved 
 to the left. The phenomenon really only depends on the relative motion 
 of the two constituents and both P and Q might move, which will in general 
 be the case. 
 
 Of course the actual motions in the electrolytes are very complicated. 
 Before the current is sent through, each molecule has its irregular heat motion 
 and the constituents may have relative motion inside the molecule. Single 
 molecules may even be already dissociated into atoms or atomic groups and 
 capable of existence as such for a short time. But in all of these phenomena 
 no direction can be chosen for which the motion has a preference. All this 
 however alters as soon as the current crosses the liquid. The atoms of the 
 constituent P will then move to the left in larger numbers or with larger 
 velocities than to the right. Similarly the atoms of Q, although moving in 
 reality in every direction, will have a slight preference for going to the right. 
 
 364. In a careful examination of all the circumstances governing this 
 phenomenon of electrolysis Faraday found the following laws to be always 
 true*. 
 
 (i) The quantity of a substance which is resolved by an electric current, 
 and thus also the quantity of each constituent which is liberated, is proportional 
 to the current strength. 
 
 (ii) If different substances are traversed by equally strong currents, then 
 the quantities resolved in each case are chemically equivalent : the same is 
 true also of the different constituents. 
 
 If for example solutions of zinc sulphate and copper sulphate are included 
 in the same circuit, then the same number of molecules of both substances 
 will be deposited and also the same number of atoms of each metal. 
 
 * Experimental Researches, Ser. 7 (1834), 662 et seq. 
 
326 Electric currents in liquid conductors [CH. vm 
 
 On the other hand from dilute sulphuric acid a quantity of hydrogen is 
 liberated which contains twice as many atoms as the quantity of copper 
 which the same current would separate from a solution of copper sulphate. 
 But then one atom of divalent copper is chemically equivalent to two atoms 
 of univalent hydrogen. 
 
 365. All of these facts tend to show that in electrolytes the motion of the 
 electricity is invariably connected with the motion of the matter, in as far 
 as that motion is conceived as a displacement of the constituent ions relative 
 to one another. There is only one explanation of this connection ; we must 
 assume that of the two parts into which a molecule can be separated (the 
 ions) the metal or corresponding part has a positive charge and the other 
 an equal negative charge. It is thus clear how under the influence of a 
 potential difference in the liquid the metal is driven to the cathode and the 
 other constituent to the anode. We must in addition assume that the ions 
 give up their charges as soon as they reach the electrodes, i.e. they are 
 uncharged as soon as they appear in a free state. 
 
 This hypothesis means that the motion of electricity in electrolytes is 
 a convection but with the characteristic that the small particles which convey 
 the current, need not receive any charge to convey, they merely give up what 
 they already have. If this convection is the only kind of electrical motion 
 involved then the quantity of electricity which goes over from the cathode 
 through a connecting wire, is equal to the sum of the charges of all the metal 
 particles which reach the cathode. The first Faraday law is thus explained 
 by assuming that in a definite electrolyte each metal-atom is combined with 
 a definite invariable charge. 
 
 As far as the second law is concerned it leads to a consequence which is 
 more than a connection between the chemical and electrical phenomena. 
 For example when two voltameters, the one with copper-sulphate and the 
 other with zinc-sulphate are connected in the same circuit, then equal 
 quantities of electricity go to the cathode in each voltameter. Since this is 
 accomplished by the deposition of the same number of atoms we see that 
 the copper atom in the one salt and the zinc in the other must have equal 
 charges. The negative charges which belong to the S0 4 ion in each molecule 
 are also equal and equal to the positive charge on a copper or zinc atom. 
 Consequently the same group in sulphuric acid has exactly the same charge, 
 so we see that an atom of hydrogen in this acid has a charge equal to half that 
 of a copper atom. Thus the atom of a divalent element carries a charge 
 twice as large as that of a univalent element. > 
 
 It is these Faraday laws which suggest that electricity like matter has 
 an atomic structure and that the ions are combinations of chemical and 
 electrical atoms; and they were first publicly interpreted in this light by 
 
364-366] The action of a voltaic cell 327 
 
 von Helmholtz*. The suggestion however was not well received at first 
 chiefly owing to the apparent lack of the necessity for it, and it was left 
 for the modern theory of electrons definitely to adopt the conception into 
 electrical theory. The remarkable success which has attended the develop- 
 ments on both the theoretical and experimental side of this theory now 
 hardly leaves any doubt as to the fundamental basis on which it is constructed. 
 
 366. The explanation of the action of a voltaic element. We can now 
 
 attempt an explanation of the action previously ascribed to the voltaic 
 element, viz. that of being able to supply a current. We know that in any 
 such element as soon as a current is flowing there is a chemical action and 
 we must therefore regard these actions or rather the forces which are in play 
 with them as the origin of the electrical motion. We are strengthened in 
 this view by the law of the conservation of energy. This connects the heat 
 developed in the current circuit with the decrease in the energy of the cell. 
 Since new chemical combinations are formed and thereby energy lost, there 
 is no doubt that we must seek for the source of the heat developed in the 
 circuit in this direction and experiment proves that the one perfectly 
 accounts for the other. We thus conclude that the chemical attraction 
 between the atoms, which combines them together, is the origin of the 
 electromotive force which drives the electricity in the element to the 
 electrodes. 
 
 How can these chemical attractions create the potential difference in the 
 circuit? How also is it that the chemical actions depend on the presence 
 of a wire outside the cell connecting the two electrodes? Or how is it that 
 a heat development can be found at a place different from that where the 
 process giving rise to it is operative? 
 
 Without being able to give a perfect answer to this question we can 
 make a fairly good representation of how the phenomena work. We shall 
 assume that in the electrolytes the ions are electrically charged and that in 
 consequence the motion of the ions and the motion of the electricity are 
 invariably connected with each other. When decomposed these ions can be 
 driven by the electric force; they may however also be moved by forces of 
 another kind : in any case then there is a motion of electricity. 
 
 When we dip a copper plate and a zinc plate in sulphuric acid solution 
 without completing the circuit the zinc attracts the S0 4 ions ; some molecules 
 of S0 4 will give way to the attraction and combine with the metal. This 
 cannot however proceed very far for the molecules as they combine give up 
 their negative charge to the zinc and so there gradually arises a repelling 
 force between the zinc and the next arriving S0 4 molecules. A condition 
 is in fact very soon attained in which the chemical attraction is in equilibrium 
 
 * Faraday Lecture, 1881. 
 
328 Electric currents in liquid conductors [OH. vm 
 
 with the electric repulsion. A similar process goes on at the copper plate, 
 although the S0 4 ions are attracted much more by the zinc than the copper, 
 so that the copper plate will have a much less negative charge than the zinc 
 when equilibrium is attained. 
 
 It is then clear that between the zinc and the copper there will be a 
 potential difference such as is actually observed. 
 
 367. We can now if we like restart the chemical action which ceases 
 when equilibrium is attained as above. We have only to put a positive 
 charge on the zinc to neutralise its negative charge. This is accomplished 
 by connecting it by a wire to the copper plate. If this is done the 
 equilibrium is broken. The negative charge on the zinc reduces and thus 
 the attraction between the zinc and the S0 4 ion is greater than the electric 
 repulsion. Thus the S0 4 ion moves towards the zinc and communicates 
 its negative charge to it and this is continually being neutralised by positive 
 charge supplied from the copper through the wire. 
 
 It would thus appear that the resultant difference in the attraction of 
 the zinc for the S0 4 and the copper for the S0 4 is the moving force in the 
 circuit. 
 
 This idea also provides us with an explanation of the heat development 
 phenomena. When a particle of S0 4 moves towards the zinc, only under 
 the influence of the chemical attraction, it acquires a kinetic energy which 
 we would notice as heat. But in the cell the attraction is practically balanced 
 by the electric repulsion so that the molecule of S0 4 reaches the zinc without 
 a large velocity, i.e. the combination takes place with only a small heat 
 production. 
 
 Similar considerations are also true for cells of other types. A chemical 
 attraction between one electrode and the negatively charged element of the 
 surrounding electrolyte always tends to create a current in a definite direction, 
 whilst an opposed current can arise through an attraction between the other 
 electrode and the positive ion. If the electrode attracts both ions it is only 
 a question as to which is the preponderating action. In the general case 
 we should also have to take into account the forces exerted by the second 
 electrode on the ions, so that the phenomena can be and is in reality, more 
 complicated than our above description. We have however been able to 
 illustrate the underlying essential physical principles and our object is thus 
 accomplished. 
 
 368. The effectiveness of the explanation of the action of the voltaic 
 cell is further substantiated by results of the application of the general laws 
 of thermodynamics to the thermal transformations which take place in them*. 
 
 * Cf. W. Thomson, Phil. Mag. [4J, 2 (1881), pp. 429, 551. The theory is originally due to 
 Helmholtz and W. Gibbs. Cf. Whetham. Theory of Solutions, p. 235. 
 
366-368] The thermal relations of a voltaic cell 329 
 
 There are several of the galvanic cell elements in which the chemical trans- 
 formations consequent on the passage of a current are perfectly reversible, 
 i.e. elements in which the reverse current produces exactly the reverse chemical 
 changes; there are others where secondary actions of a purely chemical 
 character accompany the passage of a current through the element so that 
 the actions are not properly reversible. 
 
 We shall confine our present considerations to the former type of element, 
 for which alone the general thermodynamical procedure is directly applicable. 
 
 If a reversible element at a temperature 6 has the electromotive force <f> 
 we may extract current from it until the total quantity $Q of electricity has 
 passed round the circuit. The mechanical work gained in this process from 
 the chemical reactions is </>$. In this case a definite quantity of metal 
 from one electrode has dissolved and an equivalent quantity of the other 
 metallic electrode has been deposited. The amount of heat corresponding 
 to these transformations we shall suppose to be ESQ. If then </> < H the 
 whole of the energy obtained from the chemical processes is not all transformed 
 into available electrical energy ; part of it has been dissipated as heat in the 
 circuit. Conversely if </> > H heat must have been absorbed from the circuit 
 to account for all the electrical energy. 
 
 Now increase the temperature of the cell to 6 + 86 : its electromotive 
 force will in general be different, say $ + 8$. Then at this higher temperature 
 join the element to a ceH of another type and pass in the opposite direction 
 the quantity 8$ of electricity through it. In doing this we use up electrical 
 energy from the second element of amount $Q ((/> + S</>) and an amount 
 SQ (H + SH) of energy will be absorbed on account of the reverse chemical 
 reactions taking place in the cell ; finally the amount of energy 
 
 SQ {(H + SH) -(</, + Sc)}, 
 has been developed in the cell as purely thermal energy. 
 
 Next cool the cell to the original temperature 6 ; the initial conditions in 
 it are then completely reestablished*. We have thus performed a completely 
 reversible Carnot cycle with the cell and since electrical and mechanical 
 energy are completely transformable and can therefore be regarded as equi- 
 valent we may apply the ordinary principle of Carnot to the process. But in 
 the cycle we have absorbed the quantity of heat $Q (<j> + Sc/> H 8H) at 
 the higher temperature and given up the quantity 8Q (tj> H) at the lower 
 temperature and the excess of electrical energy gained is 8Q . S^>. Since this 
 process is reversible we must have 
 
 8< .S d6 
 
 --. 
 
 * This of course assumes that the thermal capacity of the cell is so large as not to be appre- 
 ciably affected by the slight changes taking place as described. 
 
330 
 
 Electric currents in liquid conductors [CH. vni 
 
 Thus when the electromotive force of an element increases with the tempera- 
 ture the element must be cooled when a current is passed through it. 
 Conversely when heat is developed in the cell by the passage of a current 
 the electromotive force decreases with the temperature. These conclusions 
 have been completely verified by experiment. 
 
 369. Electrolytic dissociation*. The assumption made above that the 
 molecules of an electrolyte split up into ions before any current can pass 
 has been confirmed in a most remarkable manner by very different sets of 
 phenomena. It has in fact been found, for instance, that the addition of a 
 certain amount of salt or acid to a given quantity of water causes a greater 
 depression in the vapour pressure and freezing temperature than is to be 
 expected from the number of molecules thus added to the water. This fact 
 can be explained by assuming that at least some of the molecules of the salt 
 or acid ionise, that is split up into the two constituent ions, which then move 
 about separately among the molecules of the water. If we then admit this 
 explanation it is possible to calculate the extent of the ionisation in any 
 given solution simply by a measurement of the vapour pressure or freezing 
 point of the solution : it is found in this way that the proportionate 
 ionisation is bigger the more dilute the solution, and that it becomes 
 perfect in infinitely dilute solutions. 
 
 These hypotheses were first made by Arrhenius. According to his views 
 the ions of a salt or acid in a very dilute solution are no longer connected with 
 one another but are flying about in the spaces between the molecules of the 
 solvent, each with its own ionic charge, as above. If there is no electric 
 field present the motions of these separated ions are directed equally in all 
 directions so that there is no resultant flux of electricity in any direction. 
 But in the presence of an electric field impressed from without there will 
 be a tendency for each ion to move 
 in the direction of this field, the 
 positive ones in the direction of the 
 lines of force and the negative ones 
 in the opposite direction. The result 
 will be a flux of electricity through 
 the liquid. What is the velocity of 
 this average drift of the ionised 
 molecules through the liquid? 
 
 370. We imagine that a vessel in 
 the form of a rectangularparallelopiped 
 contains a very dilute solution, say, of 
 
 77 
 
 Fig. 68 
 
 sodium chloride and also that two parallel electrode plates are placed 
 against opposite sides of the vessel. 
 
 * Cf. Whetham, Theory of Solution*, ch. xn. 
 
368-371] Electrolytic dissociation 331 
 
 If the electrode on the left is kept at a higher potential than the other, 
 the sodium atom (the positive ion) will be driven towards the right side 
 of the vessel and the chlorine atoms will be attracted towards the left side 
 as a result of the electric field in the solution. Of course the charges on the 
 two ions are equal, merely differing in sign, so that the forces on them as 
 a result of the action of the electric field will be equal (and opposite). For 
 simplicity we assume that the solvent (water) is on the whole at rest. The 
 atoms of sodium will then diffuse through the water towards the right with 
 an average velocity which we may call v and the chlorine atoms will diffuse 
 to the left with an average velocity v 2 . 
 
 The first question arises : is v x = v 2 ? The answer is of course in the 
 negative. In fact when we recognise that the velocity of diffusion of any 
 given type of atom under the action of a given force depends essentially on 
 the size, shape and mass of that atom it is seen to be hardly likely that this 
 velocity will be the same for any two different atoms. 
 
 371. Now let TT be a plane in the fluid, parallel to the electrodes and/ 
 an area on this plane which lies wholly in the solution. Now let n be the 
 number of sodium atoms per unit volume in the solution; the number of 
 chlorine atoms will of course also be n : the mass of a sodium atom is taken 
 to be m^ and that of a chlorine atom m 2 . Then nv^f sodium atoms pass 
 per unit of time through the plane / going towards the right and nv 2 f 
 chlorine atoms pass through to the left in the same time. Thus if at the 
 beginning there were in all N atoms of each ion to the right of the plane 77, 
 then at the end of the first second there would be N + nvj/ sodium atoms 
 and 2V nv 2 f chlorine atoms. These latter remain neutralised by an equal 
 number of particles of the former in the solution but there is the number 
 
 n (i + v 2 )f 
 
 i. 
 of sodium atoms in excess. Thus a mass 
 
 mn r (t>! + v 2 )f 
 of sodium has appeared at the cathode. 
 
 Now if we know the resistance of an electrolyte, then we know also how 
 strong the current will be for a given potential gradient and thus also with 
 the help of a knowledge of the electro-chemical equivalents of sodium, we 
 can calculate how many atoms of sodium will be set free at the cathode per 
 unit time. We know therefore 
 
 nm l (! + v 2 )/, 
 and since we know nm^ we can also deduce from this fact the value of (v + v 2 ). 
 
 After a certain time we can also analyse the solution on one side of the 
 plane (say the right) and thus find out how much of the dissolved sodium 
 
332 Electric currents in liquid conductors [CH. vm 
 
 chloride has disappeared. According to what we have said above the initial 
 quantity on the right is $ ^ + m ^ 
 
 and after the first second it is 
 
 (N - nvj) (MI + w 2 ). 
 
 We can therefore calculate from the reduction in this quantity, the value of 
 v z and therefore also of v l . The mean velocity of each ion is thus completely 
 determined. 
 
 372. Suppose that these measurements have been made for two dilute 
 solutions, one of sodium chloride the other of potassium chloride, both with 
 the same potential gradient. Now according to the above hypotheses the 
 two constituent ions of any salt or acid in a very dilute solution are com- 
 pletely separated so that their motions will be perfectly independent : if 
 this is the case we ought therefore to find the velocity of the chlorine atom 
 in the two above-mentioned solutions to be the same. This is. actually 
 found to be the case and generally observation has shown that in all very 
 dilute solutions the mean velocity of diffusion of a given ion with the 
 same potential gradient, is always the same independently of the character 
 of the other ion. 
 
 It may perhaps be worth mentioning that these mean velocities are not 
 big : for a potential gradient of 1 volt per centimetre they are not more than 
 a small fraction of a millimetre per second. 
 
 373. Currents arising from variations of concentration in the solution. 
 
 It is obvious from what we have just said that, when v^ and v z are unequal, 
 different quantities of the electrolyte will disappear in a given time on the 
 two sides of the plane and a difference in the concentration of the solution 
 on the two sides is caused by the current. Experience has in fact shown that 
 the passage of an electric current through an electrolyte often does cause a 
 variation in the concentration of the solution round both electrodes. 
 
 Conversely a difference of concentration may give rise to a potential 
 difference and therefore also a current. To see exactly how this arises let 
 us suppose that in a solution of hydrochloric acid the concentration decreases 
 as we go down into the liquid and that in the lowest layer of solution it is s 
 small that complete dissociation of the hydrogen and chlorine atoms exists. 
 The two different ions will then begin independently of each other and with 
 different velocities, to diffuse upwards through the rest of the solution. They 
 will do this merely as a consequence of their heat motions. The velocities 
 of diffusion are not the same so that more of the one ion (hydrogen) will 
 pass upwards through a horizontal plane than of the other. A small excess 
 of hydrogen atoms is thus soon obtained above the plane, and it of course 
 carries with it a positive charge, and leaves an excess of negatively charged 
 
371-374] Conduction in gases 333 
 
 chlorine atoms below the plane. That these charges cannot increase in- 
 definitely is easily seen : for as soon as a difference of potential is established 
 in this way the electric force in the field which arises from it and which is 
 directed downwards will tend to hinder further hydrogen atoms from coming 
 up, but will on the other hand help the chlorine atoms by pulling them. 
 The diffusion of the hydrogen atoms is thus decreased whilst that of the 
 chlorine atoms is increased : when there are as many hydrogen as chlorine 
 atoms passing upwards through the plane the limit is reached and the potential 
 gradient which their relative motion tends to establish has attained its 
 maximum. If we put two electrodes in the solution one at the top and 
 one at the bottom a potential difference between them will be observed and 
 if they are joined up by a wire a current will flow. Of course the potential 
 differences between the electrodes and the liquid in their neighbourhood will 
 also have an influence in the same direction and these cannot be exactly 
 equal and opposite, even if the electrodes are of identical constitution, on 
 account of the different concentrations of the solution in their neighbourhood. 
 
 374. The conduction of electricity through gases. We must finally con- 
 sider the question of the conduction of electricity through gases. The 
 electrical conductivity of a gas in its normal state and at atmospheric 
 pressure is extremely small, and it is only within the last few years that 
 it has been established irrefutably that gases conduct at all ; it can however 
 be increased by subjecting the gas to certain influences : for instances in the 
 neighbourhood of a red hot body a gas conducts quite well*, as also do the 
 gaseous products of combustion proceeding from a red-hot flame \ : again 
 when the so-called Roentgen rays pass through a gas they render it a good 
 conductor of electricity, or to speak more accurately they increase its con- 
 ductivity ; if the rays cease to act the conductivity dies away rapidly and 
 after a few seconds the gas is no more conducting than before exposure to 
 the rays : the conductivity imparted by these rays can also be removed 
 by passing the gas through a plug of cotton wool. 
 
 The relation between the potential difference between the boundaries of 
 the gas and the current through it, when it is rendered and maintained 
 conducting by the uniform action of some agent as above described, is 
 deserving of attention : it is shown by the diagram in Fig. 69. As the potential 
 difference is increased from zero the current gradually increases, but it does 
 not increase so fast as the potential difference (as it would in a conductor 
 obeying Ohm's law) : the increase of current for a given increase of potential 
 difference, decreases as the potential difference is increased until finally no 
 increase in the current is produced by an increase in the potential difference. 
 
 * Becquerel, Ann. de Chimie et de Physique [3], 39, p. 355 (1853). 
 
 t Cf. Wiedemann, Die Lehre der Elektrizilat, iv. B; Giese, Wied. Ann. 17 (1882), p. 519. 
 
334 
 
 Electric currents in gaseous conductors [CH. vm 
 
 This limiting current, the strength of which is independent of the electric 
 intensity and depends only on the volume and nature of the gas and on the 
 rays acting is termed the ' saturation ' current. It is represented by the part 
 of the curve SS'. If the electric field be increased still further a stage will 
 be reached at which a spark will pass through the gas and the current will 
 increase once more. These facts were described and an explanation of 
 them offered by J. J. Thomson and Rutherford in 1896*. They supposed 
 that when a gas is rendered conducting there are introduced into it, by 
 some means or other, a number of particles charged, some with positive and 
 some with negative electricity. When the gas is subjected to an electric 
 field the positive particles move to the negative boundary and the nega- 
 tive to the positive : this motion of the charge constitutes the current 
 in the gas. But, if the gas is left to itself under the action of no external 
 field, the oppositely charged particles attract each other and coalesce in 
 pairs, or recombine. When once the particles have all recombined and their 
 charges are neutralised, they move no longer under the action of a field and 
 the gas loses its conductivity. However if the gas is kept under the action 
 of the rays, fresh charged particles are produced continuously : the number 
 of particles in the gas at any time is such that the number recombining in 
 
 potential difference 
 Fig. 69 
 
 one second is equal to the number produced in the same time : the gas is 
 then in a steady state, and the number of particles in the gas does not change 
 with the time. In this condition the gas shows no sign of possessing a charge 
 as a whole, and hence the total charge on all the positive particles must be 
 equal to the total charge on all the negative. 
 
 When the gas is passed through a plug of cotton wool, the charged particles, 
 which differ from the molecules of the gas, are retained, while the molecules 
 pass through. It is not necessary to conclude that the particles are larger 
 than the molecules, for the fact that they are charged, while the molecules 
 are neutral, might account for their retention. 
 
 * Phil. Mag. [5], 42 (1896), p. 392. 
 
374-376] The charge and velocity of the ions 335 
 
 375. Consider now the variation of the current with the potential 
 difference in such a mixture of molecules and charged particles as has been 
 imagined. The greater the strength of the field, the greater is the velocity 
 with which the particles move and the shorter the time that is required for 
 them to get from any part of the field to the boundary. But while any 
 particle is moving to the boundary, it is liable to encounter a particle of the 
 opposite sign and coalesce with it, losing thereby its conducting power. 
 The shorter the time of passage, the less likely will such an encounter be, 
 and the greater the number of particles which start from any part of the 
 field and arrive, still charged, at the boundary : hence the current, which 
 is measured by the number of charged particles arriving at the boundary, 
 will increase with the potential difference. But it is clear that the current 
 cannot increase indefinitely : it must reach a limit when the time occupied 
 by the particles in reaching the boundary is so short that none recombine 
 and the number arriving at the boundary per second is the number pro- 
 duced in the gas in the same time. This number multiplied by the charge 
 on each particle gives the saturation current through the gas. The further 
 increase in the current observed with still stronger fields can only be 
 attributed to an increase in the rate of production of the particles. 
 
 According to this explanation, the process of conduction is essentially the 
 same, whether it takes place in a metal or in an electrolyte or in a gas. In 
 each case the current consists of a stream of charged particles, and the great 
 difference between the laws governing the conduction in different states of 
 matter is due only to a difference in the nature of the charged particles, in 
 their mode of production and in their relation to the medium that surrounds 
 them. In view of the similarity between gaseous and electrolytic conduction, 
 the nomenclature of the latter has been applied to the former. The moving 
 charged particles are called ions and a gas, when it is rendered conducting, 
 is said to be ionised : the process of ionising a gas is named ionisation, a word 
 that is also used quantitatively to express the concentration of the ions in 
 the gas or the number per unit volume. The negative electrode is called the 
 kathode and the positive the anode. 
 
 376. The charge and velocity of the ions. The charge on each ion in a 
 gas can be ascertained if (1) the total charge on all the ions present in a 
 gas at any time, and (2) the number of such ions can be measured. The 
 first quantity presents no difficulty. Let the gas be subjected to an ionising 
 influence, such as Roentgen rays, until a state of equilibrium is reached 
 between the number of ions produced per second by the rays and the number 
 disappearing in the same time through combination. Let the action of the 
 rays be stopped suddenly and the gas immediately exposed to the action of 
 a strong electric field. All the ions of either sign present in the gas will be 
 driven to the boundary of the field, and, if the field be sufficiently strong, the 
 
336 Electric currents in gaseous conductors [CH. vm 
 
 time occupied in reaching the boundary will be so short that no appreciable 
 recombination will have taken place in the meanwhile : the number reaching 
 the boundary is equal to the number present in the gas before the field was 
 put on. If the charge received by the boundary is measured by connecting 
 it to an electrometer, the total charge on all the ions of one sign present in 
 the gas can be ascertained. It may be remarked that if the ionising agent 
 consists of Roentgen rays it is found that the charge on all the ions of 
 either sign is the same. 
 
 377. The measurement of the number of ions present is a much more 
 difficult matter, but has been effected by a most ingenious method devised 
 by J. J. Thomson and based on a discovery due to C. T. R. Wilson*. 
 
 It is known that the quantity of water that can exist in the state of vapour 
 in a given volume of gas depends greatly on the temperature of the gas, and 
 that, if a volume of gas saturated with water vapour is cooled, the excess 
 of the water, or the amount representing the difference between that which 
 the gas could hold at the higher temperature and that which it can hold at 
 the lower, is deposited in the form of rain or mist. It was found by Aitken 
 that this deposition of the superfluous water depends on the presence in the 
 gas of solid particles, or dust, which acts as nuclei for the formation of liquid 
 drops : if a gas is rendered perfectly free from dust by filtering it through 
 cotton wool, it can be cooled to a temperature very much lower than that 
 at which a cloud would form in the presence of dust without any consequent 
 condensation. The necessity for the presence of nuclei in the formation of 
 drops can be shown readily to be a consequence of a surface tension in liquids, 
 and it appears that the larger the particle of dust the more efficient is it as 
 a nucleus and the smaller the supersaturation of water vapour that it is 
 possible to produce in its presence. 
 
 C. T. R. Wilson found that ions possess peculiar properties with respect 
 to the formation of clouds in gases supersaturated with moisture. In his 
 investigations he cooled the gases by expanding them adiabatically, thus 
 producing a fall in temperature which could be calculated from the known 
 values of the initial and final volume and the ratio of the specific heats of 
 the gas. He found that it requires less supersaturation to produce a cloud 
 in air, when it is ionised, than when it is not ionised : the ions act as 
 nuclei for condensation. This does not imply necessarily that the ions are 
 larger than the molecules which are always present and available as nuclei, 
 for it can be shown on theoretical grounds that a charged body should act as 
 a more efficient nucleus than an uncharged body of the same size. In the 
 absence of ionisation eightfold supersaturation j is required to produce 
 
 * Phil Trans. A, 189 (1897), p. 265; 192 (1899), p. 403. 
 
 t By ' supersaturation ' is meant the ratio of the mass of water actually present in the gas 
 to that which it could hold without condensation in the presence of large nuclei. 
 
376-378] The charge on an iron 337 
 
 condensation, whereas fourfold supersaturation is sufficient for the same pur- 
 pose in the presence of negatively charged ions, and sixfold supersaturation 
 in the presence of positively charged ions. These values are independent of 
 the degree of ionisation and also, with few exceptions, of the means by which 
 it is produced. Further, no increase in the amount of condensation is caused 
 by increasing the supersaturation between fourfold and sixfold. 
 
 These experiments show that there is a difference in the properties of 
 positive and negative ions ; that the properties of the ions are not determined 
 solely by the method by which they are produced, and that the properties, 
 of all the ions of the same sign are the same. 
 
 378. Wilson's discovery was used by Thomson to determine the number 
 of ions in a gas. The ionised gas, in which the total charge on all the ions 
 had been determined, was expanded adiabatically and cooled to such an 
 extent that a supersaturation between sixfold and eightfold was produced : 
 a cloud formed round all the ions of the gas, but not on the molecules. As 
 soon as the cloud formed it began to fall, because the drops were heavier 
 than the air surrounding them : owing to viscosity of the air they soon 
 attained a small and constant velocity which could be measured by observing 
 the rate at which the upper boundary of the cloud moved down the vessel. 
 Now Stokes has calculated the steady velocity with which a body of known 
 dimensions moves through a medium of known viscosity under the action of 
 a constant force, such as its own weight. He finds that if F is the force, 
 a the radius of the body (supposed to be spherical), p its density, p the coefficient 
 of viscosity of the medium, v the steady velocity is given by 
 
 F 
 
 or if F is the weight of the body 
 
 2cra 2 p 
 V = Q 
 
 y n 
 
 By applying this formula to the case under consideration, where the velocity, 
 the force and the viscosity are known, a, the radius of the drop, can be 
 determined. The total mass of all the drops is the mass of the water set 
 free by the cooling, for the mass of the ion contained in each drop is so small 
 compared with the whole mass of the drop, that it may be neglected. The 
 calculation of this total mass is somewhat complicated, but it depends on 
 well known principles, and for further details the reader may be referred to 
 Thomson's account. If then, the total mass of all the drops is M, the number 
 of the drops n, which is the number of ions, can be calculated from the relations 
 
 M 
 
 n = , m = 
 m 
 
 The charge on each ion, e, is the total charge on all the ions divided by the 
 number of the ions. 
 
 L. 22 
 
338 Electric currents in gaseous conductors [CH. vm 
 
 As a final result of this research the charge on each ion in a gas is found 
 
 to be in electrostatic units 
 
 e = 3-4 . 10- 10 . 
 
 Experiments made in air and hydrogen gave results identical within experi- 
 mental error. 
 
 379. H. A. Wilson* has used a slight modification of the method, which 
 obviates the necessity for the cumbrous calculation of the total quantity of 
 water set free. The total charge on the ions is measured as before. The 
 gas is then expanded and cooled so as to give a supersaturation between 
 four and six and to cause condensation on the negative and not on the 
 positive ions. The velocity v with which the cloud falls is observed. The 
 experiment is then repeated, with the difference that the falling cloud is 
 exposed to the influence of a vertical electric field E, which exerts on the 
 charged drops a force Ee, accelerating or retarding the fall. The new 
 velocity v' is measured. 
 
 Since the velocity of the drop is proportional to the force on it, we have 
 
 v' Xe + mg 
 
 where m = i-n-pa 5 , 
 v mg 
 
 but from above 
 
 "9 
 
 97T V2 UV V -V 
 
 " 
 
 hence e = x - 
 
 E v gp 
 
 By this method Wilson found e = 3'1 . 10~ 10 . A few drops were found to 
 bear a double and triple charge but their numbers were too small to have 
 exerted any influence in Thomson's experiments. 
 
 Since Thomson measured the average charge on both positive and negative 
 ions and Wilson that on the negative ions alone the coincidence between their 
 results shows that the charge on an ion is independent of its sign. 
 
 380. The velocity of the ions can be measured in various ways, of which 
 the most convenient depends on the following principles. Suppose that two 
 plates A and B, parallel to each other, are maintained at a constant difference 
 of potential, and that ions are produced by some means close to the plate A. 
 Under the action of the electric field the ions of one sign will give up their 
 charges to A while the others will move towards B. Just as they are about 
 to settle on B let the direction of the field be reversed, so that the plate 
 which was formerly positive is now negative : these ions will then be driven 
 back towards A while those of opposite sign move towards B : as these are 
 about to touch B let the direction of the field be again reversed. If then 
 
 * Phil. Mag. (6), 5 (1903), p. 429. 
 
SJ78-4&81] Discharge tube phenomena 339 
 
 the direction of the field is reversed at constant intervals and these intervals 
 are so short that they do not permit the ions to travel from one plate to the 
 other before the field is reversed, the plate B will receive no charge : but if 
 the interval is longer, so that the ions have time to get across before the 
 reversal of the field, B will receive a charge. By measuring the time of reversal 
 for which B just begins to receive a charge, the time required for the ions to 
 pass from one plate to the other is known, and hence the velocity of the ions 
 ascertained. In this way it is found that the velocity of an ion is proportional 
 to the strength of the field in which it moves, i.e. to the force acting on it : 
 the essential condition for a diffusive motion is thus satisfied. The velocity 
 of the negative ions is slightly greater in all cases than those of the positive 
 ions. 
 
 The evidence thus far points to the fact that the carriers of the current 
 in gases are more like those in liquids than the electrons in metals. But 
 a study of the discharge of electricity through gases at low pressures has 
 thrown a new light on this question. 
 
 381. Suppose that a long cylindrical glass vessel is taken and filled 
 with a gas at atmospheric pressure, and suppose the potential difference 
 between the kathode (K) and the anode (A) is gradually increased. Only 
 an extremely small current will pass through the gas, but at a certain high 
 limit a 'spark' passes through the gas and a very considerable current 
 begins to flow. If the pressure of the gas is lower the limit at which the 
 spark passes is also lower and the spark itself is broader and more diffuse 
 than before. As the pressure is reduced the diffuseness of the spark increases 
 until finally the luminosity of the discharge fills the whole volume of the tube 
 between the electrodes; but it is seen on careful inspection that it is riot 
 continuous. Covering the kathode is a thin luminous layer, the 'kathode 
 layer' : next comes a dark space, the 'Crookes dark space' : a luminous 
 layer follows, the 'negative glow,' then another dark space, the 'Faraday 
 dark space,' and lastly, a region of light, which is sometimes divided into 
 striae, extending up to the anode and known as the positive column. As 
 the pressure is still further reduced, the Crookes dark space grows at the 
 expense of all the other regions, which diminish in extent, except the kathode 
 layer, which remains practically the same. When the dark space has become 
 so large, that its boundaries touch the glass of the walls a curious green 
 phosphorescence is excited in the glass. At first this phosphorescence 
 appears on a few isolated patches near the kathode, but by decreasing the 
 pressure sufficiently the whole of the tube may be filled by the Crookes 
 dark space, with the exception of thin layers on the kathode and anode, 
 and then all the glass walls of the tube with the exception of the parts behind 
 the kathode and anode glow with the green phosphorescence : it is in this 
 condition that the tube is emitting Roentgen rays plentifully. 
 
 222 
 
340 Electric currents in gaseous conductors [CH. vm 
 
 If the pressure be still further reduced, the current diminishes and finally 
 vanishes : at sufficiently high vacua no current can be made to pass. 
 
 382. In 1859 Pluecker* discovered that if a magnet is brought near 
 to the tube in which the luminous discharge is observed all the luminous 
 portions are distorted in some measure from their original positions. But 
 the most marked distortion was produced in the patches of phosphorescence 
 near the kathode. This discovery at once directed the attention of physicists 
 to these patches. Soon afterwards Hittorf discovered that if a solid body 
 is placed in the tube, near the cathode, the phosphorescence ceases at all 
 points which were shielded from the kathode by the solid body : a shadow 
 of the body is thrown on the walls of the tube. It thus appears that the 
 influence which caused the phosphorescence on the walls must proceed in a 
 straight line from the surface of the kathode to the walls of the tube, and it 
 was therefore described as due to certain kathode rays. It may also be 
 noted that it was found that the nature of the shadow was such as to prove 
 that the rays do not proceed in all directions from the kathode but are 
 emitted at right angles to its surface. 
 
 It was surmised by Goldstein j that these kathode rays were a type of 
 aethereal radiation, like light; but the view of Sir William CrookesJ that 
 they were streams of particles electrically charged has been proved to be the 
 more correct of the two. Before it can be definitely asserted that the rays 
 consist of charged particles some further information is required : the term 
 particle connotes a finite mass and a finite number of carrying agents and 
 to settle the point it is necessary to determine these two quantities. The 
 number is known if we can measure the charge carried by each particle : 
 hence we want to determine the mass of each particle and the charge carried 
 by it. Unfortunately no direct method has been devised for dealing with 
 individual particles in the kathode rays and neither of these quantities has 
 been determined directly, but the ratio of the mass to the charge of a kathode 
 particle can be ascertained by a method devised by Thomson . 
 
 383. Suppose that a particle of mass m carrying a charge e is moving 
 along parallel to the z-axis of a conveniently chosen rectangular coordinate 
 system with a velocity v under the action of a magnetic field of intensity H 
 parallel to the axis of y. Then as we shall see later the particle will be sub- 
 ject to a force - perpendicular to both H and v, i.e. along the 2-axis. The 
 
 c 
 
 acceleration of the body parallel to the axis of z will be /= - , and if it 
 
 cm 
 
 * Pogg. Ann. 103 (1859), p. 88. 
 t Wied. Ann. 1880-1884. 
 J Phil. Trans. 1879-1885. 
 
 Phil. Mag. [5], 44, p. 293 (1897). Cf. also Wiechert, Sitzungsber. der phys.-okon. Gesellsch. 
 zuKonigsberg, 38 (1897), p. 1; Wied. Ann. Beibldtter, 21 (1897), p. 443. 
 
381-384] The mass of an electron 341 
 
 Hevt 2 
 
 moves for a time t it will have travelled a distance |- ft 2 = -, - in that 
 
 2mc 
 
 direction. In the same time it will have travelled a distance vt parallel 
 to the x-axis. Hence the particle while travelling a distance I parallel 
 to the x-axis is deflected a distance 8 parallel to the z-axis, where 
 
 72 p 1 
 
 s _ * 77 ^ L . 
 
 *.. o . n . . , 
 
 2c m v 
 if we know the dimensions of the discharge tube we can thus find e/m. 
 
 We are not able to measure directly the charge on the kathode ray 
 particle; in fact it is only when the carriers of the current are atoms or 
 molecules of matter that it is possible for us to determine the charge on the 
 separate carrier; but since it is found that the value of this charge in all 
 cases where it can be determined is always identical, it is natural to assume 
 that the charges on the kathode particles have the same constant value, viz 
 
 4-69 . 10- 10 , 
 
 so that the mass is , n ,. 
 
 2 . 10~ 27 gms., 
 
 this is much smaller than any known molecule of matter : a molecule ot 
 hydrogen has for instance a mass of about 
 
 3 . 10- 24 gms. 
 
 The kathode particle whose existence is thus presumed is the negative 
 electron : it is identical with the negative particle thrown off by radio-active 
 substances and these facts combined with certain other evidence to be subse- 
 quently discussed confirms us in the view that these electrons are common 
 constituents of all matter. 
 
 384. While it thus appears that in the low pressure discharge through 
 gases the most important part is played by negative electricity there is no 
 doubt that positively charged particles also play some part in the mechanism 
 of the phenomena. The complicated appearance of the discharge and the 
 difference which it shows in different gases suggests that some agent other 
 than the universal negative electron must have an influence on the phenomena ; 
 but for some time after the discovery of kathode rays no such agent had 
 been directly detected. However in 1886 Goldstein working with a discharge 
 tube in which the kathode was a plate perforated by several holes, observed 
 faintly luminous streaks stretching out from the holes in the kathode into the 
 space remote from the anode. Where these streaks meet the walls of the 
 tube, they excite a slight phosphorescence, usually of a mauve colour, but 
 always totally different from the green phosphorescence excited by the 
 kathode. He imagined that these streaks represented the path of rays, 
 similar in some respects to the kathode rays, to which he gave the name 
 Kanalstrahlen. Since these rays are travelling in a direction from the anode 
 
342 Electric currents in gaseous conductors [OH. vm 
 
 to the kathode, it was natural to suppose that they are positively charged, 
 but a-t first no experiment could detect any sign of charge. However in 1898 
 Wien* showed that, if fields of sufficient magnitude be employed, a deflection 
 of the rays could be obtained, and its direction being in the direction opposite 
 to that of the electrons indicates that the charge is positive. The results 
 of the experiments were however otherwise somewhat indefinite and it 
 remained for Thomson f to attack the problem with an improved apparatus. 
 By using very low pressures Thomson was able to determine distinctly the 
 types of particles involved and he found that they were in almost every case 
 molecules of the gas in the tubes carrying the monovalent charge equal to 
 that of a single electron. They are therefore merely the atoms or molecules 
 of the substance from which an electron has been extracted. 
 
 These experiments unfortunately still leave us in the dark concerning the 
 nature of the positive electricity in the atoms and so far no definite evidence 
 is forthcoming on this point from any other branch of the work J. Crowther 
 sums up the state of affairs in his excellent little book by the statement 
 that 'the term positive electrification' remains a not too humiliating method 
 of confessing ignorance. And this is where we must leave it for the 
 present. 
 
 * Wied. Ann. 65 (1898), p. 440. Cf. also Ann. der Phys. (4), 8 (1902), p. 244. 
 
 t Phil. Mag. 13 (1907), p. 561 ; 16 (1908), p. 657; 18 (1909), p. 821. 
 
 | Cf. however p. 44, 48. 
 
 Molecular Physics (London, 1914). 
 
CHAPTER IX 
 
 THE ELECTROMAGNETIC FIELD 
 
 385. First fundamental notion : Ampere's circuital relation. We have 
 now discussed the fundamental principles underlying the theory of electric 
 flow and also of magneto-statics, but so far independently of each other. In 
 1829 Oersted* discovered however that a current exerted an action on a 
 magnet, in a transverse manner as he described it. This is the first funda- 
 mental effect connecting currents and magnetism. 
 
 The current, as Faraday would say, possesses a magnetic field, i.e. in the 
 space surrounding the wire carrying the current a magnet is acted on by 
 forces dependent on the position of the wire and strength of the current. 
 This magnetic field may be studied in the same way as we examined the 
 fields of ordinary magnets, by tracing the course of the lines of magnetic 
 force at every point. This is the rational method adopted by Faraday as 
 long ago as 1837. 
 
 Having already strongly insisted that every current flows in complete 
 circuits Faraday then showed that a small closed circuit carrying a steady 
 current possessed a magnetic field identical with that of a little magnetic 
 needle stuck normally through its middle. At least he showed that it was 
 possible to compensate the magnetic field of the current by a small magnetic 
 needle in this way. 
 
 Accepting this as a fundamental fact we .must first enquire as to the 
 moment of the little magnet which is equivalent to the current. It has been 
 shown by numerous experiments, of which the earliest are those of Ampere 
 and the most accurate those of Weber f , that it is proportional to the strength 
 of the current flowing and to the area of the elementary circuit assumed to 
 be plane. Thus if the small circuit be supposed to be filled up by a surface 
 bounded by the circuit and thus forming a diaphragm, and if a magnetic 
 shell of strength proportional to the current coinciding with this surface be 
 substituted for the current, its magnetic action on all distant points will be 
 identical with that of the current. 
 
 * "Experiments on the Effect of a Current of Electricity on the Magnetic Needle," Annals of 
 Philos. (1820), 16, p. 273. Oersted clearly recognised the fact that a current is surrounded by 
 a magnetic field. 
 
 f Elektrodynamische Maasbestimmungen [Abhandl. der koniglich Sachs. Gesellsch. zu Leipzig, 
 1850-- 1852]. 
 
344 The electromagnetic field [CH. ix 
 
 So far we have supposed the dimensions of the circuit to be small compared 
 with -the distance of any part of it from the point of the field examined. 
 The extension to finite circuits is however easily obtained in the manner 
 introduced by Ampere*. 
 
 386. Conceive any surface / bounded by the circuit and not actually 
 passing through the point P at which the field is examined. On this surface 
 draw two series of lines crossing each other so as to divide it into elementary 
 portions, the dimensions of which are small compared with their distance 
 from P and also with the radii of curvature of the surface. (Cf. fig. 5, p. 14.) 
 
 Round the boundary of each of these elements conceive a current of 
 strength equal to that of the original current to flow, the direction of circulation 
 being the same in all the elements as it is in the original circuit. 
 
 Along every line forming the division between two contiguous elements 
 two equal currents flow in opposite directions. But the effect of two equal 
 and opposite currents in the same place is absolutely zero, in whatever aspect 
 we consider them. Hence their magnetic effect is zero. The only portions 
 which are not neutralised in this way are those which coincide with the original 
 circuit. The total effect of the elementary circuits is therefore equivalent 
 to that of the original circuit. 
 
 Now since each of the elementary circuits may be considered as a small 
 plane circuit whose distance from P is great compared with its dimensions 
 we may substitute for it an elementary magnetic shell of strength proportional 
 to that of the current whose bounding edge coincides with the elementary 
 circuit. But the whole of these elementary shells constitute a magnetic shell 
 of equal strength coinciding with the surface / and bounded by the original 
 circuit and thus the magnetic action of the whole shell at P is equivalent to 
 that of the circuit. 
 
 The magnetic force due to the circuit carrying a current J at any point 
 is therefore identical in magnitude and direction with that due to a uniform 
 magnetic shell bounded by the circuit and not passing through the point, 
 
 the strength of the shell being numerically proportional to /, say . The 
 
 c 
 
 constant -, is an absolute constant depending on the units adopted. If we 
 c 
 
 define the current by its magnetic effect we can take c' = 1. This would 
 give us the absolute electromagnetic unit of current. The equivalent magnetic 
 shell has then a strength numerically equal to the strength of the current. 
 
 It is necessary to include one further remark. In order to be able com- 
 pletely to specify the shell which is equivalent to any current we must 
 
 * "Memoire sur la theorie mathematique des phenoraenes electrodynamiques," Mem. de 
 rinstitut. 6 (1820). Cf. also Ann. de Chem. 15 (1820). 
 
385-388] The potential of the field of a current 
 
 345 
 
 decide which is to be the positive side of the shell. Experiment shows that 
 in the small circuit the positive direction of magnetisation of the shell bears 
 to the direction of the current the same relation as advance to rotation in 
 a right-handed screw. 
 
 387. There is however an important limitation to the general statement 
 of the equivalence of shell and current. The 
 potential function of a magnetic double sheet being 
 obtained from a sum 
 
 Current 
 
 is essentially a single-valued function. It is true 
 that as we go round a circuit as shown by the line 
 from P to P' (near the sheet one on each side on 
 the same normal) we find that there is a drop of 
 
 potential equal to 4?r ~, but this drop is recovered 
 c 
 
 in going through the sheet from P' to P. ' The potential changes very rapidly 
 in going through the sheet. In the case of the current however there is no 
 place where the change in potential produced in going round a circuit as 
 from P to P' (which are ultimately very close together) can be recovered. 
 
 + 
 
 Direction of Magnetisation 
 Fig. 70 
 
 Fig. 71 
 
 The magnetic field of a closed current is thus a cyclic one and agrees with 
 that of the equivalent magnetic shell only at points outside the substance 
 of the shell. The agreement does not hold for points inside the shell. 
 
 388. The potential of a uniform magnetic shell of strength -, J can 
 
 
 
 easily be found; it is = f-, where Q. is the solid angle subtended by 
 
 the shell at the point in the field. Thus the magnetic field due to a closed 
 current of strength J has the multiple- valued potential function 
 
 - 
 
 where Q. is the solid angle subtended by an area closing the current circuit 
 at the point. 
 
346 The electromagnetic field [CH. ix 
 
 Thus if we take a path linked with the current circuit and, starting from 
 a place where the potential is i/f, go right round, we find that the potential 
 
 on arriving at the same place again is now /r ^- , a different value. On 
 
 going round again it becomes /r ; and so on. If the path is not linked 
 
 with the current there is no total change at all in the potential on going round 
 it; along paths not linked with the current the potential is a single-valued 
 function. The mathematical distinction between the two cases is involved 
 in the fact that the current forms a line singularity in the magnetic field. 
 The singularity is of the nature of a circulation round the wire and if we 
 consider a small circuit close up to and encircling the wire the potential must 
 change very rapidly in going round it because the total change in a complete 
 circuit is finite, viz. 4?rJ/c'. This means that the magnetic force gets bigger 
 and bigger as we approach the current. If we actually get on the current 
 itself the force is indeterminate; but this merely means that the problem 
 needs closer specification before we can examine this point. There is in 
 reality no indeterminacy in physical affairs. Of course from another point 
 of view it really means that the current cannot be confined to a geometrical 
 line. 
 
 389. This result provides us with the integral form of the characteristic 
 property of such fields ; it says that if H s is the component of the magnetic 
 force in the field along the element ds of the path drawn round the circuit 
 carrying a current J and threading it in the positive direction, then 
 
 H s ds = . 
 
 o 
 
 In other words if we have a single magnetic pole of strength m, and starting 
 from a point P go right round a circuit threading the current, eventually 
 
 arriving back at P, the potential changes by and an amount of work 
 
 
 
 The magnetic pole could thus be used to do 
 c 
 
 work at the expense of the energy of the system. This work must of course 
 come from the energy of the system or, in other words, the moving of the, 
 magnetic pole must affect the strength of the current. This is Faraday's 
 idea, by moving a magnet in the neighbourhood of a current it can be made 
 to drive an engine, the power being derived from the current. The inter- 
 action between a magnet and a current is a source of power; this is the 
 principle underlying the fact that an electric current can be used to drive 
 a motor*. 
 
 * We have spoken of moving a magnetic pole and not a magnet. We consider however that 
 a magnet has two poles and we should get no work by moving them together unless some means 
 were devised to take them along different paths relative to the current. Faraday's idea of sliding 
 contacts breaking the circuit enabled him to get a single pole through the circuit by itself. 
 
388-390] Ampere's circuital equation 347 
 
 Whether the work in the complete path is zero or not depends only on 
 whether the path encircles the current or not. This can be expressed in a 
 rather different manner. Having chosen a path we can always imagine it 
 closed over by a barrier surface. If this surface is chosen as simply as possible 
 the current will always cut through it when the path links with the current ; 
 and at the place where this happens a quantity of electricity J will flow across 
 the surface per unit time. On the other hand if the path does not link with 
 the current the surface closing it may be so chosen that the current circuit 
 does not cut through it and no electricity will flow across it. We may thus 
 state our rule in the different form 
 
 (Ho 7 s) =: (total quantity of electricity flowing through the circuit 
 
 C 
 
 s per unit time) 
 
 and now it appears to be true for the very general case if only we measure 
 the electricity flowing through the circuit by putting a barrier across and 
 finding the current flowing across the barrier ; it is the amount of electricity 
 flowing across the chosen barrier per unit time. In this form we can apply 
 the rule to a continuous current distribution of density C at any point and 
 it is 
 
 4vr /" n , ,. 
 
 Q n df, 
 
 where/ is the barrier surface bounded by the path s. Kelvin described* this 
 relation as a circuital relation between H and C. It will hereinafter be 
 described as Ampere's circuital relation, as Ampere was the first one to obtain 
 it although in a rather complicated form. We shall return to this relation 
 later. 
 
 390. On closed and open circuits : displacement currents. It is of 
 
 importance to notice that the mode of expression of Ampere's circuital 
 relation adopted at the end of the previous paragraph can have no meaning 
 whatever unless the quantity of electricity flowing through the circuit 
 measured as there specified is the same for all possible barriers; this can 
 only be true if the current is, so to speak, a stream, so that there is no 
 accumulation in the space between two barriers. This amounts to the same 
 as saying that all currents must flow in complete circuits and the funda- 
 mental significance of the relation is based on this idea of closed current 
 circuits. 
 
 If we define a current as a flow of electricity (or electrons) the discharge 
 of a condenser with a vacuum dielectric must constitute a current with a 
 break in it, so that in a case of this kind the current is an open one and the 
 preceding ideas could not possibly be applied to treat it; such cases would 
 
 * Cf. Larmor, Aether and Matter, p. 76. 
 
348 The electromagnetic field [CH. ix 
 
 be beyond a theory involving solid angles and such like, for we can attach no 
 meaning to O for open circuits. 
 
 Faraday insisted however from the beginning that all currents however 
 produced were circuital. Maxwell followed the hint and discovered the 
 wonderful simplicity thereby introduced into the analysis. Helmholtz and 
 his school, however, not willing to accept the fundamental difficulty in the 
 assumption of closed currents, constructed a theory of unclosed currents, 
 but it is terribly complicated. 
 
 In order to surmount the difficulty mentioned above, in cases like the 
 discharge of a condenser, Maxwell made the hypothesis that even in such 
 cases there must be in the aether in the gap some sort of release of strain 
 taking place which possesses the electrodynamic properties of a current or 
 a movement of electricity. It is of course not a movement of electricity. 
 
 391. The point here involved has been discussed at length in a previous 
 chapter*, but the following example will provide the general analytical form 
 of the hypothesis. Consider the process involved in charging a conductor 
 existing alone in the field. As we charge the conductor a state of 'polari- 
 sation' is gradually established in the surrounding dielectric medium (and 
 aether) being accompanied by a 'displacement' in the Maxwellian sense 
 away from the conductor. It is the essence of Maxwell's hypothesis that, 
 in addition to the true electric displacement involved in the polarisation of 
 the dielectric medium, there is some effect in the aether, an aethereal dis- 
 placement he calls it, which is not an electrical displacement but for some 
 reason or other its rate of change has the properties of a true electric 
 current. The actual measure of this current is obtained as follows : if C' 
 denote the current intensity of the true electric flow supplying the charge to 
 the conductor then the integral 
 
 taken over any closed surface/ enclosing the conductor indicates the rate at 
 which the charge on the conductor is increasing ; or if we use Q for the charge 
 on the conductor at any time t 
 
 When however there is a charge Q on the conductor the conditions in the 
 surrounding field are such that if D denotes the totality of displacement in 
 the dielectric (aethereal and true electrical polarisation) at any point, then 
 
 * Cf. p. 221. 
 
390-392] Displacement currents 349 
 
 so that we have 
 
 or 
 
 [ (C n ' + !>)#= 0, 
 /* 
 
 where we use D n for the time rate of change of D n . This indicates that the 
 vector C ' + D 
 
 is always circuital, that is always flows in closed circuits. Thus if we add to 
 the true current of electrons the time rate of change of the total displacement 
 we obtain a total current vector which is always circuital. But 
 
 and thus the total displacement current 
 
 consists of a part P depending on the presence of the dielectric which is a real 
 displacement of electric charge in the molecules of the medium. The part 
 
 T- is the part of the displacement current which must be ascribed to some 
 4?r 
 
 action in the aether; it is Maxwell's aethereal current. The only way of 
 explaining the existence of this current is by a theory of the constitution 
 of the aethereal medium, about which however we know so little. A dynamical 
 theory of its mode of action can however be described which gives some idea 
 of what sort of thing this displacement current is and of how it simulates 
 an electric flow*. 
 
 392. We must make a distinction between the true current 
 
 C' + P, 
 
 which is a true flow of electricity and this fictitious current in the aether. 
 The contrast is in reality between the total current 
 
 / E\ 
 
 and the true current. The total current always contains a part ( I which 
 
 \4arJ 
 
 is not electric flow at all but is a something possessing the electrodynamic 
 properties of a true electric flow. 
 
 The important thing is that now every current is effectively circuital. 
 Looking at it from the practical side the only case where the distinction comes 
 in is that of the electrostatic discharge. All currents of conduction are in 
 
 * Cf. Larmor, Aether and Matter. 
 
350 The electromagnetic field [CH. ix 
 
 themselves complete. In technical electrodynamics electrostatic discharges 
 are of little account although of late years the phenomena of wireless tele- 
 graphy has imparted a technical aspect even to this side of the subject. 
 
 It took over thirty years before anything like a decisive test of this 
 hypothesis of Maxwell's was obtained. At the beginning there were no 
 phenomena in which the dynamical relations of an electric discharge could 
 be investigated. Hertz however finally succeeded in discovering the electric 
 waves whose existence and theoretical relations had been deduced by Maxwell 
 as an essential consequence of his theory and thereby provided the necessary 
 experimental proof of the hypothesis on which the theory is based. 
 
 393. Second fundamental notion : Faraday's circuital relation. We now 
 
 come to the consideration of the second fundamental notion concerning the 
 interaction of magnetic fields and currents. This is the notion that under 
 certain circumstances the magnetic field may excite electric currents in the 
 closed conducting circuit : such currents are described as induction currents. 
 
 Since Oersted's discovery of the mutual action of a magnetic system and 
 a current it was a problem to find out whether magnetism could make a 
 current; and numerous attempts were made to solve it. Faraday* was the 
 first to succeed. He found that no stationary field however great could 
 make a current, but that as soon as the field was varied the conditions 
 were right. 
 
 Faraday's notion of representing a magnetic field was by lines of force; 
 as a field of flux in tubes. He then soon found that the essence of the 
 induced electromotive force necessary to drive the current observed was really 
 the change in the flux through the circuit. The rule deduced experimentally 
 by him was as follows: 
 
 Whenever the total magnetic flux through any circuit varies there is an 
 induced electromotive, force created in the circuit and the amount of it is 
 proportional to the rate of diminution of the total number of tubes of induction 
 threading the circuit. 
 
 This law is found to be universally true when either or both the magnetic 
 system and the current circuit are moving. It can be shown to be a direct 
 consequence of the previous fundamental notion, if the energy principle 
 applies to these things. The deduction will be given later. 
 
 The translation of this law into mathematics is in reality the whole subject 
 of electrodynamics. If E denotes the electric force measured in electro- 
 magnetic units the electromotive force in any closed circuit s is 
 
 E s ds, 
 * Exp. Res. n. p. 127. 
 
392-394] Faraday's circuital relation 351 
 
 and according to Faraday this is equal to 
 
 -!!/** 
 
 where B denotes the vector of magnetic induction, the integral being extended 
 over any surface / bounded by the circuit s and c is a constant, depending 
 on the units adopted. We thus have 
 
 fds- -- 
 c dt 
 
 another circuital relation forming the second fundamental equation of our 
 subject. Faraday's name is usually attached to it. 
 
 This relation of course implies that there is a definite value for the magnetic 
 induction flux through the circuit, i.e. it must be the same on whatever 
 barrier surface / it is calculated. This implies of course that B is a stream 
 vector or that ^j v g _ Q 
 
 which is of course a relation always satisfied by this vector ; so that the 
 equation so interpreted is quite consistent with our former ideas and needs 
 no extension as in the previous case. 
 
 394. The units in the electromagnetic equations. In the process here 
 employed of the construction of a mathematical theory of electricity and 
 magnetism definite sets of units have been introduced as associated with 
 the different classes of phenomena. As we now see that all the various 
 phenomena are in the most general case correlated with one another there 
 must be some definite relation between the various systems of units thus 
 adopted : the theory would otherwise not be consistent with itself. 
 
 The chief object in the choice of a system of units is to obtain the simplest 
 expression for our purposes of the fundamental physical relations on which 
 the particular theory is based. 
 
 In any relation connecting physical quantities of fundamentally different 
 characters, certain constants must occur in order to secure that the dimensions 
 of the fundamental quantities correlated are the same. For example in the 
 expression of the mechanical action between two point charges q and q 2 
 concentrated at a distance r apart we say that the force between them is 
 
 proportional to ~ a quantity of a fundamentally different kind. "We there- 
 fore write 
 
 = y ~7'. 
 
 and choose the dimensions of y so as to make the dimensions of both sides 
 of this equation the same. This constant y is then an absolute constant of 
 the theory, whose value and dimensions depend however on the choice of 
 
352 The electromagnetic field [CH. ix 
 
 unifs for the other quantities involved in the relation. In the simple electro- 
 static theory as we have developed it we choose to measure a quantity of 
 electricity so that the constant y in this expression is a simple number (without 
 dimensions) numerically equal to unity. This means that with the same 
 notation as before the dimensions of a quantity of electricity are given by the 
 symbolical equation 
 
 [Q-\=[m^t-\ 
 
 The dimensions of the electric displacement D then follow as the quotient 
 of charge by a surface ; those of the force intensity E as the quotient of force 
 by charge ; those of the potential being then the product of the dimensions 
 of E and a length ; and finally the dimensions of a current density are those 
 of a charge. divided by a time and a surface. In symbols 
 
 These are the quantities that occur in the specification of the electric part of 
 the general scheme. The magnetic quantities can be similarly examined and 
 the results are identically the same as those just given for the analogous 
 electric quantities or in symbols 
 
 395. But we have in this chapter introduced relations connecting these 
 two classes of quantities here involved : these relations however contain 
 certain universal constants c and c' so that as usual a definite choice of 
 units is not implied in the form of their expression adopted. If however we 
 interpret the equations in terms of units as defined in the earlier part of 
 this work it is found that both these constants have the dimensions of a 
 velocity. Their actual magnitudes could then be obtained by measurements 
 of the relative magnitudes of the quantities involved. We could for 
 instance determine the constant c' in the relation 
 
 f w d - 
 
 Is s * '~c r ' 
 
 connecting the magnetic field of a linear current with the strength of that 
 current by examining the field of a given current and evaluating directly 
 
394-397] The fundamental equations 353 
 
 
 the integral on the left. The exact method of doing this will appear later. 
 
 Again, the constant c in the relation 
 
 connecting the electromotive force in a circuit with the rate of change of 
 induction through it might be deduced by moving a circuit in a known 
 magnetic field and determining the current produced in it. If measurements 
 of this type, or others involving the same principles, are made it is found 
 that the two constants c and c' are exactly the same in actual magnitude, viz. 
 
 c = c' = 3 . 10 10 cm./secs. 
 
 We shall therefore in our future analysis always use the equations with c = c' 
 and imply the value just given, the fundamental importance of which will 
 subsequently appear. 
 
 396. Differential form of fundamental relations: continuous current 
 distributions. Our previous discussions have been almost entirely confined 
 to the mathematical abstractions of linear currents and the laws of the 
 magnetic field were given in a form suitable to this method of expression. 
 We have however in the previous chapters already prepared ourselves for 
 the discussion of volume distributions of current flow and we must now 
 discuss the extensions of the present ideas to such cases. The notion of a 
 linear current is obtained from the idea of a current flowing in a conducting 
 wire whose cross section dimensions are infinitely small compared with the 
 other lengths involved in the analysis : this condition was illustrated when 
 we saw how such an assumption led to difficulties in the analysis of the 
 magnetic field when we got too near the conducting circuit. The analysis 
 obtained above can however be very .easily extended to continuous volume 
 distributions of current by dividing such currents into single current 
 elements, of small cross-section, each of which can be regarded as a linear 
 current in the ordinary way. The combination of all these elementary 
 currents laid side by side constitutes the finite current. 
 
 We shall now assume that in all stationary or quasi-stationary conditions 
 the total current is a closed one and thus each of the elementary currents 
 may be regarded as closed and the previous general laws apply to it. 
 
 397. The first fundamental notion above states that in any space in 
 which stationary electric currents are flowing the line integral of the magnetic 
 
 4.77- 
 force round any closed path is equal to -- times the algebraic sum of all the 
 
 C 
 
 currents flowing through the path. This can be interpreted in terms of 
 
 continuous analysis, for if we use C as the total current density at any 
 
 L. 23 
 
354 The electromagnetic field [OH. ix 
 
 
 
 point then the sum of all the elementary currents flowing through the 
 curve is 
 
 taken over any surface abutting on the curve. Our rule thus gives 
 
 I H s ds = ^ I C n df. 
 
 J s -I f 
 
 The integral on the left can be converted by Stokes's rule and we get 
 
 and as the curve s and surface / have been arbitrarily chosen we must have 
 
 ITJ 47rC 
 curl H = -- , 
 
 c 
 
 a vector equation expressing Ampere's circuital relation in differential form. 
 The essential condition is that the current should be a stream vector, it should 
 always ftbw. in complete circuits, analytically 
 
 div C = 0. 
 
 398. The second fundamental principle is equally easily adopted to the 
 present scheme. Consider for example any closed circuit drawn in the con- 
 ducting material and count the number of lines of magnetic induction which 
 pass through it. Then according to Faraday's law there will be an electro- 
 motive force in the circuit which is proportional to the rate of diminution 
 of this number of lines of induction. The electromotive force is 
 
 and according to Faraday this is 
 
 IdN 
 c dt ' 
 
 where N = I "B^df, so that we have 
 
 - ( B n <Z/, 
 J t 
 
 c dt . 
 
 a form already quoted for the case of linear conductors. It is here regarded 
 as applying to any circuit drawn in the conducting substance, whether that 
 circuit is the path of one of the elementary currents or not. By again using 
 Stokes's theorem we can write this in the form 
 
 \) n df= \jl B n df, 
 so that if the circuit is not moving we have 
 
 ( curl EH =- ) df=Q, 
 
 V c dtj n J 
 
397-400] The vector and scalar potentials 355 
 
 or owing to the arbitrary nature of the circuit 
 
 1 dB 
 
 which is the differential form of Faraday's circuital relation. 
 
 399. These are the fundamental equations of the generalised electro- 
 magnetic theory. It is however convenient to interpret them in terms 
 of certain auxiliary vectors and scalars. In calculating N as the number of 
 lines of induction through the circuit we take the barrier surface / and 
 sum up all over it 
 
 N = E n df, 
 
 and of course this implies that the result depends only on the circuit and 
 not on the particular barrier surface/ taken to measure it on. The condition 
 for this is divB = 0, 
 
 and is of course satisfied in our case. We should therefore be able to express 
 N in terms of the circuit alone ; this is done in Stokes's theorem, for if 
 
 B - curl A, 
 
 then N = E n df = A s ds. 
 
 J f J s 
 
 This quantity A is the vector potential of the field; it enables us to abolish 
 the idea of barrier surfaces; all results can be interpreted in terms of the 
 circuit alone. 
 
 400. Now let us examine the electromotive forces. The electromotive 
 force round a circuit is, by Faraday's rule, equal to 
 
 _ IdlV_ _ld L 
 
 f* {it f* fit 
 
 and if the circuit is fixed this is 
 
 I f (dA - \ 
 
 - ( -j- . ds ). 
 cj s \dt ) 
 
 But by definition this is the same as 
 
 [(A), 
 
 and thus E=-, 
 
 c dt' 
 
 is a particular solution for the electric force at a point : to generalise it we 
 must add the gradient of any acyclic function </> which would give nothing 
 on integration round the closed circuit 
 
 F 1 dA 
 
 ~c- dt~^ d(f> ' 
 This is the general expression for the electric force in the field. 
 
 232 
 
356 The electromagnetic field [CH. ix 
 
 The first term in this expression for E is the electrodynamic part and the 
 second the electrostatic part : the first is motional and the second statical 
 or elastic. 
 
 401. The three differential equations involved in the vector relation 
 
 B = curl A, 
 are however not sufficient to determine A because if A is one solution then 
 
 obviously 
 
 A + grad x, 
 
 is another, x being any function of the coordinates. To define A more 
 completely we may therefore impose another condition. Maxwell takes 
 
 div A = 0, 
 
 and this appears to be the most convenient although it still leaves a certain 
 amount of indefmiteness. From this definition of A we deduce at once that 
 
 curl B = curl . curl A 
 
 - - V 2 A + grad div A, 
 and if we take div A = this gives 
 
 cur!B = - V 2 A. 
 
 This equation being analogous to that of Poisson we may consider A to be 
 the potential of a distribution of matter of density j > attracting according 
 to the inverse squares law. We thus see that 
 
 . If , _ dv 
 A = T curl B , 
 4-77- J r 
 
 the integral extending to the whole of space occupied by the electromagnetic 
 
 field. 
 
 Now B = H + 477-1, 
 
 1 TI ^ 
 
 and curl H = , 
 
 c 
 
 so that A = - f(C + c curl I) . 
 
 c J r 
 
 In the case of a linear conductor carrying a current of strength J and when 
 there is no magnetic matter about this reduces to 
 
 J [ds 
 
 A 
 
 c ' r 
 
 where ds is the vectorial element of the conducting line along which the 
 integral is taken. 
 
400-403] The retarded potentials 357 
 
 402. In the general case we have also 
 
 p = div D = j_ div E + div P, 
 
 or on inserting the value of E in terms of A and </> and noticing that div A = 
 we find that 
 
 V*<j> = - 477- (p - div P) 
 
 = - 477 (p + p'), 
 
 where />' is the Poisson density of ideal electrification which is equivalent for 
 some purposes of the electric polarisation. It follows then that 
 
 so that cf> is the static electric potential of a distribution of density (p + />'). 
 
 403. Most subsequent writers * have adopted a slightly different definition 
 of the vector potential which has certain advantages over that given by 
 Maxwell. Starting from the definition of the magnetic induction in terms of 
 a vector potential and the consequent derivation of the electric force in terms 
 of this same potential and a scalar potential cf> as above, it is assumed that 
 these two potentials are connected by the equation 
 
 It is then easy to verify that A and (f> satisfy respectively the equations 
 
 and 
 
 where in the former equation Cj is used to denote the density vector of true 
 electric flux. 
 
 Under ordinary circumstances and in finite regular fields the appropriate 
 solutions of these equations follow immediately from the analytical theorem 
 discussed in the introduction ( 25-29). When the conditions of the field 
 vary in both space and time we have in fact 
 
 r ,., dv 
 IP + P ] 7 > 
 
 and A = i ([^ + c curl I] - , 
 
 c J '- J r 
 
 the integrals being extended over the whole of the field ; r as usual denotes 
 the distance from the typical element of integration to the field point at 
 
 * Cf. Lorentz and others in the Encyk. d. math. Wissenscli. Bd. v. ; Macdonald, Electric Waves; 
 Lorentz, Theory of Electrons ; Schott, Electromagnetic Radiation. 
 
358 The electromagnetic field [CH. ix 
 
 which the functions are calculated and the square brackets serve to indicate 
 
 T 
 
 that the values of the functions affected are to be taken for the instant t 
 
 c 
 
 I being the instant at which the functions themselves are evaluated. 
 
 The advantage of this form of definition is that it expresses both potentials 
 directly in terms of the positions and motions of the actual charge elements. 
 The potentials themselves are usually called the retarded potentials because 
 the contributions to them due to charges at a distance r away is not due to 
 the instantaneous value of these charges but to their values at the previous 
 
 (_.\ 
 t ) . This means of course that the effect of any change in the 
 c/ 
 
 charge distribution is not felt at points a distance r away until a time r/c 
 after it has occurred, which is interpreted as implying that effects of electric 
 changes are propagated outwards through space with the 'uniform velocity 
 c in all directions. 
 
 404. The retarded potentials* are to be strongly contrasted with the 
 instantaneous potentials of Maxwell's theory, and although they are perhaps 
 more consistent with a propagation theory it is not to be inferred that the 
 instantaneous potentials of Maxwell's theory necessarily implies the instan- 
 taneous propagation of effects from all parts of the field. It must be 
 remembered that on both forms of the theory the potentials have been 
 introduced primarily for analytical simplification and they do not necessarily 
 represent directly definite physical quantities, although it may in ceriain 
 circumstances be convenient to regard them as so doing. The ultimate 
 procedure in either case involves the elimination of these potentials and the 
 expression of all necessary relations in terms of the physical quantities that 
 are propagated, without the aid of any auxiliary mathematical conceptions. 
 
 405. The retarded potentials are the most useful for the determination 
 of the field of specified charge and current distributions but in their above 
 form they are not directly suitable for numerical calculation, inasmuch as 
 the elements of charge [p] dv or current [C x ] dv\ which enters in their expression 
 are not all present in the volume element dv at the same effective instant. 
 To render them more useful we must express the integrals explicitly as 
 functions of the instantaneous distributions of the charge and current 
 elements, which are the data usually specified in any problem. 
 
 Eegarded as functions of t the densities p and Cj for a given point of space 
 may in the limit be discontinuous, as for instance when the boundary of a 
 charged conductor crosses the point; but from the nature of the case the 
 number of discontinuities or infinities which occur during any finite interval 
 
 * Cf. Larmor, Aether and Matter, pp. 111-112. 
 
 f We shall for the present drop the term in the vector potential due to the magnetisation. 
 It can be replaced quite easily at any stage or may be included in the current Ci. 
 
403-406] The retarded potentials 359 
 
 of time is necessarily finite, and for each such irregularity the aggregate 
 variation is also finite. Hence the quantities [p] and [C x ] which occur in 
 the integrals for the potential functions can always be expressed as Fourier 
 integrals. Doing this and supposing that the values of both functions at 
 any point (x e , y e , z e ) are prescribed for all values of t we get 
 
 1 [j /+/+- *(*-') drdfj, 
 
 = - dv -~ 
 
 and A :r- 
 
 27rc 
 
 where in both integrals efa> is the typical element of volume round the point 
 
 (x e , y e ,z e ) and 
 
 r* = (x - z e ) 2 + (y - y.) + (2 _ z e)2j 
 
 and p and C t are now regarded as functions of r with (x e , y e , z e ) as parameters. 
 Although these integrals appear to require a complete knowledge of the 
 future history of the field for its present determination, they in reality do 
 effectively define the field at the present instant independently of such 
 knowledge. We may in fact choose p and Cj, quite arbitrarily as far as future 
 time is concerned ; but when these values have been chosen the values of 
 [p] and [Cj] and hence also of cf> and A are quite determinate for all time: 
 they have the proper value for all past time whatever values may be selected 
 for the future. If it is desired to express the integrals explicitly in terms of 
 specified quantities only then the sine or cosine integrals must be used, but 
 this seems to be unnecessary in the present instance. 
 
 We may now in these expressions rearrange the order of integration in 
 the triple integrals ; for this only amounts to a rearrangement of the terms 
 of a triple sum which is for physical reasons known to be absolutely convergent. 
 We may therefore effect the integration with respect to v first and then the 
 difficulties of the kind mentioned above do not present themselves, because 
 the summation is that of instantaneous contributions at the time T from 
 elements all over the field. 
 
 406. The whole of the circumstances in the surrounding field can now 
 be determined from these forms of the potentials. To determine the electric 
 force and magnetic induction vectors we use the relations 
 
 1 
 
 B = curl A, 
 
 t> (t---r} 
 
 i 3 r f +o r +o i 
 
 so that E = ~ ^i M v ~ (P r i Cj) ^T^U, 
 
 27TC8?J J-oo J-oo r C 
 
 iu /'t-^-r 1 ! 
 
 i r , /+ co 
 
360 The electromagnetic field [OH. ix 
 
 i r r + 
 
 + i M [CA] 
 
 2;rC 
 
 r 
 
 wherein T 1 denotes the unit vector in the line joining the typical element of 
 integration to the field point where the functions are calculated so that 
 
 Vr = FJ'. 
 
 407. Ampere's hypothesis on the origin of magnetism. Before closing 
 this discussion reference can be made to an important consequence of an 
 obvious analogy in the specification of the magnetic vector potential of 
 a magnetic distribution and that of a continuous distribution of currents 
 just deduced above. 
 
 It has already been seen that the vector potential A of a distribution of 
 magnetic polarity of intensity I is given at points outside the magnetism by 
 the equation 
 
 but that at places inside the magnetism 
 
 + I curl I , 
 
 the previous formula being then plainly inapplicable because it integrates 
 to a quantity whose differential coefficients are infinite where r can vanish. 
 Analogously, it may be recalled that, in the ordinary statical theory of 
 magnetism, the magnetic force is derived from the potential of the actual 
 magnetic polarity only at places outside the magnets, but at places in its 
 interior is derived from the potential of the Poisson volume and surface 
 distributions of an ideal continuous magnetic substance. At a point in the 
 interior of the magnetism the magnetic force should be in fact defined as the 
 part of the force, acting on a unit pole there situated, that is independent 
 of the local polarity at the spot, it being then shown how the definite value 
 of this part can be determined. 
 
 This formula indicates that such a magnetic distribution is equivalent at 
 all points to an electric current distribution of density equal to c . curl I 
 throughout the volume of the magnet together with sheets of current given 
 by c [n, I] flowing along the interface. 
 
 This is the origin of Ampere's hypothesis that the magnetism of a substance 
 has its origin in some intrinsic distribution of currents throughout the 
 substance. Of course these currents must circulate within the molecules or 
 molecular aggregates which is the ultimate element of the aggregate magnetic 
 
406-408] Ampere's theory of magnetism 361 
 
 polarity; they are minute current whirls not involving continuous flow in 
 any direction. 
 
 The equivalence between these two systems, one a magnetic and the 
 other a current system, includes ex hypothesi, that of the vector potential 
 and therefore of its curl, that is of the magnetic induction B which is always 
 a stream vector. They are not however equivalent as regards magnetic 
 
 force; for in the one case the curl of the magnetic force is - times the 
 
 current, in the other it is null. In treating of a current system devoid of 
 magnetism, the only quantity that occurs is the magnetic induction due 
 to the currents ; the portion of the expression for this induction which forms 
 the contribution of the part of the current arising from contiguous molecules 
 or elements of volume being always negligible compared with the induction 
 as a whole. 
 
 408. For some analytical purpose it is convenient to convert in the 
 manner indicated all the magnetism associated with the medium of the 
 electromagnetic field into a volume distribution of electric currents and 
 if the surfaces of the magnetic media are replaced by continuous transition 
 layers the surface integrals disappear and thus 
 
 A f i T dv 
 A = c I curl I , 
 J r 
 
 and the distribution of current density c curl I effectively replaces the 
 magnetism by itself. 
 
 Supposing now this current distribution is added to that specified in the 
 Amperean circuital equation 
 
 47TC ,_ 
 
 = curlH, 
 c 
 
 then it is seen that the equation assumes the form 
 
 A ft 
 
 = curl (H + 47rl) = curl B, 
 
 
 
 where the current C is now supposed to include the distribution which replaces 
 the magnetism. This form is convenient as it eliminates the magnetic force 
 from the fundamental equations and thus leaves only the specification of the 
 electric current to be effected in the complete theory, and in the modern 
 theory of electrodynamics which explains the magnetism as the result of the 
 motion of electrons this seems to be the only consistent course to adopt. 
 If we thus complete the current C the definition of the vector potential 
 
 by the integral 
 
 1 [Cdv I i[G]dv 
 
 - - or - 
 c J r c 
 
 is completely effective in the most general case without the necessity for the 
 introduction of the extra term in the magnetic polarisation. 
 
362 The electromagnetic field [OH. ix 
 
 409. Maxwell's generalised theory of the electromagnetic field. So far 
 
 our discussions have been limited to electrostatic and electromagnetic pheno- 
 mena which are either actually stationary or at least so slowly variable that 
 they admit of treatment along similar lines. The restriction thus implied 
 will be more fully discussed in a more appropriate place : all that is inferred 
 is that the state of the motion adjusts itself so quickly at each instant that 
 it is practically an equilibrium one under the conditions pertaining in each 
 instant. For cases in which this restriction is satisfied the laws of Ampere 
 and Faraday provide a sufficient and satisfactory foundation but a generalisa- 
 tion is needed to extend it beyond these cases. What are the fundamental 
 laws of non-stationary electromagnetic fields in general-? 
 
 The most successful attempt to answer this question was made by Maxwell. 
 With Faraday he saw all the obvious phenomena of electromagnetics merely 
 as the terminal aspects of a variation of condition in the space (or field) 
 surrounding the apparatus. The observed actions are transmitted through 
 and by a something, the aether, in this field which is capable of varying its 
 condition. Although the aether is thus merely regarded as the definite 
 something to which we can attach the vector quantities of the electromagnetic 
 field, it is convenient to have a definite representation of its mode of action. 
 Of course any such consistent representation is simply in set terms a dynamical 
 or analytical theory of the activity of this aether. 
 
 The essential characteristics of a theory of this kind is of course that the 
 actions must be transmitted by the aether with a finite velocity. This 
 corresponds to a representation of the phenomena by means of differential 
 equations connecting the time and space variations of the vectors with one 
 another. The values of the vectors at a definite point of space are directly 
 connected only with their values at infinitely near points, and only indirectly 
 with the conditions at finitely distant points. The new departure instituted 
 by Maxwell comes, when expressed mathematically, to a statement that the 
 generalised inter-relation between the essential vectors in any electromagnetic 
 field are always exactly expressed by the two fundamental differential rela- 
 tions already established for the field of closed currents. This statement 
 of course involves the assumption that all electric discharges are effectively 
 of the nature and possess the properties of systems of closed currents, 
 being completed when necessary by the so-called displacement current in 
 free space (i.e. in the aether) and in dielectric media ; in fact the consideration 
 of unclosed circuits never arises. 
 
 The extension here implied is assumed to apply not only to systems in 
 which a part or the whole of electrical motions are those that take place in 
 ordinary conduction currents but to every conceivable phenomenon which 
 involves real electric motions, i.e. of course, motions of electrons. The true 
 electric current of the complete Amperean equation will thus include all 
 
409-41 1J Maxwell's theory 363 
 
 possible types of coordinated or averaged motions of electrons, namely 
 currents arising from conduction, from material polarisation and its convection 
 and from the convection of charged bodies. 
 
 410. The vectors necessary for a complete specification of the con- 
 ditions in the electromagnetic field at any point are : 
 
 (i) E, the intensity of the electrostatic field measured generally by the 
 ratio of the resultant force to the charge on a small conductor placed at the 
 corresponding point. 
 
 (ii) D, the total electric displacement of Maxwell consisting partly of 
 a true displacement measured by the polarisation intensity P and partly 
 
 E 
 
 by the aethereal displacement characteristic of Maxwell's theory. 
 
 vn 
 
 (iii) H, the intensity of the magnetic force in the field measured in a 
 method similar to that adopted for E. 
 
 Regarding the measures of E and H at points inside the ponderable bodies 
 it is merely necessary to refer back to the critical discussions given on their 
 introduction. 
 
 (iv) B, the magnetic induction vector which always satisfies the circuital 
 condition <jiv 3 = 9. 
 
 It is connected with the magnetic force and magnetic polarisation intensity 
 I by the general vector equation 
 
 B = H + 4rrl, 
 so that in free space outside the magnetic matter 
 
 B = H. 
 
 (v) C, the total current of Maxwell's theory. In the general case this 
 consists essentially of several distinct parts which require very careful 
 specification. These are separately examined below. 
 
 411. -The vectors thus specified in the most general case are now 
 assumed to be correlated by the two fundamental equations of the theory, 
 viz. 
 
 (i) Faraday's law 
 
 1 ^B 
 
 r = curl E. 
 c at 
 
 (ii) Ampere's law 
 
 477C ._ 
 
 - = curl H, 
 c 
 
 the two vectors B and C being however restricted to satisfy the conditions 
 
 div B = 0, 
 div C = 0. 
 
364 The electromagnetic field [CH. ix 
 
 These two relations are quite independent of the nature of the medium filling 
 the space of the electromagnetic field. They are in the nature of fundamental 
 dynamical relations between the quantities involved and for this reason 
 were adopted by Maxwell as the fundamental equations of the more generalised 
 theory. They are of course self-consistent only so long as the current C and 
 the magnetic induction B are circuital; it is therefore necessary to adopt 
 Maxwell's hypothesis if we wish to use these equations for the general case ; 
 the one condition is involved in the other. 
 
 412. There are however four vector quantities involved in these two 
 relations so that for a complete specification of the scheme two more relations 
 are required. These are easily obtained but before proceeding to this we 
 must examine the 'expression for the complete current density. This consists 
 of several distinct parts which are best examined separately. 
 
 (a) C l5 the current of conduction, is, as we have already seen, made up 
 of a drift of electrons or ions, the positive ones travelling in one direction, 
 the negative ones in the opposite direction, under the influence of the electric 
 force. 
 
 (6) C 2 , the polarisation current associated with the material medium: 
 We have already seen how the establishment of a condition of polarisation 
 in a dielectric medium is accomplished on the Larmor-Lorantz view of the 
 subject, by an electric displacement which may be caused either by turning 
 round the little bi-polar molecules, or by an actual displacement of the 
 charges relative to one another in the molecule. A time variation of such 
 a state of affairs would therefore involve an electric current in the dielectric. 
 The displacement in any position is the same as if the positive pole started 
 from the final position of the negative pole and moved up to its final 
 position. The total amount can thus be estimated and is such that 
 
 _dP 
 2 ~ It ' 
 
 (c) A material medium moving with the velocity equal at the (x, y, z) 
 point to u and having in the neighbourhood of that point a charge of amount 
 p per unit volume, clearly contributes a convection current of density pu. 
 The elements of the moving medium may be the molecules or ions as in 
 electrolysis, so that conduction currents are merely particular examples of 
 convection currents. The convection current may however also consist 
 simply of a convection of free electrons so that on such a theory a con- 
 duction current may in some respects be regarded as a convection current. 
 It is therefore in reality rather difficult to distinguish between conduction 
 and convection currents, although it is usual and convenient to use the 
 distinction provided by the applicability of Joule's and Ohm's laws. 
 
411, 412] The constituents of the current 365 
 
 The convection of a material medium merely polarised to intensity P also 
 supplies a part to the volume distribution of electric currents : but its deter- 
 mination requires more refined analyses. Consider in the first place the 
 convection of a simple type of polar molecule involving a single electron 
 + q for one pole and q for the other pole. The transfer of these two 
 electrons in company, as in the diagram, is equivalent to the transfer of 
 a positive electron round the long narrow circuit in the direction of the 
 curved arrow : and this circuit can be divided up into sub-circuits of ordinary 
 form in the Amperean manner by partitions represented by the dotted lines. 
 The distance between the two poles of the molecule is absolutely negligible 
 compared with the distance that the molecule is carried by the convection, 
 in a time which is effectively infinitesimal for the analytical theory of con- 
 tinuous currents even in its optical applications : so that the circumstance 
 
 Fig. 72 
 
 that convection round the ends of the elongated circuit is not really effected 
 is immaterial. It follows that the convection with velocity u of a medium, 
 containing such molecules polarised or orientated to intensity P gives -rise 
 to an Amperean system of currents round minute circuits, which forms 
 effectively a magnetic polarisation, or quasi-magnetism of the material, of 
 
 intensity - [Pu] per unit volume. This result will clearly not be disturbed 
 
 G 
 
 when the distribution of polarity in the molecule is more complicated than 
 that here assumed. 
 
 It will be convenient in most cases to retain this mode of specification by 
 means of a distribution of magnetisation, as it will enable us to take direct 
 advantage of the known principles governing such distributions. But we can 
 restore it directly to the form of a distribution of currents by the transformation 
 explained in the last section and this shows that the distribution is generally 
 equivalent to a current of density 
 
 curl [Pu], 
 
 the surface of the medium being replaced by a gradual transition to avoid 
 the introduction of the surface distribution. 
 
366 The electromagnetic field [OH. ix 
 
 (d) C 3 , the displacement current in the aether introduced by Maxwell 
 as an aethereal current of such amount as to complete into a single circuital 
 stream all the types of true electric flux which are associated with the 
 matter. This is shown, as above, to be measured by the density 
 
 1 dE 
 tTr'dt' 
 in the general case. 
 
 The total current density in the general case is therefore 
 dP I dE 
 
 and this is always circuital, or ^ v (j = o 
 
 always. This is the essence of Maxwell's assumption and when once granted 
 the general theory is much simplified. All the consequences of this theory, 
 some of which will occupy our attention later, confirm us in the view that 
 this hypothesis of Maxwell's is the correct one to make for the general theory. 
 
 413. The significance of the various terms in the complete expression 
 for the true electric flux density which depend on the electric [and magnetic] 
 polarisations can also be exhibited in a more analytical form, if we assume 
 that all such polarisations result entirely as the average aspects of a more or 
 less complex distribution of electrons or point charges of both signs. In 
 such a case the part of the current density depending on these electrons at 
 any point in the medium is the limiting ratio of the volume of the physically 
 small element v to the sum 
 
 extended over all the electrons inside it; v is the velocity of the typical 
 electron. In effecting the summation care must be taken to refer each 
 electron to its proper position in the matter. Having fixed on a definite 
 point in the medium moving with the velocity u we notice that the electron 
 attached to it is displaced from that point through a distance r on account 
 of the polarisation, and has therefore a velocity 
 
 df 
 u + r = u + j t + (uV) r. 
 
 The electron which is actually at the point in the matter under review and 
 which properly belongs to some other point has therefore, the velocity 
 
 u + r - (rV) u - (rV) r, 
 
 it being assumed that the velocity of an electron is a continuous function of 
 its position in the matter. This is the value that has to be taken for v in 
 effecting the summation as above. 
 
 Again we must notice that the summation is to be taken only over those 
 electrons actually in the volume element under consideration, the number 
 
412-415] The complete electric current 367 
 
 of which is a function of their displacements. In fact in setting up the 
 displacements typified by r the charge displaced across any surface is 
 
 . / 
 
 denoting a sum for the electrons in the volume element T n df. A simple 
 application of Green's lemma shows that the statistical effect of this displace- 
 ment is the same as if each electronic charge inside the surface were reduced 
 in the ratio 1 _ ^iv r : 1, 
 
 the number remaining unaltered. 
 414. It is thus the summation of 
 
 S 9 (1 - (Vr)) fu + ^ + (uV) r - (rV) u - (rV) ^ - (rV) (uV) r] , 
 |_ dt at J 
 
 or, neglecting quantities of higher order in the displacement, of 
 
 u - u (Vr) + (uV) r - (rV) u + ^ - ^ (Vr) - (rV) ^ 
 
 at at at 
 
 that is to be effected. Owing to the smallness of the volume element Sv we 
 may assume that u and its space gradients are constant throughout, whilst 
 
 where p is the density of the free charge and P the polarisation intensity of 
 the element. Thus 
 
 (uV) r = (uV) Xqi = (uV) PSv, 
 
 (Vr) = u (V, 2?r) = u (VP) Sv, 
 Zq (rV) u = (2?r, V) u = (PV) uSt>. 
 Due to these terms we have therefore a current of density 
 
 P u + d * + (uV) P - (PV) u - u (VP) + P (Vu) = pu + d * + curl [Pu], 
 which agrees with the result obtained above. 
 
 415. The remaining terms have another significance. They are 
 
 < VI > + < rv > - ~ 
 
 1 1 IV dk 
 
 curlier, ^ 
 
 where in these equations the operator V is presumed to affect all quantities 
 immediately following it. 
 
 The second term on the right-hand side of the last equation receives its 
 main contribution from those electrons which are executing rapid orbital 
 
368 The electromagnetic field [OH. ix 
 
 motions about their mean positions in the matter, the contribution of any one 
 electron being proportional to the moment of its momentum about the 
 equilibrium position. Moreover if the orbital motions are to any extent 
 permanent the sum %q (Vr)r, 
 
 for these electrons will be practically independent of the time. Thus when 
 some or all of the electrons are executing motions of the type considered there 
 is an additional term in the complete expression for the current density 
 which to the first approximation is equal to 
 
 c curl I, 
 
 1 I di 
 where ISt; = Sr, 
 
 In this form we recognise that the additional term in the current is the same 
 as would be contributed by the medium magnetically polarised to intensity 
 I, if the magnetism is regarded as equivalent to a distribution of minute 
 current whirls. This is a tentative suggestion of an electron theory of 
 magnetism, which will be further discussed in the sequel. For the present 
 we shall usually disregard terms of this type in the current expression. 
 
 416. In the whole of the above discussion it has been assumed that the 
 matter extends continuously throughout and beyond the element under 
 direct observation. If as is sometimes the case it is necessary to include the 
 effects of discontinuities in the material distribution these can always be 
 estimated by regarding such discontinuities as continuous rapid transition 
 regions throughout which the definitions given above remain effective. In 
 this way it is easily seen that a surface of discontinuity in the material medium 
 is to be regarded as the seat of a current sheet of density 
 
 where I x = I + - [Pu], 
 
 C 
 
 and n x is the unit vector normal to the surface in the direction from the side 
 1 to the side 2. The notation implies that it is the difference of the values 
 of the function on the two sides that is to. be taken as the current density. 
 Thus, for example, in any continuous piece of matter with a definite 
 boundary the electronic motions in it may be specified in their average aspect 
 as being equivalent to a distribution of body currents of density 
 
 dP 
 Cj + pu + v- + c curl Ij 
 
 throughout its mass together with the surface current sheet of density 
 
 at any point of its outer boundary. 
 
 These surface currents are important when we consider the dynamical 
 aspects of electromagnetic phenomena. 
 
415-419] The constitutional relations 369 
 
 417. Returning to our fundamental equations we now want the relations 
 connecting the vectors involved. We know already that there will be some 
 relation connecting the current intensity with the electric force, something in 
 the nature of Ohm's law only rather more general. There is also a relation 
 connecting the magnetic force and magnetic induction. These two con- 
 stitutive relations depend essentially on the nature of the ponderable matter 
 involved and must therefore be obtained by experiment. The first two 
 relations being of the nature of dynamical principles must be exact, but this 
 second pair obtained by experiment can only be approximate. As to the 
 actual form of these relations we may state the following, referring back to 
 the respective previous chapters for a discussion of their relative merits. 
 
 418. (i) In any given material medium devoid of hysteretic quality, the 
 intensity of electric polarisation P must be a mathematical function of the 
 electric force E which excites it. In ordinary cases, certainly in all cases in 
 which the exciting force is small the relation between P and E is a linear 
 one : thus in the general problem of an aeolotropic medium there will be 
 nine static dielectric coefficients. The principle of negation of perpetual 
 motions requires this linear relation to be self -conjugate and so reduces the 
 nine coefficients to six. In the special case of isotropy there is only one 
 coefficient and the relation may be expressed in the usual form 
 
 where e is the single dielectric constant of the medium. 
 
 In problems relating to moving material media the question may naturally 
 arise whether the value of e for the medium is sensibly affected by its movement 
 through the aether. When it is considered that each molecule that is polarised 
 by the electric force has effectively two precisely complementary poles, 
 positive and negative, it becomes clear that a reversal of the motion of the 
 material medium cannot alter the polarity induced : hence the influence of 
 the motion on e can only depend on square and higher even powers of the 
 velocity. 
 
 419. (ii) In cases in which the magnetisation induced in the medium 
 is of sufficient magnitude to be taken into account, similar statements will 
 apply to it. In the general crystalline medium there are six independent 
 coefficients of magnetisation : these reduce for an isotropic medium to a 
 single coefficient and the relation between induction and force is 
 
 A simple equation of this kind, representing linear and reversible magnetisa- 
 tion applies to substances such as iron only when the field is of small intensity. 
 
 24 
 
370 The electromagnetic field [CH. ix 
 
 (iii) Finally the relation between the current of conduction Cj and the 
 electric force may be taken as a linear one involving nine independent 
 coefficients of conductivity : in the case of isotropy these reduce to a single 
 one d - *E. 
 
 It has been found by experiment that coefficients of electric conduction, 
 unlike the other coefficients above considered, remain constant for all 
 intensities of the current up to very high limits,* so long as the temperature 
 and physical condition of the conducting substance are not altered. This is 
 what was perhaps to be anticipated from the circumstance that conduction 
 arises from the filtering of the simple non-polar electrons or ions through 
 the conducting medium under the directing action of the electric force, not 
 from orientation of polar complex molecules which may originate hysteretic 
 changes in their cohesive grouping in the substance. 
 
 420. For a body of compound nature at rest and in which both 
 polarisation and conduction currents can coexist the relation between the 
 total current of Maxwell's scheme and the electric force is more complicated 
 than those given above. In fact 
 
 C = C x + D 
 
 = E + T~ E 
 
 477 ^' 
 
 and this and the relation jj = jj 
 
 are the two directly required in the theory involving only media at rest. 
 
 With these four relations we have then a complete electromagnetic scheme, 
 which if presumed to apply in the whole range of electrodynamic phenomena, 
 is a sufficient basis for the mathematical development of the subject with 
 regard to media at rest. 
 
 For media in motion the additional terms arise which are directly connected 
 with the other vectors in the field by relations more of a dynamical than of a 
 constitutional character and for this reason a full discussion of their signifi- 
 cance is postponed to another chapter. 
 
 The last four chapters of this book are mainly confined to the consideration 
 of various aspects of this theoretical scheme as a basis for all electromagnetic 
 and electrodynamic phenomena and further discussion of its import will be 
 reserved for those chapters. 
 
CHAPTER X 
 
 SOME SPECIAL ELECTROMAGNETIC FIELDS 
 
 421. The magnetic field of special current distributions. It is often 
 desirable to have a complete knowledge of the magnetic field associated 
 with certain simple types of current distribution, not only for the purpose 
 of testing the validity of the theory, but also with a view to obtaining the 
 most suitable arrangements for practical purposes in electrotechnics. The 
 method is to work out the cases which are obviously susceptible of rigorous 
 mathematical treatment and then to construct the practical instruments to 
 agree with the theoretical arrangement or to attempt by a method of suc- 
 cessive approximation to obtain some idea of the working of the practical 
 case from the behaviour of the nearest representative in the theoretical 
 cases. We must therefore examine the fields of certain simple current 
 distributions with a view to obtaining for reference as many workable cases 
 as possible. 
 
 422. (a) The magnetic field of an infinitely long straight current in a 
 linear conductor. If we consider the wire in the axis of z from oo to + oo 
 then the 2-component of the magnetic vector potential of the field is the 
 only one that exists and in the case of a positive current J this is 
 
 2J . 
 
 = C - - log r, 
 c 
 
 where r is the distance of the point in the field from the wire. 
 The magnetic potential in the field is therefore 
 
 where 6 is the angle measured round the axis of z. This can also be deduced 
 in the elementary manner for the solid angle subtended at any point P in 
 the field is the area of the lune of the sphere of angle 6. 
 
 We have thus 
 
 242 
 
372 Some special electromagnetic fields [CH. x 
 
 The force is therefore of magnitude ^ . - and is perpendicular to the radius 
 
 r and the current direction, in the sense which is associated with the direction 
 of the current in the same way as rotation to advance in a right-handed 
 screwing motion. 
 
 The field of any straight current of finite length at distarices small com- 
 pared with its length is similarly determined and gives the same result. The 
 magnetic lines of force are in circles round the wire ; we should then have 
 on integrating round one of them 
 
 c 
 and this is another reason for the form of H taken. 
 
 The same result can be applied even to a closed curved current circuit. 
 If we have a current in a curved wire then very near the wire at any point 
 it may be treated as practically straight and thus the magnetic force is as 
 before. It is an interesting problem to work out the first order correction 
 due to the curvature in such a case. 
 
 423. In the above argument the curffcnt is treated as infinitely thin. 
 But very near the wire these results do not hold, the shape of the wire 
 and the distribution of the current in it being then of fundamental im- 
 portance. To treat cases of this nature we have merely to regard the current 
 as divided up into a large number of elementary current filaments, each 
 of which is straight. The total field is then the superposition of the fields 
 of each of these filaments. At any point therefore 
 
 = C - - (SJ log r, 
 
 cj 
 
 r now denoting the distance of the point in the field from the axes of the 
 typical current filament SJ. Of course if we knew the shape of the cross 
 section and the distribution of the current over it we could find A z at each 
 point of the field. As a first approximation however in slowly varying fields* 
 we may treat the current as uniformly distributed over the section so that 
 
 A z = C - -\ dflogr, 
 
 L J f 
 
 and this latter integral can be interpreted as a logarithmic potential integral. 
 If the cross section is circular and of radius a we get : 
 (i) at external points 
 
 * Cf. below, p 559. 
 
422-424] Parallel currents 373 
 
 R being now the distance from the axis ; and thus 
 
 H H H 2J l 
 
 H, = H r = 0, H, = - . 5 , 
 
 (ii) at internal points 
 
 A /"f 
 
 Ai >~ c a 2 ' . 
 
 ^ ^1 TT TT 1"T 27 R 
 
 and thus H z = H r = 0, IL = - -x . 
 
 c or 
 
 Notice that there is no magnetic potential inside corresponding to 
 
 , 2J0 
 
 / = - - outside. 
 
 424. If there are several parallel wires present in the fields and carrying 
 currents J l3 J 2 , ... J n then the vector potential of the field A is still such 
 that A z = A 3, but outside the wires 
 
 n 2J 
 
 A 0z = 2 - - log r p + const., 
 P=I c 
 
 a result which is still true right up to the wires if the wires have circular 
 cross sections and carry their currents uniformly. 
 
 The internal field of the pih wire under similar conditions is obtained by 
 superposing on its own internal field the external one due to all the others 
 and is thus determined by 
 
 const. - Av = ? + 2 - logr ff + 2 log r q . 
 
 C tt p q = \ C 
 
 From these the magnetic field strength can be determined as previously. 
 The results confirm those already obtained in the limiting case. 
 
 As a special case we may quote the results for two parallel wires of equal 
 circular section (a) carrying equal currents J in opposite directions. For these 
 in the external field 
 
 A 2 J , r, 
 
 A, = - - log- 1 . 
 c 8 r 2 ' 
 
 and in the interiors of the wires 
 
 A, r = - ( - log 
 
 c V2 a 2 
 
 in the first and 
 
 Aj-_ = H ( - - - log 
 
 in the second. The field in this case is far more concentrated into the space 
 near the wires than when the currents flow in the same direction along the 
 wires. 
 
374 
 
 Some special electromagnetic fields 
 
 [CH. x 
 
 425. (6) The magnetic field of a solenoid. A current J in a solenoidal 
 circuit can be replaced by a lot of parallel circular currents 
 round the solenoid and a uniform current J uniformly dis- 
 tributed round the solenoid and flowing along its length. If 
 the coil is very long a uniform current flowing in its length 
 is equivalent to the same current flowing up a wire in the 
 middle. Thus if we compete the circuit by a 'wire down 
 the middle the upward current would practically be cancelled 
 and we could then neglect it. It is completely cancelled if 
 the coil is very long and practically so in most other cases. 
 
 We can then treat the coil as a series of parallel currents ; 
 in any case the outstanding part is merely of the order of 
 a single convolution and is therefore negligible in a closely 
 wound coil. If we replace each circular current by its 
 equivalent magnetic shell we see that the coil is equivalent 
 to a bar magnet. The current circulating is of strength n J 
 per unit length of coil, n being the number of turns per unit 
 length : and this flow is equivalent to uniform magnetisation 
 inside the solenoid of intensity k-nnJ. The components of 
 the magnetisation would then be 
 
 1=0 1=0 
 
 * W 5 *!/ v > 
 
 The coil is thus in all equivalent to : 
 (1) an ideal volume density 
 
 81, 
 
 Fig. 73 
 
 dz 
 
 47T7 dn 
 
 ~~^dz' 
 
 (2) and magnetic polarity 
 
 dn 
 
 at the ends. 
 
 If the winding is uniform ~ = and thus the magnet is simple, with its 
 
 cz 
 
 poles right at the ends. 
 
 This of course applies only to outside space. The current in the solenoid 
 is equivalent as regards its magnetic action at external points to the magnetic 
 distribution specified. 
 
 426. The same method will however determine the internal field at any 
 point. We have merely to separate the two faces of adjacent equivalent 
 magnets so that there is empty space between them in which the force can 
 be calculated as before. The force is equal to fare, where a is the strength 
 
425, 426] 
 
 The solenoid 
 
 375 
 
 of the polarity on one side of an equivalent magnetic shell. If t is the 
 thickness of this shell, then at is the strength of the shell and this is pro- 
 
 Fig. 74 
 portional to the total current in the breadth of the shell; but the number 
 
 of convolutions in the breadth is 
 
 nt, 
 
 and if J is the current flowing, the total current round the shell is 
 
 ntJ, , 
 
 and thus 
 
 , A HtJ 
 at = 4?T . 
 
 C 
 
 Thus the strength of the field inside the solenoid is 
 
 nJ 
 H = 4?r , 
 
 C 
 
 outside we have practically 
 
 H-0. 
 
 All this is correct as long as we remain well inside the shell; as soon as 
 we get up near the ends irregularities enter and we are no longer able to apply 
 the results. Down inside the coil the effect of the ends is very small and the 
 
376 Some special electromagnetic fields [CH. x 
 
 force is as calculated, so that the lines of force are straight up, near the ends 
 they' begin to bend out and they complete themselves round in outside space. 
 The field is very intense inside compared with what it is outside, which if 
 the winding is uniform is merely due to the uncompensated polarities at the 
 ends. 
 
 We can obtain a first approximation to the behaviour of the field at the 
 ends, inside the coil ; for there the field is compounded of the part <&mJ with 
 that due to the end polarity (uncompensated). 'We thus see that if the coil 
 is densely wound the uniformity in the field remains good very close up to 
 the ends, and practically all the lines of force go right up and out at the ends 
 without cutting through the sides. 
 
 A more fundamental aspect can be given to these results. If we assume 
 that the magnetic force is uniform inside the solenoid and very small outside, 
 the integral 
 
 taken along any line of force is approximately equal to 
 
 ZH, 
 where / is the length of the coil and H the constant value of H. 
 
 But this integral is equal to - (current which threads the circuit s}> or 
 
 c 
 
 JN 
 
 , where N is the total number of turns in the coil : thus 
 c 
 
 JN nJ 
 
 H = 4*77 r = 4?7 , 
 
 d c 
 
 where n is the number of turns per unit length. 
 
 427. In this discussion we have confined ourselves to the consideration 
 of the magnetic force. This is however the same as the magnetic flux unless 
 there is iron present. When there is no iron present there is no difference 
 between the magnetic force and magnetic flux or induction. 
 
 Suppose however that the interior of the coil is filled up with an iron 
 core. This core would be magnetised by the field, the intensity of the 
 magnetisation being 
 
 nJ 
 
 K . 4?r , 
 c 
 
 due to the field of the current alone. If we could neglect the field due to 
 itself (which is very improbable in the case of iron) this is all. 
 
 The end polarities would now be - and thus the external field is 
 
 c 
 
 increased K times; the external magnetic induction is increased K times. 
 
426-428] The circular current 377 
 
 Inside the magnetic induction is 
 
 B = (1 + 47T/C) H = yuH, 
 
 and is increased in a much greater ratio. For ordinary iron undei usual 
 circumstances K is large and so /JL is about 12/c*. 
 
 We thus see that iron concentrates the magnetic flux without altering 
 the magnetic force much. This is the essential thing in dynamos, where it 
 is a great flux intensity that is required. 
 
 428. (c) The magnetic field of a circular current. The general nature of 
 the field for the case of a circular current is obvious from the analyses of the 
 preceding cases. The lines of force in any plane through the centre and the 
 axis of the circuit (i.e. the line through the centre perpendicular to the plane) 
 will be exactly similar. For this purpose it is more convenient to use 
 cylindrical polar coordinates (w, 6, z) the first two being taken in the plane 
 of the circuit and the second along its axis. There are then only the two 
 coordinates of the magnetic force, viz. H^ and H z since H 0. The vector 
 potential may thus be taken to have only the one component A and this is 
 given at once by 
 
 2Ja f 71 " cos ddd 
 
 A 9 = - - -j==. 
 
 c j o Vz 2 + ro 2 + a 2 2aw cos 6 
 
 (z, w, 0) being the coordinates of the point at which it is calculated, and 
 a the radius of the circle. 
 
 By the substitution 
 
 / 
 
 = COS -x , 
 
 this integral is transformed at once into the usual form for the complete 
 elliptic integrals of the first and second kind, viz. F and E. We obtain in 
 fact 
 
 4J 
 
 wherein the modulus k of the functions is given by the formula 
 
 4ttC3 
 1.2 
 
 2 _1_ I It \Z ' 
 
 On the axis of the circle and at infinity k = and on the wire itself k = 1 . 
 For values of k <$ 1 we can use the expansions of the elliptic integrals in 
 power series and we then get 
 
 A "^ / a 1.3 Ti ^ 2 ^ z-4 
 
 * Realty 4irH-l. 
 
378 
 
 Some special electromagnetic fields 
 
 [CH. X 
 
 From this function we obtain at once by differentiation the components of 
 the magnetic force intensity in the field 
 
 dz ' 
 
 *-^ (roA > ) ' 
 
 so that the lines of force in any plane through the axis are formed by the 
 lines Ag = const. They are exhibited graphically below in the figure. 
 
 Fig. 75 
 To a first approximation very near to the axis we shall have 
 
 and then also 
 
 (z 2 + a 2 ) 1 ' 
 
 TT> 
 
 so that the magnetic force here has components which to the same order of 
 approximation are given by 
 
 ~ T a 3 
 
 and 
 
 ca (a 2 + z 2 )*' 
 
 c (a 2 + z 2 )^ 
 
428, 429] The tangent galvanometer 379 
 
 The field in the neighbourhood of the centre of the circle is thus to a first 
 approximation normal to the circuit and of strength* 
 
 H = 
 
 ac 
 
 429. This result is made use of for the construction of the instrumentsf 
 for the measurement of currents which are known as galvanometers : the 
 more simple of these arrangements is that known as the tangent galvano- 
 meter, which consists of a circular coil of the wire placed with its plane 
 in the magnetic meridian, i.e. the vertical plane containing the resultant 
 direction of the earth's magnetic field at the point. At the centre of the 
 coil there is a very small magnet which can turn freely about a vertical axis. 
 When the magnet is in equilibrium its axis will lie along the horizontal com- 
 ponent of the magnetic force at the centre of the coil, thus when no current is 
 flowing through the coil the axis of the magnet will be in the plane of the 
 coil. A current flowing through the coil will produce in the neighbourhood 
 of the magnet a magnetic force at right angles to the plane of the coil, pro- 
 portional to the intensity of the current. Let this magnetic force be equal 
 to GJ where of course 
 
 a*.*. 
 
 ac 
 then G is called the galvanometer constant. 
 
 Let H denote the horizontal component of the earth's magnetic force at 
 the centre of the coil. Then the resultant magnetic force at the centre of 
 the coil has a component H in the plane of the coil and a component GJ at 
 right angles to it, hence if 6 is the angle which the resultant magnetic force 
 makes with the plane of the coil 
 
 a GJ 
 
 tan u = ^ff , 
 
 ti 
 
 T # tan 
 so that J = -^. . 
 
 (* 
 
 When the magnet is in equilibrium its axis will be along the direction of 
 the resultant magnetic force, hence the passage of the current will deflect 
 
 ri -I 
 
 the magnet through an angle tan" 1 and a knowledge of this angle enables 
 
 us to determine J. As the current is proportional to the tangent of the angle 
 of deflection, this instrument is called a tangent galvanometer. The larger we 
 can make G the greater will be the sensitiveness of the galvanometer. If 
 
 * When two equal parallel coils are employed the field between them is much more uniform 
 especially if they are at a distance apart equal to their common radius. 
 t Cf. Maxwell, Treatise, n. ch. xv. 
 
380 Some special electromagnetic fields [CH. x 
 
 the galvanometer consists of n terms of wire placed so close together that 
 the distance between any two terms is a very small fraction of the radius, 
 then the field of the current at the centre will be practically 
 
 ac 
 
 so that G = 
 
 ac 
 
 430. The same arrangement can also be used to measure the total 
 quantity of electricity that passes through its coil, provided the electricity 
 passes so quickly that the magnet of the galvanometer has not time 
 appreciably to change its position while the electricity is passing. Let us 
 suppose that, when no current is passing, the axis of the magnet is in the 
 plane of the coil, then if J is the current passing through the plane of the 
 coil, G the galvanometer constant, i.e. the magnetic force at the centre of 
 the coil when unit current passes through it, m the moment of the magnet, 
 the couple on the magnet while the current is passing is 
 
 GJm. 
 
 If the current passes so quickly that the magnet has not time sensibly to 
 depart from the magnetic meridian while the current is flowing, the earth's 
 magnetic force will exert no couple on the magnet. Thus if I is the moment 
 of inertia of the magnet, 6 the angle the axis of the magnet makes with the 
 magnetic meridian, the equation of motion of the magnet during the flow 
 of the current is 
 
 Thus if the magnet start from rest the angular velocity after the small 
 time t is given by 
 
 T dd f* 
 l-ji Gm Jdt. 
 at J 
 
 If the total quantity of electricity which passes through the galvanometer 
 is Q and the angular velocity communicated to the magnet CD, we have 
 therefore 
 
 la) = GmQ. 
 
 This angular velocity communicated almost instantaneously before the 
 magnet alters its position, now makes it swing out of the plane of the coil : 
 if H is the external magnetic force at the centre of the coil, the equation 
 of motion of the magnet is, if there is no retarding force, 
 
 1 Hi* + mH Sin ^ = ' 
 whence I (0a - o> 2 ) + 2mH (1 - cos 6) = 0. 
 
429-432] The linear current system 381 
 
 If a is the angular swing of the magnet then 6 = when 6 = a, so that 
 
 = 2mH (1 - cos 0) 
 
 = 4mH sin 2 - . 
 
 . . a VrnH . I 
 
 i hus Q = 2 sin H 
 
 2 mG 
 
 and thus a measurement of a would determine Q. 
 
 Galvanometers which are used for the purposes of measuring quantities 
 of electricity are called 'ballistic galvanometers.' 
 
 There are other types of magnetic galvanometer which are frequently 
 used but the principles of their action are the same as those described. A 
 common form is that known as the Desprez-d'Arsonval galvanometer : its 
 action is exactly similar to that of the tangent galvanometer but in it the 
 magnets are fixed and the coil movable, and it is the motion of the coil that 
 is observed and measured. 
 
 431. The field of a system of linear conductors the coefficients of self 
 and mutual induction. We have already mentioned the difficulties to be 
 met with in considering a finite current in a very thin wire. At a point 
 close up to the wire the magnetic field, consisting of a local part due to the 
 near element (which can be considered as straight and very long compared 
 with the distance of the point from it, if it is near enough) and that due to 
 the rest can be approximately calculated as of strength 
 
 r being the small distance from the wire of the point. If r is small this is 
 very big. The number of tubes through a small closed curve near the wire 
 can be very big. 
 
 This of course only means that the cross section of the wire is fundamental ; 
 the field in reality depends on the cross section of the wire and then of course 
 on the distribution of the current over it. Thus in order to obtain the exact 
 relations of such fields, particularly as regards the reaction between the 
 field and the current, we must proceed in a more fundamental manner ; we 
 must divide each current up into elementary filaments and estimate the 
 field more closely as due to these filaments laid side by side to form the 
 finite current. 
 
 432. Now consider one of the currents so divided into small current 
 filaments of cross sections df ly df z , ... so that the current in the first is 
 
382 Some special electromagnetic fields [CH. x 
 
 T I -If J fjf 
 
 L f \ that in the _econd 2 f , and so on ..., where /= 1,df. The total 
 
 J J 
 
 current in the wire is 
 
 1 
 
 Now apply Faraday's circuital relation to the closed curve forming the axis 
 
 of the typical current filament ~- and we 'have 
 
 1 dN f T 
 E s ds = - 7? -sr + E ds, 
 
 L at j s 
 
 where E is the electromotive force intensity in the whole field, E e that in the 
 applied external field, and N the flux of induction through the curve taken. 
 But if K is the specific conductivity of the material of the conductor at any 
 place then Ohm's law gives 
 
 - 
 
 7 = K s ' 
 
 [ J' j' 1 dN 
 
 so that fds= E e ds -^ -5- . 
 
 JsKf Js S C dt 
 
 Thus even if the wire is uniform this equation shows that a uniform distribu- 
 tion of the current over the cross section is possible only in very special 
 conditions for the distribution of the impressed electromotive force in the 
 
 field. In all the ordinary cases however fo;r which / E e ds has a value 
 
 J s 
 
 independent of the curve s inside the wire along which it is integrated we 
 can assume the uniform distribution as a first approximation provided the 
 time variations of the field are not too rapid*. Under such circumstances J' 
 is equal to J, the total current in the wire, and thus 
 
 ' Jds f _ 1 dN 
 
 .W^S. *' G~to* 
 
 78 f 
 which can also be written, using SJ = ^ , 
 
 J 
 
 or using 8k for the resistance of the filament in which ihe typical current 
 SJ flows 
 
 whence on integration over the cross section for the average 
 
 n E ldl? 
 
 E - ^' 
 
 * See below page 559. 
 
432, 433] The coefficients of induction 383 
 
 where E e denotes an average value of I E eg ds taken for the various filaments, 
 
 J s 
 
 and N an averaged value of N for the closed circuits formed by these 
 filaments ; k is the resistance of the wire so that 
 
 - 
 
 k 8k' 
 
 This is the more exact form of Faraday's relation as applied to a linear 
 current, when the dimensions of the wire are accounted for. 
 
 433. We now want to calculate N and this is done in the following 
 manner for the particular case under review when the field is that due 
 entirely to a system of currents. We can consider the most general case 
 of n linear conductors in a field in which there are no strongly magnetic 
 bodies about. We then know that N for each circuit must be a linear 
 function of the currents in the various circuits ; or 
 
 N r = a^J^ + a r2 J 2 + ... a rn j n (r = 1, 2, ... n) 
 
 a rl , a r x, ... being geometrical constants independent of the strengths of the 
 currents, and J 1} J 2 , ... these current strengths in the various circuits. 
 
 The coefficients a vv are called the coefficients of self-induction of the 
 respective circuits and the coefficients a m , those of mutual induction. 
 
 A complete determination of the field is however not necessary for a 
 determination of the coefficients a PQ , if there are no magnetic substances 
 about, an assumption we shall henceforth adopt. We know in fact that if 
 A r is the part of the vector potential A arising from the current in the rth 
 conductor 
 
 A- 2A 
 
 and cA r = 
 
 or if the cross section is constant all round 
 
 fr 
 
 wherein ds is the vector element of length of a current filament taken in the 
 conductor and r the distance of the point of integration from the point in 
 the field. 
 
 But then we know that N rs , i.e. the induction through the typical filament 
 of the 5th conductor due to this current in the rth conductor is given by 
 
 N r = ( (A r ds s ), 
 
 J s 
 
384 Some special electromagnetic fields [CH. x 
 
 and then also the average value of N rs for the current is 
 
 C/r/J J 'J r rs 
 
 and thus we see that 
 
 1 [ jf [ if f(ds r >ds s ) 
 ca rs = ^y I dj, I dj r . 
 
 J rj s J J ' ' rs 
 
 434. This result, applies of course equally well for the coefficient of 
 self-induction of the rth circuit on itself, but in the form 
 
 1 { i* ( it> [(ds r .ds r ') 
 
 1 /' ,, [ if, (\ 
 ca rr = T i df r I dfr 
 
 J r J J J 
 
 where each integration is taken once over the various filaments in the wire 
 itself. 
 
 In the case of the mutual induction coefficients, when the cross sections 
 of the wires are small compared with their distance apart we may write the 
 formula in the very approximate form 
 
 (ds r ds s ) 
 
 ' rs 
 
 whence we see that a rs = a sr ; another view of this relation will subsequently 
 appear. 
 
 This is Franz Neumann's formula for a rs *. 
 
 435. The various coefficients of induction between circuits as thus 
 denned of course only have a meaning when applied to closed current 
 circuits; but the formulae obtained show that they are composed of con- 
 tributions from each element of the circuit, and thus calculations of the 
 values of the integrals for unclosed portions of the circuits would enable 
 us to estimate the relative effects of the various parts as regards their con- 
 tribution to the total coefficients under consideration. For example we see 
 at once that the thinnest parts of the wires contribute the greatest part to 
 the integrals for the self-induction of a circuit, so that we might even in 
 some cases consider the effects of these parts as approximately the same as 
 that of the whole circuit. The greatest contribution to the integrals in any 
 case arise from those parts of the current filaments concerned which are 
 nearest together. This suggests working out the integrals for corresponding 
 portions of parallel filaments of finite but ultimately large length. 
 
 * " Die mathematische Gesetze der induzierten elektrischen Strome," Berlin Abhandl. (1845). 
 Cf. also Vorlesungen ilber elektrische Strome. 
 
433-436] The coefficients of induction 385 
 
 Firstly consider the integral 
 
 f(ds r ds s ) 
 
 I V 
 J i rs 
 
 taken along two parallel filaments of length I and distance apart. Its 
 value can easily be calculated as the mutual potential of two uniformly 
 charged rods placed along the same filaments and is 
 
 or, expanding in powers of the ultimately small quantity 
 
 -H fi 2 * 
 = 21 log -T 
 
 L 6 
 
 Now apply this result to the calculation of the mutual induction between 
 two parallel currents in the wires of length I, the cross sections of the wires 
 being small compared with their distance apart, and we get approximately 
 
 ca ra = 
 
 = 21 log. 21 - A log f #i#, + 21 
 
 The first term depends on the shape at a distance but the second term is 
 a purely local part at each point of the wires. The first term in the value 
 of a rs /2l is altered by altering the length of the wires, but the second term 
 remains constant. 
 
 Thus if we have two currents in very long parallel wires there is a local 
 inductance between them which can be reckoned as so much per unit length 
 at the place. There is another part in the inductance which depends on the 
 distant configuration, i.e. how the currents get round to complete their 
 circuits. This latter part is nearly the same all along the finite parts of the 
 wires and we could call it a constant. The distant part is so far off that its 
 influence is the same at any point in the neighbourhood. In many cases 
 it is only the local influence that is of any importance. 
 
 The same argument is still true if the two conductors are not quite parallel 
 or even if they are not quite straight ; if the radius of curvature is very large 
 and if does not change too rapidly along the wires we could reckon the 
 inductance as a constant with another part which is calculated according to 
 the above formula as so much per unit length. 
 
 436. As to the actual determination of the coefficients we notice that if 
 f flog #i#.= /i/2 log R, 
 
 *> J 
 
 L. 25 
 
386 Some special electromagnetic fields [CH. x 
 
 then R is the geometric mean distance between two points one on either 
 
 cross section and then 
 
 (21 \ 
 log ^ - 1 J , 
 
 R of course lies between the greatest and least distances between points on 
 either cross section. 
 
 Now the mean distance R of a point from a cross section/! is given by* 
 
 f 
 
 f 1 \ogR = I log ^d/!, 
 
 and the integral on the right can be interpreted as the logarithmic potential 
 of the cross section coated with electricity of density ZTT. We can therefore 
 use results calculated for such cases. When the cross section is circular 
 (radius a) we have 
 
 (i) r > a external points 
 
 /i log # = /i log r, 
 
 f = Ro, 
 (ii) r < a internal points 
 
 = ae , 
 r being in each case the distance of the point from the centre. 
 
 When the cross section is a ring whose radii are a and b (> a) we have 
 (i) Outside r > b 
 
 log # = log r, 
 
 r = R . 
 (ii) In ring a < r < b 
 
 _ b 2 log b - a 2 log r _ 1 b 2 - r 2 
 
 10 g^0- &2_ a 2 -2P^2' 
 
 (iii) Inside hollow r < a 
 
 _ b 2 log b - a 2 log a 1 
 log^o- 62 _ a2 - 2 - 
 
 From these formulae it follows at once that the mean distance between the 
 points on two rings is equal to the distance between their centres. 
 
 The logarithm of the mean distance R of a circular ring on itself follows 
 by integration of (ii) above, 
 
 a 2 . b 1 3a 2 - 6 2 
 log R = log b - ^- -- log - + - -y= -- 2 , 
 o 2 a 2 a 4 6 2 a 2 
 
 which for a circular surface (a = 0) gives 
 
 ~6e.-*i 
 
 and for a single ring (6 = a) 
 
 R = b. 
 
 * Maxwell, Trans. B. S. (Edinb.), 1871-2. Cf. also Treatise, n. ch. xiii. 
 
436-439] The coefficients of induction 387 
 
 437. We thus conclude that the mutual induction for two straight 
 wires of length I with equal circular sections is 
 
 / 21 \ 
 7Tca rs = 21 (log ^ - 1J 
 
 if d is the distance between the axes of the wires. 
 
 The same argument and results, of course, apply to the self-induction of 
 wire on itself. In the case of the straight wire with a circular section the 
 coefficient of induction is given by 
 
 o; fi 2l 3 
 a rr = 21 \ log - - - 
 
 \AJ TC 
 
 . , r 21 si 
 
 or is of amount log 
 
 & a 4J 
 
 per unit length. 
 
 438. An important case slightly different from the above is provided 
 by the previous example of two long parallel wires, but traversed by the 
 same current in opposite directions, so that they form part of the same 
 circuit. In calculating the coefficient of self-induction for this arrangement 
 we proceed just as before, but each integral with respect to s is taken up 
 one wire and down the other. The result is easily deduced that 
 
 R 2 
 
 ca rr = 21 log -^- , 
 
 wherein R l and R 2 are the mean geometrical distances for the two cross 
 sections in themselves and R 1Z the same quantity for the one relative to the 
 other. In the case when the wires have circular cross sections of radii a, 
 b and are at a distance d apart this is 
 
 The importance of this result is that the local part in the inductance alone 
 appears. The non-local part due to the distant parts of the circuits is zero, 
 or at least negligible. Thus the inductance can be reckoned to a very good 
 approximation as so much per unit length. 
 
 The return circuit localises the field, which is thus far more concentrated 
 when the return circuit exists than when it does not. The distant parts of 
 the current which are practically equivalent to equal and opposite currents 
 very near to one another do not give any effect for the field at a finite 
 distance. The field at any point in such circumstances is entirely local and 
 depends only on the conditions in parts of the conductors just near. 
 
 439. As a final example of great practical importance we must now 
 calculate the mutual inductance of two parallel circular circuits on the same 
 axis. The calculation in this case is best made directly from the formula 
 
 _ 
 12 ~ c 
 
 252 
 
388 Some special electromagnetic fields [CH. x 
 
 We introduce parallel polar coordinate systems (w lt 9-,), (t& 2 , 2 ) in the plane 
 of each circle, whose radii will be taken to be (a lt a z ) and distance apart d : 
 
 we shall have then 
 
 / 12 = a a 2 + 2 2 + d 2 2a 1 a 2 cos (0j 6 2 ), 
 
 (ds^ . ds z ) = a^dd^ . a%d6% cos (6^ 2 ), 
 
 II \A/-\(jLn COS \C^i 2/ ^tr 1 w 9. 
 
 so that ci2 = I I :=^=- , 
 
 ./ o ./ o v ttj 2 + 2 2 + d 2 2a 1 a 2 cos (0 X 2 ) 
 
 and this again is easily shown to transform in terms of the complete elliptic 
 integrals so that 
 
 where the modulus k is given by the equation 
 
 This is the general result which includes all the cases but is of rather a com- 
 plicated form. The particular problem of real practical importance concerns 
 the case when the coils are so placed that the distance between their arcs 
 is small compared with the radius of either circle. In this case k is nearly 
 equal to unity and we might deduce the expansion of the elliptic integrals 
 appropriate to this case ; the following alternative method is given by Maxwell 
 as a more direct application of electrical principles. 
 
 440. Let a, a + b be the radii of the circles and d the distance between 
 their planes, then the shortest distance between their circumferences is 
 
 = p. 
 
 We have to find the magnetic induction through the one circle due to unit 
 current in the other. 
 
 We shall begin by supposing the two circles to be in one plane. Consider 
 a small element 8s of the circle whose radius is a + 6. At a point in the 
 plane of the circle, distant r from the centre of 8s, measured in a direction 
 making an angle 6 with the direction of 8s, the magnetic force due to 8s is 
 perpendicular to the plane and equal to 
 
 sin 0Ss 
 r z ' 
 
 To calculate the surface integral of this force over the space which lies within 
 the circle of radius a we must find the value of 
 
 f"ir/2 m s { n a 
 28s -dddr, 
 
 -U Jr, r 
 
 where r l and r 2 are the roots of the equation 
 
 r 2 -2r(a + b) sin 6 + b 2 + 2ab = 0, 
 
 -.,- . 
 
439-441] The coefficients of induction 389 
 
 When 6 is small compared with a we may put 
 
 /! = 2a sin 6, r z = 6/sin 6, 
 
 and then the integral with respect to r is carried out easily so that the above 
 reduces to 
 
 28s | log (~ sin 2 6 } sin 6d0 
 
 7T 
 
 [f 9/7 QITI^ ft) f)~\% 
 
 cos 6 \2 - log - [ + 2 log tan ~ 
 
 (. o J *_k 
 
 = 28s flog -r 2 J nearly. 
 
 Thus for the whole induction we get 
 
 , 4-n-a /. 8a \ 
 
 >*= rT- 2 j- . 
 
 Since the magnetic force at any point whose distance from a curved wire 
 is small compared with the radius of curvature, is nearly the same as if 
 the wire had been straight, we can calculate the difference between the 
 induction through the circle whose radius is (a + 6) and the circle 2 by the 
 formula ^ _ a ^ _ ^ a { log b _ log r} 
 
 Hence we find the value of the induction between A and a to be 
 
 flog ^ _ 2") 
 V 6 r J 
 
 approximately, provided r, the shortest distance between the circles, is 
 small compared with a. 
 
 The coefficient of self-induction of a coil with n windings of wire, mean 
 radius a and for which R is the geometrical mean distance of the transverse 
 section of the coil from itself is 
 
 L = 
 
 a (log ^ - 2) . 
 
 441. Electromagnetic induction in conducting sheets and solid bodies*. 
 
 We have already seen what an important part is played by metallic sub- 
 stances in the theory and practice of electrostatics and steady currents. We 
 must now turn to discuss some aspects of the perhaps still more important 
 part played by the same substances in electromagnetic theory. All tech- 
 nical electrical machinery is composed largely of metallic substances and 
 the varying electromagnetic fields in their neighbourhood may thus be 
 
 * Maxwell, Treatise, n. ch. xii, Proc. R. 8. 20, p. 160. Cf. also Larmor, Phil. Mag. (Jan. 1884), 
 p. 4; G. H. Bryan, Phil. Mag. 38 (1894), p. 198; 45 (1898), p. 381; T. Levi Civita, Rend. R. 
 Ace. Lincei (5), 11 (1902), pp. 163, 191, 228, Nuovo Cimento (5), 3 (1902), p. 442. 
 
390 Some special electromagnetic fields [OH. x 
 
 considerably modified by their presence. The considerations of the present 
 section will be confined to comparatively slowly varying fields. The extent 
 of this limitation will be discussed later, but we may now assert that when 
 the motions are not too rapid we can neglect altogether the displacement 
 currents in the aether and dielectrics, so that the conduction current is 
 the only one to be reckoned with in the discussion. The electric current 
 density C at any point is then given by 
 
 C = ieE, 
 
 K being the conductivity there and E the electric force in the field. 
 The fundamental equations of .the theory are then 
 
 1 dB 
 
 ITT 
 
 - = curl H, 
 c 
 
 and if we restrict ourselves to homogeneous isotropic media for which 
 
 B = M H, 
 
 we shall have 
 
 div H = - div B - 0, 
 
 P 
 at each point. 
 
 Now from the fundamental equations we deduce that 
 
 KU, dH. , *E , __ 
 
 4rr - = 4rr curl = curl curl H 
 c 2 dt c 
 
 = V 2 H - grad div H 
 - V 2 H 3 
 
 so that H must at each point of the field satisfy the equation 
 
 . 
 
 c dt 
 
 Thus if we have any slowly varying magnetic system specified in a certain 
 way we can determine the whole problem in the following manner. We 
 solve this last fundamental equation for H, choosing the appropriate solutions 
 for the different regions of the field, and then fit them up across the boundary 
 surfaces separating the different media. The conditions to be satisfied at 
 a boundary are : 
 
 (1) The continuity of the normal component of the current or, what is 
 the same thing, the continuity of the tangential component of H. 
 
 (2) The continuity of. the normal component of B. 
 
 These conditions combined with the above equations determine the field 
 of force completely. 
 
441-443] Electromagnetic induction 391 
 
 442. Of course if there are any non-conducting media in the field a 
 slight modification of the argument is possible, for then 
 
 curl H - 0, 
 
 so that the magnetic force is derivable in such regions from an appropriate 
 potential i/r and yzj, _ Q 
 
 as well as V 2 H = 0. 
 
 The physical significance of these equations is now quite clear. The 
 adjustment of the magnetic field in the dielectric media surrounding the 
 conductors can be taken as practically instantaneous, so that the operative 
 fields are sensibly statical, although, as we. shall see later, they are in essence 
 propagated. In the conductors however the field soaks in by diffusion ; its 
 penetration is counteracted by the mobility of the electrons, whose motion, 
 by obeying the force, in so far annuls it by kinetic reaction ; and it does not 
 get very deep even when adjustment is delayed by the friction of the vast 
 number of ions which it starts into motion, and which have to push their 
 way through the crowd of material molecules ; and the phenomena of surface 
 currents thus arises. 
 
 It should thus appear that comparatively thin sheets of metal should act 
 as good screens from electromagnetic action; and this is confirmed by the 
 analysis of the following simple cases which are directly tractable as regards 
 their mathematical form. 
 
 443. (a) The easiest and most famous example of the foregoing 
 principles is afforded by the discussion of the effects of slow variation in the 
 electromagnetic field in the neighbourhood and on one side of a plane con- 
 ducting sheet indefinitely extended in all directions but whose thickness 
 may be neglected. We shall take as the central plane of the sheet the 
 coordinate plane xy and its thickness 8/2 symmetrical on either side of this 
 plane. 
 
 In the space on both sides of the sheet the magnetic force will be derived 
 from a potential ^ on the positive side and i/r 2 on the negative side. Just 
 inside the sheet which will be presumed to be non-magnetic, the magnetic 
 field agrees with that just outside on the same side but the tangential com- 
 ponents change rapidly through the thickness of the sheet. The electric 
 field is on the other hand perfectly continuous throughout the whole field, 
 as there are no typical surface infinities to introduce discontinuity. To 
 obtain the method of variation in the magnetic field we notice that at each 
 point in the interior of the sheet we must have 
 
 9z dy ' 
 
 _ 8H, _ dH x 
 c dx dz ' 
 
392 
 
 Some special electromagnetic fields 
 
 [CH. X 
 
 And then noticing that all functions vary steadily from one point of the sheet 
 to another (in the x, y plane) we get on integration of these equations through 
 the very small thickness of the sheet 
 
 i 
 
 c 
 
 4"7T/<S 
 
 1, = 
 
 -H, 
 
 from which we easily deduce by differentiation 
 
 417x8 
 
 (cur! 2 E) = 
 
 8H X 
 dx 
 
 but 
 
 curL E = 
 
 c dt 
 
 on either side of the sheet and also 
 
 div H - 0, 
 at all points of the field : inserting these values we see that 
 
 8H 2 
 
 a* 
 
 dt 
 
 dt 
 
 These last ' jlations really refer to the respective quantities just inside the 
 sheet on either side : but as the field is continuous across a conducting surface 
 we may just as well interpret them in terms of the same quantities just outside 
 and then in terms of the magnetic potentials there. In other words this 
 equation can be written in the form 
 
 c 2 dt \dz) = c 2 dt \dz) = 8z 2 1 2 ' 
 
 and provides us with surface condition connecting the potentials on either 
 side across the sheet. 
 
 444. Let us now suppose that J/T O is the magnetic potential due to the 
 external or inducing system and J/T/, / 2 ' the magnetic potentials on the 
 positive and negative sides respectively due to the currents induced in the 
 sheet, so that we have 
 
 Then /f with all its differential coefficients are continuous in crossing the 
 sheet : moreover if the sheet is very thin we must have by symmetry 
 
 Ai' (, y, z) = - ifj 2 ' (x, y, - z), 
 
 so that 
 
 ay 
 
443-445] Induction in a plane sheet 393 
 
 Thus if we write k ( ) the above surface condition may be written 
 in the form 
 
 )\ _ d /8 (0o + M\ , 
 
 ~ 
 
 dz 
 
 o as the two equations 
 
 _ 
 
 8z 2 <fe V 8 ~ 
 
 nd 
 
 8z 
 
 445. We suppose that there is no external magnetic system acting on 
 the current sheet. We may then take j/ =' 0. The case then becomes that 
 of a system of electric currents in the sheet left to themselves but acting on 
 one another by their mutual induction, and at the same time losing their 
 energy on account of the resistance of the sheet. The result is expressed by 
 the equation 
 
 8 2 /r/ d 8/>/ 
 s 8z 2 ~dt dz ' 
 
 and the analogous one on the other side of the sheet. On integration with 
 respect to z this gives 
 
 8^'_^ 
 
 8z " dt ' 
 the solution of which is ^ ^ (x> ^ z + ^ 
 
 tfj ' denoting the magnetic potential of the field of the currents at the time 
 i = 0. 
 
 The differential equation thus solved has in reality reference only to the 
 conditions at the boundary of the sheet, but the solution we have obtained 
 satisfies all the conditions of the problem as well as the boundary conditions 
 at z = and will therefore be the proper general solution for all points of 
 the field. 
 
 Hence the value of i/f/ at any point on the positive side of the sheet whose 
 coordinates are (x, y, z) at time t is equal to the original value of the potential 
 at the point (x, ?/, z + kt). 
 
 If therefore a system of currents is excited in a uniform plane sheet of 
 infinite extent and then left to itself, its magnetic effect at any point on the 
 positive side of the sheet will be the same as if the system of currents had been 
 maintained constant in the sheet and the sheet moved away in the direction 
 of a normal from its negative side with the constant velocity k. The diminution 
 of the electromagnetic forces, which arises from a decay of the currents in 
 the real case, is accurately represented by the diminution of the forces on 
 account of the increasing distance in the imaginary case. 
 
394 Some special electromagnetic fields [OH. x 
 
 446. Now let us consider a slightly different type of circumstances. 
 On 'integration of the equation for ^ we find that 
 
 If now we suppose that at first /r = ^2' = Ai' = and that a magnet or 
 electromagnet is suddenly magnetised or brought from an infinite distance 
 so as to change the value of i/r suddenly from zero to / then since the time 
 integral on the right-hand side vanishes with the time, we must have at first 
 at the surface of the sheet ^ ' _j_ ^ _ Q 
 
 We have similarly at the sheet 
 
 </' 2 ' + <Ao = 0. 
 
 Hence the system of currents excited in the sheet by the sudden introduction 
 of the system to which belongs, is such that at the surface of the sheet 
 it exactly neutralises the magnetic effect of the system. 
 
 At the surface of the sheet, therefore, and consequently at all points on 
 the negative side of it, the initial system of currents produces an effect equal 
 and opposite to that of the magnetic system on the positive side. We may 
 express this by saying that the effect of the currents is equivalent to that 
 of an image of the magnetic system, coinciding in position with that system, 
 but opposite as regards the direction of its magnetisation and of its electric 
 currents. Such an image is called a negative image. 
 
 The effect of the currents in the sheet at a point on the positive side of 
 it is equivalent to that of a positive image of the magnetic system on the 
 negative side of the sheet, the lines joining corresponding points of the two 
 systems being bisected at right angles by the sheet. 
 
 The action at a point on either side of the sheet due to the currents in 
 the sheet may therefore be regarded as due to an image of the magnetic 
 system on the side of the sheet opposite to the point, this image being a 
 positive or a negative image according as the point is on the positive or 
 negative side of the sheet. 
 
 If the sheet is of infinite conductivity k = and thus the image will 
 represent the effect of the currents in the sheet at any time. 
 
 In the case of a real sheet the resistance k has some finite value. The 
 image just described will therefore represent the effect of the currents only 
 during the first instant after the sudden introduction of the magnetic system. 
 The currents will immediately begin to decay and the effect of this decay will 
 be accurately represented if we suppose the two images to move from their 
 original positions, in the direction of normals drawn from the sheet, with the 
 constant velocity k. 
 
 447. We are now in a position to investigate the system of currents 
 induced in the sheet by any system M of magnets or electromagnets on the 
 
446-448] Induction in a plane sheet 
 
 one side (positive) of the sheet, the position and strength of which may vary 
 in any manner. 
 
 Let j/( be as before the magnetic potential from which the direct action 
 
 of this system is to be deduced, then -J- $t will be the potential function 
 
 corresponding to the system represented by 7, St. This quantity which is 
 
 the increment of M in the time &t may be regarded as itself representing 
 a magnetic system. If then we suppose that at the time t a positive image 
 
 of the system ,- St is formed on the negative side of the sheet, the magnetic 
 
 action at any point on the positive side of the sheet due to the currents in 
 the sheet excited by the change in M during the first instant after the change, 
 and the image will continue to be equivalent to the currents in the sheet, if 
 as soon as it is formed, it begins to move in the negative direction of z with 
 the constant velocity k. If we suppose that in every successive element of 
 time an image of this kind is formed and that as soon as it is formed it begins 
 to move away from the sheet with velocity k, we shall obtain the conception 
 of a trail of images, the last of which is in process of formation, while all 
 the rest are moving like a rigid body from the sheet with velocity k. 
 
 448. These general results are further illustrated by their application 
 to the well-known case when the currents are excited by a magnetic pole 
 placed at the fixed point (x y z ) on the positive side of the sheet and of 
 strength / (t), an arbitrary function of the time. Then 
 
 = 
 
 V(x-x )*+(y-y Q )*+(z-z )*' 
 
 and the potential on the negative side satisfies the equation 
 
 ~* 
 
 so that - + * 
 
 _ 
 
 dt\dz'dz 2 ~ dt dz ' 
 
 -) 0,' - T - 
 
 3zJ 
 
 -. 
 
 dt 3z V(x - x,Y +(y- </ ) 2 + (2= - * ) 2 
 
 and the solution of this may be written symbolically in the form 
 
 dr, 
 
 V(x - x,)* + (y - y,)> + (* - 
 which from a well-known result in the differential calculus easily transforms to 
 
 /'(r) 
 
 -/(O 
 
 0,' - - f ^ v</ = rfr 
 
 // \2i/ \2i/ 7/ 
 
 j GO v \^oo *^o/ " \?/ "~~ jYo/ " \^ "~~ *^o 
 
 V(x - X Q Y + (y- </ ) 2 + (* - * ) 2 
 
 ^ ,,, . 9 dr 
 
 + W (r) . 
 
 J - * dz V(x - x } z +(y~ */o) 2 +(z-z -kt 
 
396 Some special electromagnetic fields [CH. x 
 
 The first term represents the effect of a pole equal and opposite to the given 
 pole at the point (x y Q z Q ), while the second term shows that the effect of 
 the pole between the times r and T + dr is represented at the time t by a 
 magnet of strength kf(r) dr situated at the point (X Q , y , z + k (t T)). 
 
 The potential on the positive side is found from the relation 
 
 <Ai' (x, y, *)= ~ <A 2 ' (x, y, ~ z)- 
 
 When the pole is in motion the results thus arrived at enable us to plot out 
 the moving images contributed during every element of time ST of the motion ; 
 but these may be obtained more directly as follows. 
 
 449. If instead of a single magnetic pole we are dealing with a 
 magnetic distribution on the positive side of the sheet such that the volume 
 density of magnetisation of the point (x y z ) is F (x y z , t) then we must 
 write F dv for/() in the expressions for the potential and integrate over 
 the whole field on the one side of the sheet. Thus 
 
 (z - Zo) 2 + (y - y ) + (z 
 and 
 
 ~ Zn-*t - r) dr 
 
 z 
 
 V(x - x.Y + (y- y ) 
 
 If now the inducing system consist of a single pole of strength f (t) which 
 moves in any manner so that its coordinates at any time t are (t), r] (t), (t) 
 
 functions of t, we have F 
 
 -TO = u > 
 
 except when x = (T), y = ^ ( T ), z = (T), 
 
 and then F (x y z r) dv = f (r), 
 
 so that ='-^= 
 
 V(x 
 
 and 
 
 {(* - f (r)) 2 + (y - -q (r)) 2 + (2 - (r) - (I - r)) 2 }* 
 This is the expression for the magnetic potential due to the induced currents 
 on the negative side of the sheet, i.e. on the opposite side to the moving 
 pole ; and it is to be noticed that z must be taken to be negative on this side. 
 
 450. (6) As a second example of these principles illustrating further 
 aspects of the fields we consider the currents induced in a thin uniform 
 spherical metallic shell by varying magnetic fields outside the shell. As 
 before, if the variations are not too rapid, we may neglect the displace- 
 ment currents in the dielectrics so that the magnetic fields both inside and 
 outside the shell are derived from potentials tjj 2 and ^ respectively. 
 
448-450] 
 
 Induction in a spherical shell 
 
 397 
 
 To obtain the boundary conditions connecting the potentials across the 
 spherical shell it is best to interpret the general electromagnetic equations 
 in terms of spherical polar coordinates (r, 6, </>) with the pole at the centre of 
 the sphere. This is easily done by reference to the general theorem established 
 in the introduction governing such rotational transformations and the resulting 
 equations are of the type 
 
 477T 2 sin S 9 
 
 477y 2 sin0 
 
 47rr 2 
 
 , . 
 = sin 6 
 
 (H,), 
 
 In the present case there is no electric flux across the shell, C r = and 
 the other two components are proportional to the corresponding components 
 of the electric force 
 
 Substituting these values in the last two equations written down, and 
 integrating across the small thickness of the shell, neglecting the integrals 
 of all certainly continuous functions we find that* 
 
 2 
 
 a 
 
 so that 
 
 ( 9 
 
 c 
 
 4:7TK8tt 
 
 . _ . . . a . 
 
 { e) (r sm *' = 
 
 H, 
 8 
 
 , 9 . ,, 2 
 
 (rHt) (r sm 
 
 Then making use of the first fundamental equation of the second set, viz. 
 47T/cr 2 sin dH 9 , 9 . 
 
 and the conditional equation for H, viz. 
 
 div H = |- (H r ) + ~ (rH*) + I, (r sin H fl ) = 0, 
 or G(p cu 
 
 we see at once that 
 
 A 
 
 9 
 
 I <' 2H '> 
 
 the normal component H r of the magnetic force being continuous through 
 
 * a is the radius of the shell. 
 
398 Some special electromagnetic fields [CH. x 
 
 the sheet. These equations may be interpreted in terms of the magnetic 
 potentials and if we use as before 
 
 , 
 
 K = 
 
 c- 
 
 d ( 9(A,\ d f 9j/ 2 \ & a 9 / 9/ 
 
 we get -j- ( r 2 -^P = T, r ^ = IT 5~ I r ^ 
 
 <ft \ or 7 (ft \ 9r / 2 9r V 9r 
 
 Thus if we can find appropriate potentials fa and j/r 2 inside and outside the 
 sphere and these must of course be regular solutions of the equation 
 
 V 2 j/r = 0, 
 
 which agree with the implied field at a distance and if these values satisfy 
 the boundary conditions implied in the above relations, they represent the 
 complete solution of the problem from which all the circumstances of the 
 field can be deduced. 
 
 451. We may illustrate the problem by the case in which the applied 
 field is practically uniform in the neighbourhood of the sphere but is 
 oscillating with the period p : we may then assume as the applied potential 
 
 = Hr cos 6 e ipt . 
 
 We then try tentative solutions for the total field resulting from the super- 
 position on this field of the field of the currents which it induces in the shell 
 (i) outside the sphere 
 
 3 A! cos 6 t 
 
 and (ii) inside the sphere 
 
 ^2 = "Ao + A 2 r cos 6 e ipt , 
 
 the radius of the sphere being a. These forms satisfy the general conditions 
 of regularity and the outside field reduces to at a great distance. They 
 satisfy the above boundary conditions if 
 
 ip ( H + A 2 ) = ip ( H ZAj) = ak (A A 2 ), 
 
 -H 
 
 i.e. ii **-\ 'o-<^2 ' 
 
 2(1 + 
 
 
 
 If we write 
 
 we have A 9 = 
 
 = -^ , 
 2p 
 
 H 1 + ia 
 
 1-| , O 
 
 ^a 1 + or 
 
 Thus taking real parts only, we find for a field of force of the present type 
 oscillating in intensity according to the harmonic law 
 
 ifj = ~ Hr cos 6 cos pt, 
 the intensity of the field inside the shell is determined by its potential 
 
 cos pt a sin pt 
 
 Aa ~ "Ao + Hr cos v , - , 
 
 X ~|~ ft 
 
450-452] Induction in the spherical shell 399 
 
 or using tan x = P the additional part may be written in the form 
 
 j/r 2 = Hr cos 6 cos x cos (pt + x)- 
 
 This solution represents a new and opposite field with the strength reduced 
 by the factor cos x, and a phase of variation retarded by the fraction - 
 
 of a complete period superposed on the original field. This is of course the 
 part of the field due to the induced currents. 
 
 The external potential in this case is got from the above relations and is 
 
 determined by 
 
 Ha s cos 6 
 
 01 = rO o~~2 COS X COS (pt + ^), 
 
 and the part due to the induced currents is represented by the second term : 
 the induced currents thus produce the same external effect as a simple magnet 
 of moment ifj a s cos ^ cos (^ + ^ 
 
 at the centre of the sphere with its axis pointing in the stable direction along 
 the lines of force of the original field. 
 
 Ska 3c 2 a' 
 
 Now tan x ~^~ 
 
 Now for copper at C. 
 
 ^2 = 1640 > " V 
 
 and if we take the thickness of the shell to be 1 mm. and the radius 3 cm. 
 and a frequency of about 2400 alternations per second we find that half the 
 original field is suppressed inside the shell, and a new field of the same intensity 
 as this half is added whose phase of oscillation is increased by one-sixth of 
 a period. By decreasing the thickness the same effect will be produced by 
 a corresponding slower alternation. 
 
 452. It is important to notice that the electric field associated with the 
 magnetic field and which is determined by the three equations expressing 
 Faraday's rule in these coordinates, viz. 
 
 r 2 sm6dK r 8 . 
 
 3 3H r 
 
 c dt 3r o<f> 
 
 r dlL 8E 
 
 in which of course for the external field 
 
 a 3 cos x cos (pt + 
 H r = Hcosdlcospt -- - 
 
 r 3 
 
400 Some special electromagnetic fields [CH. x 
 
 . f a 3 cos Y cos (pt + Y)} 
 
 whilst H e = -HsmB \cospt- '^ ^k 
 
 and H0 == 0, 
 
 is determined by E r = E e = with however 
 8 . r 2 sine 
 
 so that E _ 
 
 1 grintf / a 3 cos x sin (pt + 
 
 / in _ a 
 \ 
 
 2 c \ r 3 
 
 The lines of electric force are circles round the direction of the lines of force 
 in the original magnetic field. The current flux in the sheet itself will be 
 in the same sense. 
 
 453. The method here adopted for the case of the thin shell is not 
 directly applicable to the more general case when the thickness is not 
 negligible but the circumstances of this more general case can be treated 
 directly from the fundamental equations of the theory* without resorting to 
 the simplifications introduced by the existence of magnetic potentials in 
 the external field. We can illustrate the method by the consideration of 
 a solid sphere in the same uniform field. 
 
 The field being uniform in the neighbourhood of the sphere and in the 
 direction of the polar axis it is obvious from considerations of symmetry that 
 the induced field must be symmetrical about the axis of polar coordinates 
 so that none of the functions depend on the coordinate < : it follows therefore 
 from the fundamental equations which in the general case inside the sphere 
 are of the type 
 
 477/cr 2 sin 6>E r 8 
 = 9 
 
 sin 0E 9 8 . 8H r 
 
 - = ~- (r sm 
 ^ 
 
 ", 
 d<f> 
 
 8H d 
 
 n r 2 sin dH. r 
 
 v 
 
 dt dr v d(f> 
 
 8E 8 
 
 dt " 30 dr 
 
 * Cf. Larmor, I.e. p. 389; Lamb, Proc. L. M. S. (1884), p. 139; C. Niven, Phil. Trans. 172 
 (1882), p. 307; C. S. Whitehead, Phil. Mag. 48 (1899), p. 165; Hertz, Dissertation, Berlin (1880), 
 Oes. Werke, i. p. 37. 
 
452-454] Induction in the solid sphere 401 
 
 E r = E e = H = 0, 
 
 477-o-rEj, _ aH, 3 
 
 ~575 5~ v **#) 
 
 c od or 
 
 r 2 sin rfH r 8 . 
 
 c dt dr 
 If therefore we use 
 
 II = r sin $4,, 
 
 r z sin 6 dH r 811 r sin 6 dH. a 811 
 
 then 3T = -5Z-. -JT = "a- ' 
 
 c r 80 c dt or 
 
 so that from the' first of the last three equations we get 
 
 47rcr cffl _ sinfl d_ I 1 811 ^ _^ 8 2 H 
 c 2 (?f r 2 8^ \sin 6 W 
 or using /z = cos 
 
 8 2 n i /u, 2 8 2 n 
 
 o n " 1 " n *^/lO 
 
 Outside the sphere the only difference is that cr = so that the appropriate 
 equation for II is 
 
 a z n i -^ a 2 n 
 
 dr 2 ~ r z dd 2 = 
 
 454. Now if the applied uniform field is varying in a simple harmonic 
 manner with the period it is easy to verify that in the neighbourhood of 
 the sphere it is defined by the function 
 
 n = ^ (1 - p*) He***. 
 
 a 
 
 We therefore take the hint suggested by this form and try solutions for the 
 internal field of the type 
 
 ^ (1 - fji 2 ) He***, 
 and for the external field 
 
 *P / i , p \ /I _ 
 
 2 
 
 ^! and ^ 2 being functions of r only and ^j in addition tending to zero as 
 r becomes infinite. On substitution of these forms in the above "equations 
 we find that R l satisfies the equation 
 
 26 
 
402 Some special electromagnetic fields [CH. x 
 
 of which the appropriate solution is 
 
 /? _ ^1 
 lf= *l' 
 
 whereas R 2 satisfies the equation 
 
 c* J 
 
 the appropriate solution of which is easily verified to be 
 
 d 
 
 " d (Xr) 
 where A 2 = 
 
 With these forms of solution all the implied conditions of the two fields are 
 satisfied but it remains to connect them up across the boundary. The 
 conditions at the boundary between the conducting sphere and the non- 
 conducting free space surrounding it are simply that the current shall not 
 become infinite at the surface, or what comes to the same thing, the tangential 
 electric and magnetic forces are continuous across this surface. This requires 
 
 OTT 
 
 that both II and -=- are continuous at the surface of the sphere (r = a) : in 
 
 other words 
 
 . / , d \ /sin Aa\ A-, 
 
 A 2 Xa . -^ = - a 2 + - -i , 
 \ d (Aa)/ \ Xa J a 
 
 k 2 /sin Xa\ A l 
 
 (Aa)/ \ Xa J a ' 
 
 455. Now as a general rule Aa is very small so that we may approxi- 
 mate not only to these relations but also to the general forms of the 
 functions for the internal field : we write in fact 
 
 d /sin Xr\ Xr X 3 r 3 
 
 o~ + 
 
 d(Xr) Xr ~ IT " ~30 
 
 and to this approximation it is easily verified that 
 
 A 2 a 2 f 1 - 
 whilst A, - 
 
 and then also the internal field is determined of the type specified by 
 
 n nftfy*** I 1 2\ ( 
 
 Tl i \ r ) 1 
 
 ~2~ "I 1 
 whilst the external field is specified by 
 
 = - _ 
 
 2 10 " 6 ' 
 
454-456] Induction in the solid sphere 403 
 
 Taking real parts only we see that to an applied uniform field determined by 
 
 n = ^(l-M 2 ) H cos ft, 
 
 ft 
 
 the internal field is given by 
 
 whilst the external field has 
 
 p (1 - ) . 
 
 II = - --{r z cospt --*- ismpt\ H. 
 
 2 { 15 . re 2 r ) 
 
 At external points the effect of the sphere is to introduce a new varying 
 field with phase increased by a greater periodj and which in other respects 
 bears the same relation to the old as the electrostatic field produced by an 
 insulated sphere of the same radius bears to the inducing field when the 
 strength of the former is diminished in the ratio 
 
 277770- 
 l5c~ 2 ' 
 
 The general form of the solution obtained for the internal field and the 
 fundamental equation from which it is deduced suggests another way of 
 looking at the problem. The induced distribution gives a field inside the 
 sphere which is determined by the function 
 
 tpq-Vlg^f, A A ^/sinj 
 2 f ^ (Ar) d(Xr)\ (Xr) 
 
 The first term gives a field equal and opposite to the inducing field at each 
 instant ; the following term represents its decay. Since 
 
 (1 "^ 2) He<\ - 
 
 2 ) 
 
 the opposite field can only be due to a surface distribution of currents. Thus 
 if at each instant we suppose a system of currents to start in the superficial 
 layer of the body which neutralises for internal space the effect of the outside 
 changes, the actual state of the body is that produced by these currents 
 soaking into it and decaying. 
 
 456. (c) The principles illustrated in the first section (a) have been 
 applied by Prof. Larmor* to more general problems of steady motions of 
 electrical systems. He starts by considering the case of an electric point 
 charge q moving with velocity v in front of the plane sheet as above. This 
 continuous motion of the charge may be replaced by a series of instantaneous 
 jerks between the positions which it occupies after successive infinitesimal 
 intervals of time S. Each such displacement of its position is equivalent 
 
 * Proc. L. M. S. (2), 8 (1809), p. 1. 
 
 262 
 
404 
 
 Some special electromagnetic fields 
 
 [CH. X 
 
 to the creation of a doublet of moment qv$t. The initial (magnetic) effect 
 of the currents induced in the sheets is to annul the field of the doublet 
 thus instantaneously created, as regards the region beyond the sheets : thus 
 the action of the currents in the sheets is, on either side of it, equivalent 
 initially to that of this doublet with sign changed, supposed to be situated 
 on the other side. The currents thus impulsively produced in succession 
 gradually die away by resistance (K) in the sheet, and in so doing each of 
 them exerts the same influence as if the equivalent doublet on the other 
 side of the sheet moved steadily away with the velocity u. The fields of 
 
 Fig. 76 
 
 force persisting from successive past intervals of time $t are thus expressed 
 as due to the sources graphically represented in the diagram ; in it the image 
 system is inserted on the remote side of the sheet, the conjugate image for 
 the near side being, of course, its reflection in the sheet, with the sign 
 changed if it is magnetic, but not if it is an electric pole. Each step 
 between one receding doublet image and the next is of length ut. At the 
 instant when the moving charge + e has reached the point marked as the 
 end of the path, the aggregate image system consists of the doublets repre- 
 
456, 457] Induction by electrical systems 405 
 
 sented by the short continuous lines, each of which is a persisting travelling 
 reflection of the instantaneous effect of the jerk representing a previous 
 element of the path of + q. 
 
 It is the change, occurring per unit time, in the conformation of this image 
 system of doublets that represents convection of electric charges and so 
 produces magnetic effect. It is in fact only while complementary charges 
 are being separated, so as to create a doublet, that a current element exists, 
 of moment equal to charge multiplied by velocity : when the doublet has 
 once been established and merely continues to exist the magnetic flux around 
 it ceases, though there is a latent accumulation of such flux which persists 
 without further electrodynamic effect and would be undone if the doublet 
 were again absorbed by its poles moving into coincidence. 
 
 Thus the magnetic effect of the induced currents in the sheet is the same 
 as that of the electric flux in the image system ; and the aggregate flux in 
 time 8t amounts to the distribution of vertical doublets, each of moment 
 qu$t, as marked by the + and in the diagram, each receding from the sheet 
 a distance uSt, and in addition' the creation of the fresh isolated image q 
 at the end of the series. It is the limiting configuration of this convection 
 as 8t vanishes with which we are concerned. 
 
 457. If the inducing charge q is at rest this image system amounts to 
 nothing, as of course it must. If the inducing charge is moving directly 
 towards the sheet with velocity v, the image system again assumes a 
 simple form ; for if -each doublet qu$t is replaced by an equivalent but 
 longer one 
 
 - quj(u + v) . (u + v) 8t, 
 
 the doublets will then form a continuous line in which adjacent poles cancel 
 and will thus represent in the aggregate the removal in time 8t of an image 
 charge quf(u + v) to infinite distance or rather to the distance appropriate 
 to the duration of the motion. Thus of the instantaneous image q there 
 remains in position, after subtracting the part thus removed, a residue 
 
 ov 
 
 - moving towards the sheet with velocity v. This reduced geometrical 
 
 u + v 
 
 qv 
 image constitutes in the present case the entire image system, as 
 
 regards magnetic field due to the disturbance excited in the sheet but as 
 regards electric field the effect of the sheet is the same as that of the complete 
 electrostatic image q. For a receding charge the sign of v is of course 
 changed. 
 
 This result may of course be generalised. When any rigid electric system 
 is approaching directly with uniform velocity from a distance towards an 
 infinite sheet of specific resistance K and small thickness 8, the magnetic 
 
406 Some special electromagnetic fields [CH. x 
 
 effect on its own side of the sheet, due to the currents induced in the sheet, 
 is the same as that of the moving optical image of the system with all charges 
 
 (Ka\~ l 
 1 H ) . 
 u) 
 
 Obviously the argument here applied to a moving charge applies equally 
 to a moving magnetic pole ; but in that case the image must be positive 
 instead of negative as here, in order to annul the normal component of the 
 magnetic field at the sheet : thus the last result can be generalised so that 
 the system may include magnets as well as electric charges. As circuital 
 electric currents can be represented by magnetic polarity, the principle also 
 extends so as to include such currents. 
 
 The representation by reduced geometrical images thus applies to any 
 steady electromagnetic system whatever which is approaching directly to the 
 conducting sheet or thin layer; and it is easy to extend it to a changing 
 system. 
 
 458. Now revert to the case of a single electron + q travelling in a 
 curved path, as represented in the diagram. In the differential interval of 
 time, St, next before the time exhibited, each of the vertical doublets in the 
 diagram will have receded a distance u&t, while a new image q will have 
 been instituted opposite to the charge, or rather transferred there from the 
 further end of the image system. To determine- the magnetic field due to con- 
 vection of these parallel doublets, we make use of the vector potential, which 
 
 for a current element qu is parallel to it and equal to . The operation of 
 moving an electric doublet of moment e along its own direction (that of z) 
 with a velocity u, thus produces a vector potential u - (- J . In the present 
 
 case quSt is the aggregate moment of these doublets, all normal to the sheet, 
 which correspond to an element Bt of the time of motion. Thus the aggregate 
 vector potential is parallel to z and is thus 
 
 A f 3 f9 u \ n * f 8 / 9 \ j 
 
 A 2 = u 5- ( - } dt or u z 5- ( -f- }ds. 
 ) oz\r J J dz \vrj ' 
 
 It is thus u z multiplied by the component along z of static attraction (towards 
 the sheet) of a line density equal to f distributed along the (elongated) 
 
 image curve, where v' is the velocity along that curve corresponding to the 
 motion of the .inducing charge. The magnetic force intensity is thus parallel 
 
 to the sheet and equal to h~, ---*, J ; in fact A z is a stream function. 
 
 The whole magnetic influence reflected from the conducting sheet is this 
 magnetic field together with the electric and magnetic. fields of an image 
 
457, 458] Induction liy electrical systems 407 
 
 q existing and travelling in the position of the optical image of the inducing 
 charge : as the latter is the instantaneous shielding image, the former part 
 represents the trail due to imperfect conductance of the sheet, which prevents 
 the currents from adapting themselves instantaneously into the shielding 
 distribution. 
 
 If the inducing charge travels uniformly in an oblique straight line, the 
 distribution of which A 2 is the Newtonian potential is a uniform straight 
 linear one : thus A 2 is expressible in logarithmic form and the solution is at 
 once completed in simple finite terms, which it is needless to express at length. 
 
 In the case of an inducing magnetic pole m, it is the actual poles of the 
 image system that produce the magnetic field, in contrast with the convection 
 which alone operates in the electric case. Thus in addition to the direct 
 instantaneous image + m (and its complement m at the other end of the 
 trail) there is a trail of decaying previous disturbance having a magnetic 
 potential 
 
 a 
 
 3 [ m j , 
 u x- ds , 
 
 dz ' vr 
 
 m 
 
 wherein the integral is the potential of a line density , distributed along 
 the receding image path in the diagram. 
 
 Similar conclusions apply if the sheet itself is in motion, everything being 
 determined by the relative motion of the inducing system with regard to it. 
 
 Thus the specification of the disturbance reflected from the infinite sheet 
 for any convected system containing charges, magnets and currents is formally 
 complete. The conducting sheet will cut off the direct action of the moving 
 system from the other side, replacing it by a decaying trail represented by the 
 receding images here investigated. 
 
CHAPTER XI 
 
 ELECTRODYNAMICS OF LINEAR CURRENTS 
 
 459. The mutual mechanical interaction between a current and a 
 magnetic system. We have so far merely dealt with the electromagnetic 
 aspect of the interaction between a magnetic system and a linear current 
 system. We now turn to an analysis of the mechanical action which is also 
 involved, basing ourselves on the elementary principles of the previous 
 chapter. There are as usual two ways of attacking the problem, the direct 
 or synthetical method and the indirect analytical method based on ordinary 
 mechanical considerations. We start with the first now and analyse the 
 interaction between a single steady linear current J and a single magnetic 
 pole m. 
 
 The potential of the field of the current at any point is 
 
 _ JO _ _ J r cos 8df . 
 c c jf r 2 
 
 the integral being taken over any barrier surface / bounded by the circuit, 
 r denoting the distance of the point from the typical element df and 6 the 
 angle between the normal to df and r. This can be written in the form* 
 
 Cjf 1 r 
 The force in the field is therefore the vector 
 
 which transforms by Stokes's theorem to the vector 
 
 the integral now being taken round the circuit s bounding the surface/. 
 
 This result admits of interpretation as the sum of effects due to each 
 element of the current. If we take a vector Jds in the direction of the element 
 ds of the current and another r from ds to the point in the field, the two form 
 the sides of a triangle of area 
 
 Jds . r sin (r, ds), 
 
 * In these expressions the operator V refers to differentiation at the point on the surface or 
 curve, these being equal but opposite in sign to the same differentiations V p at the field point. 
 
459, 460] The action of a magnetic pole on a current 409 
 
 and thus the part of the force due to ds is 
 
 x^ 
 
 Jds . sin r, ds 
 
 o ' 
 
 r 2 
 
 and is at right angles to both vectors and so as to turn positively round the 
 tangent to the circuit when it is taken positively onwards*. 
 
 
 
 460. The action of the single magnetic pole on the circuit has a 
 resultant equal and opposite to this force. The constituent forces may 
 however be applied at the elements of the circuit ; the reason being that the 
 couples 
 
 Jds sin rds 
 cr 
 
 so introduced are in equilibrium j . Thus the action of a pole m on a current 
 J may be calculated by combining the forces 
 
 1 , m 
 -[Jds, V] 
 c L r 
 
 at each element of the current. 
 
 This result admits of immediate generalisation : in fact for any magnetic 
 system we see that the force on the current element in ds is 
 
 the sum S being taken over all the elementary poles of the system; but 
 
 H = -VS = 
 
 r 
 
 is the magnetic force due to the whole magnetic system at the position of 
 ds. Thus the force on the current in any field is 
 
 \ [Jds, H], 
 c 
 
 it is perpendicular to the plane of H and ds and its actual amount is 
 
 J ' H i rft 
 ds sin Has. 
 
 c 
 
 There is however one important proviso to be placed on the above result. 
 It is restricted to the case when there are no magnetisable substances present 
 in the field; it is in fact only true in that case. The restriction is quite 
 
 * The law of Biot and Savarts, Jour, de Savants, Paris (1821), p. 221. 
 
 t The couple associated with any element of the circuit is completely represented in the 
 vector sense by the chord of the projection of this element on a unit sphere round the field point, 
 this point being the centre of projection. The circuit being closed the vector polygon of the 
 couples is closed so that they are in equilibrium. 
 
410 Electrodynamics of linear currents [OH. xi 
 
 obvious as the argument adopted is based essentially on a distance action 
 
 theory, starting from the potential function of the single current circuit , 
 
 c 
 
 which is not valid if there are magnetic substances about anywhere in the 
 field. The correct result for the most general case however will be shown 
 to be 
 
 - 
 
 C 
 
 the magnetic induction B replacing the magnetic force. This of course is 
 the same as the formula above under the conditions mentioned. 
 
 461. We can now present another aspect of these forces in basing them 
 on the energy principle. To do this we must first estimate the mutual 
 potential energy of the current and the magnetic system. 
 
 The potential energy of a single magnetic pole of strength m in the 
 magnetic field of the current is mift, where J/T is the magnetic potential (if 
 such exists) of the field of the current 
 
 W = mi/j, 
 for if we displace the pole the work we do on it is equal to 
 
 mH^ds = m ~ ds = mSi/f = $W . 
 
 3s 
 
 The work done on the pole is equal to the energy gained by it. 
 
 It is important to notice again that this result is restricted in as far as 
 the magnetic pole m must exist at a place in the magnetic field where there 
 is a potential. Every magnetic field of electric current distributions has not 
 a magnetic potential at each point of the field. Take for instance a finite 
 volume distribution of electric flow (not a current in a linear conductor). 
 We have then if C denotes the current density at any point of the field and 
 H the magnetic force 
 
 H s <fc = ^| C n df, 
 
 the first integral being taken round any closed path s drawn in the field and 
 the latter over a closing barrier across this path. Thus it is only at points 
 not internal to the current distribution that it is possible to choose the path 
 so that 
 
 which is the condition that H should be derived from a potential function. 
 A field of currents has therefore a potential only at points external to the 
 current distribution. This is the general result. 
 
460-463] The potential energy of a current 411 
 
 462. With this restriction the potential energy of the pole due to the 
 current is 
 
 jn 
 
 = m . 
 c 
 
 provided again that there are no magnetisable substances anywhere in the 
 field. But this is equal to* 
 
 W- -^ 
 
 c ' 
 
 where N denotes the number of tubes of force of the field of the pole m which 
 pass through J, or its aperture in the positive direction corresponding to the 
 circulation. 
 
 The result in this form can be immediately generalised to any number of 
 poles for we can add up the tubes of force. This is easily seen for the number 
 of tubes of force through any circuit s is 
 
 1 
 
 taken over any barrier abutting on the circuit ; but H n is obtained by simple 
 addition of the component intensities of the separate fields and so the number 
 of tubes is additive. The total number of tubes is equal to the sum of the 
 numbers belonging to each pole if existing alone. In the general case we 
 have therefore 
 
 c 
 
 where N now denotes the number of lines of force due to the whole magnetic 
 system which thread the circuit. 
 
 N is measured of course in the positive direction relative to the equivalent 
 shell. 
 
 463. In the whole of the above argument it is assumed that it is 
 possible to draw a barrier surface across the circuit entirely in free space 
 at a distance from the magnetic media. The formula must however be 
 generally true if only it is interpreted properly. 
 
 If we calculate N by means of the formula 
 
 N = 
 
 where H is the field strength in the total field the formula would still appear 
 to be correct provided only that it has a definite meaning, as it represents the 
 potential energy of the equivalent shell in the field considered as the work 
 
 * The positive direction through the sheet is the direction of increasing N and O but of 
 decreasing \f/. It is the direction in which the unit pole would be urged by the field. 
 
412 Electrodynamics of linear currents [CH. xi 
 
 done in separating the sheet of positive magnetism from coincidence with 
 the negative. Thus if the formula is to be true generally N must be the 
 same on whatever barrier we count the tubes crossing it. Thus 
 
 f 
 taken over any barrier surface must be the same. This means that 
 
 div H = 0, 
 
 or that H is a stream vector. This result is generally not true except when 
 there is no magnetism about. We know however that if there are lumps of 
 soft iron or other magnetisable substances present, the magnetic induction 
 vector B satisfies this condition, viz. 
 
 div B = 0, 
 
 always; so that it appears probable that we should count the number of 
 tubes of magnetic induction for N rather than the tubes of force. Of course 
 the two are the same in free space. The formula would otherwise be meaning- 
 less. 
 
 464. We can easily show that this surmise is correct. In fact if we 
 draw any one barrier surface / across the circuit which lies entirely in free 
 space and does not therefore cut through any magnetic substances, then the 
 formula is correct if applied to that surface by counting the tubes of 
 magnetic force cutting across it ; but if any other barrier surface /' is drawn 
 we know that 
 
 N = U n df= B n 'df 
 J f if 
 
 because the first integral is in fact the same as 
 
 B.4T- 
 
 If it is not possible to draw one single surface wholly in free space we might 
 remove a small layer of the magnetic substance over one of the surfaces 
 so that it is practically in free space. The small quantity of matter thus 
 removed would not materially affect the field of the rest*. 
 
 We must therefore in general count the induction tubes instead of the 
 force tubes : in fact the formula, if not so interpreted, is not consistent with 
 itself. We shall therefore always quote the formula with N interpreted as 
 the number of tubes of induction. 
 
 465. The potential energy of the magnetic system in the field of the 
 currents is thus 
 
 * It is interesting to notice that the H n in the thin slab of air thus made is practically the 
 same as the component B H of the magnetic induction in the medium. 
 
463-466] The forces on the current 413 
 
 But this potential energy is a mutual affair. It is in fact the mutual energy 
 of the magnetic system and the current, the energy of each relative to the 
 other. If it is a real potential it will determine not only the action of the 
 current on the magnets but also the reaction of the magnets on the current ; 
 for if the current exerts forces on the magnets the magnets must, by reaction, 
 exert forces on the currents. This principle of action and reaction, which in 
 its simplest form is Newton's third law of motion, is in its generalised form 
 a consequence of the existence of a potential energy function. This function 
 contains the coordinates of both bodies, by varying the one we get the action 
 and by varying the other the reaction. 
 
 We can thus determine the forces exerted by the magnetic system on the 
 currents. All we have to do is to vary the corresponding coordinates in the 
 energy function just obtained. The general principle is that due to any 
 displacement of the current the work of the forces exerted on it by the magnets 
 
 is -SW; 
 
 the work done by these forces is equal to the energy exhausted ; if the system 
 does work without the help of an external agency its energy diminishes. 
 The coefficients in this variation of the variations of each coordinate are 
 respectively the forces in those coordinates. We have therefore merely to 
 determine W in terms of suitable coordinates to determine the whole matter, 
 the derivation of the forces being then merely a matter of differentiation. 
 
 466. The usual method of procedure is however to determine the forces 
 exerted by the magnetic system on each element ds of the circuit carrying the 
 current. This can be obtained by supposing that the variation in the con- 
 figuration is produced by the element AB of the wire of length ds moving 
 out into a near parallel position A'B' at a vectorial distance Ss from its 
 original position. The change in W due to the displacement of this bit 
 alone is 
 
 c 
 - J (number of tubes of induction through ABB'A')/c. 
 
 We must now estimate the number of tubes through this small area ABA'B'. 
 It is obviously equal to the product of the component of the magnetic induction 
 perpendicular to the area by the area, and this is 
 
 SN = ds . Ss' Bj, sin foSs' = ([Ss' . ds] B). 
 But if F is this resultant force 
 
 (F8s')= -8TF = - (B . [Ss'ds]) 
 
 
 
 = - ([ds . B] . 8s'), 
 c 
 
414 Electrodynamics of linear currents [OH. xi 
 
 and thus the resultant force on the current element is 
 
 its direction being perpendicular to the directions of B and ds and the actual 
 
 magnitude 
 
 Jds _ . D "> 
 - . B . sm Bds. 
 
 c 
 
 This agrees with the former result deduced from elementary principles. 
 
 The whole force exerted by the magnetic system on the current is thus 
 compounded of all these elementary forces applied at the corresponding 
 elements of the circuit. Their composition is effected in the usual way.' 
 
 467. If we take the displacement of each element of the circuit alone 
 and separate from the rest the force here determined is the force acting on 
 it. But this is the weak point in the argument although the result happens 
 to be right. We cannot really have one element of the current without the 
 rest. The analysis given implies, and is only correctly applicable to, 
 continuous movements which keep the currents complete, i.e. which do not 
 break them. One is not allowed to break the current circuit or to stretch 
 the element ds. But any restriction of this kind excludes a certain type 
 of force. There might in fact be a tension in the conductor which on the 
 above assumption would do no work and would thus escape detection; ST^ 
 could not contain a part depending on such a force and so the expression 
 obtained above for the force on the element would be incomplete. 
 
 A complete investigation of this difficulty necessitates a more detailed 
 knowledge of the structure of the current. We can however state at this 
 point why it is that the result is correct. We have already been led to 
 consider a current in a wire as being made up of moving electrons, or isolated 
 charge elements. We shall prove later that if we have an electron with a 
 charge e moving with a velocity v in a magnetic field it is subject to a 
 mechanical force vectorially represented by 
 
 By simple addition over all the electrons in the current element ds we there- 
 fore obtain at once that since Jds = S. ev the current element Jds is subjected 
 
 to a force - [Jds B], according to the result stated above. There will be 
 
 C 
 
 no tension as the electrons are discrete and free entities. 
 
 The main upshot of all this is that instead of the artificial current element 
 Jds, which involves all the other elements all round and cannot exist by itself, 
 we have got a real movement of electricity, a real thing, out of which currents 
 are built up. The new current is a perfectly independent entity and might 
 be called the rational current element. 
 
466-468] Kinetic or potential energy 415 
 
 It would thus appear that the resultant reaction of the magnetic field on 
 the current can be represented as composed of forces on each element of the 
 current, that on the element ds being the vector 
 
 F = -[<fo.B]. 
 
 c 
 
 The argument however is restricted to the case when no magnetisable 
 substances are present at any point of the field. It is however obvious .that 
 the general result here obtained is true even if such substances were present, 
 for the resultant force on the current element must be the same as if all the 
 magnetism were rigid : it is only in the relations of the total magnetic field 
 to the magnetism induced and its mechanical effect on the masses in which 
 it is induced that the difference enters, and this is susceptible of treatment 
 as in the chapter on magnetism. As far as slow movements of the magnetic 
 masses are concerned the steady current may be regarded as producing a 
 rigid magnetic field. 
 
 468. Kinetic or Potential Energy? We have now seen that a steady 
 current in a linear conductor is surrounded by a steady magnetic field, 
 the maintenance of which even in the presence of other stationary mag- 
 netic systems, requires the expenditure of no work : the work done by 
 the applied electromotive force in driving the current is completely com- 
 pensated by the heat developed in the circuit. We may therefore say that 
 an electric current has a certain amount of energy in virtue of the existence 
 of its field, and by the conservation principle the amount of this energy 
 must be independent of the method adopted in setting up the field. 
 
 In the establishment of the current by the direct application of a finite 
 electromotive force in the current circuit we find that the full current is not 
 immediately produced, it rises gradually to its steady value. We may there- 
 fore well ask what the electromotive force is doing during this time that the 
 opposing resistance is not able to balance it. It is increasing the current 
 of course. But an ordinary force acting on a body in the direction of its 
 motion increases its momentum and communicates kinetic energy to it, or 
 the power of doing work on account of its motion. Thus if, as appears 
 most natural, we assume that a current has motional energy, we may say that 
 the unresisted part of the electromotive force has been employed in increasing 
 the internal motional or kinetic energy of the current. This, of course, implies 
 this definite hypothesis as to the nature of the current, and to this extent is 
 going beyond our previous range of ideas. We shall however find it most 
 convenient to regard it in this way. 
 
 But the internal motional energy of a current is identical with the energy 
 of the magnetic field associated with the current, so that if we regard the 
 energy of a current as of the kinetic type we ought also to regard the energy 
 
416 Electrodynamics of linear currents [OH. xi 
 
 in the magnetic field as kinetic energy. Moreover, as there is no essential 
 difference between the magnetic field of a current and that of a magnet, we 
 must also assume in this case that all magnetic energy is kinetic. This oi 
 course fits in with the Amperean view which regards magnetism as constituted 
 of minute molecular current whirls, but it would require a revision of much 
 of our previo'us work in so far as all quantities designated as potential energy 
 of magnetic nature would have to be classed as kinetic energy and in conse- 
 quence be reckoned with the reverse sign. 
 
 469. In general dynamics it is the Lagrangian function L = T W that 
 determines everything, so that if a part of the energy of any system be counted 
 as kinetic energy, that is, reckoned in T, it must have the opposite sign to 
 what it would have if reckoned in W. At the bottom all potential energy 
 is probably of kinetic origin but if counted as kinetic energy its sign must be 
 reversed. Calling it potential energy merely means that we do not want to 
 trouble ourselves about its actual constitution*. 
 
 The difference of sign in the energies does not of course affect the sign of 
 the forces with. which they are associated. In the most general case of 
 steady systems the Lagrangian analysis shows that the internal forces are 
 determined as the positive gradients of the function L so that for instance 
 the force in the ^-coordinate is 
 
 dL _ dT _ dW 
 
 90 "" 90 30 ' 
 
 Thus if the energy is reckoned as kinetic the force is determined as its positive 
 gradient, whereas if the energy is potential the negative gradient has to be 
 taken. The difference of sign in the gradient thus just balances the difference 
 in the sign of the types of energy. 
 
 470. In spite of the slight confusion which may thus arise we shall 
 follow Faraday's suggestion and treat all magnetic energy as of the kinetic 
 type and we shall henceforth denote it by T. It must however be particu- 
 larly emphasised that we have no definite proof that this energy is kinetic, 
 it is merely a matter of convenient choice so to regard it. 
 
 In our previous work we have always regarded the energy of magnetism 
 as potential energy and all our results were deduced on this basis. Thus if 
 we quote in our future discussions any of our former expressions for magnetic 
 energy care must be exercised to see that they are in all cases quoted with 
 the reversed sign. 
 
 * The process of "ignoring" coordinates in analytical dynamics is practically equivalent to 
 converting the energy in the coordinates from kinetic to potential. Of. Larmor, "On the direct 
 application of the Principle of Least Action to the dynamics of solid and fluid systems," Proc. 
 L. M. S. xv. (1884). 
 
408-471] The energy of the electromagnetic field 417 
 
 The distinction here involved between the two types of energy is very 
 important and must be strongly emphasised, especially in view of the fact 
 that practically all the more prominent writers* on this subject have failed 
 to appreciate it. As a result the treatment of the present subject offered 
 in the usual text-books is hopelessly confused; in several cases even a per- 
 fectly legitimate analytical procedure which properly leads to opposite signs 
 in the two cases has either been rejected as inconsistent or explained away by 
 further argument. 
 
 471. The energy of the electromagnetic field. A system of linear con- 
 ducting circuits carrying currents and existing apart from other permanent 
 magnetic systems could be said to possess a certain amount of energy 
 which would be equal to the total work required to establish the currents 
 in any order. The total amount of this work is, in fact, the measure of 
 the electromagnetic energy of the currents relative to the state in which 
 the currents are all zero and is what we now want to determine. It would 
 be transformed into other forms of energy if the currents were destroyed. 
 
 A distance action theory regards the electromagnetic conditions of the 
 system as characteristic properties of the currents in the circuits and thus the 
 currents are the most essential quantities required for the specification of 
 the system. We therefore require a definition of the electromagnetic energy 
 which makes it depend on the currents alone. This is readily obtained in 
 the following manner. 
 
 The discussion refers to a system of linear conductors carrying currents 
 Jj, J 2 , ... J n , existing alone in a field, which may however be occupied by 
 magnetisable substance and permanent magnets. Bach of these currents 
 has a magnetic field of its own and all the fields superposed form the 
 total magnetic field of the system. Now let N 1} N 2) ... N n be the 
 numbers of lines of magnetic induction of the total magnetic field which 
 thread the various current circuits respectively. 
 
 We first enquire as to the amount of work required Irom the driving 
 batteries to increase each of the currents by a small differential amount. 
 The work required to increase the current J by the amount 8J X is equal to 
 
 jWi, 
 
 where JV/ is the number of tubes of induction of the total field which are 
 ultimately enclosed by SJ^; but this is exactly the same as the number 
 enclosed by J l or 2V/ = Nj_ ; so that the work done is equal to 
 
 * Notable exceptions are Maxwell and Larmor. 
 
 27 
 
418 Electrodynamics of linear currents [OH. xi 
 
 Thus we see that the total work necessary to increase the currents in the 
 circuits by respective amounts SJ l3 SJ 2 , ^ is 
 
 ST = - (NjBJj, + NJ* + ... + N n 8J n ).. 
 c 
 
 * 
 
 This is the fundamental characteristic equation of energy of the system and 
 must be integrated somehow. 
 
 472. The test of integrability, i.e. the criterion for the existence of the 
 energy function T is that for all values of r and s 
 
 dJ s dJ 
 and we have then also 
 
 
 As a matter of fact we know that in the simplest cases the inductions 
 N!, N z , ... N n through the circuits must be linear functions of the currents 
 in them, for if we double the currents, we double the field intensity at 
 every point and therefore also the density of the tubes of induction*. 
 Thus T is a quadratic function of these quantities and by Euler's theorem 
 
 dT dT 
 
 dJ^'^ + W* ** 
 
 DHL *\ -r = ~ CtiC.. 
 
 cJi C 
 
 so that T = H- ZN . J. 
 
 2c 
 
 If we now write N r = (a rl J l + a r2 J 2 + a rS J s + ), 
 
 then we see from the above that 
 
 a rs = a sr , r=l,...,n, s = I, ..., n, 
 
 and thus 
 
 = (flue/! 2 + 22^2 2 + ... + 2a 12 J ie / 2 + 2a 23 J 2 7 3 +...), 
 
 and this is the total energy stored up in the form of electromagnetic energy 
 in the system. It is the electrokinetic energy of the electrodynamic field, 
 to use Maxwell's expression. 
 
 The coefficients a u , 22J ... a rr ... are the coefficients of self-induction of 
 the respective circuits, and the coefficients a 12 ... a rs are the coefficients of 
 mutual induction. They have already been introduced in a previous con- 
 nection. They are numerical coefficients which are functions of the forms 
 and relative positions of the circuits : a rr is the number of tubes of induction 
 
 * A certain restriction on the law of induction of the magnetisable substances in the field is 
 hereby implied. Permanent magnets are presumed to be absent. 
 
471-473] The energy of the electromagnetic field 419 
 
 that thread the rih circuit when unit current flows in that circuit and none 
 in all the others, it is therefore a function of the one circuit only ; a rs is the 
 number of tubes of induction due to the field of a unit current in the rth 
 circuit that thread the sth. 
 
 If we interpret the energy as kinetic energy these coefficients appear as 
 the inertia coefficients, they determine the electrical inertia of the circuits. 
 The currents are then the generalised velocities in the Lagrangian sense. 
 At least it looks like this and the impulse is to try it and see if it gives the 
 right results. We shall give a fuller discussion of the subject later. 
 
 473. Following Maxwell we have however continually attempted to 
 interpret all our results in the language of a medium-action theory. On 
 such a theory the motion involved in the electric current is not necessarily 
 confined to the conductors but may take place in the whole of the space 
 surrounding them, so that the current itself is merely the manifestation of 
 the varying condition in the medium occupying that space. The electro- 
 magnetic and mechanical relations of the field of a system' of steady currents 
 should therefore be expressible in terms of the medium between the circuits. 
 But for all practical purposes the only change which has taken place in the 
 surrounding field during the starting of the currents is that involved in the 
 statement that a magnetic field has been established. We should therefore 
 be able to express all the mechanical relations of the theory in terms of the 
 magnetic field vectors instead of the currents as above. 
 
 We have obtained the total energy of a system of linear currents expressed 
 generally in the form 
 
 T = - 
 
 c o 
 
 1 f J 
 
 = - S SJ 
 c J o J f 
 
 where S refers to the separate barrier surfaces / closing the various current 
 circuits. But if </> is the magnetic potential of the field at the typical stage 
 of its establishment 
 
 A J 
 
 4?T = 
 
 ds = 
 
 where s denotes any closed path which encircles once the first current only 
 and </>/!, </>/., denote the values of </> at two infinitely near points on either 
 side of the barrier /, the first being obtained on starting along s where it 
 cuts through / and the second on arriving back at /. Thus 
 
 272 
 
420 Electrodynamics of linear currents [CH. xi 
 
 or if we reckon the two sides of the barrier as different sides of the same 
 closed surface we have that 
 
 fJ o 
 
 Since each of the currents has a barrier surface the whole space is a singly 
 connected region with the two sides of each barrier and a surface at infinity 
 as boundary. We may therefore apply Green's theorem to the region and 
 convert the surface integral into the volume integral 
 
 47rT= - || (B, V)8(f>di}, 
 
 J J o 
 
 the infinite surface contributing nothing in all regular cases. Since div B = 
 and div </> = SH this is simply 
 
 = - dv 
 
 (B, V) </> 
 o 
 
 = + dv I (BSH), 
 J o 
 
 and is thus expressed completely in terms of the field vectors B and H. 
 
 The total electromagnetic energy of the currents may thus be regarded 
 as belonging to the medium occupying the field between, being distributed 
 through that medium with a density 
 
 * T-T 
 
 (BdH.). 
 o 
 
 This is the result necessitated by Maxwell's theory and is precisely the same 
 as that deduced from the special assumption in the more general case in 
 Chapter XIV. 
 
 474. If no magnetic substances are present in the field the density of the 
 energy which now must be associated with the aether, is simply 
 
 1 ~ B 1 
 
 BdB = B 2 , 
 
 47T. o OTT 
 
 but if there are magnetic media about this result is modified. 
 
 We include for generality a distribution of rigid magnetism throughout 
 the field of intensity I at any point and if we imagine this to be built up 
 with the current system an additional amount of work is done equal to* 
 
 f f j 
 
 + dv (HdI ), 
 ) . o 
 
 so that the total work done is 
 
 dv 
 
 o ' 'J 
 
 * It is now kinetic energy. 
 
473,474] The energy of the electromagnetic field 421 
 
 But if there are magnetisable media in the field and if at any stage of the 
 process of building up the system the intensity of induced polarity is I then 
 
 B - H + 477 (I + I ) 
 so that 
 
 T ={dv\ '- f(B, dE - 4-rrdI - 47rtZIo) + f /0 (HdI ) 1 
 L^ 77 " o J 
 
 - (EdE) - I (H + 47rl + 477l , dl 
 
 tTT.O .' 
 
 - |<H<Mrf{B 2 -1677 2 (I+I ) 2 }- 
 
 = - (B 2 - 167T 2 (I + I ) 2 } dv- dv (Hdl). 
 
 077V . Jo 
 
 The last term in this expression represents with sign changed the increase 
 of the intrinsic potential energy of the polarisations in the magnetic media. 
 As it stands it therefore represents the increase, consequent on the induction 
 of the polarisation, of the intrinsic elastic or motional energy of the magnetic 
 media, regarded now as kinetic energy: it is stored up in the media as 
 energy of an effectively non-magnetic nature and is mechanically unavailable. 
 The first term therefore represents the total .energy of purely magnetic nature 
 and of kinetic type stored up in the system ; on a tentative theory we could 
 regard this energy as distributed throughout the field with the density 
 
 ~ {B 2 - 167T 2 (I + I ) 2 } 
 
 at any place. Of this energy the part 
 
 - 27T (I + I ) 2 
 
 represents intrinsic energy in the magnetic media of a purely local or con- 
 stitutive nature, depending as it does only on the conditions in the medium 
 at any point. The remainder, being a distribution with density 
 
 A B 2 
 
 877*' 
 
 is then to be associated solely with the magnetic conditions in the aether. 
 
 These results are perfectly consistent 'with those deduced on the previous 
 statical basis, if allowance be made for the difference of sign in all the terms 
 arising from the different and more general conception as to the type of 
 energy with which we are now dealing*. 
 
 * This point must again be strongly emphasised, as it is the cause of all the confusion in the 
 subject. It is the origin of the inconsistency mentioned in connection with the usual treatment 
 of the energy in static fields, for that treatment gives the same sign to a quantity which in the 
 statical case is treated as potential energy and in the dynamical case as kinetic energy. Most 
 authors seem to find satisfaction in this apparent agreement of sign and the real discrepancy has 
 
422 Electrodynamics of linear currents [CH. xi 
 
 475. Of the total energy of purely magnetic nature stored up in the 
 field the part 
 
 I dv (I flfH) 
 
 J 
 
 is concerned mainly with the rigidly magnetised masses, being in fact the 
 kinetic force function of their mechanical interactions and of the interactions 
 of the currents and induced magnetism with them. Of the remainder the 
 part 
 
 [ dv.\ (I<ZH) 
 
 
 
 is concerned in a similar manner with the induced polarity, being the kinetic 
 force functions of the mechanical actions on the induced magnets. 
 
 The remainder or 
 1 
 
 [B 2 - 167T 2 (I + I ) 2 ] dv - dv (I + I , 
 
 07T J . J 
 
 is therefore concerned solely with the currents and is the force function of 
 their mechanical interactions and the interactions of the magnets with them. 
 
 Since B = H + 4rr (I + I ), 
 
 this part reduces to 
 
 [ B 
 dv (BWB), 
 
 J Q 
 
 and may therefore be regarded as distributed throughout the field with the 
 density 
 
 at each place. If as is generally the case there are no rigid magnets about 
 in the field this latter part of the total energy is the only part that is mechani- 
 cally available, provided it is not prevented from so being by friction al forces 
 tending to degrade it. The part of the energy corresponding to the induced 
 polarisations and arising from their reaction with the currents is in some 
 
 rarely been noticed. The same inconsistency is also involved in the assumption of similarity 
 between the electrostatic and magneto-static fields, which is frequently adopted as a reason 
 for the analogy between the electric potential energy and the magnetic kinetic energy. It also 
 enables Jeans and Richardson (Electron Theory of Matter, p. 103) to talk of "proofs"' of the 
 kinetic nature of magnetic energy i 
 
 Cf. J. J. Thomson, Elements of Electricity and Magnetism, pp. 267, 363 ; J. H. Jeans, Electricity 
 and Magnetism, pp. 403, 432, 433; M. Abraham, Theorie der Elektrizitcit, i. pp. 218, 253; Cohn, 
 Das elektromagnetische Feld, pp. 202, 281, 298; Schaeffer, Maxwellsche Theorie, p. 61; Classen, 
 Theorie der Elektrizitdt und des Magnetismus, II; p. 48. 
 
 Maxwell's own treatment (Treatise, n. ch. xi.) is perfectly clear and consistent. Cohn suggests 
 the present development of it but finds a difficulty in the opposite signs for the potential and 
 kinetic energies ! Cf. also Livens, " On the mechanical relations of the energy of magnetisation,'* 
 Proc. K. S. 93 (A), p. 20 (1916). 
 
475, 476] The energy of the electromagnetic field 423 
 
 respects mechanically available, but only in so far as the presence of the 
 magnetic masses increases the available energy associated with the currents 
 giving rise to the field. 
 
 A similar transformation to that employed above in the reverse manner 
 shows that 
 
 ~ dv * (KdE) = ^1, I" JdN, 
 
 where 2 denotes a sum relative to the linear circuits. This exhibits the 
 energy in its relation to the currents in a more vivid form. 
 
 476. If the induction of the polarisation I follows a linear isotropic law 
 so that 
 
 I - /*'H, 
 
 and B )U.H 
 
 where \i = 1 + 
 
 then it is easy to verify in a manner already elaborated in detail for the 
 statical case that the total energy in the magnetic field, including both the 
 energy of the aethereal field and the intrinsic energy of the material polarisa- 
 tions, can in the most general case be expressed in the form 
 
 ^V-lW^ 
 
 or in the mechanically equivalent form 
 
 1 fB 2 
 - dv. 
 
 As there is now no leakage by hysteresis this expression for the energy of 
 the field also determines the mechanically available energy of the system. 
 
 Thus in this special case the energy of the system in both senses may be 
 regarded as distributed throughout the field with the density 
 
 In the parts of the field where there is no rigid magnetism we have 
 
 B = )iiH, 
 so that the energy density there is also expressed by 
 
 These expressions now represent kinetic energy so that there is no discrepancy 
 in their having the opposite sign to the similar expressions fqr the potential 
 energy in the statical theory. 
 
424 Electrodynamics of linear currents [CH. xi 
 
 477. The most definite and consistent way to treat magnetism and its 
 energy is to consider it as consisting in molecular current whirls* ; so that 
 in magnetic media we have the ordinary finite currents, combined with the 
 molecular currents so numerous and irregularly orientated that we can only 
 average them up into so much polarisation per unit volume of the space they 
 occupy. If there were no such molecular currents, the magnetic force H in 
 the a'ether would in steady fields be derived from a potential cyclic only with 
 regard to the definite number of the ordinary circuits. But when magnetism 
 is present this potential is cyclic also with regard to the indefinitely great 
 number of molecular circuits. The line integral .of magnetic force round 
 
 each circuit is - (2U + 2J'), where /' refers to the practically continuous 
 c 
 
 distribution of magnetic molecular currents that the circuit threads. This 
 latter vanishes when these currents are not orientated with some kind of 
 regularity. If we extend the integral from a single line to an average across 
 a filament or tube of uniform cross section df, with that line as axis, by 
 multiplication by df, we obtain readily the formula 
 
 df (u s ds = 47T *- ZJ + 47T fl a dsdf, 
 
 c 
 
 in which Idv represents the magnetisation in volume dv . Thus after trans- 
 position of the last 'term and removal of the factor Sf after the average 
 has now been taken we obtain 
 
 [(H - 477-1 . Ss) = 2J. 
 
 v ^ 
 
 In other words the new vector (H 47rl) is derived from a potential cyclic 
 in the usual manner with regard to the ordinary current circuits alone. 
 
 It thus appears that H must now represent the induction vector of our 
 ordinary theory and so will hereinafter be denoted by B as usual. The new 
 vector which has a potential cyclic with respect to the finite currents only, 
 represents the 'force' and will hereafter be denoted by H, whose significance 
 is now changed. The induction B has not necessarily a potential but is by 
 constitution of the free aether always circuital; that is it satisfies the con- 
 dition of streaming flow 
 
 div B - 0. 
 
 The expression for the energy now includes terms 
 
 T 
 
 NdJ, 
 
 for the ordinary currents .7; this transforms as usual into 
 
 * Cf. Larmor, Proc. R.S.I I (1903) ; "On the mechanical and thermal relations of the energy 
 of magnetisation." 
 
477] The energy of magnetisation 425 
 
 over both faces of each barrier, which by Green's theorem is equal to 
 
 ^[<tof*(BdH) (i) 
 
 extended throughout all space. But there are also terms 
 
 N'dJ', 
 
 for the molecular currents; now taking N' to be the cross section of the 
 circuit multiplied by the component of the average induction normal to its 
 plane, and remembering that J' multiplied by this cross section is the mag- 
 netic moment of this molecular current, it appears that dJ'N' is equal to 
 the magnetic induction multiplied by the component of the magnetic moment 
 
 f J/ 
 in its direction and therefore S | N'dJ' is equal to 
 
 
 
 fdv^CBdl). 
 
 J J Q 
 
 Thus the magnetic circuits add to the energy the amount 
 
 (Off) ...(ii), 
 
 together with 2rr I I 2 dv (iii). 
 
 / 
 
 Thus the total electromagnetic energy in the field is 
 
 1 
 
 STT. 
 
 a result which differs from our previous estimate by the purely local part 
 represented by (iii) : of course when it is remembered that in our previous 
 theories the local forces and their associated energies were always presumed 
 to be negligible as regards their observable mechanical effects and were 
 therefore left entirely out of account, it is not surprising that this difference 
 occurs. In fact using the same arguments as there employed we should 
 neglect this local part and the results would then agree perfectly. 
 
 It may however be noticed that although this discussion provides us with 
 the correct allotment of the total energy in reference to its association with 
 the different parts of the system it does not indicate that the part of it 
 
 dv | (HtZI) 
 
 o 
 
 arises mostly at the expense of the internal store of kinetic energy in the 
 medium, being in fact merely the part of this energy which is temporarily 
 classed as magnetic, and is not effectively available, because the magnetism 
 cannot be annulled quickly enough to develope any considerable available 
 energy by induction. 
 
426 Electrodynamics of linear currents [OH. xi 
 
 478. Equilibrium theories in dynamics. Before proceeding further with 
 the applications of our ordinary dynamical analogies to these electromag- 
 netic phenomena it will be as well to say a few words about the limitations 
 to the theory which are necessarily implied by our analysis. 
 
 In the study of the electrodynamic relations of the field of a system of 
 currents we always attempt to interpret everything in terms of the current 
 strengths and the positions of the circuits, these being the palpable coordinates 
 of the system which are directly accessible to measurement. This of course 
 implies, not only that there is such a definite quantity as a current in each 
 circuit (the same all round that circuit) but also that the relations of the 
 surrounding electromagnetic fields can always be interpreted correctly in 
 terms of these currents. In the work so far we have treated the currents 
 as constant and the circuits as fixed and therefore the implied conditions 
 are evidently satisfied; but what if the current strengths or the positions 
 of the circuits are varying in any manner? 
 
 In the usual treatment of ordinary dynamics of rigid bodies we express 
 the energy functions, and dynamical relations generally, in terms solely of 
 the palpable coordinates of the body and the velocities in them. This 
 however implies certain restrictions on the nature of the body and its motion 
 which are in practice never actually realised. In fact we know that when 
 we hit a body so as to start it off suddenly, the actual observed motion is 
 set up by a most complicated process. Firstly a wave of compression is 
 sent off through the body from the point of application of the blow, and the 
 further parts of the body take up their motion only when this wave reaches 
 them and imparts the necessary momentum and energy. The uniform 
 motion of the body is not therefore attained until this wave disturbance 
 has been smoothed out over the whole body; in most cases of practical 
 importance however the time occupied by this process is so exceedingly 
 small that we can neglect it altogether and regard the observed conditions 
 as instantaneously 'established, and this is implied in the term 'rigid body.' 
 Of course there is a slight dissipation of the energy of the wave disturbance, 
 and it is not all transformed into the energy of the observable motion, but 
 some goes into heat ; but this quantity is usually so small that we are fully 
 justified in treating it as non-existent or, again, the body as rigid. Of course 
 there is the theory at the other end where we investigate these waves of 
 compression, but the two cases are extremes and as a rule we need not mix 
 the one with the other. 
 
 479. This is the usual sense of the term ' equilibrium theory ' as used in 
 dynamical and related discussions. The state of the motion actually 
 observed adjusts itself so quickly that at each instant it is practically in 
 equilibrium under the conditions pertaining at that instant, and the process 
 of the establishment of this condition at each instant can be ignored. 
 
478-480] Equilibrium theories 427 
 
 A simple mechanical analogy will perhaps help to elucidate the matter. 
 Consider the extension of a spring by a given weight. If we add the 
 weight gradually bit by bit the spring is at each instant practically in an 
 equilibrium condition. If however we hang the whole weight on suddenly 
 (i.e. in general terms, suddenly produce a finite change of the steady con- 
 ditions) the spring would first extend twice as far as in the above case and 
 would then oscillate up and down before settling into the same equilibrium 
 position. In this example if the forces are applied suddenly the phenomena 
 of elastic inertia come in and the problem is complicated ; but if they are 
 applied gradually the conditions at each instant are statical. 
 
 There is just the same point in the usual electrodynamic theories. We 
 have already seen how Maxwell was induced to his theoretical explanation 
 of all observed electrodynamic actions as being transmitted through and by 
 that hypothetical medium, the aether, which occupies the whole of space; 
 an essential point being that the actions are transmitted by this medium 
 with a finite velocity. It therefore follows that any disturbance produced 
 in the conditions at any part of an electromagnetic field will smooth itself 
 over the whole field by sending out an electromagnetic wave through the 
 field in a manner exactly analogous to that described above. If however 
 the variations arbitrarily produced in the conditions at any point of an electro- 
 magnetic field are slow enough, we can consider the adjustment of the corre- 
 sponding new conditions throughout the whole of the field to take place so 
 quickly as to be "almost instantaneous, so that the whole field is at each 
 instant practically in the equilibrium condition under the circumstances 
 pertaining at that instant. 
 
 Thus if we confine ourselves to the discussion of phenomena the period 
 of any change of which is large compared with the time taken by radiation 
 to get across the system and adjust any new conditions we may adopt an 
 equilibrium theory and use the results obtained for absolutely stationary 
 states as applicable at each instant to the slowly varying motion (quasi- 
 stationary state). That is, in the present case if the changes in the values 
 of the current strengths or the positions of the circuits are very small and 
 insignificant in the time taken to adjust an equilibrium condition throughout 
 the field, then we may treat the currents in each circuit as definite quantities, 
 i.e. the same all round the circuit, and the field at each instant can be deter- 
 mined in terms of their instantaneous values. 
 
 480. The velocity of adjustment of electrical conditions in air turns 
 out to be 3 x 10 10 cms. per second, so that for any ordinary sized system 
 the condition that changes in its configuration should be small in the time 
 occupied by radiation in getting across the system hardly restricts the 
 application of the results obtained except in the case of very rapid 
 oscillations, of the type in fact used to start electric waves. 
 
428 Electrodynamics of linear currents [CH. xi 
 
 In such cases we must proceed as in dynamics, to investigate the 
 process of the adjustment of electrical conditions by this wave propaga- 
 tion. This however belongs to another chapter in the general theory and 
 will not concern us at present. 
 
 Thus with the restrictions implied by the assumption of an equilibrium 
 theory we can discuss the relations of our current system in terms of the 
 currents in the circuits and the ordinary geometrical coordinates. The 
 formula obtained for the electromagnetic energy, in either form, for the 
 steady state will then apply similarly as representing the instantaneous value 
 of that quantity in the quasi-stationary system ; and the mechanical relations 
 already deduced for the steady conditions from elementary principles will 
 still be available for slowly varying ones. 
 
 481. Deduction of Faraday's law by the energy principle. We have 
 already spoken in Chapter IX of induction currents caused by variations in 
 the magnetic field surrounding conductors and Faraday's law defining them 
 more precisely was quoted as an experimentally proved fact. 
 
 The existence of these currents and their mathematical relations can 
 however easily be deduced by energy considerations from the results obtained 
 regarding the electromagnetic actions discovered by Oersted ; Helmholtz * 
 and Kelvin f were the first to indicate the possibility of this deduction, ten 
 years after the actual discovery of the currents by Faraday. 
 
 Helmholtz takes the case of a conducting circuit of resistance R, in which 
 an electromotive force E, arising from voltaic action exists. The current in 
 this circuit at any instant is J. Suppose now that this current by means 
 of its magnetic field is moving a magnet about so that it is doing work on 
 it J. The work done on the magnet may then be calculated by considering 
 the motion of the magnet as taking place by infinitely small displacements 
 so that at each instant the ponderomotive forces are determined by the 
 field and the current strength at that instant. It then follows that the 
 work of these forces during one of the small displacements may be reckoned 
 as J(ISH) dv, where I is the polarisation intensity at the typical point of the 
 matter; 8H the increase in the magnetic force at this point due to its dis- 
 placement in the field, and the integral is taken over the whole field. Thus 
 the work of the electromagnetic actions during the small instant 8t may be 
 reckoned as 
 
 Ill-} 
 
 !( dt) 
 
 dv. 
 
 But during this time the following additional changes have taken place; 
 (i) an amount of heat has been generated in the circuit equal to J 2 R8t; 
 
 * Vber die Erhaltung der Kraft (1847). 
 
 t Trans. Brit. Ass. (1848); Phil. Mag. Dec. (1851); 
 
 J The motions however all being so slow that an equilibrium theory can be adopted. 
 
480-482] Faraday's law 429 
 
 (ii) an amount of energy equal to the work of the electromotive forces in the 
 circuit in driving the current, viz. EJSt, has been added to the system; 
 (iii) the internal electromagnetic energy of the current has been altered by 
 
 .S*~ 
 at 
 
 The principle of energy thus requires that 
 
 B - - ~ 
 
 at J \ at) \ at 
 
 We have of course assumed that the magnetism of the magnet moved 
 about is rigid, i.e. the magnet must not be capable of absorbing or storing 
 internal energy from the electromagnetic field. There are difficulties about 
 this assumption but it is sufficient for the present theoretical purposes. We 
 thus see that 
 
 The integral in this last equation can as usual be reduced to the form which 
 expresses it as 
 
 1 dfl_ 
 
 c J dt ' 
 
 where N represents the number of lines of induction through the circuit 
 due to the total field comprised of the field of the magnets superposed on 
 that of the current. Thus we have 
 
 J*-*- 1 -^. 
 
 c dt 
 
 In other words in addition to the electromotive force of the battery 
 
 there is an additional electromotive force on the circuit equal to : - 
 
 c at 
 
 Thus whenever the total number of lines of induction enclosed by the circuit 
 changes there is an induced electromotive force created in the circuit and 
 the amount of it is proportional to the rate of diminution of the total 
 induction through this circuit. This is precisely Faraday's rule. 
 
 Another important point illustrated by the present example is that of 
 the induction of a current on itself, self-induction as it is called. Whenever 
 the current in a conductor is varying there is always a corresponding variation 
 in the flux of magnetic induction in its own field through its circuit,, which 
 gives rise to an electromotive force in the circuit tending to prevent the 
 variation. Induction in a circuit thus acts as a sort of electric inertia. 
 
 482. A single circuit with capacity as well as induction *. We shall now 
 discuss another important example of these matters and one which intro- 
 duces us to further new principles. We have so far always regarded our 
 * Kelvin, "On transient electric currents," Phil. Mag.[\, 5 (1853), p. 393. 
 
430 Electrodynamics of linear currents [CH. xi 
 
 currents as conduction currents flowing in complete circuits ; we shall now 
 consider the case of the current discharge of an ordinary parallel plate con- 
 denser of high capacity when its plates are connected by a wire conductor. 
 As soon as we connect the plates there is a rush of electricity round the 
 wire and we want to investigate the nature of the current so produced. 
 Although the theory of Maxwell which we shall eventually adopt states that 
 the current is even in this case closed by an aethereal displacement current 
 in the space between the condenser plates, we shall not make much direct 
 use of the idea because we can avoid the difficulty here by imagining that 
 the plates of the condenser are so very near together that we may practically 
 regard the circuit as a complete one, in that the small distance between the 
 plates is too small to make any difference to such quantities as the resist- 
 ance and self-induction coefficients, which may in any case be considered 
 as experimentally determinate. 
 
 In Ampere's time the question was tested experimentally to see if it was 
 an ordinary current circuit, the method being to make the connecting wire 
 into a solenoid and to examine whether an iron needle stuck in it during the 
 discharge became magnetised. This was found to be the case but there was 
 a certain irregularity in the phenomena in that the magnetism induced along 
 the needle was sometimes in one direction and sometimes in the other. This 
 irregularity was quite a mystery until Helmholtz suggested that the discharge 
 was an oscillation. There was in the circuit a considerable amount of inertia 
 and so the rush in one direction always overdid itself and got past the 
 equilibrium position and thus had to swing backwards and forwards until 
 killed by damping. In such a case it would of course be largely a matter of 
 chance whether the first, second, third or any subsequent swing gave the 
 preponderating magnetisation to the needle. Kelvin developed the idea 
 more precisely a few years later. 
 
 483. We shall adopt an equilibrium theory, of course, so that at each 
 instant the current will practically have settled down so that it is the same 
 across every cross section of the wire and corresponds to the equilibrium 
 value if all the other conditions pertaining at that instant could be main- 
 tained invariable. We shall thus be able to define our theory in terms of the 
 current J in the circuit. The above simple experiment shows that such a 
 current has associated with it a magnetic field in which there will be a certain 
 amount of energy measured by 
 
 loJ^ 
 ~2 c ' 
 
 a being the coefficient of self-induction of the circuit. This of course again 
 implies that the magnetic field at each instant is that steady field corre- 
 sponding to the current at that instant. 
 
482-484] The single circuit with capacity 431 
 
 We have now a condenser in the circuit and if its capacity b is large the 
 electrostatic energy in the system is no longer negligible. In fact if Q is 
 the charge at any instant on the condenser plates (+ Q on one and Q on 
 the other) the electrostatic potential energy W of the system is practically 
 
 w IQ2 
 W = 2~b> 
 
 which is that of the condenser. The potential energy of the whole system 
 is practically confined to the condenser, the other parts being of small 
 capacity and carrying also a small charge. 
 
 The current at any instant along the discharging wire may be expressed 
 as the rate of diminution of charge on the condenser or 
 
 r *Q 
 
 " dt> 
 
 so that the kinetic energy in the circuit is 
 
 la (dQ\* 
 
 2 c (dt ) ' 
 
 If also there is a resistance Jc in the circuit there is a dissipation of energy 
 
 at a rate j2 
 
 i 
 per unit time. This can be introduced into the general dynamical scheme 
 
 by using the dissipation function which for this case is 
 
 F = 
 
 On the mechanical analogue of these things the inverse capacity r appears 
 
 as a coefficient of elasticity, the self-induction a as a modulus of electric 
 inertia, and the resistance Jc as a coefficient of electric resistance. 
 
 484. As there is but one variable in the system, viz. Q, it is sufficient 
 to apply the generalised energy principle to solve the problem ; it gives 
 
 dt 
 
 the total amount by which the internal energy of the system falls is equal 
 to the energy dissipated, which reappears in the form of heat ; this is Joule's 
 result. Thus 
 
 dt |_2 c \ dt ) r 2b 
 
 The equilibrium theory assumption now asserts that both a and 6 are constant, 
 a statical distribution being attained at each instant, so that the equation 
 reduces to 
 
 dt z a dt ab 
 
432 Electrodynamics of linear currents [CH. xi 
 
 the solution of which determines the complete circumstances of the affair. 
 This equation, first obtained by Kelvin, is easily solved, for if 
 
 Q - Q e pt 
 
 Jcc c 
 
 is a solution then v 2 -\ p H 7 = 0,, 
 
 a r ab 
 
 kc lie k z c z \ 
 
 ~2a V ~\ab~ 4^2 )' 
 
 or p _!_ * / , , 
 
 2a V \ab 
 
 say p-i and jt? 2 , so that the complete solution of the equation is of a type 
 
 Q = Q-^e^ 1 + Q 9 e v ' 2t . 
 There are two distinct types of solution of an equation of this kind. If 
 
 ab 4a 2 / 
 
 / 4-^7 
 
 is positive, i.e. if k < A/ j-, imaginary values are obtained for p and the 
 
 solution is an oscillatory one. In this case we can in fact write the general 
 solution in the form 
 
 _kct 
 
 
 
 the integration constants Q ti and x being obtained from the initial conditions. 
 
 485. If the resistance R is practically negligible or the conduction 
 nearly perfect, the solution reduces to 
 
 and represents a permanent oscillation of period 
 
 /ab 
 
 V 7* 
 
 Even if k is sensible its effect on the period is practically always negligible ; 
 in fact in the general case 
 
 / c k z c z Vc f k 2 cb\ 
 
 q \/^f 75 = -7= 1 ~ approx., 
 
 V ab 4cr vab \ a / 
 
 so that the period of the oscillation is increased in the ratio 
 
 the effect of k thus being of the second order and therefore negligible unless 
 k is very big. This is a general result in dynamics, when the dissipation is 
 comparatively small its effect on the period is of the second order of smallness ; 
 the main effect of resistance is in damping the amplitude, which gradually 
 decreases to zero. 
 
484-486] The oscillating discharge 433 
 
 Sometimes however the resistance is so very considerable that it actually 
 destroys the periodic nature of the motion altogether. This is the case 
 
 /2a 
 
 when k > \/ 1 , when the solution is of the type 
 V be 
 
 so that the discharge falls off in one direction only dying down to zero. A 
 swinging pendulum in a liquid settles down at once if the viscosity is very 
 big. 
 
 Helmholtz's suspicions are therefore entirely corroborated. The 
 theoretical possibility of these oscillatory discharges in a condenser was 
 however first recognised by Kelvin. The formula obtained above for the 
 period of the undamped discharge is in fact known after him as Thomson's 
 formula. In order that the period may be large we must have a or 6 or both 
 very big. We might for instance increase a by bringing iron into the field : 
 practically however this method leads us into further difficulties because iron 
 behaves so erratically that the simplicity of the above solution is spoilt. 
 The iron cannot respond quick enough to such rapid oscillations. 
 
 486. To illustrate the various points more vividly let us take a special 
 case given by Abraham*. The capacity b is that of a spherical condenser 
 with radii 10 cm. and 10'2 cm. with a dielectric medium of constant 5. In 
 the electromagnetic units adopted this gives a capacity 
 
 2550 
 ~9xl0 20 ' 
 
 This condenser is discharged through a circuit of resistance 1 ohm and 
 coefficient of self-induction a = c . 10 7 cm. In absolute units k = 10 9 so that 
 in this case 
 
 *** in 7 
 
 -g- ~ t 10 approx. ; 
 
 the influence of the resistance on the frequency is therefore extremely slight 
 and we can use Thomson's formula. This gives for the period 
 
 /ab , 
 
 ZTT x / = * . 10~ 4 sec. approx., 
 V c 
 
 and the corresponding wave length is 
 
 A = 10 6 cm. 
 The damping constant is 
 
 ^ = 50, 
 2a 
 
 so that in ;02 of a second the amplitude has been decreased by - = n. 
 
 * Theorie der Elektrizitat, i. p. 292. 
 L. 28 
 
434 
 
 Electrodynamics of linear currents 
 
 [OH. xi 
 
 but during this time about 600 oscillations have occurred so that the rate of 
 reduction of the amplitude from period to period is slow. 
 
 These results are of the order of those first experimentally determined 
 by Feddersen*, who examined the spark of the discharge by a revolving 
 mirror. In such a case the equilibrium theory certainly does apply because 
 the dimensions of the whole system are very small compared with the wave 
 length of the oscillation, and the state of affairs is at each instant practically 
 the smoothed out equilibrium one. The actual distinguishing characteristics 
 of Maxwell's theory are therefore not appreciable in such cases so that it 
 was necessary to get beyond the limits where this analysis applies by creating 
 much faster electrical oscillations. 
 
 487. From Thomson's formula it follows that we can theoretically 
 reduce the wave length by decreasing the capacity or self-induction. This 
 was accomplished experimentally by H. Hertzf, who was the first to demon- 
 strate the existence of electric waves of sufficiently short wave length. The 
 original form of exciter adopted by Hertz was in the form of that shown 
 diagrammatically in the figure, this form having very little capacity and 
 induction. When the current in the wire is concentrated 
 into a small cross section, when the strands, as it were, 
 of the current are close together the induction in the 
 circuit is very large, but if they are spread out a bit the 
 inductance is smaller. A similar argument applies to 
 the electrostatic phenomena and shows that the energy 
 is large if the charges are concentrated together, but 
 small if they are spread out over large areas far apart. 
 This is the distinction between the form of circuit adopted 
 by Hertz and that discussed above. In our case the 
 elastic spring is almost entirely confined to the space 
 between the condenser plates and the inertia is in the 
 actual circuit wire connecting them so that both the 
 electrokinetic and electrostatic energies are very large. 
 In the Hertzian form things are much more spread out. 
 The theory above however does not apply to a case of 
 this kind because the quasi-stationary conditions are no 
 longer satisfied. It was in fact this electric oscillation 
 with tremendously short period that sent out the electric 
 waves detected by Hertz. In such cases an appreciable 
 part of the energy of the system is dissipated in the form 
 of radiation : this loss is not accounted for by the present theory where the 
 
 * Pogg. Ann. 103, p. 69. 
 
 | Untersuchung iiber die Ausbreitung der ehktrischen Kraft. English translation by D. E. 
 Jones (London, 1900). Cf. below, ch. xii. 
 
 Fig. 77 
 
486-489] The general theory 435 
 
 only dissipation is in the currents driving against the friction and converting 
 their energy into heat according to Joule's law. 
 
 On the Maxwellian theory the oscillation of the concentrated electric 
 field between the plates of the condenser represents a violent disturbance 
 in the aether, which if quick enough should send out electric waves of dis- 
 turbance through the aether : such waves carry energy away with them. 
 An analogy is provided by the vibrations of a tuning fork in air. If the 
 vibrations are slow the fork does not send out sound waves, because the 
 air has time to get round the moving prong and settle down at each instant : 
 the motion of the fork is thus independent of the air because such slow 
 alternations cannot get hold of the elasticity in the air. If however the 
 vibration is rapid the air has no time to get out of the way and there is a 
 compression caused which relieves itself away in a sound wave. 
 
 488. The general electrodynamical theory of currents*. The develop- 
 ments in the previous paragraph illustrate in the various simple cases the 
 application of ideas of ordinary dynamics to the solution of electrodynamic 
 problems concerning the action and interaction of linear current systems, the 
 results obtained being entirely consistent with the experimental facts. We 
 have not however got beyond applications of the energy principle and this 
 is not sufficient for the general discussion of any system with more than one 
 degree of freedom. It is therefore very desirable that we should have, if 
 possible, a general principle from which all the results, statical or dynamical, 
 can be deduced for the most general system. We shall now show that we 
 can adopt the more general results of analytical mechanics into a scheme 
 which will enable us to deduce from the expressions for the kinetic and 
 potential energies already obtained all the equations of a system of linear 
 currents with any number of degrees of freedom. By applying Lagrange's 
 dynamical equations to the more general problem expressed in a definite 
 manner we are again led to results which are in entire agreement with the 
 experimental facts. 
 
 Following Maxwell we assume that in the field of any system of electric 
 currents there is some motion of which we cannot take direct cognisance. 
 The kinetic energy of this motion is that energy which we have already 
 obtained as the electromagnetic energy in the field; Maxwell calls it the 
 electrokinetic energy. For a system of stationary or quasi-stationary 
 currents it appears as a quadratic f unction f of the current strengths in the 
 various circuits, the coefficients in which depend merely on the forms and 
 relative positions of the various circuits. 
 
 489. Whatever mechanical analogy we adopt in the application of 
 dynamics to electric currents we are evidently always involved in the class 
 
 * Cf. Maxwell, Treatise, n. chs. v-vii. 
 f If there is no iron about. 
 
 282 
 
436 Electrodynamics of linear currents [OH. xi 
 
 of motions which Helmholtz described as cyclic. The parameters which de- 
 termine the instantaneous position of such a system are of two kinds : (i) the 
 parameters of the first kind which are of the general type of coordinates in 
 mechanics. These parameters occur in general in the expression of the kinetic 
 energy with their differentials with respect to the time. To this group belong, 
 in the present instance, those geometrical parameters which determine the 
 position of the circuits : (ii) the parameters of the second group on the other 
 hand, which are the coordinates of the cyclic motion, do not themselves 
 appear in the expression for the kinetic energy, only their time rates of change 
 being involved. If the kinetic energy is known to be correctly expressed 
 in terms of the coordinates and velocities explicitly we have therefore no 
 difficulty in separating the coordinates into their respective classes. But in 
 systems where the internal connections are only partially known, a difficulty 
 may occur, in as far as in obtaining the expression for the kinetic energy, it 
 may have been convenient or even necessary to introduce the generalised 
 momenta in the cyclic motions, in order to obtain a usable expression. For 
 example in the hydrodynamical analogue, in determining the forcives between 
 cores in problems of cyclic motion, the circulations in terms of which the 
 energy is usually expressed, must be treated as generalised momenta. It is 
 therefore necessary and essential to have a clear view of the circumstances 
 which determine whether the various quantities which enter into the specifica- 
 tion of the energy are to be classed as velocities or momenta. The basis 
 of the distinction between these two classes of quantities is of course funda- 
 mental ; it is to be found in the way in which they occur in the Hamiltonian 
 analysis of the dynamical problem. The essential property of a velocity is 
 that it is a perfect differentia] coefficient with respect to the time; any 
 function involving rate of change of configuration, which enjoys this property, 
 so that its time integral is a function of position only may be taken to be 
 a velocity; provided we, if need be, contemplate also a corresponding force. 
 On the other hand, any such function of the rate of change of configuration, 
 even though it be a perfect differential with respect to the time, must be treated 
 as a momentum, if it is known to remain constant with time while no external 
 forces are applied to it ; for if it were a velocity, linked up with other velocities, 
 its constancy in the free motion could not usually fit in with the analytical 
 theory. 
 
 490. In the theory of cyclic fluid motion, the circulations being constant, 
 must thus be taken as momenta, and, when the energy is expressed in terms 
 of them, it must be modified before the forcives can be derived from it in the 
 manner of Lagrange and Hamilton. In the theory of electrodynamics, on 
 the other hand, the electric currents are not unalterable with the time, even 
 if no applied electromotive forces are applied, and as they are the differential 
 coefficients with respect to the time of definite physical quantities, the charges 
 
489-492] The energy functions 437 
 
 of electricity, they may be taken as the velocities, provided we recognise 
 the play of the corresponding (electromotive) forces. 
 
 In the electrodynamics of complete circuits however there is no reason, 
 in that theory taken by itself, why the functions defined hereinafter as the 
 electrokinetic momenta should not be taken as the velocities instead, if so 
 desired, for they satisfy all the above conditions, though of course the 
 corresponding forcives would be of quite different types from the usual ones. 
 This remark is in illustration of the fact that the distinction between momenta 
 and velocities is to a certain extent one of convenience. We shall however 
 adopt the method indicated on account of the enormous difficulties underlying 
 this suggested alternative. 
 
 491. The system we shall treat will consist of n-conducting circuits 
 carrying currents J t , J z , ... J n , We have therefore n-cyclic coordinates 
 in which the velocities are J l9 J 2 , ..., in addition to a certain number of 
 ordinary geometrical coordinates l5 6 2 , ... 6 m which determine the relative 
 configuration of the system. The actual cyclic coordinates may be taken 
 as the integrals of the currents with respect to the time reckoned from a 
 definite instant, i.e. the quantities Q 1} Q 2 , ... Q n of electricity which since 
 that instant have crossed any cross section of the respective conductors 
 
 r _dQ 1 _dQ, 
 
 Jl ~ dt' J2 ~~dT'- 
 
 The generalised force components corresponding to the cyclic-coordinates are 
 the electromotive forces which work on the currents' flowing. The work 
 function of the forces applied in these coordinates would thus be 
 
 - 8W Q = S 
 
 r=l 
 
 if we assume impressed electromotive forces E^ , E 2 , . . . E n in each circuit 
 respectively, since the work of the electromotive force E in any virtual 
 increase of the coordinate Q defined as above is ESQ. 
 
 We have also to include the virtual work of the applied mechanical forces 
 if there are any. In the general Lagrangian method the generalised force 
 component which corresponds to the coordinate 6 is the coefficient in the 
 work done when that coordinate is alone altered. Thus 
 
 SW e - 20S0. 
 
 If the forces are applied from without 8 W would represent energy added to 
 the system. If they are merely forces exerted by one part of the system on 
 another or against the external system the work in them would come from 
 the energy of the system and must therefore be taken as SW e . 
 
 492. We have to take into accpunt the resistances to the flow of the 
 currents. These may be introduced either by including them in the 
 generalised impressed or external force components corresponding to the 
 
438 Electrodynamics of linear currents [OH. xi 
 
 cyclic coordinates, or by the more general method involving the intro- 
 duction of a dissipation function. If the resistances in the circuits are 
 R 1} R 2 , ... then on the first method the impressed electromotive forces in 
 the separate circuits would have been respectively diminished by R^J,, 
 R Z J Z , ... R n J n , in order to obtain the resultant force components. The 
 more general method consists in the introduction of the function 
 
 7 -Mfik/i 1 + */.+.-), 
 
 into the general dynamical scheme. 
 
 In order to obtain complete generality we shall assume that each circuit 
 has included in it a condenser, or an appreciable capacity for storing energy. 
 It is only in this case that the potential electric energy is at all comparable 
 with the magnetic kinetic energy. If the original charges in these condensers 
 were Q 0l , Q 0z , ... then the general potential energy function for the system 
 would be 
 
 W - &b rr (Q Qr - QrY + S6 r . (Q gr - Q r ) (Q 0g - Q,), 
 
 and the product terms would be negligible in most cases if the condenser 
 in each circuit is of such a form as to concentrate its field sufficiently to 
 prevent mutual influence with the others. 
 
 We shall leave out of account any so-called permanent magnets or 
 magnetisable substances in which the magnetisation is not capable of following 
 the field without hysteretic loss. 'Permanent' magnets are in reality far 
 from permanent and are indeed very erratic things; their properties are 
 very indefinite and a theory including them becomes largely an empirical 
 subject. We 'may thus regard the coefficients of induction of the circuits 
 to be dependent merely on the geometrical configurations in the circuits. 
 
 493. The electrokinetic energy of the system is, as before, given by 
 
 dt dt 
 
 This is kinetic energy of some kind; we do not as before need to know of 
 what kind. We only want its amount in suitable terms to enable us to 
 apply general dynamical methods ; this is the great advantage of the present 
 line of attack. 
 
 We must now also include the kinetic energy of the movement of the 
 material conductors because the material conductors involved may possess 
 very considerable masses. The positions and general configurations of these 
 masses are as before specified by the generalised coordinates. X , d z , ... 6 m 
 and the kinetic energy corresponding to them will be denoted by T^ so that 
 the total kinetic energy is given by 
 
 2\ is of course a quadratic function of l} 6 Z> ... m in which the coefficients 
 are functions of X ... m . 
 
492-494] The application of Lagrange's equations 439 
 
 For absolute generality we should include in the complete expression for 
 the kinetic energy terms involving such things as (Q r d s ) but this would be 
 getting beyond our theory. In all realisable cases and certainly in all 
 those cases where an equilibrium theory is applicable the electric changes 
 adjust themselves so quickly compared with the slow motions of ordinary 
 matter that the general electromagnetic system is at each moment sensibly 
 in an equilibrium condition ; so that there is practically no interaction between 
 the kinetic energies of the electromotive and material systems such as would 
 arise from mixed terms in the energy function involving both their velocities 
 a fact verified experimentally by Maxwell. The expression for T thus 
 represents completely the energy of the system as far. as electromotive 
 disturbances are concerned, whether the system is in motion or not. It is 
 therefore sufficient for the determination of the electrical conditions. The 
 other part T l is solely the ordinary kinetic energy of motion of the conductors 
 and is alone necessary for the determination of the mechanical relations of 
 the electrodynamic field. 
 
 494. We can now proceed to apply any of the usual methods of 
 obtaining the equations of the motion in the various coordinates, electrical 
 and geometrical. The most general method involves a use of the principle 
 of least action which represents the most general principles of dynamics 
 in their most condensed form. If the system is governed by dynamical laws 
 at all, we have merely to obtain the energies in their most compact form 
 and then to substitute them in this principle. Lagrange's principle of least 
 action is in fact the infallible method to apply to all dynamical systems and 
 is the one which avoids the investigation every time of the conditions 
 peculiar to each case. We shall however find it more convenient for the 
 present case, in which we have actually determined the form of the functions, 
 to take the slightly less general form of the principle which is contained by 
 the expression that the motion is determined by the ordinary Lagrangian 
 equations in dynamics. 
 
 If we apply the general dynamical equations to the electrical coordinates 
 first we find that there is an equation for each circuit of a type 
 
 d a T diT+Tj dw a* 1 
 
 dtdQ, dQ r 8Q r dQ r 
 
 but since T } is a function of the 0's and 0's and does not contain either Q r 
 or Qr this reduces to 
 
 d fdT\ _ 8T dW _ W _ E 
 
 dt\dQ r ) a& s&. dQ r ' 
 
 There are as many of these equations as there are circuits .and so they are 
 sufficient to determine the electrical motions in the circuits. 
 
440 Electrodynamics of linear currents [OH. xi 
 
 495. On the analogy with the ordinary Lagrangian equations the term 
 
 is the applied force acting in the coordinate. The term on the other side 
 
 d_(dT\_ 8T 
 dt\dQ r ) dQ r ' 
 
 which, since the actual coordinates Q 1} Q 2 ... Q n corresponding to the cyclic 
 velocities J 1} J 9 , ... do not explicitly occur in any of the functions involved 
 so that for each circuit 
 
 d dT 
 
 reduces to -j- ^--7- , 
 
 dtdJ r 
 
 represents what Kelvin called the 'kinetic reaction' of the system, taken in 
 D'Alembert's sense. In the electrical case the applied electromotive force 
 is balanced by the kinetic reaction of the changing current. 
 
 But 
 
 dT 
 
 fl~T a rl" 1 + a r2" I T a rnJ n i 
 
 and differential coefficients of T with respect to the cyclic velocities are the 
 momenta in the respective coordinates ; we call them the cyclic momenta. 
 We see at once that they are identical for each circuit with the magnetic 
 induction flux through the circuit; we can therefore describe them, after 
 Maxwell, as the electrokinetic momenta of the circuits. The electrokinetic 
 reactions to the variation of these momenta are determined as usual by 
 
 d /dT\ d . 
 
 -dt(dJ r ) = -dt (a ^ Jr+ - } ' 
 
 so that they correspond exactly to the induced electromotive forces in the 
 circuits. We thus deduce that Faraday's law is quite consistent with the 
 general dynamical hypothesis, even in the most general case when the current 
 circuits are in motion. 
 
 496. We have above limited ourselves to the application of the 
 dynamical analysis to the cyclic coordinates, thereby determining the elec- 
 trical motions only. We must now apply the same method to find the 
 equations corresponding to the geometrical coordinates which determine 
 the configurations of the conducting circuits. For each geometrical coordi- 
 nate 6 r we have an equation of type 
 
 d_ d(T+ T t ) _ 9 (T + TJ dW _ d_F_ = 
 
 ~ ~ 
 
495-497] The electrodynamic potential 441 
 
 rlT 1 
 
 which since -- = can be put in the form 
 
 l = Q , 
 
 dd r dd r de r de r ' 
 
 The two terms on the right represent with signs changed the kinetic reaction 
 of the circuits to ordinary motions. The terms on the right 
 
 represent the mechanical force exerted by the system in the coordinate 6 r . 
 The second part arises from the electrokinetic energy of the system and the 
 first from the electrostatic part. We call the former the electrodynamic 
 forces and the others the electrostatic ones. 
 
 We thus see that the electrodynamic forces of stationary currents always 
 tend to increase the electrokinetic energy T. We then speak of T in 
 this sense as the electrodynamic potential of the system ; it plays the part 
 of the force function of the electrodynamic forces. This is Neumann's 
 result*. 
 
 Prof. Larmorf criticises the above argument on the ground that the 
 mechanism that links the mechanical and electrodynamical systems together 
 is too complicated to be treated otherwise than statically. Such a procedure 
 is however quite sufficient for our purposes because the mechanical changes 
 in the conductors, as already explained, have usually a purely statical aspect 
 compared with the extremely rapid electric disturbances. Larm.or puts the 
 argument in the following manner. A small displacement of the system 
 increases T by 8T ; this increase must come from some source ; if we suppose 
 for the moment, to avoid complications, that there is no dissipation, we see 
 that this energy must come from the energy of the material system. During 
 the displacement the electromotive system is at each instant sensibly in an 
 equilibrium condition so that somehow, by means of unknown connecting 
 actions, the displacement alters the mechanical energy by ST and of this, 
 considered as potential energy, the mechanical forces are the result. The 
 expression T given above with its sign changed thus appears as the potential 
 energy of the mechanical electrodynamic forces acting between the material 
 conductors which carry the currents. This is the result as deduced above. 
 
 497. The general conclusions thus arrived at are in perfect agreement 
 with the actual facts of the phenomena. Consider for example the case of 
 two rigid conductors, the one fixed and the other moveable about a fixed 
 point, and suppose the currents maintained constant or that there is no 
 
 * "Ueber ein allgemeines Prinzip der mathematischen Theorie der induzierten Strome," 
 Berlin. Abhdg. (1848). 
 
 t Phil. Trans. A, 185 (1894), p. 761. 
 
442 Electrodynamics of linear currents [CH. xi 
 
 resistance in' the circuits to cause dissipation. The electrokinetic energy 
 of the system then alters by an amount which is determined solely by the 
 alteration of a lz J^J^\ but this is proportional to 12 /i, the number of tubes 
 of induction in the field of the first conductor which passes through the 
 second. The second moveable conductor will thus always tend to turn so 
 as to enclose as many as possible of the lines of induction in the field of the 
 first current. 
 
 The deductions are also true for non-rigid or flexible circuits, wherein the 
 coefficients a n , a 22 , ... of self-induction are variable. 
 
 The success of these investigations points to the conclusion that the laws 
 of electrodynamics can be all deduced from the general equations of mechanics, 
 and a conclusion of this kind is of immense importance for our future specula- 
 tions. Whatever view we may hold of the actions underlying such physical 
 phenomena we must nevertheless admit the importance of the discovery that 
 the motions of ponderable bodies and electrodynamic phenomena are subject 
 to the same laws. We need not necessarily regard this close connection 
 between mechanics and electrodynamics as providing any evidence of a 
 mechanical basis for electrical phenomena. We might just as well say that 
 it favours the view that mechanical laws have an electrodynamic basis. This 
 latter point of view is in fact characteristic of the general trend of modern 
 physical speculations. 
 
 498. On the solution of the equations for circuits with capacity and 
 induction. We can now proceed to discuss the numerical solution of the 
 equations representing the conditions in a set of w-circuits with capacity and 
 induction. We have then the kinetic and potential energies and the dis- 
 sipation function in the form 
 
 9W Ytt r 
 
 = s -- 
 
 and in addition there are the impressed electromotive forces E 1} E 2 , ... E n 
 in the respective circuits. 
 
 The general equations are then of the type 
 
 dtdQ r dQ r 9 ~ 
 
 from which one or two important conclusions can be directly inferred. The 
 complete solution of the equations in all their generality will be examined 
 at a later stage in the discussion. 
 
497-500] Particular cases 443 
 
 If the solution shows that the currents are periodic forced vibrations with 
 a frequency n then all the quantities may be treated as dependent on the 
 time by the factor e int and thus we can have 
 
 d /3T\ 8T 
 
 I _ f\ Aft . _ 
 
 ~T7 I i v'v * 9 
 
 dt\dQ T J 8Q,' 
 and when n increases indefinitely we must have 
 
 in each circuit. Thus in the case of enormously rapid vibrations of the 
 currents,, their distribution in the various conductors is independent of the 
 resistances and is determined by the fact that the kinetic energy (and not 
 the dissipation) function is a minimum. 
 
 A similar remark applies when the question under consideration is one of 
 initial impulse effects. 
 
 This explains why it is that when a rapidly alternating current is sent 
 along a wire, the current really only travels in the outer layer of the wire ; 
 the mean distance between the various filaments in the current being thereby 
 increased and their mutual inductance and the kinetic energy of the field 
 decreased to their minimum values. 
 
 499. When W = it is convenient to express everything in terms of 
 the currents J r = ~. Thus in the problem of steady electric flow when all 
 
 the quantities E r representing impressed electromotive forces are constant, 
 the currents are determined directly by the linear equations 
 
 which express the condition that the function 
 
 (F - 2E r J r ) 
 
 is a minimum. If all the U's are zero this is Joule's law of minimum 
 dissipation for steady currents. The above is the more generalised form 
 including impressed electromotive forces. 
 
 500. The general problem dealing with forced vibrations in circuits 
 with potential energy can easily be reduced to the simpler case when W 
 if attention is directed to the following point. 
 
 In the most general case in which 
 
 W = Z,ib rr Q r *+Z,b rs Q r Q s , 
 the equations for each circuit are of the form 
 
 rlJ? 
 
 Ojj/j + a l2 J z + ... a ln J n + b ll Q l + 6 12 Q 2 + &mQ + gjr = ^i> 
 
444 Electrodynamics of linear currents [CH. xi 
 
 but when vibrations proportional to say cos pt are in progress in the circuit 
 
 and thus the equation above reduces to 
 
 t> 
 
 and then it is of precisely the type obtained without a potential energy 
 function. 
 
 This remark allows us to simplify our equations by omitting the potential 
 energy part altogether. When the solution is obtained we may at any time 
 
 generalise it to include these cases, by the introduction of a rs -- ^ in place 
 
 of any induction coefficient a rs . In following this course we must be prepared 
 to admit negative values of these coefficients a rs . 
 
 We can now illustrate these general principles by the exact solution of 
 the equations in a few special cases*. 
 
 501. A single current circuit with resistance and induction only : here 
 we have 
 
 2T = afr, 
 
 2F = RQ Z , 
 
 W = 0, 
 and the equation of motion assumes the form 
 
 where E is the impressed force. We use J for the current strength in the 
 circuit and then 
 
 a J 4- RJ = E, 
 
 and the integration of this equation determines completely the time variation 
 of the current intensity J. Two cases present themselves. 
 
 (1) E constant. In this case we have, since a and R are constants, 
 
 F Rt 
 
 T ** n -- 
 
 J = ~ Ce a . 
 K, 
 
 The integration constant C is determined from the initial conditions. If at 
 the time t = the circuit is closed and the constant electromotive force 
 applied we shall have J = initially and thus 
 
 E 
 
 c = l 
 
 * These examples are given by Maxwell and Rayleigh, "The Theory of Sound," Phil. Mag. 
 [5], 21 (1886), p. 369; Proc. R. S. 48 (1891), p. 203. Cf. also J. J. Thomson, Recent Researches 
 in Electricity and Magnetism (Oxford, 1893). 
 
500, 501] The single circuit 445 
 
 E 
 
 or J = 
 
 R 
 
 the constant , called the ' time constant,' determines the rate of increase 
 
 d 
 
 of the current up to its steady value J = ^ which corresponds to Ohm's law. 
 
 (2) E periodic : say 
 
 E = E sin pt, 
 we have then aJ -f- RJ = E sin pt ; 
 
 the solution of this equation will, as is well known, be the same as the imaginary 
 part of the solution of the simpler equation 
 
 aJ+RJ = E e ipt . 
 We try a solution of the fprm 
 
 which will satisfy if J (aip + R) = E , 
 
 if T E 
 
 1.6. II e/ft 
 
 || R + aip 
 
 The solution of our first equation is then the imaginary part of the function 
 
 Tf. aipt f, (ft /W<n\ p*J>< 
 
 i!/Qt. JLVQ \JLii ill/JJ) K 
 
 R + aip (R 2 + a 2 p 2 ) 
 
 T?._ (ipt6) 
 
 or of 
 
 where tan 6 = -. 
 
 R 
 
 We have therefore 
 
 J = - == = sin (pt - 0). 
 VR Z + a V 
 
 The complete solution is therefore 
 
 E _^ 
 
 J = . sin (pt - 6) + Ce a , 
 
 the integration constant C being again determined by the initial conditions ; 
 if these are again such that J = when t = then 
 
 C= 
 
 E 
 
 / - \ 
 
 sin(j^-0) +e~a~sin0 . 
 
 a z p z \ J 
 
 Thus J= 
 
 The second part of the solution is however unimportant except for a considera- 
 tion of the initial establishment of the steady oscillating current finally 
 
446 Electrodynamics of linear currents [OH. xi 
 
 established. If we consider the conditions started at some remote past time 
 the solution is represented correctly by the one term 
 
 E sin (pt - 0) 
 
 The phase of the current therefore lags behind the phase of the impressed 
 force by an amount = tan -1 -^ , which amounts to a quarter phase when 
 p is infinitely large. 
 
 We also see that the ratio of maximum current to maximum impressed 
 force is 
 
 this ratio in the steady case being 
 
 1 
 R- 
 
 7D 
 
 Thus if p is very much bigger than the current is greatly reduced by the 
 
 CL 
 
 self-induction. The quantity VR Z + a z p 2 is called the 'impedance' in the 
 circuit for the given periodic disturbance. 
 
 When we introduce an appreciable capacity into the circuit (perhaps 
 a Leyden jar) the current is again increased. This case is easily solved on 
 the above lines. 
 
 502. Two detached circuits without capacity influencing one another 
 only by induction. 
 
 We shall examine the effect on the second circuit of the instantaneous 
 establishment and subsequent maintenance of a current J x in the first circuit. 
 At the first moment the question is one of the function T only, where 
 
 2T = a u e/! 2 + 2a 12 J 1 / 2 + a 22 J 2 2 , 
 
 and the solution is to be obtained by making T a minimum under the condition 
 that J l has the given value. Thus initially 
 
 T 
 
 ^ 20 
 
 and accordingly after a time t 
 
 #2 is the resistance of the circuit. The whole induced current as measured 
 by a ballistic galvanometer is 
 
 
 
 in which a 22 does not appear. 
 
501-503] Two circuit in parallel 447 
 
 The current in the secondary circuit due to the cessation of a previously 
 established steady current J^ in the first circuit is the opposite of the above. 
 
 The general equations for two detached circuits as above may be obtained 
 in the usual manner from the above form of T and 
 
 2F = R^^ + R 2 J 2 *. 
 Thus ]!/! + a lz j z + Jj^Jj = E 1} 
 
 &12" 1 ~t~ ^22"2 ~^~ "2*' 2 = "% 
 
 If a harmonic electromotive force 
 
 EI = Ee ipt , 
 
 act in the first circuit, and the second circuit be free from imposed forces 
 (E 2 = 0) we have on elimination of J z 
 
 showing that the reaction of the secondary circuit upon the first is to reduce 
 the inductance by 
 
 and to increase the resistance by 
 
 503. Let us now consider two circuits in parallel. 
 
 Firstly it is not necessary to include the influence of the leads outside the 
 points of bifurcation; for provided there be no mutual induction between 
 these parts and the remainder, their inductance and resistance enter into the 
 result by simple addition. 
 
 Under the sole operation of resistance the total current J would divide 
 itself between the two conductors (of resistances R l and R 2 ) in the parts 
 
 7? 7? 
 
 -p -^-p- J and ^ ^-p- J, 
 
 and we may conveniently so choose the second coordinate that the currents 
 in the two conductors are in general 
 
 Z> i 7? 7? i Z? 
 
 /ij + n. 2 /ij + /i 2 
 
 J still representing the total current in the leads. The dissipation function 
 found by multiplying the squares of the above currents by ^R l and %R 2 is 
 
448 Electrodynamics of linear currents [CH. xi 
 
 also a u , a 12 , a 22 being the induction coefficients of the two branches 
 _ 1 a nJ R 2 2 + 2a 12 fl lJ R 2 + azzRj 2 J2 
 
 ~~ 
 
 (a n - a 12 ) # 2 + (a 12 - a 22 ) ^ 
 
 Thus in the notation 
 
 J ,. 2 
 
 (a n - a 12 ) # 2 + (a 12 - a 28 ) R l 
 
 a 22 a 1T 2a 12 22 , x ^ _ 
 
 we have 2J? = p x J 2 + p 2 J' 2 , 
 
 2T - ct u J 2 + 2a 12 JJ f + a 22 J' 2 , 
 
 and we are then in the same case as previously discussed. We thus see at 
 once that the effective resistance R and the effective inductance a of the 
 combination* are obtained as 
 
 j) 2 a 12 2 /> 2 
 
 2* 
 
 2 
 
 By substitution and reduction we find that 
 
 - 2 
 
 22 + 2 flu - 
 
 ("i ~H " 2 ) ~^~ P (^11 -"^12 ~^~ ^22) 
 
 (^ + .R 2 ) 2 + p 2 (a n 2a 12 + a 22 ) 2 
 
 in which (a n 2a 12 + a 22 ) and (aii 22 ~~ a i2 2 ) are both positive by virtue 
 of the nature of T. 
 
 504. As p increases from zero, we see that R continually increases and 
 that a continually decreases. 
 When p is small 
 
 T? 1 2 ~ 11 2 ' 12 1 2 i^ -2 1 
 
 J? + R ' (.R 4- .K ) 2 ' 
 
 in this case the distribution of the main current between the conductors is 
 determined by the resistances. 
 
 On the other hand when p is very great 
 
 JB = 5 '" 
 
 (a u - 2a 12 + a 22 ) 2 
 
 a ii a 22 ~~ a i2 
 CL = 5 
 
 a u 2a 12 + a 22 
 * Defined as the effective constants in the" solution for the total current J. 
 
503-505] 
 
 The (jeneral case of u-rirctiits 
 
 449 
 
 in this case the distribution of the currents is independent of the resistances, 
 being determined as usual so that the ratio of the currents in the two con- 
 ductors is 
 
 a 22 # 12 
 
 a n a 12 
 
 When the conductors exert no mutual induction the formulae are simpler : 
 
 12 = and so 
 
 (Ri + R 2 ) 2 + p 2 (n + 22 ) 2 
 
 a = 
 
 (R, 
 
 (o n + a 22 ) 2 
 
 505. The more general form of these results applicable to the case of n 
 current circuits in parallel can be similarly treated. Let J be the current 
 in the leads, J 1 ,J. it , ... J n the currents in the wires ; we may assume, for sim- 
 plicity, that there is no mutual induction between the wires and the leads. 
 Let a rr be the self-induction and k r the resistance of the wire through which 
 the current is J r , a rs the coefficient of mutual induction between this wire 
 and the wire through which the current is J s . Let a be the self-induction, 
 r the resistance of the leads, < the electromotive force in the external circuit ; 
 we shall suppose that this varies as e ipt . The current through the leads and 
 those through the wires in parallel are connected by the relation 
 
 </ - (e/i + J 2 + ... + J n ) = 0, 
 
 so that these variables are not all' independent. Thus in forming the general 
 equations we may treat these currents as independent if we introduce an 
 undetermined multiplier in the usual manner in connection with this relation : 
 we then get 
 
 (a ip + k )J + A - c , 
 
 + kj J 1 + a 12 ipJ z + ... - A = 0, 
 ) J 1 + (a. Z2 ip + k 2 ) J 2 + ... A = 0, 
 
 + - A - 0, 
 
 solving the last w-equations linearly we find 
 
 J, 
 
 where 
 
 An + A lt 
 
 + ...A ln A 21 + A 22 +...A 2n 
 
 As 
 
 ttnip + k 1} o, 12 ip, ... ^m^p 
 
 
 a 2l ip, 
 
 
 , 7, 
 
 29 
 
450 Electrodynamics of linear currents [OH. xi 
 
 and A VQ denotes the minor of A corresponding to the constituent in which 
 
 a vq occurs. Since 
 
 VQ 
 
 :s. Since 
 we have from the above equations 
 
 X 
 
 A' 
 
 substituting this value of A in the first equation for the currents we find 
 
 ( a ip + k ti + 
 
 where S = A n + A 22 + ... + 24 12 + ..., 
 
 whence the self-inductance and impedance of the leads can be deduced; 
 their expressions are however in general very complicated, but they take as 
 usual comparatively simple forms when ip is either very large or very small. 
 
 506. When ip is very large 
 
 A _ . D k, (A u f + A 12 ' + ... A ln ') z + 2 (A^ + ... A 2n ') 2 + ... 
 
 ~~ 
 
 8 
 where 
 
 and A VQ ' is the minor of D corresponding to the constituent a PQ , while 
 
 S' = A 11 ' + A 22 '+... + 2A 12 '+.... 
 Thus the self-inductance of the wires in parallel is in this case 
 
 Z) 
 
 S" 
 while the impedance is 
 
 {*! (A u ' + ... A ln 'Y + k 2 (A 2l f + ... A 2n 'r + ... }/S' 2 . 
 When ^ is very small 
 
 (a t] a 22 2a ]n \ 
 
 %i + 5i+- + %F,+ -; 
 
 p ' " 
 
 So that in this case the self-induction of the wires in parallel is 
 
 11 I 22 i i 12 I 
 
 1 2 1 2 
 
 ~7i 5~~ 
 
 (F + I + - 
 
 \/Vl A/g i 
 
505-507] The general case of ^-circuits 451 
 
 and the resistance is 
 
 1 
 
 1 2 
 
 When there is no induction between the wires in parallel, a 12 , a 22 . ... all 
 vanish ; hence when ip is very large the self-induction is 
 
 1 
 
 _L _L _JL' 
 
 a ll a Z2 a nr* 
 
 and the impedance 
 
 k_ + k z 
 a n z a 22 2 
 
 + ~)' ' 
 
 These are the general results. 
 
 507. We shall now briefly consider the general case of any number 
 of circuits : the investigation will apply whether the circuits are arranged 
 so as to form separate circuits or whether some or all of them are metallically 
 connected so as to form a network of conductors. 
 
 Let /!, J 2 , ... J n be the variables required to fix the distribution of 
 currents through the circuits ; let T, the kinetic energy due to these currents, 
 be -expressed by the equation 
 
 while the dissipation function F is given by 
 
 Let us suppose that there are no external forces of types J 2 . J 3 , ... and that 
 (f> l , the external force of type / 15 is proportional to e ivt . The equations 
 giving the currents are 
 
 (a n ip + k n ) J + (a l2 ip -f k lz ) J 2 + =<f>i, 
 
 =0. 
 
 From the last (n 1) of these equations we have 
 
 J\ /2 _ "3 
 
 TJ D ~ D ~~ > 
 
 **WL -^12 -"13 
 
 where B v<1 denotes the minor of the determinant 
 
 corresponding to the constituent a PQ ip + r vq ; we shall denote the deter- 
 minant by A. 
 
 292 
 
452 Electrodynamics of linear currents [OH. xi 
 
 Substituting the values of ./,, J 3 , ... in the first equation, we have 
 
 (a u ip + r n ) /i + D~ {(*? + *ia) B i* + ) J i = X i> 
 n 
 
 which may be written in the form 
 
 A 7 4, 
 ST/ 1 -* 1 ' 
 
 If b e written in the form Lip + R, where L and R are real quantities. 
 
 " 
 then i is the effective self-induction of the circuit and R the impedance. 
 
 508. We have also 
 
 If an electromotive force </> 2 of the same period as </> a acted on the second 
 circuit, then the current J 1 induced in the first circuit would be given by 
 
 -fi-Jl =<f>2- 
 *M 
 
 Comparing these results we get Lord Rayleigh's theorem, that when 
 a periodic electromotive force F acts on a circuit A the current induced in 
 another circuit B is the same in amplitude and phase as the current induced 
 in A when an electromotive force equal in amplitude and phase to F acts 
 on the circuit B. 
 
 When there are only two circuits in the field 
 
 A . , (a l2 ip + fc 12 ) 2 
 
 -- a^p- ' 
 
 il22 
 if the circuits are not in metallic connection & 12 = and we have 
 
 A 
 
 A / 2 a,,a 12 2 \ . 
 
 5- = tt ll -- 2 . 7 2 1 
 
 5 U \ a 2 2 P + ^22 / 
 
 Tnus the presence of the second circuit diminishes the self-induction of the 
 tirst by 
 
 while it increases the impedance by 
 
 3> 2 & 22 a 12 2 
 
 ^ "r "-' 
 
 509. On the propagation of waves along a cable. We now turn to a 
 final illustration of the principles of this chapter, which analyses from a 
 simple point of view a problem to be subsequently discussed in greater 
 detail. The subject hardly belongs to the title of the present chapter but it 
 is convenient to include it in the general discussion at the present stage. 
 
507-510] The propagation of waves along a cable 453 
 
 The problem concerns a very long cylindrical metallic conductor 
 surrounded by a coaxial cylindrical metal sheath (somewhat like a cable but 
 in this case the outer conductor is water) . If we create an electrical distur- 
 bance in the space between the metals at one end of the conductor, say by 
 discharging a condenser into it, the complementary plates- being connected 
 to the inner and outer metallic coatings, then an electric disturbance in the 
 form of a pulse will run along the conductor, in the sense that a disturbance 
 of the otherwise statical electrical conditions will appear to run along the 
 conductor much as a wave, representing a disturbance of the statical conditions 
 of the liquid surface, runs along a water trough when started at one end. 
 The general case discussed is one that really involves the application for 
 a short time of a periodic disturbance at one end so that a group of electric 
 waves runs along the conductor with a definite velocity of propagation and 
 wave length. 
 
 If we send a disturbance along the inner conductor these will be, as we 
 have seen, a complementary disturbance in the outer conductor and the two 
 are connected across with one another by the electrical field in the dielectric 
 shell between. The apparent current in the inner conductor at any point 
 produced by the instantaneous disturbance of the field is thus accompanied 
 down the cable by the opposite and equal complementary current in the 
 outer conductor. Thus the energy in the wave or disturbance is locally 
 distributed and does not spread much, the outer conductor preventing this 
 spreading. This means that we can calculate the induction and capacity of 
 the cable as so much per unit length, because they arise merely as local 
 effects between neighbouring elements of charge and current. We shall thus 
 assume that our cable has a self-induction a per unit length at the point 
 distant s from the one end where the disturbance is started, a capacity b per 
 unit length and also a resistance k. There may also be a certain dissipation 
 by leakage across between the conducting core and the outer sheath, but we 
 shall for the present neglect any effect of this kind. 
 
 510. The current in the inner conductor will be measured by the amount 
 of charge crossing the typical section per unit time, and may be denoted by 
 
 dQ 
 dt' 
 
 so that the kinetic energy of the current at this place will be of amount 
 
 1 afdQ\ z 
 
 2 ' c \dt ) 
 per unit length, or in all 
 
 _ / ~ ' L ct I (lty\ j 
 T = \ - [-ji } ds, 
 
 integrated along the cable of length Sj 
 
454 Electrodynamics of linear currents [OH. xi 
 
 The charge on an element of length of the cable will be then equal to the 
 difference of the charges which have crossed the sections bounding this length 
 and will therefore be 
 
 QH, 
 
 j U/d, 
 
 as 
 
 and since the capacity of this element is bds the potential energy of this 
 charge element is 
 
 1 
 
 26 
 
 or in all the potential energy will be 
 
 There is then the dissipation function which is similarly seen to be 
 
 The coefficients a and 6 may change along the cable, that is they may be 
 functions of s, but for the present analysis to be valid their rate of change 
 must be slight in a length comparable with the wave length of the dis- 
 turbance ; the properties of such cables are not to change perceptibly in a 
 length equal to that of the shortest wave transmitted along it. This 
 implies that the cables must be uniform in a length large compared with the 
 dimensions of their cross section (200 to 300 times is practically sufficient). 
 The values of a, b and k will of course depend on the distribution of the 
 current in the cross section. It appears that if the alternations are slow, 
 as in telegraphy, the current practically goes full bore, or is uniformly 
 distributed over the cross section, but if the period is very small the current 
 is confined to a very thin layer at the surface of the conductors. 
 
 511. There is only one electric variable and we may use any of the 
 usual methods of obtaining the equation of motion. The general method 
 involves a use of the principle of least action, which, for a system with no 
 dissipation and for which the potential and kinetic energies are W and T 
 respectively, is expressible in the form 
 
 If however there is a dissipative function for the system it must be introduced 
 as follows. If the generalised coordinates of the system are 6 1} 6 2 , ... and 
 the corresponding velocities are 1} O z , ... then the equation of action is 
 modified by adding 
 
510-512] The equation of propagation 
 
 so that generally it assumes the form 
 
 455 
 
 * 
 
 - 0. 
 
 In our case this equation is 
 81 dt J 
 
 The variation of the first integral with respect to the 'single independent 
 variable Q is 
 
 'a dQ d ^^ I dQ d 
 
 ji (v) r ~j~ T 
 It o as CLi 
 
 wherein we have the two different variations -y- (SQ) and -y- (&Q), which are 
 
 not however independent. We must therefore get rid of them by integration 
 by parts : this part of the variation is then equal to 
 
 dt 
 
 ti -' \c 
 
 a dQ 
 
 c dt 
 
 SQds 
 
 2 2 j< f d a d Q\ snj 
 dt -=- - -- ] SQd 
 
 i Jit J 8 dt Vc dt j 
 
 and the total variation is thus equal to 
 
 o \c t as \b ds J dt 
 
 and this has now to vanish whatever 8Q may be. The arbitrary nature of 
 8Q not only ensures that the terms at the limits vanish independently but 
 that at every point of the conductor and at any time 
 
 dtc'dt ds\b ds dt 
 
 This is the differential equation which tells us how the disturbance travels. 
 The terms at the limits give us the conditions at the ends of the cable and 
 for the initial and final displacements and need not further trouble us. 
 
 512. If the induction and capacity are constant along the cable the 
 general equation becomes 
 
 or 
 
 c dt 2 
 d 2 Q 
 
 b ds 2 
 
 kc dQ 
 a dt 
 
 d 
 
 _ 
 
 ' ' 
 
 dt 2 a dt ab ds 2 * 
 
 This method of deriving this equation is that consistent with a strict dynamical 
 theory of the subject. The argument can however be interpreted in purely 
 
456 Electrodynamics of linear currents [CH. xi 
 
 electrical language and the same equations are obtained. Our method of 
 deduction from the principle of least action is however independent of any 
 special argument at all. 
 
 The solution of this equation is obvious. It is identical in form with 
 the well known equation of propagation of damped waves along a string. 
 The general type of solution is obtained as 
 
 Q = Q e i(nt + ms \ 
 
 where n is real, being the period of the wave, and m and n are connected 
 
 fv 
 
 by the equation 
 
 a ( 
 
 _ . _ I r 
 
 I . 
 
 c \ a J 
 
 so that m is partly real and partly imaginary ; say 
 
 ab ( iknc\ 
 
 mt _ . _ I ry\ t I 
 - 
 
 so that Q=Q e-"e i( * t+ . 
 
 The current is thus damped out as it goes along the wire, by the frictional 
 
 resistance. 
 
 nf* 
 
 513. If the resistance is small, or at least if the period is such that - 
 
 a 
 
 is very small then the imaginary part of m, is negligible, and the wave will 
 travel along without any appreciable decrease in its amplitude. This con- 
 dition is satisfied if n is large enough, that is if the oscillations are very fast. 
 We have already seen the significance of this statement : the inertia of the 
 current is then so large that the ordinary resistance, depending on p only, 
 is comparatively ineffective. In such a case the equation assumes the 
 simpler form 
 
 d*Q [ = c^^Q 
 
 At* ~ ab ds 2 '. 
 of which the general solution is 
 
 f* 
 
 where c, 2 == -,-. 
 
 ab 
 
 The electric disturbance then travels along the cable in a simple permanent 
 wave form with velocity 
 
 ' 1 = V ~al' 
 
 514. If the resistance and the dissipation depending on it are not too 
 big t it is possible to approximate to the general form of solution of the 
 equation 
 
 9 2 ^Q_ 1 WQ 
 
 8*' ~ kb Tt~ 
 
512-514] The solution of the equation 457 
 
 but the most general case requires more careful treatment. The most 
 elegant solution is obtained by a method due to Riemann*. We first trans- 
 
 form the equation by writing 
 
 Q = qe-^it, 
 
 then, if 2X 1 = &6c x the equation for q reduces to 
 
 d * q 192(? + A^-0 
 to* tfW^ 
 
 We next notice thatf 
 
 is a solution of this equation which we can temporarily denote by q'. It 
 follows then that , 
 
 d ( ,dq dq'\ _ 1 8 / , dq dq' 
 
 ~ 
 
 ( ,dq dq'\ _ 1 8 / , q q\ 
 
 ( q ds~ q d^)~c^dt( q di~ q to)' 
 
 so that the integral 
 
 '\ ds 
 
 taken round the boundary of any region in the (s, t) plane, in which q, q' 
 are continuous functions with continuous derivatives, must vanish. Such a 
 region is that bounded by the axis of s and the lines 
 
 s c-^t = s x c 1 t 1 , 5 + 0^ = $! + c^! 
 
 and on the two latter lines q' = 1 and ds = c^dt. We conclude therefore 
 by taking the integral in its three parts that 
 
 ,dq 
 ~ 
 
 This formula determines q as a function of the time ^ and position s t on the 
 cable in terms of the values of the function at the time ^ = between the 
 points s a + c^ and Sj c^ x . If the time ^ = is the initial instant of 
 starting the signal and the conditions then are specified at all points by 
 
 Q-t-fW, !-<>. 
 
 it follows that at the s r point at time t the function q is determined by 
 
 /i4c!< r \ f> f (j "i 
 
 2? =/(*i - c,t) +f( Sl + c,t) + q(s) + -f(s) 7 (z) ds, 
 
 where now z ^Vc^t 2 (s Sj) 2 . 
 
 * Riemann-Weber, Die partielle Differentialgleichungen dr mathematischen Physik, Bd. n, 
 (4th Ed.), p. 322. 
 
 f 7 (z) is the zero Bessel function with imaginary argument. 
 
458 Electrodynamics of linear currents [CH. xi 
 
 515. To obtain some insight into the nature of the solution thus obtained 
 let us take the particular case given by Heaviside where 
 
 f(s} = 9(s} = Q for -co <s<0 
 f(s)=g(s) = Q for < s < + oo 
 
 so that 2Qe^ =Q Q + Q f [~/ (z) + ^ ~1 ds, 
 
 si-dt L z az J 
 
 provided s cj < ; for all other values Q = 0. This means that the wave 
 disturbance extending backwards originally from the origin expands itself in 
 the positive direction of the cable with velocity c 1 . The front of the wave 
 is at all times distinctly marked but the amplitude of the disturbance in it 
 is gradually being reduced on account of the resistance. 
 
 The case of an impulse of finite length initially can be obtained by a 
 combination of such solutions slightly displaced relative to one another. It 
 appears in the general case that any definite wave figure travels along the 
 cable with a general distinctness as regards its terminations fore and aft, but 
 the resistance causes dissipation and distortion of the signal as it proceeds. 
 In addition to this the wave leaves behind a general trail of disturbance of 
 small but finite amplitude which itself gradually dies away by dissipation. 
 
 516. It may however happen that the resistance is so very large or the 
 period of the disturbance so very long that the damping term in the equation 
 is of considerable importance and perhaps more important than the other part. 
 The affair does not then travel in a wave at all because it is damped out 
 very quickly. The wave characteristic is thus eliminated by the resistance ; 
 in such a case the electric inertia in the circuit is comparatively inoperative 
 and the elasticity works against the resistance ; the circumstances are those 
 of the diffusion of a charge along the cable and no wave motion exists. A 
 sharp well-defined disturbance sent in at one end of the cable would then 
 arrive at the other end in a very weak and distracted form. This is of course 
 not desirable in signalling, where a sharp signal should turn out sharp and 
 distinct at the other end. 
 
CHAPTER XII 
 
 ELECTROMAGNETIC OSCILLATIONS AND WAVES 
 
 517. The general problem with electromagnetic waves. In the previous 
 chapter we have confined our attention entirely to stationary or quasi- 
 stationary electromagnetic systems, i.e. systems in which the time of 
 variation is small compared with the time taken by radiation to cross the 
 system and we have in consequence found it unnecessary to consider the 
 process by which the varying conditions established in one part of the field 
 are smoothed out over the whole of the field by radiation. We shall now 
 turn to the other side of the matter and make a special investigation of 
 the radiation processes by which a given state of affairs is transferred from 
 one part of the field to another. We have already had cause to investigate 
 in a previous chapter a possible source of a very rapidly oscillating field 
 (viz. that associated with a condenser discharging through an induction) in 
 which we can no longer neglect the time taken to smooth out the field, and 
 as this case has an important theoretical as well as practical bearing, we 
 shall examine it more closely by more general methods. 
 
 The general problem is the investigation of the conditions in any electro- 
 magnetic field consequent on a rapid alteration of the conditions in any one 
 part of it ; perhaps by discharging one conductor in the field by connecting 
 it through an induction to another conductor in the same field. Complete 
 generality will be obtained by the disposition of dielectric and other conducting 
 bodies throughout the field, the whole being then included in one scheme. 
 
 518. The generalised scheme adopted by Maxwell for the treatment of 
 these cases has already been set out in detail, the underlying idea being 
 to try and explain the electromagnetic phenomena by means of some 
 action, mechanical or otherwise, transmitted from one body to the other 
 by means of a supposed medium, the aether, occupying the space between 
 them; the mode of action of this medium being completely specified 
 by the two fundamental circuital relations of the theory. 
 
 The essential point of this scheme involves the assumption of a quasi- 
 current in the aether of density 
 
 JL^? 
 4^ dt' 
 
460 Electromagnetic oscillations and waves [CH. xn 
 
 which is equal to the time rate of change of the aethereal displacement. This 
 current in addition to the real displacement current in the dielectrics of 
 density* 
 
 dP^d^ (e- 1)E 
 
 dt ~ dt TT 
 
 being sufficient to secure that all currents flow in complete, cycles. We can 
 thus adopt the two circuital relations of electrodynamics as descriptive of 
 the general state of affairs. In their differential form they are 
 
 477C 
 
 - = curl H, 
 c 
 
 1 dE 
 
 -j- = curl E. 
 c dt 
 
 These equations represent the simplest conception of a general electromagnetic 
 theory of these things. They could not of course be right unless in addition 
 
 div C = 0, 
 and also div B = 0, 
 
 the first being secured by Maxwell's hypothesis and the second indicating 
 that it is B and not H that must be used to count the flux of induction. 
 
 519. These two dynamical equations expressing exact physical principles 
 are independent of the constitution of the substances in which the action 
 takes place. As however they involve four vectors they are not sufficient 
 for a complete scheme and we must again introduce the constitutive relations 
 depending on the nature of the media occupying the field. We have 
 already had these relations; the first one expressing the total current as a 
 function of the electric force is of the form 
 
 C = aE + -i- ~ (eE), 
 
 4:77 dt 
 
 the first term, representing the conduction current, expressing an exact 
 relation as far as experiment can follow it, but the second expresses the best 
 we can do in our theory; it represents however a fairly good approximation 
 to the facts. The second relation between the magnetic induction and 
 magnetic force also assumes the form 
 
 and in the simplest cases //, is constant. This relation is however not so exact 
 as the above. 
 
 * Throughout this chapter where not otherwise specified we shall assume that linear iso- 
 tropic relations hold between the electric and magnetic forces and the induced polarisations 
 respectively. 
 
518-521] The general field equations 461 
 
 These four relations represent Maxwell's complete scheme : adopting the 
 latter we can write the first two in the form 
 
 47ro-E e dE 
 curl H = - - + - -=-, 
 c c at 
 
 , _ u dH. 
 
 curlE = - " =-. 
 c at 
 
 We shall now limit the complete generality of our scheme by the assumption 
 that there are no magnetic bodies present so that we can take /.,= ! every- 
 where. 
 
 520. We deduce at once that 
 4770- dE 
 
 But we have 
 
 = div eE, 
 
 _ 
 
 = ~dt 
 
 = c curl curl E 
 
 = cV 2 E - c grad div E. 
 
 /eE 
 and yet = div (7- + 
 
 -l+'divE, 
 at a 
 
 or 3C + - div (eE) = 0, 
 
 at eo- 
 
 dp 4?rp 
 or again - -\ -- Z = 0, 
 
 at <y 
 
 _4vt 
 or P = Po e ~~^> 
 
 This equation shows that the changes in p are independent of the external 
 electromagnetic influence, so that even if there is an initial electrical volume 
 charge distribution it will decrease very rapidly except in the very improbable 
 case when ea is a very large quantity. We may thus consider that p = 
 always, for we may consider the origin of time to be chosen when p = 0. 
 The equations then become 
 
 * = 
 
 a dt dt* 
 
 eE 
 
 since now div = 0. 
 
 4?r 
 
 521. In general we can neglect the displacement current in the metallic 
 conductors in comparison with the conduction currents. Our equation thus 
 reduces to the form 
 
 4 ^E 
 c 2 ' dt ' 
 
462 Electromagnetic oscillations and waves [CH. xn 
 
 which exhibits the propagation of the electromagnetic disturbances into the 
 conducting substances as a simple process of diffusion. 
 
 In the dielectric parts of the field a = and the equation becomes 
 
 c" 2 dt* ' 
 
 which shows that the electric field in the dielectric can be propagated as 
 a simple wave motion in the medium with a velocity 
 
 c 
 
 vv 
 
 A similar discussion easily shows that the magnetic force is propagated 
 in an exactly similar manner, the equations satisfied by it in the separate 
 media being 
 
 (i) in the conductors 
 
 (ii) in the dielectric 
 
 c*' dt' 
 
 dt* 
 
 These equations represent the characteristic differential equations of the 
 theory. In attacking any problem where there are different regions (con- 
 ducting or dielectric) we of course have to solve the different equations 
 for each region and the corresponding solutions have then to be fitted together 
 or connected by the appropriate continuity conditions at the boundary. 
 Before proceeding we must therefore obtain the boundary conditions. 
 
 522. We first notice that any discontinuities in crossing the surface at any 
 point clearly arise from the distribution over a surface element 8f surrounding 
 that point, for the disturbances propagated from the more distant parts are 
 virtually the same at points on the two sides of the surface whose distance 
 apart is infinitesimal compared with the linear dimensions of S/. Moreover 
 if this is the case the discontinuities will be the same as in the corresponding 
 static or stationary condition of that part of the boundary S/, because the 
 field from it produces almost instantaneously its effect at an infinitely near 
 point. We may therefore conclude at once that, in the general case, 
 
 (i) the tangential electric force must be continuous unless a double sheet 
 distribution exists on the surface. This case is excluded. 
 
 (ii) the normal magnetic induction is also continuous : this follows also 
 as a consequence of the general circuital property of that vector. 
 
 (iii) the total normal electric current component is also continuous. 
 
521-523] Discussion of a particular case 463 
 
 The first and second of these conditions are however not independent for 
 
 we know that 
 
 1< ,_ 
 
 - = curl E, 
 c at 
 
 so that if the tangential components of E are continuous the normal component 
 of B must also be continuous. 
 
 Also since 
 
 477-C 
 
 = curl H, 
 
 c 
 
 we see from the third relation that unless there are surface current sheets 
 at the surface the tangential components of the magnetic force must also be 
 continuous. 
 
 There are thus in all two independent boundary conditions which have 
 to be satisfied. 
 
 The previous general equations with these boundary conditions provide 
 us with a complete scheme of equations for all electromagnetic wave problems. 
 Before however proceeding to the consideration of particular problems we 
 will apply these results to the general case discussed above with the additional 
 assumption that all the conductors in the field are perfect. 
 
 523. In this case we have a = for all the conductors and therefore inside 
 them 
 
 V 2 E - 0, 
 
 and also V 2 H = 0, 
 
 which combined with the general results 
 
 div E = and div H - 0, 
 
 show that inside the conductors 
 
 E = H = 0; 
 
 there is no field inside the conductors. In external space on the other hand 
 we have still 
 
 c 2 dt* ' 
 
 and also V 2 H -^ ^-^5 
 
 c 2 at 2 
 
 but if the alternations of the field are not too fast the time variations on the 
 right-hand side containing the very small factor - 2 is negligible and the 
 
 C 
 
 equations can then be written in the simpler form 
 
 V 2 E = 0, 
 and V 2 H = 0. 
 
464 Electromagnetic oscillations and ivaves [OH. xn 
 
 We know moreover from the boundary conditions that the electric force 
 just outside the conductors is normal to their surface but the magnetic force 
 is tangential even in the general case. 
 
 524. The physical explanation of these solutions is now obvious. The 
 electromagnetic field exists only in the dielectric between the conductors. 
 The lines of electric force go across from one conductor to another and the 
 magnetic ones are round about. The positive and negative charges on the 
 surfaces of the conductors are the terminations of the tubes of electric force 
 in the intermediate field. The real propagation of the effects thus takes place 
 in the dielectric medium, a given field being propagated through that medium 
 with a velocity depending only on the medium. Wherever the electric force 
 arrives at a conductor it pulls the electric charges on the conductors (the 
 electrons) about until the statical force due to their rearranged distribution 
 counterbalances the electric force in the field on any one of them, i.e. until 
 there is no resultant electric force inside the conductors. The conductors are 
 full of charges (positive and negative) more or less free which slightly 
 adjust themselves, concentrating on the surface so as to get the necessary 
 field in the interior of the conductors which cancels that of the oncoming 
 wave. If the conduction is perfect the redistribution of charge at each 
 instant takes place instantaneously and thus the field in the conductor right 
 up to its surface is annulled ; the charge on the conductor creating the induced 
 electric force is entirely on its surface. If the conduction were not so good, 
 the electrons would not be so free and the electric field would at each instant 
 penetrate into the conductors a little way before being annulled by the reaction 
 of the field due to the electrons which it pulls about. If the conduction is 
 good the cancelling takes place instantaneously at the surface. 
 
 This is the general idea of the phenomena. Before electrons were 
 discovered one was however not able to put the matter so definitely. As 
 early as 1884 however Poynting, Hertz and Heaviside emphasised the point 
 that where a current is used to transmit the power the energy travels in the 
 dielectric round about, which is the real elastic thing, the conductors only 
 acting as guides to prevent the disturbance spreading. 
 
 525. There is a rough mechanical analogy in the propagation of waves in 
 an elastic medium with holes in it. In this case the transverse waves or waves 
 of shear travel along through the elastic material adjusting itself by material 
 deformation of the surfaces of the holes so that there is no elasticity inside 
 the holes. In the electrical case the dielectric is the elastic medium through 
 which the field is propagated by wave motion ; this field (or the elasticity 
 in it) is annulled at the surfaces of the conductors (the holes) by the pulling 
 about of the mobile electrons on their surface. The conductors thus appear 
 as places where there is no elasticity, where the electrical elasticity of the 
 aether is annulled by the mobile electrons. 
 
523-526] The solution of the field equations 465 
 
 The property of perfect conductors thus appears to be merely a negative 
 one, viz. that of cancelling the elasticity of the aether. In imperfect con- 
 ductors there is a damping action and the elasticity is only partially annulled. 
 
 We have thus the general idea that the whole affair of electric current 
 phenomena is actually in the field outside the conductors, the current merely 
 providing a convenient mode of describing the changes taking place. All 
 the energy is to be found in the dielectric medium surrounding the conductors ; 
 on this view, the heat developed in the conductors is energy which has 
 soaked in as it were from the store in surrounding field. The old method o| 
 describing the phenomena was to say that the energy was that of the moving 
 charges; we know now that the energy of any charge, however small, is 
 in the electromagnetic field which we have learnt to associate with it ; thus 
 ultimately all electric energy is in the aether. The electric charges (or 
 electrons) which are the nuclei with which the fields are associated merely 
 provide the means of communication of the energy from the aether to the 
 matter. 
 
 526. The fundamental equation of wave propagation. We have so 
 far tacitly assumed that any quantity which is determined mathematically 
 by certain scalar or vector component quantities which satisfy an equation 
 of the type 
 
 1 ^ 2A d$ 
 
 V( ?-~2 P+ c ft 
 
 is essentially propagated by a wave motion throughout the field. That this 
 is so follows from our knowledge of such phenomena as, for example, accom- 
 pany the propagation of sound through any elastic medium; but it may be 
 inferred directly as a mathematical consequence of the implied condition 
 involved in the characteristic equation. 
 
 The general problem in the present aspect of the theory is to determine 
 how any electromagnetic disturbance is propagated across space filled with 
 dielectric and conducting masses in any specified configuration, and to see 
 how the conditions at any one point of this field are affected by those occur- 
 ring at any other. The complexity of the conditions involved naturally 
 excludes the determination of a simple solution for the general problem and 
 we must therefore be content with the examination of simpler problems 
 with restricted circumstances. 
 
 We first examine the general case of the propagation of effects from a' 
 specified type of disturbance located in a finite region of an infinite homo- 
 geneous isotropic dielectric with zero conductivity. The disturbance will 
 for simplicity be assumed to be of a continuous character and to have been 
 in operation for an indefinite period previous to the instant at which the field 
 is examined*. 
 
 * More general cases are examined by Love, Proc. L. M. 8. (2), vol. i. (1903), p. 37. 
 
 L. 30 
 
466 Electromagnetic oscillations and waves [OH. xn 
 
 Suppose we enclose the origin of the disturbance by any closed surface/: 
 then at all points in the region outside this surface the field vectors will 
 satisfy an equation of the type 
 
 The function <t> satisfying this equation must necessarily be regular at all 
 points outside / even at an infinite distance and it therefore follows from the 
 general problem analysed in the introduction ( 25-29) that the appropriate 
 form for the function at time t at the typical field point outside / is 
 
 where each integral is taken over the surface / bounding the origin of the 
 disturbance; r is the distance of the element of this surface from the point 
 of the external field where the conditions are examined and square brackets 
 as usual indicate that the functions affected are to be taken for the time 
 
 t-- 
 
 V 
 
 527. Now let us see what this formula means. The conditions at any 
 point in the dielectric medium outside the surface / depend only on the 
 conditions of the field on the surface itself, so that any alteration of condition 
 in the disturbing system inside / affects the external field only through the 
 medium of the field on the arbitrary separating surface. Moreover the con- 
 ditions existing on any element df of this surface at a given instant are not 
 effective at any external point distant r from it until after the time r/c. This 
 suggests the view that the conditions originated at any point in the field 
 travel out from that point into the surrounding field, traversing each part 
 of the intervening field in turn and proceeding from point to point with the 
 velocity c. 
 
 This is the essence of a radiation theory and is exactly analogous to the 
 phenomenon with which we are familiar in the theory of sound, and although 
 it will appear that the type of radiation is essentially different from that met 
 with in all such material phenomena, it is convenient to talk of electro- 
 magnetic waves and radiation in the same sense as we talk of waves of sound. 
 
 528. The analytical formula under review has an important physical 
 significance which it is worth while examining in detail. The potential 
 propagated from a point source variable with the time and of strength f (t) 
 is with the same characteristic equation 
 
526-529] Huyghevts principle 467 
 
 The potential propagated from a doublet consisting of simple sources / (t) 
 and f(t) separated by an interval 8n is consequently 
 
 s d (1 ,/ r 
 
 on -j- \ - f (t 
 
 dn (r j \ c 
 
 Thus for a doublet of strength F (t), the equivalent of/(Z) Sw, it is 
 
 < r 
 
 in which the function jP comes under the differentiation. 
 
 The formula quoted above for </> thus implies that each differential 
 element of the surface / acts, as regards the point P inside it as a complex 
 radiating element consisting of a simple source of strength 
 
 and a normal doublet of strength 
 
 The wave disturbance originated by these elements travels out into the space 
 inside / as a simple spherical wave* propagation. The disturbance at P is 
 thus just the same as if the surface itself acted as a sort of secondary radiator 
 and this is the essence of Huyghens' well-known principle in physical optics. 
 The above mode of deduction of the formula, not free from analytical 
 difficulties, can hardly however be said to throw much light on the character 
 of the simple principle which is thus demonstrated. In this connection 
 however the following discussion due to Prof. Larmor* is of special interest 
 as indicating the exact amount of precision in the specification of the secondary 
 disturbance thereby introduced. Reference may also be made back to the 
 discussions of Green's theorem and the equivalent stratum. 
 
 529. Consider a potential specified throughout all space as follows. It is 
 a function </>, single valued and continuous as to itself and its first gradient, 
 and satisfying V 2 </> = 0, in all the space outside a boundary /, and as a conse- 
 quence diminishing towards infinity according to the law r~ l or higher inverse 
 power : it is zero everywhere inside the boundary. What is the distribution 
 of attracting mass to which this belongs? This distribution is as usual in 
 Green's manner determined by the singularities and discontinuities of the 
 potential function. It consists of a surface density a over / and a double 
 sheet r over / also ; where 
 
 1 8<i 1 
 
 a = T~ ^ > r= ^r ( P> 
 47rdn 4-n-^ 
 
 8w is an element of the outward normal. For it follows by the usual procedure 
 that if <f>' is the potential of this distribution then (<f> (f)') is a potential 
 * Proc. L.M.S. (2), vol. i. (1903), p. 1. 
 
 302 
 
468 Electromagnetic oscillatiQns and waves [CH. xn 
 
 function which has no singularities, or discontinuities throughout all space 
 and is therefore identically null. Expressed analytically the potential at a 
 point in space is 
 
 1 f ia f,7/- , l d 
 
 - o "J + 7- 
 ' J 
 
 in other words the formula gives the value of a potential function $ in the 
 free space outside the surface in terms of the values which it and its gradient 
 assume on the surface, it constitutes in fact the analytical continuation of 
 the function outward from the surface, while inside the surface the value of the 
 expression is everywhere null. In the case of a closed surface as well as that of 
 an open sheet either side may be called the outside for the present purpose. 
 This continuation of the function is necessarily unique and determinate, but 
 the form of the integral expressing it is far from being so. We may in fact 
 generalise the formula immediately in Green's manner. Consider a function 
 cf) which is the potential throughout' space of any assigned distribution of 
 mass. Draw any surface dividing space into two regions A and B each of 
 which contains part of the mass, these parts being represented by M A and M B . 
 What distribution of masses and of surface densities and normal doublets 
 on the surface /is required to produce a potential equal to cf> in the region 
 A and equal to zero in the region .B? Clearly M A together with 
 
 1 d<f> 1 
 
 a = -r- ~, T = (h. 
 
 4:7T OH 4:7T^ 
 
 What distribution is required to make the potential zero in the region A and 
 <f> in the region 5? Clearly M B with the same distribution on the surface 
 but with the sign changed if 8n is measured in the same way. Thus a distri- 
 bution of surface density and normal doublets is found which exactly cancels 
 the effect of M A on the other side of the dividing sheet /; moreover an 
 infinite number of such distributions can be found, for in determining it M B is 
 entirely arbitrary. 
 
 530. The same procedure can now be extended to a scalar potential 
 propagated in time, i.e. which satisfies a characteristic equation involving 
 the time as a variable. 
 
 Consider first the simplest case of a velocity potential </> satisfying 
 
 It is necessary to ascertain what distribution of sources on a surface/ will 
 create given discontinuity in the values of <f>, and of its normal gradient, in 
 crossing the surface, it being clear that such discontinuities in <f> and dfi/dn 
 constitute the most general type, involving only first differential coefficients 
 of <J>, that can exist. 
 
529-531] Huygheris principle 469 
 
 We have already seen that the velocity potential propagated from a 
 doublet consisting of simple sources / (t) and / (t) separated by an interval 
 8n is 
 
 s d I 1 tf t r 
 dn -T- \-t[t 
 
 dn (r j V c 
 and that for a doublet of strength F (t), the equivalent of / (I) Sn, it is 
 
 A. 
 
 dn 
 
 F{t- r - 
 
 in which the function F comes under the differentiation. Within a region of 
 such small extent that the functions f(t) and F (t) do not sensibly change in 
 the time required for the disturbance to pass across it, these potentials are 
 
 of tvpes^-^ and F (t) -^- (-}, so far as thev relate to sources inside the region 
 r dn \rj 
 
 of which / (t) and F (t) are the strengths at this interval of time ; for this 
 modification only neglects lower inverse powers of r than those retained. 
 Thus for such a region enclosing an element 8f of the surface /, and at times 
 for which the functions f(t) and F (t) do not change there abruptly, the 
 potentials, subject to exceptions to be presently encountered in the case of 
 double sheets, take, throughout any time of the order above specified, the 
 form of simple gravitational potentials, the circumstances of propagation not 
 sensibly inferring. 
 
 We are therefore invited to follow the procedure of Coulomb and Laplace 
 for the ordinary potential and investigate the discontinuities arising from 
 a surface distribution of simple sources and one of doublets orientated normally 
 to the surface. The discontinuities, on crossing the surface at any point, 
 clearly arise from the distribution over a surface element S/ surrounding 
 that point; for the disturbances propagated from the more distant sources 
 are virtually the same at points on the two sides of the surface, whose 
 distance apart is infinitesimal compared with the linear dimensions of S/, sb 
 that, as regards their effect, no discontinuities can arise. 
 
 531. Taking first then, the case of a simple 'surface density a (t) spread 
 over S/, which we may take to be uniform all over it at each instant, its effect 
 is to transmit towards both sides a train of plane waves with fronts parallel 
 to S/, which remain plane until the distance n to which they have travelled 
 becomes comparable with the linear dimensions of S/. For them the value of 
 
 / <M\ 
 
 d<f)/dn at a distance n at time t is Sna it - - j , but with different sign on the 
 
 \ C / 9 
 
 two sides; such a surface distribution a (t) of simple sources thus accounts 
 for a discontinuity in d^fdn of amount krra (t), but introduces no discontinuity 
 in (f> itself. 
 
470 Electromagnetic oscillations and waves [CH. xn 
 
 This result now assists us to analyse the circumstances of a sheet of 
 normal doublets of strength r (t) per unit area, for we can replace it by two 
 simple parallel sheets of densities CT X (t) and - CT, (t) at an infinitesimal distance 
 Sw apart, such that CTJ (1) n = r (t). At a point at a distance n from the 
 
 sheet of density + o-j (t) the values of -^ at time t arising from these two sheets 
 
 / n\ ( , n + 8n\ . , . , 
 
 are, as above, 2w<7,. ( ^ ) and T ^TT^ (t - - , m which signs are 
 
 \ c/ \ c / 
 
 to be determined by the sides of the respective component sheets on which 
 this point lies. If the point is not between the sheets, the signs are opposite 
 
 Bn d A n\ . . . . ,, , d(t> , 
 
 and the sum for both is 2?r -- -r a^ (t - - L which is the value of -f- due 
 
 C Cvt \ / CLYl 
 
 to the element rBf; but it has the same sign on both sides of the double sheet : 
 so that in crossing the double sheet there is no discontinuity in the value of 
 
 - , though the element rSf of the double sheet contributes -- -=- S/ to that 
 on c dt 
 
 quantity on each side. But between the sheets the value of -~- arising from 
 them is of a higher order of magnitude, being a sum instead of a difference, 
 
 / f fki\ 
 
 and is 47ro- 1 1 1 - - , or simply kira^ (t}, when CTJ is not discontinuous in the 
 
 \ c / 
 
 time ; and this value integrated across the interval Sn gives a discontinuity 
 in </> itself, on crossing the double sheet, of amount 4778^^ (t) ; that is km (t). 
 
 Collecting these results we see that a discontinuity in cf> over a surface/ 
 
 JSI 
 
 of amount % (x, y, z, t) and a discontinuity in ^- equal to t/r (x, y, z, t) over the 
 same surface are accounted for respectively by a double sheet on the surface 
 
 of strength r equal to r- Y and a single sheet of density a equal to - iL. 
 4-n- A 4?7 r 
 
 532. We are thus in a position to proceed exactly as in the first instance. 
 Consider any system of sources, and let </>, a function of (x, y, z, t) be their 
 potential function in infinite free space. Assign any surf ace / x dividing space 
 into two regions A and E and let m^ and m B stand for the sources as divided 
 between the two regions. What distribution of sources would give rise to 
 a potential equal to </> in region A and equal to zero in region -B? Clearly 
 the sources m^, together with a distribution (cr^r) over /given by 
 
 a^ 
 
 47TCT = ?r~ , 477T = <p ! 
 
 dn 
 
 for if cf>' is the potential arising from this distribution and O is a function 
 equal to (/> in region A and to zero in region B, then O $' will be a potential 
 having no singularities or discontinuities throughout infinite space, and must 
 therefore be null by simple physical intuition, or analytically by the usual 
 
531-533] The solution of the equation with dissipation 471 
 
 type of theorem of determinacy based on the energy of the relative disturbance 
 being of necessity essentially positive. As the total effect within the region 
 B is zero, we can say thus that <f> A the part of it arising from the sources 
 m A outside the region is given at time t by 
 
 r [dn] Jf J an 
 
 fOUL~] 
 ~ are the values of these quantities for the element 8/ at 
 on] 
 
 time t - - and in forming j- , the variation of [ft with regard to the 
 coordinates is calculated only in as far as it involves them implicitly as a 
 
 / T \ 
 
 function of It- - j*. This formula expresses the vibration potential due to 
 
 sources m A within the surface/, throughout the region B outside, as determined 
 by the values which it and its gradient assume on that surface. It is so to 
 speak an analytical continuation beyond the surface of a function satisfying 
 the aforesaid characteristic differential equation. Such a continuation must 
 be unique, and it is determined by the value assumed by <j> alone on the surface : 
 
 as therefore - 7 is determined by a knowledge of </> over the surface, the data 
 an 
 
 for the formula here given are redundant; if arbitrarily assigned they will 
 usually be self-contradictory and the formula thus nugatory. Moreover the 
 formula determines $ A in terms of 'the surface distribution of <f>, equal to 
 $A + ^B j where 4> B is due to an entirely arbitrary distribution of sources 
 within the region B to which the formula relates. Thus the quantities 
 integrated in it are very widely arbitrary and the element of the integral 
 corresponding to 8f in no sense represents any influence actually propagated 
 from that part of the surface. The formula is purely analytical and in no 
 degree a mathematical formulation of the principle of Hughens, relating to 
 propagation of actual disturbance. In fact if m B vanishes the formula 
 represents a distribution of surface disturbances, which does not radiate at 
 all into the region A. 
 
 533. In the more general case when the uniform medium possesses 
 conducting qualities the above analysis is no longer applicable. The general 
 character of the solution in this case can however be demonstrated by another 
 method which is also suited to the simpler problem. 
 
 In this case the characteristic potential equation assumes the form 
 
 * This point was overlooked by Prof. Larmor in the original paper. 
 
472 Electromagnetic oscillations and waves [OH. xn 
 
 If we transform this equation to a spherical polar coordinate system, then 
 multiply it by r z da>, where da> is the element of solid angle at the polar origin, 
 and finally integrate over the unit sphere it reduces immediately to the 
 form 
 
 8 2 d> , 80 
 
 T f 
 
 where O = 7- 4>daj, 
 
 *7T J r 
 
 Let us now examine the propagation of conditions from an initial disturbance 
 specified by 
 
 <f>-f(x,y,z), --^ = g(x,y,z) 
 
 at the time t = ; the propagation is assumed to take place, in a uniform 
 isotropic medium possessing dielectric and conducting properties corre- 
 sponding to the equation chosen. 
 
 We first transform the functions / and g to the same spherical polar co- 
 ordinates and then use 
 
 so that F (r) and G (r) are respectively the initial values of <I> and -^- . 
 
 We are of course concerned only with the positive values of r so we can 
 choose the functions F and G for negative values as we please. We choose 
 them so that 
 
 F (- r) = - F (r), G (r) = - G (- r). 
 
 We then have, by applying the formula obtained at the end of Chapter XI, 
 the general solution for O in the form 
 
 fr+ctr Xrf fJ~] 
 
 - ct) + \G(s) + XF (s) + F (s) ~ 7 (z)ds 
 
 Jr-ct\_ % Zj 
 
 where again Z is the Bessel function of zero order and imaginary argument 
 and 
 
 , 
 
 z = XVc*i 2 -(s- r) 2 . 
 
 We conclude that the average conditions propagated from the disturbance at 
 any point in the field travel outwards radially from that point in exactly the 
 same way as a signal travels along a telegraph cable. In other words if 
 there is no friction the propagation is like that of a simple undamped wave 
 form with the velocity c, but if there is appreciable conductivity rapid dis- 
 tortion and dissipation occur to destroy these simple propagation effects. 
 
 The value of the more general function ^ can be easily obtained from the 
 
533, 534] The electromagnetic theory of light 473 
 
 value found above for O by determining the b'miting value of the ratio O/r 
 as r tends to zero. The result is that* 
 
 + A I \G (s) + Ac* F (s) - r [ - -=- ^ sdsl 
 J o ( z dz} z dz 
 
 = e 
 
 where now 
 
 This formula determines completely the way in which the conditions at any 
 one point in the field depend on those at the other points, and is in complete 
 accord with our physical conception of these things. 
 
 534. The electromagnetic theory of lightf. We have thus far been at- 
 tempting to generalise an explanation of electromagnetic phenomena by 
 ascribing it to some mechanical action transmitted from one body to another 
 by means of a medium, the aether/ filling all space; the dynamical or 
 analytical theory of the activity of this medium being expressed in the repre- 
 sentation of the mode in which electrodynamic action is propagated across 
 free space outlined in the previous paragraph. 
 
 Starting from the general scheme, we showed that it is an essential conse- 
 quence of the fundamental concept of displacement currents introduced by 
 Maxwell that electrodynamic disturbances are propagated through the 
 medium as a wave motion through an ordinary elastic solid with a velocity 
 
 equal to - : we shall soon prove that these electromagnetic waves are 
 
 ye 
 
 necessarily transverse waves of the type with which we are familiar in optics. 
 
 When Maxwell formulated his electrical theory, any such waves of purely 
 electrical origin were entirely unknown, so that the introduction of the concept 
 of displacement currents was then a pure hypothesis, unsupported by any 
 experimental evidence. 
 
 It was soon found however that the theoretical behaviour of these electro- 
 magnetic waves was governed by exactly the same laws as had been found 
 through many years of combined theoretical and practical investigation to 
 apply in the corresponding phenomena in physical optics, which was the one 
 great branch of physical science, for the description of the phenomena in 
 which it was found essentially necessary to presume the existence of transverse 
 waves. The analogy thus suggested between the two sets of phenomena is 
 moreover more than a mere qualitative one; as Maxwell soon found, it is 
 quantitative as well and to an extent that led him to formulate the 
 
 * Cf. Riemann-Weber, Die partielle Differentialgleichungen, etc. n. pp. 299-312 (4th Ed. 1901). 
 t Cf. Maxwell, Treatise, i. Ch. xx. f 
 
474 Electromagnetic oscillations and waves [en. xn 
 
 opinion that the waves in light are in fact identical in type with the electro- 
 magnetic waves which his theory predicts. This is the now famous electro- 
 magnetic theory of light, about the correctness of which there are now hardly 
 any doubts. 
 
 The velocity of electromagnetic waves in a vacuum is equal to a constant 
 c introduced originally as a physical constant in the fundamental equations ; 
 and the value of this constant which can be determined by purely electrical 
 measurements turns out to be 
 
 c = 3.10 10 cms./sec., 
 
 which is identical within the very close limits of experimental error with the 
 velocity of light in vacuo. This identity was known previous to Maxwell's 
 time. 
 
 535. In any other medium than a pure vacuum the velocity of radiation 
 is 
 
 c 
 
 GI =. -ji , 
 
 Ve 
 
 where e is the specific inductive capacity of the medium. Now in optics 
 the ratio 
 
 is defined as the index of refraction of the medium under consideration relative 
 to a vacuum ; and thus if Maxwell's surmise is correct we must expect the 
 square of fhe index of refraction of a medium to be equal to its dielectric 
 constant, properly defined. It was usually inferred that the proper value 
 of the dielectric constant to be used with this relation was the simple statical 
 one, and it was then found that except in the case of gases and a few other 
 substances which show but little or no dispersion, the relation was by no 
 means, verified in actual experience. For instance water has an index of 
 refraction of about V33 and a dielectric constant of 81. But it must be 
 remembered that the dielectric constant introduced as a physical constant in 
 the relation between the complex electric displacement D and the electric 
 force E producing it, viz. 
 
 D= ^E, 
 
 4:7T 
 
 can only be a definite constant in the statical theory when it depends only 
 on the simple internal statical forces tending to annul the polarisation induced 
 in each molecule or molecular group by the external field. In the more 
 general case when it is a question of rapidly varying fields such as those in 
 radiation, it is necessary to consider what effect the inertia of the molecules 
 will have on the setting up of the state of polarisation in them, and the rela- 
 tion between the displacement produced and the force producing it will be 
 of a more complex type fc A 'priori, we might expect that the value of e will 
 
534,535] The dielectric constant for oscillating fields 475 
 
 be a function of the rate at which the field is varying and the value for steady 
 fields may therefore be quite different from that for rapidly changing 
 fields. This point may be illustrated in further detail by an appeal to 
 the ideas briefly discussed above in Chapter V where the conception of 
 dielectric polarisation was defined in terms of the electron constants of the 
 molecules of the substance. As there explained the polarisation consists in 
 the small relative displacements of the negative electrons in the atoms or 
 molecules and in the statical theory it is only the quasi-elastic forces holding 
 the electrons which are effective against the action of the applied field ; but 
 in the more general case the inertia of the electrons themselves will become 
 effective and the relation is then necessarily of a more complex type. If we 
 denote by m the mass of the typical electron with charge e and use all the 
 other notation as in the previous discussion the more general equations of 
 motion of this electron will now be of type 
 
 mx = e (Ej. + aP x ) kx, 
 
 and it is only in the case that E, and therefore also P, are independent of the 
 time that the solution is obtained under steady conditions by the equation 
 
 - e (E x + aP x ) - kx. 
 If we assume that the applied electric field is varying in a simple harmonic 
 
 manner with a period . as would, for instance, be the case were it the electric 
 
 P 
 part of an applied radiation field, all the functions may be taken to be 
 
 dependent on the time by the imaginary exponential factor e ift and then we 
 
 shall have 
 
 (- mp 2 + k) x = e (E x + aP x ), 
 
 
 
 so that x = T g (Ej. + fflPa), 
 
 fc *~~ iwip 
 
 the displacement of each electron is therefore proportionately larger in the 
 present case in the ratio k: k mp 2 . The dielectric constant is then just 
 as before 
 
 z 
 
 4:7T2j 
 
 k mp 2 
 
 c-i-f -f- 
 
 1 -2 
 
 k mp 2 
 
 and is therefore in general a function of p, as is in fact required by experience. 
 If we write 
 
 then it is easily seen that - is the period of free vibration of the typical 
 electron about its position of equilibrium in the molecule. 
 
476 Electromagnetic oscillations and waves [CH. xu 
 
 If the period of the incident light is very long compared with the different 
 periods of free vibration of the internal electrons this formula reduces in fact 
 to the statical form deduced above, viz. 
 
 536. Now the formula just determined for e is precisely of the type 
 which has been found necessary to account for all the phenomena of dispersion 
 in solid, liquid and gaseous bodies, and although the theory cannot be said to 
 be more than a descriptive one, it indicates the general lines along which the 
 explanation of the phenomena associated with dispersion must be sought. 
 
 In those bodies which show an exceedingly small dispersion we might on 
 this general theory expect some agreement in Maxwell's relation interpreted 
 with the statical dielectric constant. The gases are the best examples of 
 such bodies and here the agreement is remarkable ; for instance in air under 
 ordinary conditions we have 
 
 Vt = 1-000295, 
 whilst the index of refraction is 1 '000294. 
 
 It has also been found that the results of experiments with long Hertzian 
 waves can be similarly interpreted. For instance it is found that for these 
 the index of refraction of water is 9*, whose square is exactly equal to the 
 dielectric constant 81, statically determined. 
 
 The theory in its most general form is thus perfectly consistent with our 
 experience, and there can therefore be no doubt whatever of the fact that 
 light is an electromagnetic phenomenon of the specified type. A very 
 extensive branch of physics is thus brought into line as a chapter in electric 
 theory; its complete development would however take us beyond the scope 
 of the present work, but reference may be made to any of the standard works 
 on optics. 
 
 The one great advantage possessed by the electromagnetic theory of light 
 over the older elastic solid theories is that it definitely excludes the possibility 
 of the existence of longitudinal waves, the absence of which was rather difficult 
 to account for as a pure matter of elastic solid theory. It also provides simple 
 physical foundations for the necessary surface conditions governing the 
 phenomena of reflection and refraction which will be discussed in detail at 
 a later stage. 
 
 537. The Hertzian oscillatorf. We have now seen that it is an essential 
 consequence of Maxwell's theory that electromagnetic disturbances are pro- 
 
 * A. D. Cole; Wied. Ann. LVH (1896), p. 290. Cf. also Fleming, Principles of Electric Ware 
 Telegraphy, p. 320. 
 
 t Ann. d. Phys. xxxvi. (1889), p. 1 ; Electric Waves, p. 137. 
 
535-538] 
 
 Hertz's oscillator 
 
 477 
 
 pagated as a wave motion through dielectric media. We have also seen in 
 Chapter XI that the discharge of a condenser of capacity & through an 
 
 induction a is oscillatory with a period 
 
 It follows therefore that 
 
 electric waves of a simple harmonic type must be passing through the dielectric 
 medium surrounding a condenser in the act of discharging. In 1887 Hertz 
 succeeded by a wonderful series of experiments in demonstrating the existence 
 of these waves, and thus established the general validity of Maxwell's as- 
 sumption. There were two difficulties which Hertz had to overcome in his 
 work. The first one was to construct a suitable form of condenser to give 
 reasonably short waves, the second was to discover a suitable means of 
 detecting electrical and magnetic forces reversed some million times per second 
 and only lasting for an exceedingly short time. Tire means by which Hertz 
 overcame the first difficulty has .been fully explained above in another con- 
 nection. He used a form of condenser similar to that exhibited in the figure. 
 A, B are two metallic plates which in Hertz's original experiments were of 
 
 Fig. 78 
 
 zinc and about 40 cms. square : to these were soldered brass rods C, D 
 (30 cms. long) terminating in brass balls E, F. Such a condenser has very 
 little capacity and induction. In order to excite the waves the balls E, F 
 are charged to very different potentials by connecting C, D to the terminals 
 of an induction coil. When a sufficient potential is attained sparks cross the 
 air gap between E, F which then becomes a conductor and the charges on 
 the plates can then oscillate backwards and forwards like the charges on 
 the coatings of a Leyden jar. 
 
 538. To obtain some idea of the radiation field in the dielectric surrounding 
 a condenser of this type we may notice that the main and most vigorous 
 part .of the electrical motions occurs in the wires C, D and across the discharge 
 gap EF, so that to all intents and purposes the field should be symmetrical 
 round an axis along the wires and also about the origin mid-way between 
 the balls E, F with the magnetic force in horizontal circles round this axis. 
 To obtain a solution of the fundamental equations of this type we shall find 
 it most convenient to refer the field to a system of spherical polar coordinates 
 with the pole at the origin and axis along the axis of symmetry of the field : 
 we shall then have jj _ -a _ jj __ Q 
 
478 Electromagnetic oscillations and waves [CH. xn 
 
 and thus, assuming for the present that the dielectric medium surrounding 
 the apparatus is a pure vacuum the remaining equations of Ampere give 
 
 1 rfE, 13 
 
 c dt r 2 sin 080 
 
 1 dE 13 
 
 sin 6], 
 
 sm 
 
 r = a a 
 
 c dt r sm or 
 
 whilst the third equation of the Faraday type becomes 
 
 Thus if we write = rH^ sin 0, 
 
 1 80 1 dE e 1 
 
 c dt r 2 sin 80 ' c dt ~ r sin 30 ' 
 and on using these values in the last equation we find on putting /x = cos 
 
 _ 
 
 dt 2 " dr 2 ~^ fy*' 
 
 A simple solution of this equation is obtained by putting 
 
 = JK (1 - ^ 2 ), 
 
 and regarding R as a function of r only : this function satisfies the equation 
 
 1 d z R d*R 2R 
 c 2 dt z ~ dr z " r 2 ' 
 of which the general solution is easily verified to be 
 
 _ d (F 1 (ct - r)\ d (F 2 (ct + r 
 
 **' - ' J 1 I ' ~T~ I 
 
 dr \ r J dr\ r 
 
 Of the two parts of this solution the first will represent an expanding wave 
 whilst the second represents a condensing wave : it is with the first alone that 
 we shall be concerned and we shall also find it more convenient to write 
 
 so that t/j = sin 2 6 if" (ct - r) + l -f (ct -r)l. 
 
 It is then easily verified that 
 
 sin0 
 
 ) + f (ct - 
 
 dashes being used to denote differentiation of the function / with respect 
 to its argument. 
 
538-541] Hertz's oscillator 479 
 
 539. Near the origin, that is near the oscillator itself, the electric field 
 reduces to 
 
 2/(cOcos0 /(cQainfl 
 
 E r = T3 > ^ = -^ ' 
 
 the other terms being very small compared with these : the magnetic field 
 reduces to 
 
 /' (ct) sin 
 
 **tf> ~ "" 9 ' 
 
 ipm 
 
 Here then we have the type of solution obtained. In the immediate 
 neighbourhood of the system the electromagnetic field is identical as regards 
 its electric part with the electrostatic field of a simple doublet of strength 
 at any time given by / (ct), while as regards its magnetic part it is identical 
 with the field of the current produced in the changing of this doublet. It 
 is only so far as the actual electrical motions in the vibrator described above 
 approximate to this simple specification that the solution obtained will 
 represent the field actually investigated by Hertz; in any case however it 
 indicates the type of solution to be expected. 
 
 540. At a great distance from the origin it is the other terms of the 
 solution which become appreciable and if we assume that the function / 
 and its differential coefficients are of the ordinary type of regular function 
 then we may put 
 
 f"(ct-r)sme 
 
 & r = U, *A 9 = - - , 
 
 /" (ct - r) sin 6 
 
 whilst E* -, 
 
 r 
 
 and this is of course probably more representative of the conditions in the 
 actual case than the field near the oscillator is likely to be. 
 
 541. Although it is hardly representative of the conditions realised in 
 actual practice we may for the present assume that the dissipation of the 
 energy in the system is so slight that the oscillations are maintained for a 
 considerable time and as the motion is oscillatory, we may represent it in 
 such a case by taking f (x) = A sin px, 
 
 so that the surrounding field is determined by the force vectors "in it which 
 at the point (r, d, </>) are given by 
 
 2 A cos 6 , 
 E r = 3 {pr cos p (ct r) + sm p (ct r)}, 
 
 E 9 = 3 {pr cos p (ct r) + (1 p z r 2 ) sin p (ct r)}, 
 
 Ap sin 6 , 
 H,*, = - ^ { pr sm p (ct r) + cos p (ct r)} , 
 
 all the others being zero. 
 
480 
 
 Electromagnetic oscillations and waves [CH. xn 
 
 Thus the intensity of the field at any point is oscillating in full accord 
 
 with the vibrator with the period , The actual conditions are best 
 
 P 
 
 exhibited by plotting the lines of force in it. The equations to these lines 
 are easily obtained, being in fact in any meridian plane the lines along which 
 ifj is constant. Several of these curves have been drawn by Hertz and are 
 
 Q 
 
 depicted below ; they correspond to the instants t = 0, - , = , - - when r is 
 
 T: ^ * 
 
 half a complete period. From these figures we see that the lines of force 
 running between the opposed parts of the vibrator gradually expand in all 
 directions. This expansion goes on continuously but the ends on the vibrator 
 itself slowly close up and finally coalesce; the line then breaks itself away 
 and travels out into space as a closed line of force. This breaking away 
 of the field and its travelling to a distance is the essence of the radiation 
 emitted by a vibrator of this type. 
 
 542. The distant field in this case is specified by 
 
 _ p 2 A sin 6 sin p (ct r) 
 
 ~ r ~> 
 
 p 2 A sin sin p (ct r) 
 
 so that the surfaces over which the phase of the vibration is constant are 
 the spheres 
 
 r const. 
 
 To use the ordinary phraseology we may call these the wave front surfaces : 
 each of them is advancing outwards with a velocity of linear expansion 
 equal to c. 
 
 Fig. 79 
 
541, 542] 
 
 Hertz's oscillator 
 
 481 
 
 We see that in the present case also the directions of both the electric 
 and magnetic vectors at any point in the field are tangential to the wave 
 front, and they are in addition of equal magnitude and perpendicular to one 
 another. The electromagnetic waves emitted by a vibrator of this type are 
 therefore transverse waves. We shall see presently that properties of the 
 
 Fig. 80 
 
 Fig. 81 
 
 waves here illustrated by a special case are characteristic of the waves 
 produced under all circumstances and, combined with the fact that the velocity 
 of their propagation in a vacuum is c, a velocity identical in magnitude with 
 the velocity of light, they point to the conclusion that electromagnetic waves 
 however they are produced, differ from light waves only in the magnitude 
 of their wave length." . 
 
 L - 31 
 
Electromagnetic oscillations and waves [CH. xn 
 
 Fig. 82 
 
 543. We must now say a few words* regarding the method employed by 
 Hertz to examine the radiation fields thus generated. The difficulty involved 
 in the detection of these fields was at once removed by his discovery that the 
 small rapidly alternating electric forces would produce small sparks between 
 pieces of metal very nearly in contact of sufficient regularity to be used as an 
 indicator of the presence of such forces and to investigate their properties. 
 In his first experiments therefore Hertz used as a detector a piece of thin copper 
 wire bent into a circle, the ends being furnished with two balls, or one ball 
 and an adjustable screw-point. The radius of the circle for use with the 
 vibrator described above was 35 cms. and was so chosen that the period of 
 free electrical oscillations in it (when it is used as a condenser) might be the 
 same as that of the vibrations in the oscillator, so as to take full advantage 
 of the resonance effects described in the previous chapter; for this reason 
 also it is desirable to have as small a resistance in this detector circuit as 
 possible. 
 
 Now as to the explanation of the action of the detector. It acts in the 
 way that all resonators do by a sort of induction process, where the on-coming 
 wave motion continually releases the natural elastic restraint of the system 
 and thereby induces into existence the oscillations of which it is capable. 
 In order to start the oscillations in any electric circuit of the type under 
 consideration in which there exists a small air gap it is first necessary to 
 create an electric field (i.e. a state of strain) across the small air gap between 
 its metallic ends and then to release it. The usual method of doing this is 
 to charge the metallic ends oppositely until a discharge is effected across the 
 gap ; but if it is possible to move the necessary condition of field up through 
 
 * Cf. the article 'Electric Waves,' by J. J. Thomson, in the Encyclopedia Briiannica, where 
 a more detailed description of the apparatus and methods is given. 
 
543-545] Other types of oscillators 483 
 
 the dielectric medium the same result is produced. Thus in Hertz's experi- 
 ments sparks should appear in the detector as soon as the line of centre of 
 the balls is placed along the direction of the electric force and they will be 
 the more vigorous the greater the intensity of the field. This was actually 
 found to be the case and rough as these experiments necessarily were the 
 results obtained were found to be in general agreement with the predictions 
 of theory. 
 
 544. On some other types of electrical oscillators. Although they are of 
 no real practical importance it is of interest theoretically to consider certain 
 other means of generating oscillating fields of the type of those just considered ; 
 but in cases which are more susceptible of rigorous mathematical examination. 
 Some of these cases have, it is true, been submitted to experimental examina- 
 tion with results in agreement, as far as it is possible to follow them, with 
 the theoretical predictions. 
 
 If the distribution of electricity on a system in electrical equilibrium is 
 suddenly disturbed, the electricity will redistribute itself so as to go back 
 to the distribution it had when in electrical equilibrium. If for instance the 
 original distribution is induced by a field which is suddenly altered or removed 
 altogether the induced distribution will have to readjust itself, being no 
 longer in equilibrium. This readjustment will be accomplished very rapidly 
 owing to the free mobility of the charges, and the new distribution will soon 
 be attained; but in it the charges will have considerable motional energy which 
 will take them beyond the new distribution and an oscillation will result. 
 If the damping is not too big this oscillation may continue for some time. 
 
 545. As an example of the principles here involved we may consider the 
 oscillation of the charge induced by a uniform field of force of intensity E* 
 on a perfectly conducting sphere (radius a) when this field is suddenly 
 removed. The original density of the charge is 
 
 3E cos 6 
 
 n = i 
 
 4rr 
 
 d being the polar angle from the radius of the sphere parallel to the lines of 
 force of the original field. When the inducing field is removed this distribution 
 will tend to annul itself by currents flowing over the surface of the conductor ; 
 by the symmetry of the whole affair it follows that this surface current flux 
 will be entirely confined to the meridian planes and the external field will 
 
 * Cf. J. J. Thomson, L.M.S. Proc. xv. (1884), p. 197; Recent Researches, p. 361. Other 
 cases have been examined by A. Lampe, Wien. Ber. cxn. (1903), p. 37; P. Koh^ek, Ann. d. 
 Phys. LVIII. (1896), p. 271; J. J. Thomson, Recent Researches, p. 373 et seq. ; J. Lannor, Proc. 
 L. M. S. xxvi. (1894), p. 119; Rayleigh, Phil. Mag. XLIH. (1897), p. 125; E. H. Weber, Ann. 
 der Phys. vin. (1902), p. 721; Riemann- Weber, Partielle-diff. Gleichungen, etc. n (1901), p. 348; 
 M. Abraham, Ann. d. Phys. LXVI. (1898), p 435, Math. Ann. LII (1899), p. 81. 
 
 312 
 
484 Electromagnetic oscillations and waves [on. xn 
 
 at every instant be symmetrical about the polar radius and of the type of 
 field just investigated where the magnetic force is in circles round the radius 
 and the electric force in the meridian planes. 
 
 We refer the whole system to a spherical polar coordinate (r, 6, c/>) frame 
 with the pole at the centre of the sphere and the axis along the direction 
 of the original field and we then follow the hint suggested by the analysis 
 of the previous paragraph and try solutions of the fundamental equations 
 for the external (vacuum) field 
 
 E* = (f" (ct -r) + -f (ct-r) + -J(ct-r)} t 
 
 f \ f f% J ^ ' > 
 
 The corresponding vectors for the internal field are of course all zero, since 
 the sphere is perfectly conducting (a oo ). 
 
 The type of solution here assumed corresponds of course to the case where 
 the field is an expanding one and it is thus tacitly assumed that there are no 
 reflectors anywhere to send it back again, thus producing the corresponding 
 condensing wave. 
 
 546. The material of the sphere being perfectly conducting there will be 
 nothing except the very slight inertia of the electrons themselves to prevent 
 the adjustment of any specified conditions in the sphere and the inertia itself 
 is entirely occupied in interaction with the adjusting field. This means that 
 the distribution of charge on the surface of the sphere at any instant is 
 precisely that which would exist on the sphere in a statical field identical with 
 the instantaneous electric part of the varying electromagnetic field in the 
 neighbourhood of the sphere. The electric force in the field must be 
 therefore strictly normal to the surface of the sphere at any instant. This 
 is the essence of the assumption of a perfect conductor. 
 
 We must therefore have 
 
 /" ( C t - a] + i/' (ct - a) + ~f (ct - a] = 0, 
 for all values of t. This means that the function / (x) must be of the type 
 
 where a 2 A 2 + aA + 1 = 0, 
 
 ~ 
 
545-548] The establishment of radiation fields 485 
 
 We may then verify that the real part of the solution for the general field 
 vectors corresponding to the root with ( + -~- J has 
 
 2 A cos 6 f/1 
 
 A sm# 
 
 - ( ( i + I cos v ;- 
 2ar \ rj 
 
 \ x ' 
 
 ct r 
 
 Jfl + -^ cos V3 -Vs(l- ") sin \/3 01 e - @ , 
 
 where we have used = . 
 
 2a 
 
 The external field and therefore also the conditions at the surface of the 
 conductor are therefore simple oscillatory in type but with very large damping 
 so that the oscillations soon die out in any case. This of course arises in 
 a physical way from the tendency of the oscillating field to expand outwards 
 and thus include more and more of the inertia of the surrounding aethereal 
 field. 
 
 547. The conditions in this problem are probably more like those 
 realised in the actual experiments of Hertz so that for a complete analysis of 
 that problem it would be necessary to take into account the damping of 
 the field. 
 
 This has in fact been done in a tentative manner by Pearson and Lee* 
 who assume that the more appropriate form of vibration function defined 
 above in a general manner is such that 
 
 / (x) = Ae~ KX sin px, 
 
 the constant K, which can only be empirically determined except with such 
 simple circumstances as those just examined, determines the rate of decay 
 of the vibrations. The question has been experimentally examined by 
 Bjerknesf who found that with the vibrator used by Hertz the amplitude of 
 
 the vibrations fell to - of the original value after a time it n , where t n is 
 
 ,3 
 
 the period of the vibrations. The vibrations would thus become inappreciable 
 after a few alternations. 
 
 548. On the mechanism of the establishment of radiation fields J. We 
 
 have so far confined our discussions only to the continuous propagation of 
 electromagnetic disturbances in homogeneous radiation fields of indefinite 
 extent. It remains therefore to examine the mode of generation of such 
 
 * Phil Trans. A. cxcni. (1900), p. 159. 
 
 f Ann. d. Phys. XLIV. (1891), pp. 74, 92, 513; LV. (1895), p. 121 
 
 $ A. E. H. Love, L. M. 8. Proc. (2) I. (1903), p. 37. 
 
486 Electromagnetic oscillations and waves [CH. xn 
 
 radiation fields. It is the essence of a propagation theory that the conditions 
 of the field at any point P are affected by the conditions at any other point Q 
 
 PO 
 
 only after thf time - ~ after the establishment of the conditions at Q. To 
 c 
 
 illustrate the point in more detail let us consider the simple case of a Hertzian 
 oscillator at the beginning of its oscillations. If we assume that the charge 
 distribution on the oscillator before it collapses has been held there for an 
 indefinite time previously the field surrounding the oscillator will be identical 
 with the simple electrostatic field appropriate to the charge distribution 
 involved. Now suppose that at the time t = the discharge takes place and 
 the consequent series of oscillations started. The radiation field which now 
 begins to be generated does not however instantaneously cover the whole 
 field because the conditions at any point in the field at a distance r from the 
 oscillator will remain unaffected by the changes produced by the discharge 
 
 T 
 
 for the time - after the instant t = when the discharge takes place, in other 
 c 
 
 words the conditions which existed there at the time t = remain unaltered 
 until the disturbance in the radiation field in the surrounding aether which 
 travels outwards in all directions with the velocity c has reached the point. 
 This means that the new radiation field is at the instant t confined within 
 the sphere r = ct surrounding the oscillator : outside this sphere which is 
 a wave surface for the advancing waves the old electrostatic field remains 
 undisturbed, although of course the extent of the field covered by it is gradually 
 diminishing. 
 
 549. The first question that naturally arises is as to the manner in which 
 the two essentially different fields thus involved are connected across the 
 advancing wave-front: the radiation field has resulted mainly from a collapse 
 of the initial electrostatic field and must therefore be connected with it in 
 some way or other ; by some boundary conditions applicable at the surface 
 of the advancing wave front. These conditions were first directly formulated 
 by Prof. Love* by applying the fundamental equations of the theory to 
 small circuits at the wave front. 
 
 Let us assume quite generally that the wave front in the neighbourhood 
 of the point under investigation is practically a plane surface advancing 
 through the field with the velocity Cj in a direction parallel to the axis of 
 a; in a conveniently chosen coordinate system. Draw a small rectangle 
 parallel to the plane Oxz and through which the wave front cuts (dotted in 
 the figure) : the dimensions of this rectangle which are assumed to be small 
 compared with the radius of curvature of the wave front surface and the 
 
 * They were however previously known to Heaviside (Electrical Papers, 11. p. 405) and 
 Duhem (Comptes Eendus, t. cxxxi. (1900), p. 1171). An elegant analytical proof is given, by 
 Bateman in Electrical and Optical Wave Motion (C. U. Press, 1915), p. 20. 
 
548, 549] The conditions at the wave-front 
 
 487 
 
 wave length of the radiation, are such that the sides parallel to Oy are of 
 length I and those parallel to Ox of length I' ' , which is extremely small compared 
 with I and at the instant t is divided by the wave front into portions of lengths 
 
 Fig. 83 
 
 and I' . On the positive side of the wave front the field is determined 
 by the vectors E+, H + and on the negative side by E_, H_. Now the dis- 
 placement current through the small rectangle is the rate of change of the 
 quantity 
 
 I . 
 
 477 X 
 
 or since ^ c 1} this current is 
 
 v ,JL ,. _Z 
 
 y _ (V - 
 
 U (*' - 
 
 which on account of the extreme smallness of and I' i; compared with 
 the wave length of the radiation, is practically equivalent to 
 
 Ic 
 
 But by Ampere's relation this will be proportional to the line integral of the 
 magnetic force round the small circuit which on account of the smallness of 
 I' is practically equal to 
 
 I (H z+ - H..), 
 
 and thus 
 
 ? (H z _ - H z+ ) - 
 
 - B 
 
 An application of Faraday's relation in a similar manner gives 
 
 c l 
 
 c 
 
488 Electromagnetic oscillations and waves [CH. xn 
 
 If the rectangle is taken parallel to the (x, z) coordinate plane instead of the 
 (x, y) plane as above two further conditions are obtained, viz. 
 
 _ H z _). 
 
 C 
 
 These four conditions break up into two pairs for they are equivalent to the 
 condition that 
 
 and also (E v - H a )+ = (E,, - H,)_, 
 
 The propagation of the wave front is thus verified to be with the velocity c 
 and the new field inside the front is connected with the old field outside by a 
 boundary condition which expresses that the tangential component of the 
 vector 
 
 is continuous across the surface : c denotes a vector defining the direction 
 and magnitude of the velocity of transmission of the conditions in the radiation 
 field. 
 
 The condition expressed in this form which has been deduced on the 
 assumption of an approximately plane wave front is not necessarily restricted 
 by this assumption and it will apply to all sufficiently extended wave front 
 surfaces of ordinary type without discontinuity. 
 
 550. The first case examined by Love is that of the oscillations on the 
 perfectly conducting sphere (of radius a) of the charge distribution induced 
 by a uniform field, when that field is suddenly removed. The initial state of 
 the aethereal field outside the sphere is that expressed by the electrostatic 
 vector components 
 
 x 
 
 A"O && 
 
 H = 0. 
 
 At the instant t = the cause which previously maintained the field thus 
 expressed is supposed to cease to operate. It is required to determine the 
 subsequent state of the field to agree with this initial field at time t = and 
 to be such that the tangential electromotive force is continuous at the wave 
 front which at the time t will be the sphere r = ct + a and the tangential 
 electric force at the sphere is zero. The form of solution which suggests 
 itself is naturally of the type previously obtained in which 
 
549, 550] Oscillations on a sphere 489 
 
 
 
 
 rf(ct-r + a)}, 
 
 E = ~T- (f( ct - M- a) + rf (ct-r + a) + r*f" (ct - r + a)}, 
 H * = Of (<* - r + a) + r/" (<0 - r + a)}. 
 
 This solution for the field can only apply inside the sphere r = ct + a : 
 outside it the old electrostatic conditions still prevail. 
 
 At the surface of the sphere the tangential electric force is zero always 
 80 that / (ct) + af (ft) + a*f" (ct) = 0, 
 
 which provides the differential equation for the arbitrary function / : it leads 
 to the solution 
 
 and this will apply for values of x = ct r + a greater than zero. The 
 conditions at the wave front imply that 
 
 E = E 
 
 TI * J r 2 j 
 
 giving on substitution 
 
 E 0i ~ HA, = E fl 
 
 C 
 
 /' (0) = 0, 
 E 
 
 and thus we must have A - 
 
 sma 
 
 where tan a = A/3, a = , 
 
 and the problem is completely determined. With this form of /it is easily 
 verified that the expressions for the force vectors in the field can be put in 
 the form 
 
 _ 
 
 r /-r o 
 
 . ct r + a 
 
 wherein Q = - 
 
 a r ,/5 r 
 
 tan p r = - r~ , tan p e = V 3, tan j8* = ^ - , 
 
 a 2r a r 2a r 
 
 results which differ from those given in a previous paragraph only by the 
 phase of the motion. 
 
490 Electromagnetic oscillations and waves [OH. xn 
 
 It is important to notice that the field of the radiation is strongest close 
 up near the wave front and just inside it; in fact the intensity of the field 
 diminishes exponentially as the distance from the front is increased. In 
 the next chapter we shall examine the energy in this field and will there show 
 that by far the greatest proportion of the total energy radiated out is con- 
 centrated close behind the wave front. 
 
 It appears from this solution ^hat the damped harmonic wave train can 
 advance into a region in which the electric field is the statical one described 
 above. It is also clear that it cannot advance into a region free from electric 
 and magnetic forces. , 
 
 551. Aided by the solution thus obtained for a simple mathematical case 
 Prof. Love* attempted to specify an appropriate solution for the more practical 
 case, the Hertzian oscillator, taking into account the damping which is really 
 existent. In this case also the original field is the electrostatic field of a 
 doublet at the origin giving a field in which 
 
 2cos0 ffsinfl 
 
 E r = -75' E '= -73 > H 4>=> 
 
 the radiation field which originates on the collapse of the distribution giving 
 this statical field (presumed to take place at the time t = 0) will be of the 
 usual Hertzian type in which 
 
 a specification which will at the time t hold at all points inside the wave 
 surface which to a first approximation may be treated as the sphere r = ct. 
 
 We then try a solution of the appropriate type in which 
 / (x) = Ae~ KX sin p (x + a). 
 
 We have to connect this with the external statical field by the boundary 
 conditions deduced above which in this case are 
 
 E ri = E r2 , E ei H0J = E0 2 , 
 and initially we must have / = E. 
 
 The boundary conditions give 
 
 * Proc. R. S. LXXIV. (1904), p. 73. I am greatly indebted to Prof. Love and the Royal 
 Society for permission to reproduce some of the diagrams illustrating this paper: they are 
 appended to this section. 
 
550-552] Hertz's oscillator with damping 491 
 
 so that /' (0) = 0, f(Q) = E: 
 
 this means that 
 
 E = A sin pa, = p cos pa. K sin pa, 
 
 whence tana = -. 
 
 K 
 
 Thus the arbitrary function starts in a definite initial phase which is equal to 
 ^ if there is no damping. 
 
 The field is thus in the general case defined by the vectors 
 E r = ^^ Ae~ K *-'> [(1 - KT) sin + pr cos 0], 
 
 E 8 = Ae-*< et -" [(1 - KT + T Z K Z - p z ) sin + pr (1 - 2/cr) cos 0], 
 
 sin 6 
 
 Ae~ K (ct-r) [( K _ r/c 2 _ ^2) sin - p (1 - 2/cr) cos 0], 
 
 r 
 
 where we have used = p (ct r + a). 
 
 These formulae apply only for r < ct ; for regions beyond the ordinary 
 electrostatic field remains valid. The radial electric force is continuous at 
 the front of the wave, i.e. at r = ct. The discontinuity of the transverse 
 component of the electric force at the front of the wave is 
 
 - . A (K Z + p 2 ) sin pa, 
 
 and this is equal as it should be to the magnetic force at the front of the 
 wave. 
 
 552. The curves of electric force in this case are easily obtained for we 
 know that 
 
 c dt ~V sin 090' c dt 
 where 
 
 r sin 
 
 Thus if we write th = -- =-, 
 
 c dt 
 
 B E 
 
 * r ~r~^irT0a0' ^~rsin0 dr 
 
 and thus the curves of intersection of the surfaces 
 
 Q = const., 
 
492 Electromagnetic oscillations and waves [CH. xn 
 
 with planes through the axes of the doublet are the lines of electric force. 
 Now- it is easily verified that 
 
 m 
 
 Ae~ K (ct - r) [(1 - KT) sin 6 + pr cos 
 
 sin 2 6 
 
 A. sin pa, 
 
 Q = _ 
 
 when ct < r and 
 
 when c > r. 
 
 Some of these curves have been drawn in a special case by Prof. Love 
 and are depicted below. The case taken is that for which 
 
 - = -25 approx., 
 P ' 
 
 and if r is the period the curves are plotted for the times 
 
 t = -26r, -385r, '51r, -635r, '76r, 'SSSr, 1'Olr, and I'lSSr, 
 
 after the initial instant of starting. In the figures the fine continuous circle 
 represents the wave front at the time t. The discontinuity of the field there 
 is shown by the change of direction of the lines of force at this circle : the 
 lines of force themselves are shown by the heavy dotted, heavy continuous, 
 fine dotted and fine continuous lines respectively. The dotted circles that 
 lie within the wave front are curves at which Q = or the electric force has 
 
 Fig. 84 
 
 Fig. 85 
 
 no radial component. It appears that no spherical surface of the set Q = 
 is the front of the advancing wave train but one of these surfaces tends to 
 coincidence with this front as the wave train advances. 
 
552] 
 
 Love's figures for the Hertzian field 
 
 493 
 
 Fig. 86 
 
 Fig. 87 
 
 Fig. 88 
 
 V 
 
 s, 
 
 Fig. 
 
 Fig. 90 
 
 Fig. 91 
 
494 Electromagnetic oscillations and waves [OH. xn 
 
 553. Plane waves. We shall next turn to the consideration of certain 
 cases' of wave motion where the circumstances are much simpler than those 
 just analysed. If the radiating system is at a very great distance from the 
 part of the field under investigation the expanding wave front surfaces have 
 become so large that in the part of the field where they are investigated they 
 may be treated as practically plane surfaces. We then realise the idea of 
 plane waves and the consequent rectilinear propagation of electromagnetic 
 disturbances, and the analytical problems become much simplified. 
 
 Analytically the conception is obtained by a simple type of solution of 
 the fundamental equations of propagation, which directly suggests itself. 
 Let us choose our rectangular axes so that the z-axis is along the direction 
 of propagation and the other two conveniently in the perpendicular plane 
 which is parallel to the front of the wave. The general solutions for this 
 case are then of the form 
 
 El 
 
 _ A pint- (a+ib) z+id 
 
 Hj 
 which represents plane waves of period -- advancing in the medium with 
 
 n 
 a velocity c x = = along the positive direction of the z-axis. The solution for 
 
 any vector is of course represented by the real part of the general solution 
 thus obtained for it. 
 
 If we assume, for the isotropic medium, that both the two simple con- 
 
 stitutive relations 
 
 e dE 
 
 and B = H, 
 
 are valid even for the general case of electric waves under consideration 
 then both vectors E and H satisfy the equation of the previous paragraph 
 
 ' 
 
 dt c 2 dt* ' 
 and thus the above-mentioned forms for E and H are valid if 
 
 - ew 2 + i&rna = c 2 (a + ib) 2 , 
 or -ew 2 = c 2 a 2 -& 2 , 
 
 Thus if a is different from zero, a is so also and thus the amplitude A of each 
 vector contains the factor e~ az , which means that as the waves are propagated 
 along the positive direction of the z-axis, the amplitudes of the two vectors 
 gradually decrease as the wave proceeds. Conductivity in the medium 
 implies a damping of the waves, the energy being absorbed by the medium 
 and converted into heat. 
 
553-555] Plane waves 495 
 
 554. We can now show that these electric waves are transverse. To do 
 this we merely prove that in the most general aeolotropic medium any vector 
 which has the property of the ordinary stream vector of hydrodynamic 
 theory has its direction parallel to the wave front ; for if V is any such vector 
 
 ,. 8V, dV y 9V, 
 
 div V = -5-= + ~ + -5-= = 0, 
 
 ox oy oz 
 
 and since all quantities in a plane wave specified as above only depend on 
 the z-coordinate we must have 
 
 3V* = 3V, = 
 dx dy 
 
 8V 
 
 and therefore also -*-* = 0, 
 
 oz 
 
 which implies that V 2 = 0. 
 
 This is a general property of a train of plane waves whatever the constitu- 
 tion of the medium, whether it be crystalline or not. It means in any case 
 that the total current and magnetic induction vectors are in the plane of 
 the wave front. Thus in this sense all such electromagnetic waves are 
 transverse, the fluxes associated with them being both transverse to the 
 direction of propagation. The electric and magnetic force vectors are of 
 course in the general case not in the wave front ; this is true only when the 
 medium is isotropic as already assumed. 
 
 555. We now assume that the magnetic and electric forces are in the 
 wave front and thus are expressed by their components 
 
 (H x , H,, H z ) = (4 ra cos</, TO , A m sm<f> m , o)e*- <"+>+, 
 (E., B v> E z ) = (A e cos #., A e sin< e5 o) '*-<+<+., 
 
 and these must satisfy the fundamental equations of the field. Faraday's 
 relation implies that 
 
 ^ nt TT \ ^ na PA 
 
 ~cdt (Ux ' r &** ~ EX}> 
 A 
 
 so that -in cos </> TO = A e sin <f> e (a + ib) e i(6 e - e m\ 
 
 c 
 
 A 
 
 -in - sin <f> m = A e cos (f> e (a + ib) e i(e e- e J. 
 
 C 
 
 Whence we deduce directly the two important conclusions, 
 (i) By division we see that 
 
496 Electromagnetic oscillations and waves [OH. xn 
 
 Thus the electric and magnetic force vectors in the wave front are perpen- 
 dicular to one another. 
 
 (ii) Using the fact that (f> m = <f> e + ~ we see a l so that 
 
 tt 
 
 - inA m = c (a + ib) A e e i(6e ~ e ^. 
 Now from Ampere's circuital relation we can deduce similarly that 
 
 A m c (a + ib) e'PM-tJ = A e (a + ieri), 
 so that since A e and A m are real we must have 
 
 % 
 
 where Cj = j- is the velocity of propagation in the medium. 
 
 r A _ 
 Also A m = ^Va*+b*. 
 
 Thus there is in the general case always a phase difference (6 m d e ) between 
 the electric and magnetic force vibrations, this phase difference vanishing 
 only for the case when a = 0, i.e. a = or for a perfect non-conductor. The 
 amplitudes of the waves are also different in the general case of absorption. 
 
 The velocity of the wave =- is given by 
 
 which reduces in the case of non-absorbing media to 
 
 c 
 
 556. [*If we now choose the real part of the general solutions and also 
 write 
 
 Q = nt-bz + 6 e , 
 then we can put 
 
 E = A e cos @e- az , 
 
 H = A m cos (0 + 6 e - d m ) e~ az . 
 
 The electric energy per unit volume at a place is 
 
 ,372 ,A 2 
 W = ^- = ~^c^Qe-^, 
 
 O7T O7T 
 
 and the mean value at the place taken over a whole oscillation is 
 
 alt) o 
 
 * The results deduced and given between brackets thus [...] depend on the forms of the 
 energies in the field which are not properly deduced until chapter xiv. 
 
555-557] The dissipation factor 497 
 
 similarly the mean kinetic or magnetic energy is 
 
 2ir 2ir 
 
 f = 
 
 27T 
 
 
 
 
 167T 6 2 - a 2 
 
 Whence it follows that the mean magnetic energy is in general larger than 
 the electric energy, equality occurring only in the non-conducting substances 
 when a = 0. This result also exhibits clearly the way in which the energy 
 in the wave is absorbed as it progresses, the mean total energy of the wave 
 at any place being the sum of the electric and magnetic energies, viz. 
 
 _ 
 
 STT 6 2 -a 2 'J 
 
 557. These results enable us to explain in greater detail the behaviour 
 of metallic conductors in a radiating dielectric field discussed in the previous 
 paragraph. They show that the propagation of the waves in the dielectric 
 takes place without any absorption at all. As soon, however, as a distur- 
 bance reaches a conducting surface and starts off through the conducting 
 medium the damping factor e -az 
 
 at once enters into the expression. If the conductivity is big a is large and 
 the wave is practically damped right out before it gets far into the metal. 
 The larger the value of a the shorter the distance the waves penetrate into 
 the conductor, and by sufficiently large values we may neglect the penetration 
 altogether. We can also increase a by increasing n, so that for very rapid 
 oscillations the conducting material will always act as if it were a perfect 
 conductor. Let us take an example to illustrate the matter further. For 
 
 c 2 
 copper a = TT an d e i g negligible. Now consider the incidence of waves 
 
 27T x 3 x 10 10 
 n = 2~ = 2 x 10 9 approx. 
 
 = c 2 (a + ib) 2 , 
 
 of length 100 cms. : then 
 
 n = 
 and since e is negligible 
 
 whence - V4:TTna = c (a + ib), 
 
 a 
 
 or 2ca = 2cb = 
 
 2 x 10 x 477 
 
 2 c 2 V 16 x 10 2 
 
 = 2 x 10 3 approx. 
 
 32 
 
498 
 
 Electromagnetic oscillations and waves [CH. xu 
 
 Thus at a depth 5 . 10~* cms. the amplitudes are reduced to l/e times their 
 initial value. The penetration is therefore very slight in a real case of this 
 kind. 
 
 558. In many cases it is convenient to make one further simplification 
 in the above scheme. In the analyses given above, A e , </><, and A m , <j> m may 
 vary with the time at any one place, the only conditions to be satisfied at each 
 instant being those relations connecting them which we have obtained. We 
 might however now imagine the medium and the original disturbance so 
 adjusted that in the region in which we examine them the electric force vector 
 has the same direction at every point of the field. This direction we may 
 take as the axis of x in our coordinates ; it then follows that the magnetic 
 force vector is always parallel to the ^/-axis at every point of the same field. 
 In the language of optics this means that our ray is polarised, although the 
 plane of polarisation is undetermined by the analogy. Future evidence 
 however points to the conclusion that the plane of polarisation is determined 
 by the vector H and the wave-normal or direction of propagation. The 
 electric force is normal to this plane. 
 
 559. On experiments with electric waves. Having definitely established 
 in the manner briefly described above the existence of the electromagnetic 
 waves required by Maxwell's theory Hertz proceeded at once to the further 
 problem of proving by certain simple experiments the complete analogy which 
 is suggested by Maxwell's analysis between these waves and waves of light. 
 He first examined their reflexion and refraction at the interfaces between 
 two dielectric media or between one dielectric medium and a metallic medium. 
 
 Fig. 92 
 
 For this purpose however he used another form of apparatus. The vibrator 
 (see fig.) consisted of two equal brass cylinders 12 cm. long and 3 cm. in 
 diameter placed with their axes coincident and in the focal line of a large 
 zinc parabolic mirror about 2 metres high with a focal length 12 '5 cm. This 
 arrangement should on the optical analogy produce plane polarised waves 
 
557-561] Hertz's experiments 499 
 
 by reflection from the mirror. The detector which was placed in the focal 
 line of an equal parabolic mirror, consisted of two lengths of wire, each having 
 a straight piece about 50 cm. long and a curved piece about 15 cm. long 
 bent round at right angles so as to pass through the back of the mirror. 
 The ends which came through the mirror were connected with a spark micro- 
 meter, the sparks thus being observed from behind the mirror. 
 
 560. To show the reflexion of the waves Hertz placed the mirrors side- 
 by side, with their openings looking in the same direction and their axes 
 converging to a point 3 metres from the wire. No sparks were observed in 
 the detector when the vibrator was in action. When, however, a large zinc 
 plate was placed at right angles to the line bisecting the angle between 
 the mirrors sparks became visible, but disappeared when the metal was 
 twisted through an angle of about 15 to either side. This experiment 
 showed that electric waves are reflected by the metal sheet and that approxi- 
 mately at any rate the angle of incidence is equal to the angle of reflexion. 
 
 To show refraction Hertz used a large prism made of hard pitch with an 
 angle of 30. When the waves from the vibrator passed through this the 
 sparks in the detector were not excited when the axes of the two mirrors 
 were parallel, but appeared when the axes of the mirror containing the detector 
 made a certain angle with the axes of the containing vibrator. When the 
 system was adjusted for minimum deviation the sparks were more vigorous 
 when the angle between the axes of the mirrors was 22, corresponding on 
 the optical analogy to an index of refraction for pitch of 1'69. 
 
 561. If a screen be made by winding wire round a large rectangular 
 framework so that the turns of the wire are parallel to one side of the frame, 
 and if this screen be interposed between the parabolic mirrors when placed so 
 as to face each other there will be no sparks in the detector when the turns 
 of the wire are parallel to the focal lines of the mirror ; but if the frame is 
 turned through a right angle so that the wires are perpendicular to the focal 
 lines of the mirror the sparks will recommence. If the framework is sub- 
 stituted for the metal plate in the experiment on reflexion of electric waves, 
 sparks will appear in the detector when the wires are parallel to the focal lines 
 of the mirrors and will disappear when the wires are at right angles to these 
 lines. Thus the framework reflects but does not transmit the waves when 
 the electric force in them is parallel to the wires while it transmits but does 
 not reflect waves in which the electric force is at right angles to the wires. 
 The wire framework thus behaves towards the electric waves exactly as a 
 plate of tourmaline does to waves of light. 
 
 When light polarised at right angles to the plane of incidence falls on 
 a refracting substance at an angle tan" 1 ^, whereat is the refractive index 
 of the substance, all the light is reflected and none refracted. Whereas when 
 
 322 
 
500 Electromagnetic oscillations and waves [CH. xn 
 
 light is polarised in the plane of incidence, some of the light is always reflected 
 whatever the angle of incidence. Trouton* showed that similar effects take 
 place with electric waves. From a paraffin wall 3 feet thick, reflection always 
 takes place when the electric force in the incident wave was at right angles 
 to the plane of incidence, whereas at a certain angle of incidence there was 
 no reflection when the vibrator was turned, so that the electric force was 
 in the plane of incidence. This shows that on the electromagnetic theory 
 of light the electric force is at right angles to the plane of polarisation. 
 
 562. The surest test of any wave theory of propagation of disturbance 
 of any kind is obtained, if it is possible to produce some effect which depends 
 essentially on the well-known phenomenon of the interference of two trains 
 of the wave disturbance ; it is in fact by this means that the essential wave 
 character of the disturbance is usually exhibited. The simplest effect of this 
 kind is the production of ' stationary ivaves ' by the interference of a direct and 
 reflected train of disturbance. Hertz attempted to get stationary waves by 
 reflection at a metallic surface. His experiments were made in a room about 
 15m. long : a vibrator of the type described above was placed at one end of 
 the room with its plates parallel to the wall and a large sheet of zinc was 
 placed vertically against the wall at the other. The circular ring detector was 
 held with its plane parallel to the plane of the plates of the vibrator, its centre 
 on the line perpendicular to the zinc plate bisecting at right angles the spark 
 gap of the vibrator. The following effects were observed when the detector 
 was moved about. Close up to the plates there were no sparks but they 
 began to pass feebly at a little distance from the plate and increased rapidly 
 in brightness up to a distance of about 1*8 m. from the plate, where the 
 maximum was attained. When the distance still further increased they 
 diminished in brightness and vanished again at a distance of about 4 m. 
 from the plate. When the distance was still further increased they reappeared, 
 attained another maximum and so on. The most obvious explanation of 
 these experiments was the one given by Hertz that there was interference 
 between the direct waves given out by the vibrator and those reflected from 
 the plate, this interference giving rise, in the well-known manner, to stationary 
 waves. The places where the electric force was a maximum were the 
 places where the sparks were brightest and the places where the electric 
 force was zero were the places where the sparks vanished. On this explanation 
 the distance between two consecutive places where the sparks vanished would 
 be half the wave length of the waves given out. 
 
 Sarasin and De la Rivef however showed that this explanation could 
 not be correct since by using detectors of different sizes they found that 
 the distance between consecutive places where the sparks vanished depended 
 
 * Nature, xxxix. p. 391. 
 
 t Comptes Rendus, cxv. p. 489. 
 
561, 562] Stationary waves 501 
 
 mainly upon the size of the detector, being in fact proportional to its linear 
 dimensions (i.e. using similar shapes), and very little upon that of the vibrator. 
 This is due to the very large damping of the oscillations in the vibrator and 
 the very small damping of those of the detector. The rapid decay of the 
 oscillations of the vibrator will stifle the interference between the direct and 
 reflected wave, as the amplitude of the direct wave will, since it is emitted 
 later, be much smaller than that of the reflected one and not able to annul 
 its effect completely, while the well-maintained vibrations of the detector 
 will interfere and produce the effects observed by Sarasin and De la Rive. 
 To see this let us consider the extreme case in which the oscillations of the 
 vibrator are absolutely dead beat. In this case an impulse, starting from the 
 vibrator on its way to the reflector strikes against the detector and sets it 
 in vibration; it then travels up to the plate and is reflected, the electric 
 force in the impulse being reversed by reflection : it then again strikes against 
 the detector which is still vibrating from the effects of the first impact; 
 if then the phase of this vibration is such that the reflected impulse tends to 
 produce a current round the detector in the same way as that which is 
 circulating from the effects of the first impact, the sparks will be increased, 
 but if the reflected impulse tends to produce a current in the opposite direction 
 the sparks will be diminished. Since the electric force is reversed by reflection 
 the greatest increase in the sparks will take place when the impulse finds, 
 on its return, the detector in the opposite phase to that in which it left it ; 
 that is, if the time which has elapsed between the departure and return of 
 the impulse is equal to an odd multiple of half the time of vibration of the 
 
 ct' 
 detector, the distance between two spark maxima would then be -^-, where 
 
 Z 
 
 c is the velocity of radiation and t' the period of vibration of the detector. 
 
 Thus although these experiments bring out the essential wave charac- 
 teristic of the phenomena and the possibility of producing interference of 
 electromagnetic wave trains the circumstances in them are much more 
 complicated than was at first supposed and the phenomenon of stationary 
 waves is not in reality exhibited in them at all. It is possible however 
 to produce stationary wave motion in other ways as we shall see 
 presently. 
 
 We shall in the next chapter analyse the circumstances exhibited in these 
 experiments in order to show that the results obtained are in accordance 
 with theory. As the results of the experiments can, with the possible exception 
 of the tourmaline analogy, all be explained in terms of any wave. theory 
 with the proper laws of reflection and refraction, we shall confine ourselves 
 to the illustration of these laws by the examination of the comparatively 
 simple circumstances connected with the reflection and refraction of plane 
 waves at plane boundaries. 
 
502 Electromagnetic oscillations and waves [CH. xn 
 
 563. The mechanism of radiation. The modern theory of electrical 
 actions ascribes all electrodynamic effects of electric currents solely to the 
 motion of electrons : every disturbance of the aether including radiation as 
 one type of disturbance is originated by translatory motion of electrons 
 through the aether. Thus if we can obtain complete expressions for the 
 aethereal disturbance initiated by and propagated from a single moving 
 electron, we should be in a position tq attempt a theory of the mechanism 
 of radiation. 
 
 Now except at places whose distance from the nucleus of the electron is 
 so small as to be comparable with the linear dimensions of the nucleus, it 
 is usually sufficient to consider the electron as a point charge; and the 
 aethereal disturbance arising from the motion of the electron is to be obtained 
 by simple superposition of elementary disturbances arising from its transit 
 over the successive elements of its path. Suppose then that an electron is 
 at the point Q and after a short time 8t is at Q' ' , where QQ' = vS, v being 
 its velocity ; the effect of its change of position is the sajne as that of the 
 creation of an electric doublet of moment e\8t, with its axis along $Q'. Thus 
 we have only to find the disturbance produced by the creation of such a doublet 
 and then integrate the result along the path of the electron. But we have 
 already determined above in 537 the complete effective field for a doublet 
 whose' moment is varying in any arbitrary manner and thus the solution for 
 the present case will be obtained by choosing the function / thus involved 
 
 80 that /(cO = when t < t , 
 
 rn, 
 f(ct) = e\ vdt t n <t<t Q + St 0) 
 
 J o 
 / (at) = t + 8t < t. 
 
 564. Before however making this transformation we may notice that 
 the field specified in the previous paragraph can be regarded as defined in the 
 ordinary way by a scalar static potential 
 
 together with the kinetic vector potential whose components are 
 
 cos 6 , 
 A r = / (at - r), 
 
 sin 6 ,, 
 f^ = f (ct-r). 
 
 These expressions are simplified by reference to a general rectangular 
 coordinate system when they may be written in the form 
 
 r 
 f (ct - r) 
 
563-565] The mechanism of radiation 503 
 
 f denoting in the complete vector sense the moment of the doublet; the 
 
 \ 
 
 magnitude is of course /. The differential - denotes the gradient of the 
 function in the direction of the axis of the doublet. 
 
 The conditions at any point in the general field at the time t thus depend 
 
 T 
 
 essentially on the conditions of the vibrator at the time t' = t -- previously ; 
 
 c 
 
 this is the effective time for the relative position of vibrator and field-point. 
 This result is of course the essence of a propagation theory. 
 
 A slight change in the charge distribution on the vibrator, specified by 
 the function /, during the small interval of time 8t results in a spherical shell 
 of disturbance travelling out with the veloqity c of radiation so that at any 
 time t after the instant of its generation it lies between the concentric spheres 
 r = ct and r = c (t + &t) . Moreover the total field integrated across the small 
 thickness of the shell is, except as regards the static electric part which is not 
 really propagated, independent of the thickness of the shell which of course 
 depends on the time taken to establish it. 
 
 565. Returning now to our moving charge we see at once that the slight 
 displacement of it during the small time St results in a shell of a spherical 
 disturbance which travels out with the velocity of light from the centre at 
 which it was generated and in which the field is precisely of the type just 
 investigated, in which however the function / is as specified above ; this 
 field is thus determined by 
 
 - ~- - 
 os \r 
 
 A g & V 
 
 " ~ - J 
 
 cr 
 
 where however it must be remembered that as regards the field at a distance 
 r from the electron at time t these values are effective only at the instant 
 
 T 
 
 t = t n + - , or in other words the effective field at the time t arises from the 
 c 
 
 T 
 
 motion and position of the electron at the previous time t = t - -, r being 
 
 c 
 
 the radial distance from the field point to the effective position of the electron 
 at this previous instant. We denote this by enclosing the quantities con- 
 cerned by square brackets so that we may write 
 
 .f 
 
 eSy 
 
 cr 
 
504 Electromagnetic oscillations and waves [CH. xn 
 
 It follows therefore by integration along the whole of the path from the last 
 effective position backwards to infinity that the two potentials 
 
 define completely the field of the single moving charge. 
 
 566. These are the so-called retarded potentials of electron theory. It is 
 however usual to express them to the next order of approximation. The above 
 values are calculated on the assumption that the velocity of the electron is 
 very small compared with the velocity of light, so that the change of position 
 of the electron during the effectively small time St is negligible. If we proceed 
 to the next order of approximation and take into account the motion of 
 the electron during this interval of time 8t we shall see that the shell of dis- 
 turbance sent out by the electron during it will not be uniformly thick but 
 will in reality lie between two eccentric spheres with their centres in the 
 initial and final effective positions of the electron, i.e. at a distance [vSf ] 
 apart. The thickness of the shell along any radius r will now be approxi- 
 mately [(c v r ) Bt], v r being the component of the electronic velocity along 
 that radius. But it is clear that the aggregate of the field sent out in any 
 direction in the shell will be independent of the thickness and therefore of 
 whether it is on the whole of uniform thickness or not, so that the field will 
 be intrinsically stronger in the thinner parts of the shell and correspondingly 
 weaker in the thicker parts : this means that the field vectors in the 
 part of the shell along the radius r will be increased from the values in the 
 uniform shell in the ratio 
 
 ^C 1 
 
 It follows therefore that to the second order of approximation the field is 
 given by* 
 
 e[vl 
 
 1-5 
 
 c 
 
 and these are the values usually quoted. 
 
 * A. Lic'nard, Ueclairage electrique; E. Wiechert, Arch, neerlandaises (2) v. (1900), p. 54; 
 K. Schwarzschild, Gott. Nachr. (1903). Cf. also Schott, Electromagnetic Radiation, p. 22; 
 Bateman, Electrical and Optical Wave Motion, Ch. vm; De la Rive, Archives de. Geneve (1907), 
 p. 433. 
 
565-567] The potentials for a point source 505 
 
 567. The mode of deduction of the last expressions for the retarded 
 potentials is perhaps not so rigorous as is desirable with such fundamental 
 formulae. They can however easily be deduced in a more rigorous manner 
 from a slight modification of certain general formulae already obtained. In 
 Chapter ix we have obtained general expressions for the scalar and vector 
 retarded potentials in any electrodynamic field and using the expressions 
 there obtained we see that if the total charge distribution in any field is of 
 density p and if the current flux is due to the motion with velocity V of this 
 charge then the scalar and vector potentials of the field may be written in 
 the form 
 
 + 00 
 
 . 00 J 00 
 
 A 
 
 where dQ = pdv is the charge element in the small volume dv at the time r. 
 When the charge is of amount e and concentrated in a small volume element 
 round the point (x e y e z e ] these expressions reduce to 
 
 
 - 
 
 torcJ-vJ.* r 
 
 Now change in these integrals the variable r to r' where 
 
 r = r + - 
 c 
 
 so that dr' = dr (l + - ^\ = dr 1 1 - }, 
 
 \ c dr) \ c J 
 
 then we get, for instance, 
 
 i _ e /"+ f +co ili(i _ 
 
 ^Tt J x . oo 
 
 r l - "' 
 
 since the limits for r are the same as those for r when (v) < c. But the double 
 integral in this last case is a proper Fourier integral whose value is 
 
 277 
 
 so that (f> = 
 
506 
 
 Electromagnetic oscillations and waves [CH. xn 
 
 Similarly we find that 
 
 A = 
 
 cr l 
 
 where in the last two expressions square brackets serve to indicate the values 
 
 (f\ 
 t - - j . These are the same expressions 
 
 as found above. 
 
 568. The whole circumstances in the field of the moving charge can 
 now be readily deduced and the expressions for the electric and magnetic 
 force vectors at any point obtained by simple differentiation of these poten- 
 tials. It is however easier to deduce them from the general results obtained 
 in Chapter ix on the introduction of the retarded potentials. In fact if we 
 again put in these expresssions C x = pV and carry out the volume integration 
 and then transform the integrals as we have just done for the potentials we 
 find that 
 
 E 
 
 whilst 
 
 er d 
 
 X r >-^~ 
 
 
 ^1 H 
 
 <& 
 
 cr 
 
 (i.-*) 
 
 
 r e [VTj] -| 
 
 , d r e [vij] -, 
 
 flr'fl-*')] 
 
 *-iB 
 
 / v r \ 
 
 /'*-** 1 
 
 L V C/ J 
 
 v c yj 
 
 wherein TJ denotes the unit vector in the direction of the radius r. 
 
 We must now bear in mind that the quantities inside the large square 
 
 brackets are functions of T = t as well as (x, y, z) explicitly ; but 
 
 G 
 
 dr 1 
 
 dt~ 
 
 _ 
 c di 
 
 and hence 
 
 and 
 
 Hence finally we have 
 
 = 
 
 1 - 
 
 1 _ Ir = i _ 
 j- i -- 
 
 c 
 
 (vr) 
 
 c - 
 
 c dt 
 
 r dt 
 
 c 2 r~ (dt) ' 
 
 cr 
 
 \dt 
 
 (dt 
 
567-569] The field of a moving electron 507 
 
 569. We first notice that H - ftE] 
 
 so that the magnetic force is everywhere perpendicular to the electric force 
 and to the radius from the field point to the effective position of the charge 
 element at the instant. On the other hand 
 
 r e (c 2 - v 2 ) -i 
 
 so that the electric force is not transverse to the radius vector unless c = \ v 
 but the deviation from perpendicularity becomes smaller and smaller as the 
 distance from the charge increases. 
 
 v r (vr,) / v 
 
 Again since = r, , r, -- 
 
 c c V c 
 
 it is easy to verify that* 
 
 Also since H = [ijE] 
 
 we have [HrJ = - [ij [ijE]] = E - ^ 
 
 and thus also E = [HrJ + (c 2 - v) ( 
 
 The part of the electric force not depending on the acceleration and 
 the predominant part in the field near the electron is 
 
 ^i e 
 
 E, = -r 
 
 whilst the corresponding part of the magnetic force is 
 
 r v~i 
 
 Now the vector r x - is parallel to the direction of the radius from the 
 
 field point to what would be the instantaneous position of the moving charge 
 (as distinct from its effective position) if it be assumed that it has moved 
 from its effective position with the constant velocity [v] that it then had. 
 
 If the motion of the particle is with constant velocity in a straight line 
 this is the whole field and as such will be more fully reviewed in the sequel. 
 We are at present more fully concerned with the part of the field depending 
 on the acceleration, which is far more dispersed than the present part which 
 is concentrated close up round the electron. 
 
 * Cf. Introduction, 6. 
 
508 Electromagnetic oscillations and waves [en. xu 
 
 570. If the motion of the particle is accelerated the part of the field 
 depending on the accleration which predominates at large distances from the 
 particle and is insignificant in the rest of the field is determined by 
 
 and E = 
 
 Thus in this part of the field the electric and magnetic forces are both per- 
 pendicular to the radius vector from the effective position of the particle to 
 the field point under review ; they are also perpendicular to one another and 
 are therefore of equal magnitude. 
 
 At large distances from the particle therefore the field, which is completely 
 specified by these components alone, is like a proper field in radiation ; the 
 wave front surfaces are eccentric spheres each having its centre at the position 
 of the particle when it was generated and the two force vectors lie in these 
 surfaces. 
 
 If the velocity of the motion is small then we have practically 
 
 E=^ 
 re 2 ' 
 
 and E = [Hrj] 
 
 so that the electric force is tangential to the meridian circles and the magnetic 
 
 force to the parallels of latitude, the polar axis being parallel to the effective 
 
 acceleration. 
 
 We shall see presently that this means that there is a flux of energy 
 normal to the wave front and away from the electron : this flux amounts to 
 
 2 
 
 sin 2 
 
 per unit area per unit time so that on the whole there is the amount 
 
 2e 2 
 
 3 c^ \_dt 
 per unit time that travels away and is lost to the electron. 
 
 When an electron is put into motion it sends out a stream of radiation 
 which lasts as long as its velocity is being accelerated : in the process of 
 setting up the velocity v from rest, there is a total loss of energy by 
 radiation equal to 
 
 2 e 2 
 
 1 dt ' 
 
 When the velocity has become constant there is no more radiant energy 
 sent out from it, though the previous sheets of radiation will continue to 
 travel on into the more distant stagnant aether, leaving behind them the 
 
 * 6 is the latitude of the place. 
 
570, 571] Natural radiations 509 
 
 steady magnetic field of the uniformly moving electron : but that field, 
 which thus becomes established as a trail or residue of the shell of radiation 
 arising from the original initiation of the motion of the electron,, does not 
 itself involve any sensible amount of energy except in the immediate neigh- 
 bourhood of the electron. 
 
 571. This analysis has an important application to the explanation of 
 the radiation from an incandescent body. We have seen generally how long 
 Hertzian waves can be produced by a process which consists essentially in 
 the production of a rapid oscillatory motion of electrical charges. Now we 
 have already been led to the conclusion that each element or molecule of a 
 material body contains as an essential part in its constitution a number of 
 electrons and positive charges which under ordinary conditions take up a sort 
 of equilibrium configuration inside it : if we can produce a disturbance in this 
 steady configuration the individual electrons will emit radiation of a type 
 depending on their motion. Thus if we knew the motions of the electrons we 
 could specify completely the type of radiation emitted by the body ; but this is 
 just what we do not know. We are still unable to specify completely the type 
 of mechanism governing the electronic motions inside an atom, and we can 
 therefore only offer tentative suggestions such as that given above on page 475. 
 We can however infer from an examination of the radiation itself certain 
 details concerning its mode of generation and it is on this evidence that the 
 suggestive mechanisms are being constructed. It is for example found that 
 the radiation from a gas whose density is not too big consists mainly of a 
 limited number of distinctly separated harmonic constituents with periods 
 and intensities characteristic of the substance of which the gas is composed. 
 This suggests that the vibrations of the electrons giving rise to the radiation 
 must be very nearly simple harmonic ; and this suggests again that th e electrons 
 are vibrating in the molecule about certain definite positions of equilibrium 
 to which they are bound by certain quasi-elastic forces proportional to their 
 displacement from the position. This is of course the simplest possible 
 idea and has already been introduced on a previous occasion, but further 
 evidence seems to indicate the impossibility of its validity : it would for 
 instance seem to imply that one electron cannot be responsible for more 
 than three of the harmonic constituents of the radiation, and an almost 
 incredibly large number of electrons would thus have to be assumed to exist 
 in the molecules of certain substances. It appears however that no com- 
 pletely satisfactory explanation of these difficulties has yet been offered and 
 we shall therefore content ourselves with this simple explanation. 
 
 In striking contrast with the radiation from a gas, the radiation from an 
 incandescent solid or liquid presents as a general rule nothing of a periodic 
 character, for it arises from the independent and irregular disturbances of 
 countless molecules and electrons : it thus has the appearance of a formless 
 
510 Electromagnetic oscillations and waves [CH. xu 
 
 mass of radiant disturbance advancing with the velocity of light : it is possible 
 however even in these cases to have certain more or less predominant con- 
 stituents of a definite period, but at high temperatures these are completely 
 covered by the irregular radiation which is of a purely thermal character. 
 It is a problem in these theories to determine how, if the thermal radiation 
 of a substance is resolved by a prism into its harmonic spectrum, the energy 
 of the total radiation is distributed among the harmonic constituents thus 
 separated out : some aspects of this problem will be discussed at the end of 
 chapter xiv. 
 
 572. Before concluding this paragraph reference must be made again to 
 the so-called Roentgen or X-rays. These rays were first observed by Roentgen 
 in the neighbourhood of a discharge tube when the vivid green phosphorescence 
 is exhibited on the walls of the tube, and they have been found also as an 
 important constituent (the y-rays) of the radiation emitted by radio-active 
 substances. These rays exhibit a remarkable resemblance to light. Their 
 rectilinear propagation, as evidenced by the sharp shadow thrown by bodies 
 which intercept them, their power of affecting a photographic plate and their 
 power of passing through solid bodies are obvious examples of this resemblance. 
 But there are equally striking differences between Roentgen rays and rays 
 of light. They are not refracted in their passage from one medium to another : 
 they show some sort of reflexion, but the laws governing it are totally different 
 from those of the corresponding phenomenon in light. 
 
 The generally accepted view of this radiation as originated from the 
 discharge tube is that first proposed by Stokes : it is composed of thin 
 spherical sheets of disturbance sent out into the aether by the sudden impacts 
 of the rapidly moving electrons of the kathode stream against the walls of 
 the tube : it may also in part be due to the shocks imparted to the molecules 
 forming the walls of the tube. The similar radiation from radio-active 
 substances would then have its origin in part in the sudden generation of 
 the rapid motion in the electrons thrown off from those substances and again 
 in part in the readjustment of the remaining molecule to its new conditions 
 after the electron has left. 
 
 In so far as these sheets of radiation are due to sudden but transient 
 disturbance of the electrons in the molecules themselves, the magnetic force 
 belonging to them alternates in direction in crossing each thin shell of pulse 
 so that the average value taken across it is null. In so far as they are due 
 to the sudden start or stoppage of the kathode particles or electrons, each of 
 which is a single moving electron, this balanced alternation of magnetic force 
 across the thickness of the sheet does not hold ; the force may be in the same 
 direction all the way across. As during the progress of the emission or impact 
 the accelerations of the kathode particles arrested or emitted and of the 
 disturbed electrons of the molecules will be presumably of the same order of 
 
571-574] X -radiation 511 
 
 magnitude, we would naturally conclude from the formula expressing the 
 radiation in terms of the acceleration of the electron that these are both 
 concerned in the emission of radiant energy, and the fact that the radiation is 
 found to contain certain constituents characteristic of the substances on which 
 the particles collide or from which they are thrown off supports this view. 
 
 In addition to the thin pulse arising from the sudden shock imparted to 
 the molecules of the substance stopping or starting the electron, we would 
 expect to find also more continuous radiation due to their state of vibration 
 which would ensue : this would be represented by the phosphorescent light 
 which accompanies the phenomenon. 
 
 573. As regards the X-radiation the present explanation has received 
 remarkable confirmation in the last few years in a wonderful series of experi- 
 ments originated by Laue and extensively developed by many workers, 
 among the most prominent of whom are Prof. Bragg and his son*. In these 
 experiments beams of carefully sifted homogeneous X-rays are reflected or 
 refracted by crystalline media, and it is found that the regular arrangement 
 of the molecules of such substances makes them behave towards the radi- 
 ation more or less like a three-dimensional optical grating. The radiation 
 passing across each molecule sets the electrons in that molecule in rapid 
 vibration and these in their turn emit the secondary radiation which is 
 observed as the transmitted or reflected beam, the regular arrangement of 
 the molecules or vibrating centres giving rise to a measurable regularity in 
 the radiation. It has thus been found possible not only to discover the 
 arrangement of molecules in the crystals but also to establish definitely the 
 periodic character of the X-radiation, even to the extent of obtaining an 
 accurate estimate of its frequency. The wave length of the radiation is 
 characteristic of the exciting substance but is much shorter than the radi- 
 ation in the visible spectrum. 
 
 574. The previous discussion suggests that we have to deal in actual 
 practice not with the single electrons but with whole groups of them more or 
 less tightly bound to the elements of the ponderable matter or moving about 
 freely in the interstices between these elements : and since the formulation 
 of this more general problem brings out further aspects of the radiation from 
 incandescent bodies, it seems desirable to give at least its barest outlinesf . We 
 first suppose that the motion of the electron under consideration is confined 
 to a certain very small region v, one point of which is chosen for origin of 
 coordinates. Eef erred to the axes of coordinates thus chosen let i e be the 
 
 * Cf. W. L. Bragg, Proc. Camb. Phil. Soc. xvn. (1913), p. 43; Proc. R. 8. A. LXXXVIH. 
 (1913), p. 428; M. Laue, Munchen. Ber. (1912), p. 363; Ann. d. Phys. XLI. (1913), p. 989; 
 XLII. (1913), p. 397; P. P. Ewald, Phys. Zeitschr. (1913), p. 465; L. S. Ornstein, Amsterdam. 
 Proc. (1913); M. Born u. T. von Karman, Phys. Zeitechr. (1912), p. 297. 
 
 t Cf . Lorentz, The Theory of Electrons, p. 55. 
 
512 Electromagnetic oscillations and waves [OH. xn 
 
 position vector of the electron so that its velocity is T e and its acceleration 'i e . 
 We shall regard all these quantities as so small that we may neglect any 
 terms involving their squares and products. Next let r denote the coordinate 
 vector of a point P at some distance from the origin of coordinates, outside 
 the small surface considered, for which we want to determine the field. Now 
 if Q is the effective position of the electron as regards the field at the point 
 P at time t, the distance PQ will differ from r only by terms of the first order, 
 
 f 
 
 and the effective time t Q will differ from the time t - - in the same way. 
 
 c 
 
 The changes of position and motion of the electron in a very small time being 
 infinitely small of the second order we may regard Q as the effective position 
 
 V 
 
 at the instant t - - and the velocity there as the velocity at this time. 
 
 C 
 
 Moreover 
 
 because the difference between the distances PQt and PO is equal to the 
 difference between the vectors r and r e taken at P or Q. The square brackets 
 of course serve to indicate the values of the quantities affected at the instant 
 
 Thus if we use also 
 
 we have <j> = -~ - - ([r J V) - + 
 
 - 
 
 477 (r ' r cr 
 
 Now as regards the last term in the expression we may write 
 
 cr 
 
 cr 
 and this is = div [r e ] , 
 
 , 9 \x f ] d \x f ~\ dt n or 
 
 since for example ~^~ = k~ ^ ^r 
 
 dx ot or ox 
 
 r , - / 1 x\ 
 
 = \^f] - 
 
 eJ V c rj 
 
 Thus we have finally for the scalar potential at the external point of the 
 field 
 
 The expression for the vector potential is similarly deduced and is 
 
 * In this formula the operator V is presumed to affect all quantities following it. 
 
574, 575] The radiation from a group of electrons 513 
 
 The radiation field which predominates at large distances and in which we 
 find the flow of energy of which we have already spoken, is determined by 
 the second term in <j> and by the vector potential. At smaller distances it 
 is superposed on the field represented by the first term of (/>, which is the 
 static potential of the electron at rest. 
 
 575. Now suppose that there are a number of electrons and elements of 
 charge inside the small volume v under consideration. The field of each of 
 these charge elements will then be exactly of the type thus specified and the 
 total field of them all together will be simply obtained by superposition of 
 their separate fields. Thus if 2 is used to denote a sum over all the 
 elements of charge, we have in this total field 
 
 / ^ri \ 
 
 < = ( ^ - di 
 r 
 
 and to the first order the vector potential is given by 
 
 A _ [ 2ev J [MJ 
 r r 
 
 But if the volume element under consideration is very small and contains 
 only the constitutional electrons and charges of the molecules of matter 
 inside, so that the total charge is zero Ee = 0, and then 
 
 is the vector quantity which we have previously recognised as the polarisation 
 of the volume element : we may denote this by 
 
 Pdv 
 
 rpi 
 
 and then we see that d> div - dv , 
 
 r 
 
 cr 
 
 These relations would also hold in the case of a single uncharged molecule 
 if the appropriate value of the vector P is implied. They show that the single 
 molecule or small volume element of a material body will be a centre of 
 radiation whenever the polarisation P is changing : the distant field will 
 again be determined by 
 
 with the usual spherical polar frame of reference with the pole at the centre 
 of the element and axis parallel to the direction of polarisation. Thus however 
 rapidly the individual electrons may be rushing about inside the molecule 
 or element of volume there will be no radiation if the acceleration of the 
 polarisation or the vectorial sum (Sev) taken over them all is constantly zero*. 
 
 * Cf. Larmor, Aether and Matter, p. 228. 
 L. 33 
 
514 Electromagnetic oscillations and waves [OH. xn 
 
 The importance of this result is that it shows that certain groups of rapidly 
 moving electrons may, in spite of their rapid individual motion, yet be very 
 permanent as a self-contained dynamical system. The condition for per- 
 manency is that there should be no dissipation of the energy of the motions, 
 and if the individual motions are subject to the above restriction this condition 
 is certainly satisfied. For example, the orbital motion of two electrons of 
 equal inertia and opposite charge, round each other, the accelerations reinforce 
 each other instead of cancelling, so that this simple type is not a permanent 
 molecular conformation, though it is easy to construct other types that would 
 be possible. 
 
 576. Although it has little or no connection with the present aspect of 
 the subject it is of fundamental importance in the general theory to con- 
 sider one other special case of the analytical results of the present section. 
 Suppose that the charges in the element of volume under consideration are 
 moving so that the polarisation of the element is continually zero. The 
 vector potential of the field of the element is then determined solely by the 
 term 
 
 f r \ 
 
 the time being however [t - instead of t. Again we have 
 \ cj 
 
 Xe (Vr e ) r e - curl i Se [i e i e ] - i | Zq {(Vr e ) r e } 
 
 so that it is the electrons moving in permanent orbits that are mainly effec- 
 tive. Moreover for these the second sum on the right is negligibly small, 
 so that if we use 
 
 the vector potential of the field of the element is practically 
 
 A - [curl I] *? 
 r 
 
 which is just the same as if the element were magnetically polarised to 
 intensity I. 
 
 This result suggests an assumption, which we have already hinted, that 
 all magnetism is the result of small whirling motions of the electrons con- 
 tained in the ultimate molecules or molecular groups of matter, and it has 
 in fact been interpreted in this way by Larmor and Lorentz. The connection 
 of the idea with Weber's theory of magnetism is obvious. 
 
 Although this view of the matter is now generally accepted as providing 
 a sufficient account of the general phenomena of magnetism, it cannot yet 
 be regarded as definitely established in fact. We shall however generally 
 assume that it is a sufficient theoretical basis for our future discussions. 
 
CHAPTER XIII 
 
 THE REFLEXION, REFRACTION AND CONDUCTION OF 
 ELECTRIC WAVES 
 
 577. On the reflexion and refraction of electric waves at an interface 
 between two different dielectric media. We have so far merely treated analy- 
 tically of the propagation of electromagnetic waves in homogeneous bodies. 
 But what happens when such a wave reaches the boundary between two 
 different media ? The interface between the two media then becomes the seat 
 of fresh sources of disturbance of a fictitious nature, of course, as they do not 
 bring in any new energy. Thence arises the new effect, viz. the reflexion : 
 not the whole but only a part of the wave disturbance enters the new medium, 
 the rest is thrown back and forms the reflected beam. The disturbance 
 outside the second medium is thus due to the superposition of the primary 
 and secondary waves and if there be just the one interface, this state of 
 things continues. This secondary wave may itself be very complex because 
 the disturbance going into the second medium may reach the interface 
 again at other parts and there suffer reflexion and transmission anew. It 
 is however sufficient to treat the simpler case, the conditions for which are 
 secured by making the second medium extend indefinitely beyond an infinite 
 plane face which it has in common with the first medium. 
 
 The general equations of 520 combined with the conditions which must 
 hold between the fields on either side of the boundary at the surface will 
 enable us to determine the problem in this case. For the present we shall 
 however slightly modify the notation of the previous chapter and put the 
 z-axis along the normal to the boundary surface between the two media, 
 which we shall assume to be a surface of indefinite extent lying in the (x, y) 
 plane of coordinates. The plane of incidence, i.e. the plane containing the 
 normal to the surface and the direction of propagation of the plane wave 
 falling on the surface, is taken as the (y, z) plane. The direction of propagation 
 makes an angle a with the normal; this is the angle of incidence, to adopt 
 an optical term. From the part of this surface where this wave-train falls 
 there arise two other plane wave trains, one of which goes on into the second 
 medium in a direction making an angle /3 with the normal; the other is 
 reflected back on the other side of the normal at an angle a'. 
 
 578. We shall confine ourselves to the case when the two media are 
 absolutely non-conducting and non-magnetic. We first treat the case when 
 the magnetic oscillations are perpendicular to the plane of incidence. The 
 
 332 
 
516 Electric ivdve problems [CH. xra 
 
 electric force is then in this plane, so that E,,, E z , H,,. are the only components 
 of the two vectors differing from zero ; the dynamical equations are thus 
 
 c dt dz ' c dt dy ' 
 
 c dt oy dz 
 We now assume that the incident wave is a simple periodic train specified by 
 
 (E, , E v H isc ) =(Ai cos a, - A t sin a, - A t Ve) e ilnt - b{ysina+zcosa>} , 
 which must satisfy the equation 
 
 V'E - d ^ 
 c 2 dt* ' 
 
 n* n c 
 
 so that b = sr, 7- = p-. 
 
 c 2 b Ve 
 
 For the reflected wave we must have 
 
 (E ry , E rg , H rx ) = (^ r cosa', - A r sin a', - A r Ve) e * <<-&' <v sin a'- z cos a',, ? 
 
 which also satisfies the above equation so that 
 
 Moreover as the two waves (incident and reflected) must agree on the 
 boundary as regards variation along it we must have a = a. 
 
 For the refracted wave we have 
 (E, v , E Sz , H Sx ) = (A s cosj3, -A s& mf3, - H s Ve) c (*-,rinfl-Mco./i)). 
 
 Certain assumptions are_of course already involved in the forms here 
 adopted. The functions chosen all satisfy the fundamental electromagnetic 
 equations between themselves. Moreover we have assumed the frequency of 
 the vibration for each wave to be the same; but this must be so because 
 we are dealing with forced vibrations, and so the frequency n remains constant 
 under all conditions. We have taken the polarisation in each beam to be 
 similar, this is of course necessitated by the general theory, particularly as 
 regards the boundary conditions which compares corresponding components 
 of the fields on either side of the surface. 
 
 579. The boundary conditions state that the tangential components of 
 the electric and magnetic force vectors in the total field must be continuous as 
 we go through the surface ; but in order that these conditions may be satisfied, 
 the exponential function in each beam must be the same for 2 = 0. This 
 requires 
 
 (i) as above a = a' ; the angles of incidence and reflexion are equal and 
 (ii) also fcj sin j8 = 6 sin a, 
 
578-581] FresneV s reflexion formulae 517 
 
 6, sin a /e, C, 
 nr / 
 
 n \/ ; 
 
 o smp v c 
 c and Cj being the velocities of propagation in the two media. This is the 
 
 exact analogue of Snell's law in optics : the relation 
 
 / 
 ri _ Ai 
 
 r ~ V ~> 
 V e 
 
 is Maxwell's relation between the dielectric constants of the two media and 
 the index of refraction for light between them (c^c), 
 
 With these the boundary conditions imply that 
 
 (Ai A r ) cos a = A s cos j8, 
 AfVe + A r Ve = A s 
 
 . /cos a sin B\ , /cos a sin B\ 
 whence 4 r -- - 5 + ^- } = AA- 
 
 \cos sin a/ Vcos sin a/ 
 
 or 
 
 - 5 ^- - 
 
 p sin a/ Vcos p 
 
 tan (a 
 
 tan (a + p*) 
 
 which is another formula of theoretical optics ; it is Fresnel's formula* for 
 the reflected amplitude. 
 
 580. If ^ = S ^ or s i n 2 a = sin 2i8, 
 
 cos p sm a 
 
 then A r = and for this particular direction there is no reflected light ; it 
 is when 
 
 2a = TT 2jS or a + j8 = ^ , 
 
 i.e. when the refracted ray is perpendicular to the reflected wave direction. 
 The angle a which satisfies this condition is called the polarisation angle ; it 
 is such that 
 
 ye, sin a 
 -i= -.a = tana. 
 smp 
 
 When the angle of incidence is tan" 1 */ and the magnetic force is 
 
 perpendicular to the plane of incidence all the light passes through into the 
 second medium, none whatever being reflected. 
 
 581. If the magnetic oscillation were in the plane of incidence then we 
 should use 
 
 int-ib 
 
 x z e 
 
 (E rx , &r y , "Rr t ) = (A r , - A r V^ COS a, - A r Vt sin a) e int ~ ib (vsina- 2 cosa)^ 
 
 and , 
 
 (E Sx , K Sy , H 8z ) = (A,, 
 
 * Mfm. de I'Acad. xi. (1832), p. 393; Ann. chim. XLVI. 1843. Cf. GEuvres, I. p. 767 
 
518 Electric wave problems [OH. xm 
 
 The boundary conditions again give 
 
 b sin a = b sin f$, 
 so that Snell's law is still true. Moreover 
 
 A ( + A r = A S) 
 
 and VI (Ai - A T ) cos a = A s Ve l cos j8, 
 
 whence again we deduce Fresnel's second formula for the reflected amplitude 
 
 sina-rg 
 *sina + j8' 
 
 Thus in the reflected wave the electric force at the surface is reversed, the 
 magnetic one however retains its direction. 
 
 These formulae of Fresnel have been shown to be in good agreement with 
 experiment if in the first case the plane of polarisation is perpendicular and 
 in the second parallel to the plane of incidence. The magnetic force must 
 therefore lie in the plane of polarisation and the electric force perpendicular 
 to it. Although this theory thus determines this question of the plane of 
 polarisation quite definitely it still leaves open the question as to which is 
 the actual working force in the propagation of the wave. According to the 
 modern view of these things the electric force appears as the most probable 
 direct working agent in the electromagnetic propagation. 
 
 If in this second case a = then j8 = also and 'we get then, neglecting 
 the sign 
 
 A ^-A 
 A < a + p- A " 
 
 j J.T- Cj a 
 
 and since then = , 
 
 c 
 
 s* _ st 
 
 this gives A. A* 
 
 G! + c 
 
 a formula which is also true under the same circumstances in the first case 
 treated. 
 
 In this case it is impossible to make the reflected amplitude zero because 
 
 sin (a /?) 
 
 sin (a + jS) 
 can never be zero. 
 
 582. Iie 1 <e then 
 
 Bina 
 
 n <- i, 
 smj8 
 
 so that when a gets beyond a certain limit the equation 
 
 can no longer be satisfied by any real value of /?, since for such 
 
 sing^ 1. 
 
581-584] FresneTs formulae 519 
 
 If however we put 
 
 sin [3 = p, cos j8 = Vl p 2 , 
 then for such cases as mentioned 
 
 cos j8 = Vjt> 2 1, ^ > 1. 
 
 The exponential factor which occurs in both cases in the refracted wave 
 is then of the form 
 
 int - ib^ (py iz v^ 2 - 1 ) 
 
 > 
 
 and we must take the lower sign because the quantities would otherwise 
 increase with increasing z. The oscillations in the second medium thus 
 decrease rapidly on account of the factor 
 
 g-zv/pTTi^ 
 
 583. If the magnetic oscillations are parallel to the plane of incidence 
 then the boundary conditions give in this case 
 
 (At A r ) cos a = A s i Vp 2 1, 
 
 A t (cos a + i <J- Vp 2 1\**'A, (cos a-i J -Vp 2 - 1 ) . 
 \ V e^ / \ V 6^ / 
 
 or 
 
 * \ 
 
 T , cos a Vp 2 I 
 
 It now we put -=- = p cos 6, -= = p sin 0, 
 
 Ve Ve, 
 
 /e Vp 2 -! /cos 2 a j 2 -l 
 
 then tan^-=A/ --- -, p \f7=- 7= 
 
 V e t cos a > Ve Vx 
 
 and then ^jg ie = J. r e- ifl , 
 
 or A r = ^ie 2i9 . 
 
 Thus the amplitude of the reflected light is equal to that of the incident 
 light, although the one is out of phase with the other by 26 where 
 
 2 
 
 e x V cos" a 
 
 584. If on the other hand the magnetic oscillation is in the plane of 
 incidence 
 
 At + A r = A g , 
 
 (Ve A t - Vej A r ) cos a = - A s iVp 2 - 1 Ve l 
 
 = -(A t + A r )iVp 2 - lVe 1? 
 here we put 
 
 Ve cos a p cos 6 l Ve x , Vp 2 1 = p sin 6 l , 
 and have then again A r = A^ 6 ^ 
 
 1 Ve, 
 
 where 9 l = tan 
 
 Ve cos a 
 
520 Electric wave problems [GIL xm 
 
 The difference of phase in the two cases is 
 
 2 (0 - 0j), 
 
 but the combination of oscillations which are perpendicular to one another 
 with a definite difference of phase results in an elliptically polarised beam. 
 Thus in the case of total reflexion of a general plane polarised train of waves 
 the reflected wave is elliptically polarised. 
 
 These are the principal results for the passage of a beam across the interface 
 between two dielectric media : they are in all cases identical with the well- 
 known results of physical optics and in this connection have been fully 
 verified by experiment; in the more difficult circumstances with the much 
 longer electric waves the tests as mentioned above are entirely favourable 
 in a general way to the theory, although the same amount of accuracy is 
 not attainable. 
 
 585. On the reflexion of electric waves at a conducting surface. In the 
 
 previous paragraph we have treated of the reflexion and refraction of electric 
 or light waves at the plane surface separating two different dielectric media : 
 the case when one or both media have conducting properties can be similarly 
 discussed and leads to similar conclusions, the only essential difference being 
 due to the damping action now introduced. We shall therefore content 
 ourselves by giving the analysis for the simplest case of this kind, which is 
 the one from which Hertz tried to get standing waves. We consider the 
 incidence normally of a plain wave travelling in a dielectric medium on to 
 an infinite block of metal with a plane face. 
 
 We may take the incident wave to be specified by 
 ( E ^' H (tf ) (Af, A z Ve) e l{r - 62) , 
 
 2 2 
 
 in which 6 2 = - 
 
 c 2 
 
 and then the reflected and refracted beams are determined by 
 
 and 
 
 (E.,, By - (A., - A 
 
 , . na 
 
 wherein 4-77- -r- = 6, 2 , 
 
 c 2 
 
 provided of course the displacement current can be neglected, 
 
 19 -A na i I i / . na 
 
 V= -^~2> &! = -y- V 47 V' 
 
 The wave in the metal has therefore the exponential factor 
 
 as a damping factor to its amplitude. The amplitude therefore diminishes 
 rapidly, as previously explained, as we go down in the metal. 
 
584-586] Metallic reflexion 521 
 
 586. If we write a = . / **? ; 
 
 V C 
 
 then &! = a (1 i) and the exponential factor for the wave in the metal is 
 
 g az gi(nt az)^ 
 
 The boundary conditions give besides that 
 
 At + A r = A,, 
 
 A A - ^ A 
 
 n-i -n-r 7 -"-si 
 
 -so that 
 
 b + b, 
 
 f r i t 
 
 + 1 
 
 Thus the amplitudes of the various beams are in the ratios determined by 
 
 Al A* A? 
 
 46 2 ~ (6 - a) 2 + a 2 " (b + a) 2 + a 2 ' 
 
 but they are different in relative phase determined by the imaginary parts 
 of the amplitudes as defined. The total electric force intensity in the field 
 just outside the metal is 
 
 
 and in the general case of short waves is very small because 6/6 x is in general 
 small if the conductivity is big. We may even approximate and put this 
 equal to 
 
 26 , i(nt+bz) 
 
 The same form holds for the electric force in the internal field with the proper 
 exponent 
 
 E26 . -az i(nt-az) 
 s ~ = 1 "-iG 6 
 
 X r\ 
 
 ^ A *~ az i(nt-az) 
 
 /I -\ * *- V^ 
 
 a (1 - i) 
 
 n 
 
 / c\ A Cf* 
 
 = V2 - A t e e 
 
 -az i 
 e 
 
 /2n 
 = - 2 A/ 
 
 V CT 
 
 in the metal 
 
 H = a \ * E s z dx = SntA? I e 2az cos 2 (nt + az + j) dz, 
 J Jo \ 4 / 
 
 [The heat developed in the metal in such cases reckoned per unit area is 
 
 * 
 
522 Electric wave problems [OH. xm 
 
 where we have chosen the real part of the expression for E s as representing 
 the field. On the average this reduces to 
 
 a 
 
 The energy degraded into heat thus forms a negligible fraction of that incident 
 on the conductor. Thus in the general case the waves are turned back 
 without sensible loss by degradation, and for an ideal good conductor the 
 surface layer is at a node of the electric force. There is a superficial current 
 in the conductor but there is no sensible electric resistance, the small electric 
 force near the node establishing the necessary current without production of 
 heat.] 
 
 587. It can also be seen that it is impossible to reduce the amplitude of 
 the reflected beam to zero : in other words it is impossible to have a perfect 
 absorber or absolutely black body bounded by an abrupt interface ; but that 
 does not preclude the possibility of a molecular constitution of the interface, 
 of a loose and gradual kind such as may exist in lamp-black for example, 
 which might absorb light as soft porous bodies absorb sound*. 
 
 We may notice also that the total field in the dielectric is completely 
 specified as regards its electric part by the force component 
 
 which to the first order in 7- is the same as 
 
 &1 
 
 _ ^ .gint (gibz _ Q ibz\ 
 
 = Ai6 int sin bz, 
 
 so that in this case the characteristic stationary wave condition is produced ; 
 but of course we have assumed that there is no damping so that the amplitude 
 A t of the incident wave remains constant in time. If this were not the case 
 the solution would not be of the simple form here specified. 
 
 588. The experimental investigation of the optical properties of metals 
 has led to results of a far reaching importance in the general theory of metallic 
 conduction. The theory of the measurements usually made is rather too 
 complicated for us to enter into in this work, but a few of the results may 
 suffice to explain the points which they bring out. The indices of refraction 
 of metals have been determined by Kundt by using very thin metallic prisms 
 of small angle, and although the method is not capable of any great exactness 
 his results agree well with those deduced by more elaborate arrangements : 
 the values of the following three metals are quoted : 
 
 Silver ......... '18, 
 
 Gold ......... -37, 
 
 Steel ......... -. 2-41. 
 
 * Rayleigh, Theory of Sound, Ed. 2, 350-1. 
 
586-589J Metallic reflexion 523 
 
 The reflecting powers of metals, i.e. the ratio of the reflected to incident 
 energy for normal incidence has been determined by Hagen and Rubens* : 
 they find for 
 
 Silver ...... 95'3 % 
 
 Gold ...... 85-1% 
 
 Steel ...... 58-5%, 
 
 and from these the coefficient of absorption is determined from the formula 
 
 n* + K *+I-2n 
 
 ~ ' 
 
 so that for the above metals K is given by the values 
 
 Silver ......... 3'67, 
 
 Gold ......... 2-86, 
 
 Steel ....... "... 3-4. 
 
 But the dielectric constant of the metal on the ordinary definition is given by 
 
 so that in each of the above cases it will be negative ; but a negative dielectric- 
 constant has no sense. The clue to the difficulty is provided in a further 
 result obtained by Hagen and Kubensf : the absorption coefficient of the 
 metal is determined by the formula 
 
 lima 
 
 == \/ 9~ 5 
 
 V c 2 
 
 and a knowledge of K would enable us to determine a : in this way it was 
 found that for long heat waves the absorption is completely determined by 
 the ordinary ohmic resistance but that for shorter waves the value of a 
 decreased as a function of the wave length ; this points to the conclusion 
 that the constitutional relation of the theory, viz. the relation between the 
 electric force and electric conduction current, has to be modified for application 
 to rapidly alternating fields. 
 
 589. To find what is to replace this relation we must refer back to our 
 previous general discussion of the mechanism of metallic conduction. It was 
 there found possible to interpret the phenomena of conduction in metals in 
 terms of the ordinary conceptions of the electron theory and a formula for the 
 conductivity of the metal was obtained by statistical considerations involving 
 the ordinary electron constants of the metal. In this deduction we assumed 
 however that the applied electric field driving the current is stationary. If 
 the field is varying the deduction must be slightly modified but the general 
 principles of the analysis remain : we shall therefore content ourselves by 
 
 * Ann. d. Phys. i. (1900), p. 352; vra. (1902), p. 1. 
 t Ann. d. Phys. xi. (1903), p. 873. 
 
524 
 
 Electric wave problems 
 
 [CH. xin 
 
 just indicating the essential steps of the analysis which differ from those 
 adopted above in the case where the field is varying harmonically with 
 
 j 
 period 
 
 P 
 
 The function which determines the distribution of motions among the 
 electrons satisfies quite generally the differential equation 
 
 F * dj + F "87? + Fz dl + ^ + - + dt + ^ = ^' 
 
 where the notation is exactly as previously and 
 
 With the approximation adopted in the previous case and on the assump- 
 tion that the physical conditions of the metal are uniform throughout its 
 volume we have 
 
 t\=t-r 
 
 _ 
 
 2? e r m 
 
 ~ 
 
 the suffix 1 serving to indicate the values of the functions at the time ^ . 
 For the present purposes it will be sufficient to use 
 
 F - 
 
 /yi 
 
 m 
 
 i 1 ^ = F 2 = 0, 
 
 so that 
 
 ml- ipr m \ ' 
 
 590. It follows then by exactly the same argument that the electric 
 current density is given by 
 
 T Ne*l m 
 
 C = 
 
 . 
 
 L % 
 
 .pl 
 
 -*Jr 
 
 V O7T 
 
 mu 
 
 e~ z dz 
 
 - a^E e ivt + ipa 2 E e ipt , 
 
 where 
 
 l+-2= 
 
 
 a 2 CT O / 
 
 . 
 
 o 1 
 
 * Cf. Livens, PM. ifagr. xxx. (1915), p. 112. 
 
589-591] The diffraction of electric ivaves 525 
 
 Thus the relation between the current density and electric force is 
 
 And this is the more general relation of which we have spoken. 
 
 If we use this relation instead of the simpler one given above we shall 
 find in exactly the same way that 
 
 , 
 P 
 
 and now we have a completely effective account of the facts observed by 
 Kundt and Hagen and Rubens. The inertia of the free electrons provides 
 a contribution to the dielectric constant of a. negative amount and which 
 may therefore in a real case be the predominating part. Moreover the more 
 correct formula for the conductivity is one that decreases with the wave 
 length from the ordinary Ohmic form to which it reduces in the case of p = 0. 
 
 591. The diffraction of electric waves. We have so far treated 
 of the reflexion and refraction of electric waves only at the indefinitely 
 extended plane interface between two different media. The cases where the 
 bounding surface is neither plane nor of indefinite extent are of equal 
 importance both from the theoretical and from the practical point of view 
 and, in spite of the fact that the analysis involved in it is much more than 
 ordinarily complex it seems desirable to indicate the details of one or two 
 problems. The only tractable cases are of course those in which the bounding 
 interface between the different media assumes some simple geometrical 
 shape : we shall here deal only with those cases in which it is either in the 
 shape of a circular cylinder of indefinite extent in both directions in its axis 
 or of a sphere. In each case the medium outside the enclosed surface will 
 be assumed to be free aether and the medium inside a simple homogeneous 
 isotropic dielectric with constant e. 
 
 The first case to be examined is that in which a simple harmonic plane 
 polarised train of waves in the aether is incident normally on a circular 
 cylinder composed of the simple dielectric medium*. Problems of this type 
 naturally divide themselves into two distinct classes, according as the 
 direction of the electric force vector in the incident radiation is parallel or 
 perpendicular to the axis of the cylinder ; any other case can be made up of 
 a combination of two such cases. The analysis of the two types of problem 
 is similar and we examine the first one in detail. 
 
 * Rayleigh, Phil. Mag. xn. (1881), p. 81; Seitz, Ann. d. Phys. xvi. (1905), p. 746; Ignatow- 
 sky, Ann. d. Phys. xvm. (1906), p. 495. Debye, Phys. Zeitschr. (1908), p. 775; Schaefer, Ann. 
 de Phys. xxxi. (1910), p. 462; Nicholson, Proc. L. M. S. (2), xi. p. 104. 
 
526 Electric wave problems [OH. xin 
 
 592. The field is referred to a cylindrical polar coordinate frame (r, 8, z) 
 with the z-axis in the direction of the cylinder axis and the polar plane (6 = 0) 
 in the direction of the propagation of the incident waves. The general 
 equations of the field are then of the type 
 
 8H, 8 r dEL,. _ 8E ? _ 8 
 
 dd dz c dt dd dz 
 
 _8H r _8H z _l^ = !?r_? E i 
 
 c '' dz " ' dr ' . c dt ' ' dz " dr ' 
 
 4wrG, 8 , 8-H r T dH., d 8E~ 
 
 _? = (rH.a) , ~5~ = (?"Ee) - 
 
 dE 
 where, inside the cylinder, 47rC = e ^- , 
 
 tfK 
 
 and outside 
 
 If the electric force in the incident wave is parallel to the axis of the cylinder 
 it is natural to try a general solution of the field equations which satisfies 
 the same condition ; we put 
 
 E r - E 9 - 
 
 and then it immediately follows from the third of the second set of equations 
 
 that 
 
 H 2 = 
 
 in addition. The general equations thus reduce to 
 
 r dR 8E 1 dE 8E 
 
 with, 
 
 ^ _ = _ 
 
 c dt '' = 30 ' c dt dr 
 
 8 . 9H r 
 
 47T dC z 1 a / 8EA 1 8 2 E 2 
 
 T =" '- V I r 1? ) + 5 "aSa 
 
 c 2 a^ r dr \ dr J r 2 dv 2 
 
 It follows that 
 
 z z 
 
 where, inside the cylinder 4?r -jg = e -jrp , 
 
 whilst outside the same relation holds with e = 1 . 
 
 593. The main problem is now, just as always before, to solve the 
 characteristic equations for the interior and exterior fields and to fit the 
 solutions up at the boundary interface and at a large distance away. The 
 necessity for agreement at the interface suggests that the type of functions 
 are the same in the two fields, while the types themselves are determined 
 by the fact that the external field must agree at very distant points with 
 the undisturbed field. 
 
592-594] Diffraction by a cylinder 527 
 
 The undisturbed field is that of a simple harmonic wave train in which 
 the vibrations are plane polarised in the principal axial plane of reference. 
 Thus in this field we may put 
 
 and the total external field must agree with this value at a large distance 
 from the cylinder. The form of this function is not however suitable to the 
 problem in hand and consequently it must be expanded in the more appro- 
 priate equivalent form* 
 
 E z = E&v ct J (pr) + 2i n I, J n (pr) cos nd 
 
 wherein each term is known to be a suitable normal solution of the funda- 
 mental characteristic equation. J n is Bessel's function of the first kind and 
 nth order. 
 
 594. The method is now obvious. The general type solution of the 
 characteristic equation of the external field, viz., 
 
 1 _a / 9E 2 \ J. 9% = 1 d*E z 
 r dr\ dr / r- dd 2 c 2 dt 2 
 
 is e ipct [A n J n (pr} cos nd + B n K n (pr) cos n6] 
 
 where K H is Bessel's function of the second kind which is regular at infinity. 
 
 The same form of solution applies to the internal field but with p l = Vep 
 in place of p. 
 
 Now the total conditions in the field must be regular except for the imposed 
 conditions of the incident radiation field; it follows that the internal field 
 can only involve Bessel functions of the first kind, while the additional part 
 of the external field can only involve those of the second kind. We may 
 therefore generally assume for the external field the form 
 
 E 2 - EeW |/ (pr) +A K (pr) + 2 {J n (pr) + A n K n (pr)} cos ftf] 
 L =i J 
 
 whilst for the internal field we assume that 
 
 \ B J ( Pl r) + 2i S B n J n ( Pl r) cos n01 ; 
 
 L n = l J 
 
 and these two fields must now be fitted up at the surface of the cylinder 
 whose equation may be taken to be 
 
 r = a. 
 The surface conditions are that the tangential electric and magnetic 
 
 r)E 
 forces are continuous across the boundary. This means that E z and -^ 
 
 or 
 are continuous at r a or that 
 
 J n (pa) + A n K n (pa) = B n J n (p t a) 
 J n ' (pa) + A n K n ' (pa) - B n J n ' ( Pl a) 
 
 * Heine, Kugelfunktionen, t. i. p. 82; Whittaker, Modern Analysis (2nd Ed. 1915), p. 351. 
 
528 Electric wave problems [CH. xm 
 
 for all values of n. These equations determine all the unknown coefficients 
 of the field functions so that the whole circumstances in the fields are at 
 least theoretically determinate. In practice however the coefficients thus 
 deduced are much too complex to be of any real service so that resort has to 
 be had to approximate values, depending on the relative magnitudes of the 
 wave-length of the incident radiation and of the radius of the cylinder. We 
 consider here only the case when the radius of the cylinder is so small com- 
 pared with the wave length that all terms above (ap) 2 can be neglected. 
 
 595. To the second order of approximation when (pa) is small 
 
 P 2 a 2 2 
 
 'o=l- V' * = 1 
 
 The equations to determine A Q and B Q are then 
 
 - A eB 
 
 T A 'a~ ^^' 
 
 Thus A = 
 
 e - 1 
 
 a 
 
 2 & ypa 
 = *-l 2ft2 
 
 approximately, whilst 
 
 1 e 2 
 
 .Bo = 1 p 2 a log - : - . 
 
 2 ypa 
 
 Thus to this order of approximation the internal field is exactly identical, 
 except as regards velocity of propagation, with the original field reduced in 
 the ratio 
 
 l:l-(l_ 6 )?-log 
 
 The external field consists mainly of the original field, on which however is 
 superposed the field of the scattered radiation in which 
 
 E 2 = ( e - 1) ^~ EK (pr) e' et . 
 
 The field of this scattered radiation is, to the present order of approximation, 
 symmetrical round the axes of the cylinder. 
 
 596. The case in which the electric force in the initial radiation field 
 is perpendicular to the axis of the cylinder is similarly treated; but in ap- 
 proximating to the values of the coefficients it is now necessary to proceed 
 to the next higher power of the product (ap) and also to include the next 
 
594-597] Diffraction by a sphere 529 
 
 term in the expansion of the field functions in their normal form. In this 
 case there is only one component of the magnetic force, viz. that paralle? to 
 the axes of the cylinder and it is most convenient now to regard this com- 
 ponent as the fundamental vector of the theory. We now find just as above 
 that the external field is determined by the original undisturbed field on 
 which is superposed the field of the scattered radiation in which the magnetic 
 force is 
 
 H 2 = a V ~ p eW HKi (pr) cos 
 
 and the electric force analogously. 
 
 In this case the distribution of the scattered radiation is not symmetrical 
 round the cylinder. The intensity in it is greatest in the positive and negative 
 directions of propagation and it vanishes entirely in the perpendicular direc- 
 tions. The scattered radiation thus appears to be partially polarised. 
 
 597. The case* in which the radiation is incident on a sphere is quite 
 similarly treated but the analysis is necessarily more complicated even than 
 that of the cylindrical cases. We shall give the main outlines for the case 
 when the sphere is composed of uniform dielectric material with constant e. 
 
 In this casef spherical polar coordinates are used, with the polar axis in 
 the direction in which the radiation is incident on the sphere, and the origin 
 at the centre of the sphere. The general field equations are then of the type 
 
 4rrr 2 sin 6 d . d 
 
 -C r = 3 0(rs 
 
 4vrr sin 6 dH r 
 
 r 2 sin0dH r 8 . 3 . 
 
 and _____ (f sm Q E,) - ^ (rB,), 
 
 _rj^dH W d ^ 
 c dt d<f> dr 
 
 ^-i'la.i-.S 
 
 c~3T dr (r * ti) W ' 
 
 _ e rfE 
 wherein C = , 
 
 4:7T dt 
 
 inside the sphere and C = T 
 
 4:77 dt 
 
 outside. 
 
 * Rayleigh, Phil. Mag. XLI. (1871), pp. 107, 274, 447; xn (1881), p. 81; Collected Papers, i. 
 pp. 87, 104, 518. 
 
 f Rayleigh, Phil. Mag. xuv. (1897), pp. 28-52 ; Collected Papers, iv. p. 321 ; Proc. Roy. Soc. 
 LXXXIV. (1910), p. 25; xc. (1914), p. 219. Cf. also Love, Proc. L. M. S. xxx. (1899), p. 308. 
 T,. * 34 
 
530 Electric wave problems [CH. xm 
 
 598. Again the solutions are divided into two classes: in the first the 
 radial component of the electric force vanishes and in the second the radial 
 component of the magnetic force vanishes. If these two cases are respec- 
 tively denoted by 1 and 2 then it is easy to verify by examining each case 
 separately and superposing the results that the general solution of the 
 equations for internal space may be written in the form 
 
 _ 
 
 r ~ ~~dr* ' c 2 dt 2 ' r ~ dr* 
 
 1 8 2 n 2 i 8 2 n t lj!5i^ _* ?!5 
 
 r8r30 cr sin 8</> 8* ' r dr .30 cr sin0803r 
 
 E, = JLi a _ 2 -52 + 1 ?!5i H,, : - a -- n i = - - na 
 
 r sin 8</>8r cr 808^ ' r sin 8r3(/> re 808^ ' 
 
 where II x and II 2 are both solutions of the characteristic equation 
 
 8r 2 r 2 sin 9 80 V 80 / r 2 sin 2 edf~^ 2 ~W' 
 Outside the sphere the same type of solution holds but now e = 1. 
 
 The main problem is now to obtain the appropriate regular solutions of 
 this equation in the two parts of the field and then to make them agree with 
 each other at the boundary of the sphere, and with the applied field at a 
 large distance away, where the disturbance created by the sphere is necessarily 
 inappreciable. 
 
 The typical normal solution* of the fundamental equation is of the type 
 
 n - R n S n e^ ct 
 
 where S n is a surface harmonic of order n and R n is a function of r which 
 satisfies the equation 
 
 2 n(n+l) 
 
 ' " 
 
 
 wherein p^ = ep 2 . The general solution of this equation is known to be of 
 the type 
 
 r n + l (1 d _] n (R 
 
 (rdr) V7 
 where *5> + P1 *R = 
 
 so. that # = A sin (p^r) +B e- ipir . 
 
 The type solution of the general equation is therefore 
 R n = A n R n . + B n R no 
 
 /I d \ n smp.ri , /I d\ n (e-'*'* 
 
 = A n r n +i \--j-] y + B n r n +* [ - 3- ) 
 
 \rdrj r j \r drj ( r 
 
 the former part being regular at the origin and the latter part at infinity. 
 
 * Whittaker, Modern Analysis (2nd Ed.), Ch. xvni; Bateman, Electrical and Optical Wave 
 Motion, Ch. in. 
 
598-600] Diffraction by a sphere 531 
 
 599. We consider the case of the incidence of a simple harmonic wave 
 train which is polarised so that the electric force is in the plane < = : the 
 functions defining it will be assumed to depend on the time and position by 
 the exponential factor 
 
 gip(ct->rrros(>) 
 
 or what is the same thing* 
 
 .*- r 2 (2. + 1).*" flA ' (cosfl) i 
 
 L=o r J 
 
 where P n is Legendre's function of order n, and 
 
 zdz z 
 
 The radial component of the electric force is then of the form 
 E = E sin 6 cos (> e^ ct e j >' -cos e 
 
 90 
 
 and the other components are analogous. Thus the corresponding II 
 functions for the initial field are 
 
 _ S&*a*4 r 2n+l 8P. 
 
 ' 
 
 
 600. These functions suggest the typical forms to be assumed in the 
 more general circumstances. In fact, following the usual method, we can 
 now try a general solution for the external field determined by functions II j 
 and n a which differ from those just given by having 
 
 and R ni (pr) + A n "R m 
 
 in place of R ni in the two functions respectively. For the internal field on 
 
 the other hand we replace R n . (pr) by E n . (p^) in both functions, but with 
 
 additional constant multipliers B n ', B n " to distinguish them in the two 
 
 cases. 
 
 The internal and external fields thus specified have now to agree at the 
 boundary of the sphere (r a). The conditions there are that the tan- 
 gential electric and magnetic forces are continuous, and they are easily seen 
 
 * Rayleigh, Theory of Sound, n. p. 272; Heine, Kugelfunktionen (1878), i. p. 82. 
 
 342 
 
532 Electric ivave problems [CH. xm 
 
 to be satisfied if ell l5 II 2 , -^ and ~.- 2 are continuous at r = a. From the 
 
 continuity of II j we have that 
 
 R n . (pa) + A n 'R nt> (pa) = B n 'R n . ( Pl a) 
 
 and j- a {R ni (pa)} + A n ' A {R no (pa)} = B n ' ^ {R ni ( Pl a)} 
 
 so that A n ', B n ' are completely determinate. The equations derived from 
 the continuity of II 2 determine in a similar manner the constants A n ", B n " . 
 
 It is easy to prove that the new series obtained to represent the internal 
 and additional part of the external field are themselves convergent so that 
 a completely effective representation of these fields is obtained. Again 
 however the solutions, although theoretically complete, are too complex to 
 convey any practical meaning and we have to resort to approximations. 
 One particular aspect of these approximations will alone be considered. 
 
 601. If the radius of the sphere is small compared with the length of 
 the wave then the argument (pa) of all the functions is small : thus we can 
 use the approximations 
 
 R (x)- -* n+1 - R fa)- 1 - 8 '"* 2 *- 1 ),.-* 
 
 *"< w -l'.3...(2n+l)' x" 
 
 which are true for small values of x. To the first order in the product (pa) 
 the scattered wave is represented therefore by the first term in each of the 
 series with the unknown constants determined as 
 
 e - 1 2a 3 
 
 Thus in this case the scattered wave is specified as regards its electrical con- 
 stituents by the components 
 
 e- a . 
 
 E e = - - - - - - sin <f) cos 6 
 
 e- 
 
 The disturbance in it is zero in the direction 
 
 which is parallel to the direction of the electric force in the undisturbed 
 incident radiation. Thus when a plane train of non-polarised waves falls 
 upon a small dielectric sphere, the light scattered in any direction perpen- 
 dicular to the direction of incidence will be completely polarised. 
 
600-603] The colour of the sky 533 
 
 It is also important to notice that the intensity of the scattered radiation 
 depends on the wave length of the light (^Tr/p) and under similar circum- 
 stances the radiation of short wave length is much more intensely scattered 
 than that of long wave length. 
 
 602. An interesting application of these results has been made by 
 Rayleigh* to explain the blue colour of the sky. The light coming from the 
 sun arrives at our atmosphere as a parallel beam of radiation : in traversing 
 the atmosphere it is scattered by reflexion at the numerous small particles 
 which are present there. The particles may be the molecules of the atmo- 
 spheric gases themselves or also small globules of water or other vapour; in 
 any case the majority of them are probably of sufficiently small dimensions 
 to justify an application of the above results. We should thus expect on 
 theoretical grounds that the blue constituents of the white light (the short 
 wave lengths) would be more strongly scattered than any other. The light 
 from the sky, which is none other than the light scattered from the small 
 particles of matter in the atmosphere, should therefore be blue. The 
 theoretical polarisation phenomena have also been very satisfactorily veri- 
 fied by the facts, although here it is necessary to go to the next order of 
 approximation in the theory to obtain complete agreement. 
 
 The more general theory of the scattering of incident radiation by a 
 spherical obstacle with arbitrary optical properties which can be developed 
 on the above lines also admits of some very interesting applications in the 
 study of the colours exhibited by metal glasses, metallic films and colloidal 
 solutions or suspensions of metalsf. The problem is also of interest in its 
 dynamical aspects in connection with the theory of comets' tails. 
 
 603. It may be remarked in conclusion that the general methods of 
 analysis used in this section are not necessary if only the approximate results 
 finally obtained are required. In fact the form obtained for the approximate 
 solutions shows that they could have been written down at sight. Consider 
 the case of the sphere just examined. The wave' length of the radiation is 
 very large compared with the radius of the sphere, so that at any instant 
 the sphere may be considered as existing in a uniform field; the statical 
 effect of such a field is to induce polarisation in the spherical dielectric which 
 is equivalent, as regards outside points, to a doublet of strength 
 
 * E a 9 e ipct 
 
 e + ^ 
 
 * 1. c. p. 529. 
 
 f Maxwell Garnett, Phil. Trans. A. ccm. (1904), p. 385; ccv. (1905), p. 237; G. Mie, Ann- 
 d. Phys. xxv. (1908), p. 377; R. Gans, ibid. ; xxix. (1909), p. 280; xxxvn. (1912), p. 881. 
 Other problems of a similar type have been worked out by J. J. Thomson, Recent Researches, 
 p. 437; J. W. Nicholson, Proc. L. M. 8. (2), ix. (1910), p. 67; xi. (1912), p. 277; Debye, Ann. 
 d. Phys. xxx. (1909), p. 57. Full references are given by Bateman, Electrical and Optical Wave 
 Motion. 
 
534 
 
 Electric wave problems 
 
 [CH. xin 
 
 at the centre at time t. The radiation field due to this varying doublet, 
 which, as such, is equivalent to a simple Hertzian oscillator, is identical with 
 the field determined above as associated with the radiation scattered from 
 the sphere. 
 
 604. On the propagation of electric waves along conductors : Rayleigh's 
 method. We have so far discussed the radiation in fields of unlimited 
 extent, so that any local electromagnetic disturbance could' be transmitted 
 throughout the whole of the field subject merely to the conditions of 
 reflexion and refraction at interfaces of discontinuity in the transmitting 
 medium. Unfortunately however this unlimited nature of the field 
 necessarily implies cubical expansion of the volume covered by the radiation 
 field and the consequent very large diminution of the intensity of the field 
 in any part of the field under observation, a process which is admirably 
 illustrated by the example discussed in 544. It thus becomes necessary 
 
 Fig. 93 
 
 both from a practical and theoretical point of view to introduce some 
 means of preventing this indefinite spreading of the field. This is easily 
 accomplished in many ways. Hertz discovered that by placing a long 
 straight conducting wire with one end near the radiator, he was able as 
 it were to induce the field of radiation to flow out in the direction of the 
 wire : the wire acts as a sort of guide for the radiation which then travels 
 out along it and does not spread out sideways. In his successful experiments 
 he therefore conducted the waves from the vibrator described above by a 
 long straight wire soldered perpendicularly on to one of the condenser plates 
 in it; the other plate and the end of this wire were then earthed. The 
 oscillating current in the conducting circuit thus flowed through the condenser 
 out along the wire and back through the earthed connections, which in the 
 
603-606] The physical problem of conduction 535 
 
 actual case were the walls of the room in which the experiments were per- 
 formed : owing however to the fact that the complementary or return circuit 
 was provided in the walls, the results were still somewhat irregular. A. still 
 better plan was adopted by Lecher* by discharging the second plate not 
 directly to earth but first through a second long wire leading out from it 
 close to and parallel to the wire from the other plate. By this means the 
 field in the air is still more localised, the lines of electric force going from 
 one conductor to the other, instead of a long way off to the walls of the 
 room. The results of the experiments with this apparatus were consequently 
 much more regular and precise. 
 
 605. The aether is a thing that can vibrate somewhat like a jelly and the 
 electromagnetic condition is propagated through it in waves. The function 
 of the conductors is to annul the elasticity so that in the above form of 
 apparatus the vibrations are like those in a jelly filling the whole of space 
 with two straight cylindrical holes in it (the conductors). The two holes 
 have a complementary function, the strain caused by the one being released 
 on the other, instead of having to extend to infinity before being relieved. 
 Thus the waves which would otherwise spread out in all directions are guided 
 by the wires so that the activity is confined to the medium immediately 
 surrounding them. 
 
 There are of course other equally obvious and some still more effective 
 ways of restricting the radiation fields ; they may for example be enclosed by 
 conducting media, which if the conductivity is good are practically imperme- 
 able. This method is of practical importance in connection with the pro- 
 pagation of electromagnetic wave signals along cables or in telephone wires. 
 A cable for our purposes consists essentially of a cylindrical metallic conductor 
 surrounded at a small distance by a coaxial cylindrical metallic sheath (in 
 submarine cables this conducting sheath is the water itself), the space 
 between being occupied by a uniform dielectric substance. According to our 
 theory any disturbance created in the dielectric medium between the metallic 
 surfaces will run along in that medium without however spreading very 
 much because it cannot penetrate far through the metallic boundaries. The 
 field and propagation are therefore confined to the dielectric shell. 
 
 606. Let us now examine some of these cases in greater detail. We shall 
 commence by an examination of the general circumstances of the propagation 
 of electromagnetic disturbances of the type just discussed, the propagation 
 taking place only in the direction of the conductors which are all parallel 
 and cylindrical. We shall simplify the problem as much as possible by 
 neglecting the complicated state of affairs in the field up near the vibrator. 
 
 * Ann. d. Phys. XLI. (1890), p. 850. Cf. also D. Mazzotto, II Nuovo Cimento (4^. vi. (1897), 
 p. 172. 
 
536 Electric wave problems [OH. xm 
 
 We shall in fact treat all the conductors as very long and consider only the 
 circumstances in them at some distance from either end. The vibratory 
 discharge of condenser at one end of the conductors will then make oscillatory 
 currents run along the conductors,, but of such amounts that the total current 
 crossing any surface perpendicular to the direction of propagation is zero. 
 If this latter condition were not fulfilled some other conductor outside the 
 system (the return circuit) would have to carry the complementary current 
 and the irregularity observed in Hertz's experiments would present itself. 
 The wave length of the disturbance is presumed to be large compared with 
 the dimensions of the cross section of the conductors and also with their 
 distance apart. 
 
 It then follows by symmetry that the conduction current is directed along 
 the length of the conductors, there being no cross flow in a steady state; 
 and if the disturbance is periodic,, but not too fast, this conduction current 
 is by far the more important part of the total current flow, and compared 
 with it the others may certainly be neglected. Of course in the surrounding 
 dielectric field there is no conduction current and the displacement current 
 is all there is, but this is excessively small compared with the conduction 
 current in the metallic parts. If therefore we interpret everything in terms 
 of the vector potential in the field, there will only be one appreciable com- 
 ponent of that, quantity, viz. that parallel to the axis which is equal at any 
 point to 
 
 C l denoting the volume density of the electric flux in the wires. From this 
 the electric force is determined by 
 
 1 dA 
 
 E= -c^- gradj/f ' 
 and the magnetic force by 
 
 H = curl A. 
 
 It follows that the magnetic lines of force of the field round the conductors 
 are round about in planes perpendicular to the direction of the wires. The 
 electric force has its main electrokinetic part along the direction of the wires ; 
 but its static part is practically in the perpendicular planes, because it is 
 contributed in the ordinary way by the charges accumulated ; and we have 
 assumed that the wave length of the disturbance is long compared with the 
 distance apart and dimensions of the wires so that the distribution is uniform 
 along at any place ; or at least the gradient of along the direction of pro- 
 pagation is small compared with its value perpendicular. 
 
 607. We have next to investigate the field in the different regions. In the 
 interior of each medium the total current of Maxwell's theory, comprised of 
 
606, 607] The mathematical theory of conduction 537 
 
 the conduction and displacement currents, is connected with the magnetic 
 field in that medium by the universally valid circuital reaction of Faraday, 
 
 C = curl H, 
 
 c 
 
 and in the general case C consists of two parts (i) the conduction current 
 which is proportional to the electromotive force 
 
 C, = crE, 
 
 the constant a being the specific conductivity of the medium, (ii) the dis- 
 placement current which is 
 
 Thus curlH = 47r-E + -E, 
 
 c c 
 
 or interpreting it in terms of A, 
 
 a dA e d 2 A ( . a d e d 2 \ 
 
 curl curl A = - 4?r -r -j- -^ - KTT -= -57 + -= -^ ) grad y. 
 c 2 dt c 2 dt z \ c 2 at c 2 dt 2 / G 
 
 Since now there is only one appreciable component of the vector potential, 
 viz. that along the direction of propagation, and since moreover the gradient 
 of tfi in this direction is negligible this leads to an equation for the one com- 
 ponent of the form 
 
 z denoting the coordinate along the direction of propagation. This equation 
 has of course different forms in the different regions. 
 
 In the conductors the displacement current is negligible and the equation 
 is of the form 
 
 tf 2 A- a2A -4^^ 
 
 V A oo ** 9 Jj 
 
 dz 2 c 2 dt 
 
 In the dielectric region between the conductors on the other hand there is 
 no conduction current (a = 0) and the equation reduces to 
 
 - 
 
 dz 2 c 2 dt 2 ' 
 Of course in the general case the total current is 
 
 so that in these problems the utmost generality is obtained by the substitution 
 of complex coefficients for real ones. 
 
 We conclude that in the conductors the quality expressed by the quantity 
 A is propagated, like heat conduction, by diffusion, not by waves ; instead 
 
538 Electric wave problems [CH. xm 
 
 of a wave pulse going through it merely soaks in. In the dielectric regions 
 on the other hand this quality is propagated in simple undamped wave forms 
 with a velocity c/Ve, e being the dielectric constant of the medium. 
 
 608. If the metal were a perfect conductor a would be- infinite, but if a 
 is large dA/dt must be very small and actually zero if the current is alter- 
 nating. This means that A is constant at all points in the conductors ; but 
 in this case the current density is zero at all internal points so that we arrive 
 at the conclusion that in a perfect conductor the current is confined to the 
 surface of the metal, it does not soak in at all. In this case and when the 
 alternations are not too rapid the equation satisfied by A in the dielectric 
 reduces to 
 
 32A P52A 
 
 + ^ = 
 
 **\ o \^ > o v * 
 
 ox* cy* 
 
 the axes being chosen so that the z-axis is along the direction of the field. 
 This indicates a very simple state of affairs. A is constant in all conductors 
 and satisfies the equation for the ordinary electrostatic potential in the dielectric 
 medium between: the determination of A is thus reduced to an electrostatic 
 problem. 
 
 Moreover the current density on any sheet multiplied by c/e corresponds 
 to the charge density of the electrostatic problem : this follows at once from 
 the form for A which expresses it as the Newtonian potential of the instan- 
 taneous current distribution. The distribution of current would thus be 
 analogous to the distribution of charge in the electrostatic problem : the 
 total current flowing over the cross section of any conductor in the plane 
 under consideration would for example be 
 
 c f 9A , 
 
 J r = -. w ds, 
 far j r on 
 
 the integral being taken round the boundary of the section. 
 
 609. The energies also correspond, in fact, if <f> is the electrostatic 
 potential of the charged conductors, the static potential energy per unit 
 length of conductors is 
 
 where Q r = - 
 
 whilst the corresponding energy in the electromagnetic case, which is got 
 from the general formula 
 
 T = ~ [(AC) <fc, 
 
 is practically S/ A 
 
 2c 
 
607-610] Rayleigh's simplification 539 
 
 where J r is given as above : this can be written in the form 
 
 so that 'the energies are in the ratio e : 1. In the case of a single conductor 
 whose capacity per unit length is b and induction a we have that 
 
 A r 
 
 V : "S' 
 
 whilst Q r = b(f) r . 
 
 The electromagnetic and electrostatic energies for corresponding cases are 
 
 thus respectively 
 
 IV b 
 
 2ac 2 ' 2 9r ' 
 
 c 2 1 
 whence we see that 
 
 e ab 
 
 1 c 
 
 or = == c l ^, 
 
 Vab Ve 
 
 which verifies that the velocity of propagation is the velocity of light : this 
 is the result obtained by KirchhofTs method on a former occasion*. 
 
 610. The special condition of perfect conductivity might at first sight 
 appear to destroy the essential wave propagation characteristic of the 
 phenomena since it leads to a characteristic equation for A of the form 
 
 82A 8 2 A_ 
 
 ~dx* + fy 2 ~ 
 
 It is however very easy to see that even in this special case the field is still 
 in essence propagated. We have in fact under the assumptions mentioned 
 
 E - -?* E -?* 
 
 ** dx' y ~ dy' 
 
 9A _ 3A 
 
 and H -= % H -= --& 
 
 Thus from the first of Ampere's circuital equations we get 
 
 e dtfi _ dA 
 
 c dt dz ' 
 
 and from the second of Faraday's equations we get similarly 
 
 _ IdA^ty 
 c dt dz' 
 whence it immediatel follows that 
 
 and 
 
 c 2 dt 2 dz z ': 
 * Cf p. 456. 
 
540 Electric wave problems [OH. xin 
 
 Confining ourselves to the case of progressive waves these equations are 
 solved by = ^J ( Cl t - z), 
 
 where c, = -- and ^ and A are appropriate functions of x and y satisfying 
 
 Ve 
 the equations 
 
 
 
 and 1? ^ 
 
 This gives the essence of the affair ; the conditions in any plane perpen- 
 dicular to the direction of propagation are similar to those which would 
 hold in a statical theory : the lines of electric force running out normally 
 from the conductors beginning on positive charges and ending on negative 
 charges, just as in the electrostatic problem ; the direction of the gradient of 
 the magnetic vector potential at each point of the field is identical with that of 
 the line of force there, so that the lines of magnetic force are the orthogonal 
 trajectories of the lines of electric force. This state of affairs then travels 
 through the dielectric medium with the velocity Cj. The field of radiation 
 is therefore of the typical transverse type with the electric and magnetic 
 forces of equal intensity and perpendicular to one another in the wave front. 
 
 611. As an example we may consider the very simple case of two con- 
 centric cylinders (radii a and 6) with the dielectric field between them. The 
 total currents along the conductors are equal and opposite, the one being 
 the exact complement of the others just as the charges are in the statical 
 case. The vector potential in the field is then 
 
 and the magnetic force is in concentric circles round the axis of the cylinders 
 and the electric force is radial. 
 
 As a second example we may quote the results for Lecher's* modification 
 of Hertz's original apparatus for the production of electric waves. Let us 
 assume that the two wires are circular. A cross section of the field is repre- 
 sented in the figure and we know that if we put in the limiting points of the 
 coaxal system of circles determined by the two circular sections and denote 
 by r x and r z the distances, in the plane of the paper from these points the 
 electrostatic potential of these conductors uniformly charged oppositely is 
 
 T 
 
 c lg ^ ' tnus tne vector potential in the present radiation field has the one 
 component 
 
 * Ann. d. Phys. xu. (1890), p. 850. 
 
610-612] 
 
 The general theory for cables 
 
 541 
 
 In this case the lines of electric force are the arcs of circles through the limiting 
 points and the lines of magnetic force are the orthogonal circles. 
 
 This simple method of deducing results for radiation problems was 
 suggested by Rayleigh* but of course it only applies for the case of perfect 
 
 Fig. 94 
 
 conductors. The results will be true as approximations when the conduction 
 is not perfect, provided that the oscillation is quick enough to ensure that 
 the current is confined to a thin layer near the surface of the conductor. 
 The more general problems are soluble in certain cases but involve rather 
 complicated analysis. 
 
 612. The propagation of waves along cables: general theory. In the 
 
 previous paragraph we have considered the propagation of waves along wires 
 and cylindrical conductors under the assumption of perfect conductivity in the 
 metal : the cases where this condition is not fulfilled are much more difficult 
 to analyse but it is essential that we should have some idea of the circumstances 
 in such a case in order that we may form an estimate of the degree of 
 approximation of the results obtained above. We shall now consider the 
 general problem in the one particular case which is of real practical importance, 
 viz. the propagation of waves along a circular cylindrical cable consisting of 
 
 * Phil. Mag, LIV. (1897), p. 199. 
 
542 
 
 Electric wave problems 
 
 [OIL xni 
 
 an inner and outer conductor with a shell of dielectric between them. The 
 analysis for this case is rather complicated and we shall find it more convenient 
 to approach the problem tentatively by first examining a very simple problem 
 which brings out most of the important additional points of the theory and 
 is easily analysed. 
 
 We first consider therefore the propagation of electric waves in a 
 dielectric slab between two parallel conducting slabs extending on both sides 
 
 Fig. 95 
 
 to infinity. This would represent fairly well the circumstances of the pro- 
 pagation in the dielectric shell of the cable as above provided the thickness 
 of the shell at any place is not too large compared with the diameter of a 
 cross section. 
 
 613. We choose axes as in the figure. The 2-axis is normal to the plane 
 surfaces and the x-axis in the medial plane of the slab in the direction of 
 propagation of the waves. The magnetic force is thus parallel to Oy. 
 The general equations of the theory thus reduce to the form 
 
 , dE x \ = ^ dH.y 
 dt ) dz ' 
 
 - [47TO-E, 
 
 c \ 
 
 dt 
 
 dx ' 
 
 and 
 
 1 dH, 
 
 9E, 
 
 c dt dx dz ' 
 whence "EL y satisfies the usual fundamental equation 
 
 ~\~ : ) == ~T~ 
 
 dt ' dt 2 J dx 2 
 
612-614] A special case 543 
 
 which assumes different forms in the various regions. In the conducting 
 regions the displacement current is negligible and H y satisfies the equation 
 
 c 2 dt ' dx 2 ' ^ 2 ' 
 and in the dielectric medium a = so that H v satisfies the equation 
 
 Following the usual plan an appropriate solution of these equations can be 
 specified by 
 
 (i) in the dielectrics 
 
 H yi = A l e i (nt+a cosh bz, 
 
 if 
 
 where 5- = a 2 + 6 2 
 
 c 2 
 
 (ii) in the conductor on the positive side 
 
 H A f>Hnt+ax)-bi(z-d) 
 1/2 -"-Z K > 
 
 where 2d is the thickness of. the slab and 
 
 (iii) in the conductor on the other side a similar symmetrical form holds, 
 it need not further be specified. 
 
 The surface conditions of continuity of the tangential electric and magnetic 
 forces show that 
 
 A 2 = A v cosh bd, 
 
 and -7 - = A, sinh bd, 
 
 so that -r-^~ = tanh bd. 
 
 we 
 
 The usual assumptions made above that the length of the wave is long 
 compared with the thickness of the slab enable us to approximate to the 
 value of tanh bd and we can write 
 
 or 
 
 614. Let us examine these results in a particular case. In copper 
 
 c 2 
 
 a = =- 7: and if we take a wave length A 100 cms. then 
 IbOO 
 
 n = 3. 10 8 . 
 
544 Electric ivave problems [OH. xm 
 
 and thus since a turns out to be a fraction of the order 10~ 2 we must have 
 
 -,2 ' 
 
 this part of 6 X 2 being by far the largest. Thus 
 
 ft i = ~ ST~ \/ ~72~ ' 
 
 V ^ 
 
 which gives precisely the same damping factor as previously discussed on 
 more than one occasion. The propagation of the field directly into the 
 conductors is therefore damped off in precisely the same rapid manner as 
 previously exhibited. 
 
 In this case 
 
 ib, we 
 
 1 - 'i / w 2 e 
 
 2 V 4 2 7T 2 CT 2 d 2 ' C 2 
 
 1 i nc /- 
 
 = ~~?> 7~ ~~j v 47rno-, 
 2 47Tacd 
 
 and is thus of the order 10~ 10 , and is extremely small. Again 
 
 and since & 2 is small compared with - r we can write 
 
 c 2 
 
 a = 
 
 c 
 
 nVe 
 
 + a, (1 - i), 
 
 U 
 -i 
 
 where a t = , 
 
 and is of the order 2 . 10~ 6 . 
 
 615. The wave motion in the dielectric is thus propagated in the 
 specified direction with a velocity 
 
 n 
 
 , __ ^ /i *" 
 
 nVe V x e\ wVV 
 
 h j 
 
 which differs from ^ but very slightly. The velocity of propagation in the 
 
 Ve 
 
 medium is hardly affected by the presence of the conductors. The chief 
 influence of these is however in their damping effect which is no longer 
 negligible. There is an exponential factor in the wave form in the dielectric 
 
614-616] The general theory for a cable 545 
 
 which defines the mode of decay of the disturbance as it is propagated. 
 The energy is gradually used up against friction in the conductors, even 
 the supply located in the dielectrics being called upon. In the particular 
 problem quoted the damping comes out to be such that in a distance of the 
 
 order about 10 4 metres the amplitude is reduced to - of its initial value. 
 
 In the general case however we see that with given materials the shorter 
 rapid oscillations are the least damped : very rapid ones are not appreciably 
 damped at all. The reason for this is obvious for in such cases the current 
 is confined to a very thin layer at the surface of the conductors. There is 
 no current in the body of the conductor, simply charge oscillation at its 
 surface. We do not of course mean that there is no resistance at all to a 
 surface current of this nature; it is merely the excessive rapidity of the 
 oscillations that makes the resistance unimportant. 
 
 If we consider the real part of the above solutions only it is easily verified 
 that the energy dissipated in the conductors at any place (per unit breadth 
 along the ?/-axis) is equal to 
 
 and this again illustrates the above remarks very vividly. 
 
 616. An approximate solution for the actual circumstances in the cable 
 could now be obtained by wrapping this solution round a cylindrical shell as 
 already explained, the results here obtained applying per unit length round 
 the cable thus formed; owing however to the intrinsic importance of the 
 problem it seems advantageous at least to indicate the various steps in the 
 analysis*. .For this purpose we first examine the case where there is no 
 outer conductor. In this case and in fact in any case involving coaxal 
 circular cylinders it is most convenient to refer the field to cylindrical polar 
 coordinates (r, 6, z) with the axes along the axes of the cylinders. 
 
 Under the usual circumstances of propagation along the cylindrical 
 conductor the field is symmetrical round the axis and in the case of the type 
 above examined where the magnetic force is in circles round the axis the 
 general field equations assume the form 
 
 dE 47TCT Id 
 
 dE 47TO- _ 
 
 r 
 
 c dt c r dz 
 
 3E E8 
 
 = 
 c dt dz dr ' 
 
 * Cf. Rayleigh, Phil. Mag. (5), xxi. (1886), p. 381; Heaviside, Elect. Papers, n. pp. 39 and 
 168; J. Stefan, Wien. Ber. xcv. (1887), p. 917; Ann. d. Phys. XLI. (1890), p. 400. 
 
 L- 35 
 
546 Electric wave problems [OH. xin 
 
 These equations are satisfied by the forms 
 
 1 3 / 9IT 
 
 __ 
 
 ~ ~ 
 
 _ _ 
 r ~ 
 
 47TO- an 
 
 c drdt c dr ' 
 when II satisfies the equation 
 
 8 a n 
 
 " c 2 dt 
 
 i a / gn\ 
 
 r dr V dr ) ' 
 
 The equations in the interior of the wire are obtained as usual by neglecting 
 the displacement current (i.e. by putting e = 0) whereas in the field outside 
 the conductor there is no conduction current (or a 0). We shall also 
 assume for simplicity that e =1 in the external medium. 
 
 At the surface of the conductor the boundary conditions imply the con- 
 tinuity of the tangential components of the electric and magnetic forces. 
 
 617. To represent the propagation of simple periodic waves along the 
 cylinder we try the solution 
 
 II = e ip(ct ~ lz) x (r}> 
 
 wherein Us & constant to be determined and which determines the velocity 
 and damping of the wave propagation. The equation for % is the simple 
 Bessel equation 
 
 d 2 y 1 d x 
 
 _ A _i ___ * _i_ v = 
 
 dx* + x dx + x 
 
 where the independent variable x has the respective forms 
 x = prVk z - I*, 
 
 lt . _ 
 
 x - pr V 1 - l z , P c 
 
 in the solutions corresponding to the conductor and the dielectric medium. 
 We shall write 
 
 l 2 , VI = P Vl ^l 2 . 
 In the interior of the conductor we must choose the particular Bessel function 
 which is finite on the axis, that is we must take 
 
 X = CJ Q (i/ r), 
 where J x 1 " e i*cosa ^ a 
 
 Outside in the dielectric we must use the second type of function and take 
 
 x - DK Q ( Vl r), 
 i f iao r 00 e ix d? 
 
 where K ( v ,r) = * e ixcosa da = 
 
 ^Jo J i 2 - 1 
 
616-618] The cable problem 547 
 
 618. Thus inside the conductor the field is completely determined by 
 the vectors 
 
 E z = Vo *CJ far) eW*-**, 
 
 E r = - ilv CJ ' far) e*^-^, 
 
 whilst outside in the dielectric field they are 
 
 E, = V!*DK O ( Vl r) e ij > (ct - l *>, 
 E r - - ilviDKJ far) e ip(ct - lz \ 
 
 The boundary conditions that the force components E 3 and H e are continuous 
 across the surface of the cylinder give 
 
 and CJ ' (v a) = i 
 
 c 
 
 whence on elimination of C and D 
 
 cv Jp faa) 
 
 J ' faa) ip K ' faa) ' 
 
 which is the transcendental equation for the constant I which determines 
 the complete circumstances of the propagation. This equation cannot, of 
 course, be generally solved, but the approximate solution can be obtained in 
 most cases of real practical importance. The algebraical processes involved 
 are however rather complicated although they are quite straightforward, and 
 since the results obtained are precisely of the character of those obtained 
 above in the simple problem it is perhaps not necessary to give them out 
 in full here. They are very fully discussed in an elaborate paper on this 
 subject by Sommerfeld*. The propagation takes place mainly in the dielectric 
 with a velocity nearly equal to that of radiation, and there is a slight penetra- 
 tion into the conductor and consequent dissipation of energy, the main 
 effect of which is to introduce a slight damping effect in the wave propagation. 
 
 The complete problem f of the cable where the cylindrical conductor above 
 discussed is enclosed by a second coaxial one can now be directly solved. 
 
 * Ann. d. Phys. LXVII. (1899), p. 233. Cf. also J. J. Thomson, Proc. L. M. S. xvn. (1886), 
 p. 310; Recent Researches, 259; Abraham, Encyklop. d. math. Wiss. Bd. v. 2 (1910), p. 526; 
 Larmor, Proc. 5th Int. Congress of Math. i. (1912), p. 206. 
 
 f Cf. J. J. Thomson, Proc. R. 8. XLVI. (1889), p. 1; Recent Researches, p. 262; F. Harms, 
 Ann. d. Phys. xxin. (1907), p. 44. Other cases have been examined by Mie, Ann. d. Phys. n. 
 (1900), p. 201 ; W. B. Morton, Phil. Mag. i. (1901), p. 563; J. W. Nicholson, Phil. Mag. (1909 ) 
 and Phil. Mag. xix. (1910), p. 77; H. C. Pocklington, Proc. Camb. Phil. Soc. ix. (1897), p. 324; 
 D Hondros, Ann. d. Phys. xxx. (1909), p. 905; Dissertation, Munich (1909). 
 
 352 
 
548 Electric wave problems [CH. xin 
 
 The field in the interior of the core will involve a Bessel function of the first 
 kind and that in the metallic sheath, which is presumed to extend to infinity 
 all round, will depend on a function of the second kind. The field in the 
 dielectric shell between will involve both kinds of functions. The two 
 boundary conditions at each surface will determine the four constants thus 
 involved. The details of the method are now obvious and they need not 
 further detain us. 
 
CHAPTER XIV 
 
 GENERAL ELECTRODYNAMIC THEORY 
 
 619. The energy in the electromagnetic field. Since we became con- 
 vinced of the impossibility of perpetual motion there has always been 
 connected with our conceptions of natural phenomena the idea of that 
 something which we call energy. The kinetic and potential energies of 
 matter were the kinds of energy first recognised and for this reason it is 
 customary to try to associate any new form of energy which turns up with 
 something which in its properties is akin to matter. Thus arose for example 
 the idea of the material aether which formed the basis of the older wave theory 
 of light and which ascribed to this aether elastic and inertia properties and then 
 regarded the energy of the light waves as composed of the kinetic and potential 
 energy of the medium. The main object of Maxwell's electric theory is to 
 adopt the idea of an elastic aether to explain the electrodynamical actions of 
 electromagnetic systems, although care has been taken not to attribute to 
 this aether any nature analogous to that of ordinary matter and the theory 
 is in fact quite independent of any definite assumptions or hypotheses we may 
 make as to its constitution. Thus far therefore we merely use the word 
 aether as a convenient means of describing those properties of space which 
 are concerned in electromagnetic phenomena, those properties being mathe- 
 matically expressed by the general equations of Maxwell's theory, which 
 contain in themselves not only the totality of the older laws of electrostatic 
 and electrodynamic phenomena, but also the laws for the propagation of light 
 and electric waves in space. 
 
 In reality the universe consists of matter and the electromagnetic field in 
 the aether. If the electromagnetic field were not present our eyes would not 
 indicate to us the presence of the matter. In fact natural phenomena in 
 general appear to us as a result of the interaction between matter and the 
 aether. A simple materialistic conception of things regards the interaction 
 of the material constituents as the essence of the affair and regards the 
 electromagnetic theory merely as an auxiliary means of formulating the laws 
 of these interactions ; matter is the only actuality. We may however with 
 equal justification regard the matter from the other point of view. We can 
 regard the electromagnetic aether as the only actuality and matter as a 
 special manifestation of a * condition ' in this aether. Such a view is of course 
 one-sided but it is helpful in preventing us from going to the other extreme. 
 There is in fact one point in its favour ; our knowledge of electromagnetic 
 phenomena is much more precise and extensive than our knowledge of matter. 
 
550 General electrodynamic theory [CH. xiv 
 
 For the present however we shall not definitely take up either view, 
 although we shall as before often fall back on particular analogies which point 
 to certain definite conclusions as to the nature of this aether. In any case 
 we have as before a definite something to which to attach the energy which 
 is associated with any electromagnetic field, and our present object is to 
 discuss the distribution and variation of this energy location. The method 
 to be pursued involves a simple application of the energy principle with 
 as few additional hypotheses as possible. 
 
 620. This energy principle seems to suggest that energy is to be considered 
 as a definite substantial something. In any self-contained electromagnetic or 
 mechanical system the total quantity of energy is always the same, just like 
 the quantity of matter in a self-contained material system. About the matter 
 we know moreover that it can only move continuously in space, a jump from 
 one place to another without crossing the intervening space being excluded. 
 A sudden discontinuous translation of energy from one point of space to 
 another is however a priori not impossible. A distance action theory of 
 gravity would, for example, conceive it as possible that one body can accelerate 
 another, i.e. give it energy, without reference to the intermediate space. 
 Maxwell's theory on the other hand does not admit of any discontinuous 
 energy translations of this nature, its underlying idea being in fact that the 
 energy in the electromagnetic field is transferred continuously from point to 
 point even with a finite velocity. 
 
 The general theory of the energy streaming in the electromagnetic field 
 was first developed by Poynting on the basis of Maxwell's theory and we shall 
 now go through the important points of the subject although on rather 
 different lines from those adopted by Poynting*. We shall proceed tentatively 
 and attempt to get as much information from our few hypotheses as possible. 
 
 621. The energy of the electromagnetic field must be continuously 
 distributed throughout that field in the aether ; the only way of getting away 
 from this is by assuming action at a distance and this practically asserts that 
 we cannot trace the energy. Now for all dynamical purposes it is necessary 
 to know not only the total amount of the energy in the field but also how 
 it is distributed in . the field. For example in a dynamical theory of an 
 elastic solid it is always necessary to know the potential energy w per unit 
 volume at any point (a quadratic function of the strains in the element dv at 
 the place) as well as the total potential energy 
 
 W = Iwdv, 
 
 before using the results in any mechanical discussion. The important point 
 
 * Phil. Trans. A, CLXXV. (1884). The present treatment follows the lines sketched by 
 Larmor in Phil. Trans. A, cxc. (1897), p. 285, and developed in further detail in his lectures. 
 
619-622] Energy 551 
 
 is that we must be sure that this represents the actual distribution and not 
 only the total amount. We might, for example, in the process of obtaining 
 W, have integrated by parts and so got 
 
 W = f w n 'df + Iw'dv, 
 '/ J 
 
 where the surface integral is taken over a surface/ bounding the system. If 
 this surface is indefinitely extended we can neglect this part and thus 
 
 W = 
 
 and w' =|= w. But by integrating by parts we always mix up the energy 
 from different parts of the system to get that at the typical volume element; 
 the energy in any element would then depend on all the distant elements 
 and we should not then have a proper local distribution. In the case of media 
 like the aether this is one of the complexities to be met with and we have to 
 find out as best we can the proper distribution of energy; but in any case 
 we are never absolutely certain that our simply obtained energy distributions 
 have not after all been obtained by some such process as integration by parts 
 and do not therefore represent the true distribution required. 
 
 622. With these preliminary remarks let us now consider the conservation 
 of the total energy in any electrodynamic field. In this case we interpret the 
 general principle in the form that the diminution of energy inside any closed 
 surface in the field is equal to the flow of energy outwards across the surface. 
 In other words if E is the total energy inside, dE/dt is equal to the flux 
 of energy outwards over the surface ; and we ought to be able to express this 
 flux as a surface integral in the form 
 
 S n denoting the outward normal component of the vector determining the 
 energy flux, which will of course have magnitude and direction like the flux 
 of anything else. 
 
 The total energy E of course represents the available electrodynamic 
 energy in the field. If W represents the potential energy and T the kinetic 
 energy then 
 
 dE dW AT 
 
 wherein F represents the electromagnetic energy dissipated in the space 
 considered per unit time, either directly into heat or in the performance of 
 mechanical work on the masses with free charges or polarisations. 
 
552 General electrodynamic theory [OH. xiv 
 
 623. In this relation we are fairly sure of the form of two of the quantities 
 involved. The potential energy is very generally expressed by 
 
 W = ^(E.8P) + Q - Wdv, 
 
 J V ' 07Tj v 
 
 where P is the polarisation intensity of the molecules of the medium produced 
 by the electric force E. The first term represents the work done against the 
 material reactions to the setting up of the polarisation and is stored up as 
 internal potential energy in the polarisation of the medium being simply the 
 organised part of the energy of elastic stress. The second part represents 
 the purely electrical part of the potential energy associated with the polarisa- 
 tions in the molecules representing energy of strain in the aethereal medium 
 due to the presence and configuration of the electrical charges. We have 
 of course excluded the existence of hysteretic effects in the polarisation of 
 the medium so that it is generally reversible. The polarisation is presumed 
 to be an elastic affair involving no dissipation. 
 
 Again we know that the energy dissipated per second is 
 
 dv, 
 
 Cj representing the part of the total current depending on the motion of the 
 free electrons. This current consists of two essentially distinct parts con- 
 cerned respectively with the true conduction electrons and with those giving 
 rise to the convection currents. In both cases the electric force acts on the 
 free electrons and increases their velocities, but in the former case this 
 increase is dissipated by collision into irregular heat motion whereas in the 
 latter it is converted into mechanical energy of effectively non-electric 
 nature. In the most general case the convection current will arise in the 
 motion of charged bodies and polarised media. 
 
 The expression for the kinetic energy is not considered as so certain as 
 the above two expressions which appear to be very generally valid. We can 
 however leave it over for the present as the theory determines possible forms 
 for it. 
 
 624. We now have 
 
 or = (E . D) dv, 
 
 J V 
 
 and F = [(E . CJ dv 
 
 = |(E.C - i> 
 
 = l(E.G}dv~ I (ED) 
 
 dv. 
 
623-625] Poynt ing's theory 553 
 
 Therefore we have - + F = [(E . C) dv, 
 
 and thus we must have 
 
 AT dE 
 
 and then if Maxwell's ideas on the nature of the electromagnetic actions are 
 
 dE 
 
 dE 
 
 tenable we must be able to express as a surface integral as above, so that 
 
 - dt =-S n df- 
 
 Various possibilities are now open to us. This is all we can learn about the 
 distribution of the kinetic energy from the energy principle alone. We can 
 however make further simple hypotheses and thereby gain additional insight 
 into the matter. 
 
 625. (i) The most natural hypothesis is obtained by transforming the 
 last integral by the substitution 
 
 4:77 
 
 - C = curl H. 
 c 
 
 We get then 
 
 f(E . curl H) dv = - f[EH] n d/ + [(H . curl E) dv 
 
 S0 that - 
 
 And now we might take 
 
 and then we should have 
 
 dT^l^l 
 
 or T = j-dv (HdB), 
 
 477.' J o 
 
 provided of course that the integrand is a perfect differential, otherwise we 
 could attach no meaning to it. This means that this is a possible form for 
 T if the induced magnetism is reversible ; if the magnetisation were reversible 
 and (H.dB) were not a complete differential then we could have perpetual 
 motion. 
 
554 General electrodynamic theory [CH. xiv 
 
 We might thus take the kinetic energy as distributed throughout the space 
 with a volume density 
 
 = ~ (B 2 
 
 477 J 077 
 
 which in empty space gives a density 
 
 This is Maxwell's form of the kinetic energy in the electromagnetic field ; 
 it thus regards all currents and magnetism as having associated with them 
 kinetic energy. This suggests that the magnetic induction is a type of velocity 
 
 in the aether and that ^- is a coefficient of inertia. This would be a natural 
 
 O77 
 
 assumption to make and brings the theory into line with our ordinary notions 
 of elastic bodies. 
 
 This form for T has the one great advantage of expressing the energy at 
 a place directly in terms of the field vectors specifying the condition of the 
 aether at that place; it is therefore the natural specification for T. 
 
 626. On this hypothesis we have 
 
 7,Tt 
 
 for this is what remains when we identify -j- with the other part of the 
 complete expression. This means that the vector 
 
 represents the flux of energy; the integral of the flux of this vector across 
 any surface represents the rate of change of energy inside the surface. The 
 resultant flux of energy at any point is 
 
 ^-(H.EsinHE), 
 
 477 
 
 and is directed perpendicular to both H and E ; the energy flows perpendicular 
 to both forces in the field. 
 
 This is Poynting's result and this vector S is usually called after him. 
 It is however only right on the hypothesis that the kinetic energy is distri- 
 
 l -B 
 buted in the medium with a density ^ (HdB) per unit volume; but even 
 
 then we are not absolutely sure that it is right because we might have added 
 to it some other vector quantity which would however integrate out when 
 taken all over the surface /. However, following a usual practice in physics, 
 
625-628] MacdonaMs theory 555 
 
 it is best to adhere to the simplest hypothesis. This idea of Poynting's is the 
 simplest certainly, and the addition of anything else is merely a gratuitous 
 complication which is not, after all, necessary. The actual phenomena 
 strongly suggest that the flux of energy is correctly represented by this 
 vector. 
 
 627. (ii) There is however no definite and precise reason why we should 
 take the matter this way; we might have adopted some other scheme. The 
 only other one of any importance is obtained by performing the first in- 
 tegration by parts in some other way. We found that 
 
 f = f-|,E.C,*, 
 
 fA/t/ U>(/ J 
 
 4:77 
 
 and we integrated by the substitution -- C = curl H. We might however 
 
 C 
 
 follow another course and introduce the vector potential A by the substitution 
 
 E= - - 
 
 c dt 
 
 and then we have 
 
 |(E . C) dv = - - [f^P .C\dv- f(C . grad if,) dv 
 c J \dt ) J 
 
 r ( 
 
 v + \ iff . div C dv djC n df, 
 
 J Jf 
 
 _ 1 
 
 c J V dt ' 
 
 and since div C = 0, 
 
 we see that 
 
 f(E . C) dv = - - i(c -} dv - I *liV n df, 
 J c j\ dt / J f 
 
 so that now we have 
 
 i 
 
 dt . dt 
 
 dT dE [ . 1 |7- dA\ j 
 
 -Hi = -j- + 0C n d/+ - C -rr }dv. 
 f n J c J\ dt 
 
 We might now take 
 
 dE 
 
 T =- IdvT (GdA). 
 
 ' 
 
 and then we should have 
 
 628. This is the general form of a result which has received very 
 influential support from some quarters* and there is a good deal to be said 
 for it. Consider the case of a conduction current flowing in complete circuits. 
 If we consider that what goes on is the electric force pulling the electrons then 
 the work done per unit time is precisely this result This is to a certain 
 * Gf. Macdonald, Electric Waves, Chs. iv, v, vm. 
 
556 General electrodynamic theory [CH. xiv 
 
 extent a reasonable hypothesis ; but it entirely neglects the aether ; it says 
 that the electrons are the things to which the energy is attached and the work- 
 is that done by the electric force pulling the electrons about. On the aether 
 theory however the energy of an electric particle is really distributed in the 
 field all round it. We thus want to distribute the energy in the aether and 
 take the electrons merely as the key, the singular point which binds or locks 
 up a portion of the energy; on this idea the electron appears merely as a 
 nucleus of strain which locks up a portion of the energy ; the actual nature 
 of the process or mechanical action by which this is accomplished is at present 
 unknown ; but we do not need any more precise knowledge of it. 
 
 There is another disadvantage to this scheme ; there is no obvious physical 
 explanation of the functions involved in it. In fact the mathematical 
 definition of the functions involved is not complete, so that the functions 
 themselves cannot, without further arbitrary restrictions, represent definite 
 physical entities. Even if we did choose, say, the instantaneous potentials 
 of Maxwell's theory then their definition in the integral form shows that the 
 value of each of them at any point depends on all the other elements of the 
 field, so that if we take a distribution of the kinetic energy like this it is 
 practically importing action at a distance into the theory. In a theory of 
 action in and through a medium any quality at one place in the medium 
 depends only on the conditions at the place and not on the other places 
 remote from it, the quality being propagated from place to place with a finite 
 velocity. In such a theory therefore this form for T is an unlikely one. 
 
 However we cannot say that either form is wrong ; it all depends on the 
 point of view adopted. The chief point to be noticed is that we get different 
 distributions of magnetic energy according to the assumptions made. The 
 particular value adopted is after all merely a matter of preference, not proof. 
 We propose however to adopt in our future investigations the form first 
 given as this is in agreement with the work of most authors. 
 
 629. It is important to notice that in the above argument very little has 
 been assumed respecting the constitutive properties of the material media 
 distributed throughout the field. We presume merely the non-existence of 
 hysteretic effects in the electric and magnetic phenomena. The results 
 admit therefore of very general application. Of course the form 
 
 assumes a linear relation between B and H. 
 
 In the case when there are no magnetic substances about the kinetic 
 energy is distributed throughout the field with a density 
 
628-630] The magnetic energy 557 
 
 and in this case it belongs entirely to the aether, because it is independent 
 of the nature of the substances present. The inertia of the electromagnetic 
 field thus appears to be in the aether. 
 
 If there are magnetic substances about and we are dealing with slow 
 electric changes so that the magnetism can follow the field the total available 
 magnetic energy is 
 
 - Idv [ B (HcZB) = ~ f(B 2 - 167T 2 ! 2 ) dv - Idv [ 
 r / J o ST .' J J o 
 
 It thus consists of the total magnetic energy of the electromagnetic field 
 less that part of it which is concerned with the bodily forces on the magnetic 
 media and which is mechanically available only in so far as the presence of 
 these media increases the available energy associated with the currents and 
 charges of the system. 
 
 Even when magnetic substances are present they have generally nothing 
 whatever to do with radiation phenomena. The magnetisation of the medium 
 being an affair of the molecules takes a comparatively considerable time to 
 establish, and the alternations in radiation are far too quick for it to follow*. 
 This is the reason why even iron may be treated as non-magnetic in the 
 electromagnetic theorv of light. 
 
 v O 
 
 It used to be argued against this theory, that according to it, the reflexion 
 of light from the surface of an iron magnetic cannot be altered by altering 
 the magnetisation in any way, a fact which is in direct contradiction to 
 experiment. However the magnetism in this case does come in, but in another 
 way, as a second order effect, and the rotation of the plane of polarisation 
 on reflexion from a magnetised mirror is a second order phenomena involving 
 the magnetisation of the metal. 
 
 The potential energy of course always depends to a certain extent on the 
 matter present in the field, the energy of polarisation of the medium always 
 forming an essential part in its complete specification. 
 
 630. The physical significance of the energy flux or Poynting vector may 
 be illustrated by a simple application of practical importance. A long straight 
 cylindrical conductor carries a current J . The conductor is surrounded by 
 an electromagnetic field and we require the nature of the flow of energy in 
 this field. 
 
 The magnetic lines of force in the surrounding field are in circles round 
 the wire. As regards the electric force we know that just inside the conductor 
 it is directed along the axis and is just sufficient to drive the current, this 
 being secured by a slight initial accumulation of charge on the conductor. 
 But this force cannot change in going across the surface of the conductor 
 
 * This applies only to the appreciable cases of ferromagnetism ; the ordinary intramolecular 
 phenomena are far too small. 
 
558 
 
 General electrodynamic theory 
 
 [CH. xiv 
 
 so that just outside it is also tangential and equal to the internal value. Thus 
 at a place just outside the conductor the flux of energy is into the conductor 
 normally and is equal per unit area to 
 
 cH .E. 
 
 Thus the total energy flowing into unit length of the conductor from the 
 external field is the integral 
 
 c I H . Eds, 
 
 s 
 taken round a ring on the conductor. By symmetry this is equal to 
 
 cE \Hds, 
 and 
 
 so that the energy flowing in per unit length is 
 
 EJ, 
 
 which is the energy dissipated into heat in the unit length of the conductor, 
 by Joule's law. We thus see that when a current is flowing along a wire 
 energy flows in sideways from the external field in just sufficient quantities 
 to account for the energy converted into heat. From this point of view the 
 energy appearing in the form of heat is supplied from the aether. 
 
 
 1 1 
 
 f 
 
 \. ; 
 
 
 I . - 
 
 
 Fig. 96 
 
 631. Let us now assume that the wire is a long straight cylindrical one 
 with a circular cross section of radius a. Then if the current flow is uniform 
 all along the wire the magnetic force will be in circles round its axis and the 
 electric force directed along the axis. To obtain a closer insight into the 
 field thus specified we shall find it convenient to refer the field to a system 
 of cylindrical polar coordinates (r, 6, z) with the axis along the axis of the 
 cylinder. The only components of the field vectors at any point which are 
 not zero are E 2 and H# and these are symmetrical round the axes of the 
 field. In the conductor these two components are connected by the relations 
 
 = ld (rH) 
 r dr 
 
 = m, 
 
 dt ~'~~ dr' 
 
630-632] The energy of an electric current 559 
 
 to which the fundamental field equations of Ampere and Faraday reduce 
 under the special circumstances of the present problem. It is of course 
 assumed that it is possible to neglect the displacement current in comparison 
 with conduction current ; this is justified in most cases of any real importance. 
 
 It follows then that 
 
 dE z _ 1 d ( dE z \ 
 
 " ~ T~ I f T 1 
 
 c 2 dt r dr \ dr /' 
 A particular solution of this partial differential equation is* 
 
 E 2 = Ce^Jo (x), 
 where we have written 
 
 J ( x ) = - e ixcos *da. 
 
 and where J (x) satisfies Bessel's equation 
 
 ,7 " (x) + - J Q ' (x) + J = 0. 
 
 x 
 
 Since the field cannot become infinite inside the cylinder J must be taken 
 as the Bessel function of the first kind, viz. 
 
 1 
 
 o 
 
 The density of the current flux at any point in the interior of the cylinder 
 is given by 
 
 C.lJ 1 fr(~^ S) il)t ~f ( rv*\ 
 - \jMmA~ C/L/ 1/ tf A \JU ) f 
 
 and its distribution over the cross section is thus determined. Remembering 
 the particular approximate forms of the function J when its argument is 
 small and large we notice that for very slowly alternating currents the 
 distribution is practically uniform across the section but for very rapid 
 oscillations it is confined to a very thin layer at the surface. 
 
 632. To excite this field in the interior of the conductor we must apply 
 along its outer surface a field of strength 
 
 a being the radius. 
 
 The total current flowing through a section of the cylinder is 
 
 fa 
 J = 277 aE z rdr 
 
 J o 
 
 dr )r =a 
 
 or J = - {xJr! (#)}z=xa- 
 
 2ip 
 
 * Of. Bayleigh, Phil. Mag. (5), xxi. (1886), p. 381; 0. Heaviside, Electrical Papers, n. 
 pp. 39, 168; J. Stefan, Wien. Ber. xcv. (1887), p. 917; Ann. d. Phys. XLI. (1890), p. 400; 
 Abraham, EncyM. d. Math. Wissensch. v. 18, p. 514. 
 
560 General electrodynamic theory [CH. xiv 
 
 We may thus write the relation between the complex expressions for J and 
 E _ in the form 
 
 whence it follows that 
 
 R + i ?L= ^{-i 
 
 C 2 C 2 (xJo (X}) x = v a 
 
 From this relation we can get the two real quantities R and L. 
 
 The physical meaning of these coefficients R and L can now be directly 
 deduced by an application- of Poynting's theorem as above explained. If we 
 now denote by E 0z , H , J the real parts of their respective complex representa- 
 tions as above, we shall find for the energy which enters per unit time and 
 length into the conductor from the external field is 
 
 But from the above this is 
 
 Suppose now we integrate this equation over a complete oscillation; the 
 energy entering the conductor must then just be equal to the heat developed 
 in the circuit as Joule's heat : this is given by the first term on the right of 
 the above equation. The second term on the right of this equation which 
 disappears on integration over the whole oscillation must then give the 
 increase of the magnetic energy of the field of the current inside the wire. 
 According to this explanation it is usual to call R the effective resistance 
 and L the effective self-inductance (per unit length) of the conductor for the 
 particular period of the oscillating current. 
 
 633. If the period of oscillation is long or the radius of the wire small va 
 will be small and we may use the approximate form for J which expresses 
 it as an ascending power series ; it is then found that 
 
 . (^ 2 ) 2 (0 2 ) 4 , 
 
 1 ( (da*)* 13 
 
 V 24 4320 
 where we have used 6 = ^ R n = . 
 
 n 
 
 The effective resistance only departs slightly from the value for a uniform 
 current. 
 
632-635] The energy flux in radiation fields 561 
 
 If the period of the oscillation is very big or the radius of the wire large 
 then we must use the asymptotic representation of the Bessel function by 
 semi-convergent series. We then get the so-called Rayleigh-Stefan formula 
 
 = bV 
 
 L_4-? 
 
 2<TC2 + 4 ! 
 
 2 
 **"& 
 
 In this case the current only very slightly penetrates into the interior of the 
 conductor so that the resistance and self-induction are both small. This is 
 of course merely a particular example of a general principle established in 
 connection with more general types of alternating fields in a previous 
 chapter. 
 
 634. On the flux of energy in radiation fields. The physical significance 
 of the energy flux or Poynting vector may be further illustrated by another 
 application of practical importance. When energy travels by radiation the 
 direction of the flux is along the ray, so that this vector gives not 
 only the direction but also the intensity of the ray (the intensity of a ray 
 being measured by the energy that passes along it per unit time). In the 
 ordinary propagation of plane optical waves in an isotropic medium the ray 
 of light is perpendicular to the front of the waves, because the electric and 
 magnetic force vectors are both in the wave front and the energy flux is 
 normal to both. The energy in this case travels normal to the wave front 
 along the direction of propagation. In crystalline media on the other hand 
 it is the two stream vectors, the electric and magnetic fluxes, that are in the 
 wave front; but the electric and magnetic force vectors are not coincident 
 in direction with the corresponding fluxes and do not therefore in general 
 lie in the wave front. The ray in this case is therefore not normal to the 
 wave front and the energy which flows along it thus crosses the front 
 obliquely. It may be regarded as providing the relation between the ray 
 (i.e. the energy flux) and the wave front. We define the ray in the general 
 case as the path of the energy and its direction at any point will therefore 
 be that of the Poynting energy flux. 
 
 635. Returning however to the case of isotropic media let us consider in 
 further details the circumstances involved in the propagation of a train of plane 
 waves travelling parallel to the axis of x in an absorbing medium, the waves 
 being polarised so that the magnetic force is parallel to the axis of z, and the 
 electric force parallel to the axis of y in a system of rectangular coordinates. 
 The equations of propagation are then just as before 
 
 = _ = 
 
 dt dx ' c dt dx 
 
 36 
 
562 General elevtrodynamic theory [OH. xiv 
 
 It follows that 
 
 which is the equation of propagation. Considering the case of radiation of 
 
 period - - so that 
 
 P E = E P ivct-(a+^)x 
 
 Jy - -L^Ot ? 
 
 this gives - e/r = (a + *o) 2 , 
 
 pc I 
 On separating the real parts we have 
 
 Ej, = E e~ ax cos (pet bx). 
 corresponding to 
 
 E A 
 
 H, = - (a 2 + fe 2 ) 5 e~ ax sin (pet - bx + 0), 
 
 where tan 6 = - , 
 
 a 
 
 -#o i 
 
 and C,, = (a 2 + b 2 r e~ ax sin (pet bx + 20). 
 
 P 
 
 Thus as we have already seen the magnetic flux is in a different phase from 
 the electric force, involving a diminution in their vector product which deter- 
 mines the energy transmitted across any plane. 
 
 636. The energy per unit volume of the radiation at any part of the wave 
 
 11 H 2 
 
 consists of an electric part ^- EJX. or eE 2 and a magnetic part = : 
 
 STT 877 STT 
 
 and the ratio of the time averages of these is exactly as before 
 
 or (1 + 
 
 which is constant, but not unity except for transparent media. The time 
 rate of propagation of energy is, by Poynting's theorem, 
 
 dE THE] E n 2 
 
 e -zax i\ 12 _|_ periodic term). 
 at 477-c -i -i 
 
 Across the plane x = it is therefore on the average 
 
 which corresponds to a density of energy equal to the mean square of electric 
 force travelling with the speed of the waves. This involves the result* 
 that only the fraction 
 
 
 * Cf . Larmor, Aether and Matter, p. 1 35. 
 
635-638] The energy flux in the Hertzian field 563 
 
 of the total energy of the wave system can be considered as propagated; 
 in the case of an undamped wave train this is only the purely aethereal part. 
 The aethereal wave train, passing across the material medium, sets its mole- 
 cules into sympathetic independent vibration : the energy of these vibrations 
 constitutes a part of the total energy per unit volume, but that part is not 
 propagated. This remark applies equally to all optical theories in which 
 change of velocity of propagation is traced to the influence of sympathetic 
 vibrations of the molecules, in fact it applies to all cases in which velocity 
 depends upon the wave length. 
 
 637. We must however leave these considerations and return to the 
 discussion of further aspects of the general flux of energy in radiation fields. 
 We first consider the flux of energy in the field surrounding the ideally simple 
 type of vibrator discussed in 538 of chapter xn. It was there shown 
 that the field of a small vibrating electric doublet of moment f (ct) at time 
 t and placed at the origin and along the axis of a system of spherical polar 
 coordinates reduces at a large distance from the vibrator to the simple 
 radiation field in which the electric and magnetic forces are simply 
 
 _ sin0/"(c-rK 
 
 &e = <!> = - - ' 5 
 
 and the wave front surfaces are the spheres 
 
 r = const. 
 
 It follows therefore by direct application of Poynting's theorem that the 
 flux of energy at the point (r, 6, (f)) is radially outwards, i.e. in the direction 
 of propagation, and of density per unit area 
 
 sin 2 0(/"(ct -r)) 2 
 
 The total flux over the sphere of radius r is thus 
 
 2ir fir 2 
 
 /y . n f\ 1 f\ 1 I 
 
 47TC 
 
 r2ir TTT 2 
 
 sin 3 d dd dd> = =- (f" (ct r)) 2 . 
 
 / > _ * *j \ / / 
 
 ' n ' A OG 
 
 638. In the particular case when the vibrations are periodic so that we 
 may take 
 
 / (ct r) = A sin p (ct r), 
 
 it is on the time average 
 
 _ 
 
 3c ' 
 
 or if A is the wave length this is 
 
 1677-M 2 
 
 3cA 4 ' 
 
 This energy which is radiated outwards from the vibrator is of course lost 
 to the system, and it must have been drawn from the store of the energy 
 
 362 
 
564 General electrodynamic theory [CH. xiv 
 
 in the original vibrations. Thus unless the oscillations of such a system can 
 be maintained by external agency they will gradually decay owing to the 
 dissipation of their energy by radiation. It is important to notice that the 
 rate of dissipation increases rapidly as the wave length is decreased. 
 
 We must therefore conclude that however ideal the conditions may be 
 there must necessarily be dissipation accompanying any electrical vibrations 
 and that therefore it seems absolutely essential to take such dissipation 
 into account in the complete theory of such vibrations : and then of course 
 it is necessary to go still further and treat the problem along the lines 
 suggested by Prof. Love, as one which involves a finite time extent, so that 
 the conditions on the initiation of the field require specification. In this way 
 other aspects of the dissipation process arising from radiation are brought 
 out: we can illustrate them by two simple cases* analysed in detail in 
 chapter xn. 
 
 639. In the case of the charge oscillating on the perfectly conducting 
 sphere the field inside the sphere r = ct + a is specified by 
 
 4:E cos 6 
 
 2E sin 6 
 
 
 
 2E sin 6 r r z >> 
 
 H * - - Vl ^ V l ~ a + if e 
 the notation being exactly as before 
 
 ct r + a 
 
 "2aT 
 In the initial state the aether in the region between the spheres r and r .+ dr 
 
 1 2 
 
 possesses electric energy of amount 5- 47rr 2 drE 2 - 6 or r~*E 2 dr and the total 
 
 energy of the field is ^E 2 a~ s . In the subsequent state of wave disturbance 
 the same portion of the medium possesses magnetic energy of amount 
 
 2 . ~ sin V30 - - sin V30 - } 
 
 STT ' 3 r '3r% 2 
 
 # 
 
 Thus as soon as the wave front has travelled to a distance from the conducting 
 surface which is at all large compared with the radius of this surface the factor 
 r - 20 w jji k e small except in the immediate neighbourhood of the wave front : 
 the energy of the wave motion will be accumulated near the wave front. 
 Also when r is large compared with a the above expression may approximately 
 be replaced by * 
 
 2 # 2 
 
 9 a* " 
 
 * Both cases are worked out in detail by Prof. Love in the papers there cited. 
 
638-640] The energy flux in Love's problems 565 
 
 We may calculate the energy between the wave front and a spherical surface 
 within it and not far from it by integrating this expression. Consider the 
 case where the inner of these surfaces is at a distance of half a wave length 
 behind the front, i.e. at a distance 27ra/\/3. The magnetic energy between the 
 surfaces is approximately 
 
 O J72 fir/VS ( I A/^ ] 
 
 ~ I e- 20 l + cos 2 \/30 - sin 2 V30 2ad, 
 
 which is E z a~* l - 
 
 640. If we had taken the first wave length of the advancing wa va instead 
 of the first half wave length, we should have found 
 
 I 
 
 5 
 
 o 
 
 as the approximate value of the magnetic energy between the surfaces. 
 If we calculate the electric energy in the same way and to the same order 
 of approximation, we find the same values, so that the total energy between 
 the two surfaces is approximately equal to f 
 
 when the surfaces are half "a wave length apart and 
 
 when they are a wave length apart. The terms omitted in the calculations 
 
 
 are small compared with those retained in the order - and higher powers of 
 
 -, r denoting the radius of the wave front. It appears therefore that the 
 T 
 
 energy of the electrostatic field is propagated outwards with the wave in 
 such a way that the energy that was initially within a spherical surface 
 surrounding the conductor is the energy of the wave motion when that 
 surface is the wave front and it is gathered up close behind the wave front. 
 When the wave front is at a great distance from the conductor the accumula- 
 tion of energy at the front is so great that all but about 7 2 F of the total initial 
 energy of the field is gathered up in the first half wave length and all but 
 y^o of it is gathered up in the first wave length. Thus as the wave advances 
 it transforms into electromagnetic energy the excess of the statical energy 
 of the initial field over that of a free charge distribution of the same total 
 amount (in this case zero) on the same conductor; and this electromagnetic 
 energy is transferred continually towards the front of the advancing wave 
 in such a way that at a distance from the conductor the wave practically 
 
566 General electrodynamic theory [OH. xiv 
 
 passes as a pulse or in other words as the wave front of the disturbance 
 travels out over the field the energy in that field is picked up almost entirely 
 by the first portions of the disturbance which travel across it. 
 
 Similar considerations of course apply to the case of a Hertzian oscillator 
 and they show very vividly how unsuitable such an arrangement really must 
 be for the generation of a continuous train of waves unless it is possible in 
 some way or other to maintain by forcing the vibrations of which it is capable. 
 
 641. The results obtained above concerning the loss of energy by radiation 
 from a Hertzian oscillator can be applied to deduce the loss experienced by 
 an electron moving with an acceleration. The method is obvious and need 
 not be *given in full; it is found that the amount of energy per unit time 
 that is lost to the electron moving with a comparatively small velocity is 
 
 3c 3 [dt 
 
 so that the total amount lost during the time between the effective instants 
 t l and t% is 
 
 2e 
 
 dt. 
 
 < 
 
 This constant draining of .energy from the electron in accelerated motion is 
 often interpreted as implying the existence of a resistance to its accelerated 
 motion and in many cases this is a convenient way of describing the dissipa- 
 tion action of the radiation. The idea is obtained from the fact that the 
 total loss in the effective interval stated may be written in the form 
 
 2 e 2 f * fdv 2 , 2e 2 
 
 , 
 -77 I at = 
 
 3c 3 
 
 Ife 92 rt-> 
 
 v.v) (v.V) A. 
 
 3c 3 
 
 Now the first term on the right disappears, if, in the case of a periodic motion 
 the integration is extended to a full period ; also, if at the instants ^ and t^ 
 either the velocity or the acceleration is zero. In either of these cases 
 2e 2 ffe /dv\* ' 2f 
 
 so that the energy dissipated is exactly the same as if the force 
 
 2e 2 .. 
 
 I q "I*, 
 
 o c d 
 
 acted on the electron for the period under consideration. 
 
 This conception of a retarding force on the electron has been used by 
 Planck to account for the dissipation of the energy of the electronic vibrations* 
 inside a material atom and the consequent absorption of energy from incident 
 radiation fields which is observed : on this view of the matter the energy of 
 the incident radiation is merely absorbed by the vibrating electrons to be 
 immediately re-emitted as radiation, of a dispersed or scattered type. It 
 
 * Berlin JSer. 1902. 
 
640-643] The principle of least action 567 
 
 would appear* however that this representation of the matter is not effective 
 in accounting for the absorption that is really observed in many cases ; but 
 as no satisfactory explanation yet appears to be forthcoming concerning the 
 mechanism of this inter-molecular absorption the suggestion offered by 
 Planck is at least illustrative of the possibilities of a strict theory. 
 
 642. General electrodynamic theory f. The tendency of the physical 
 investigations outlined in the previous chapters has been towards the con- 
 struction of a dynamical theory which shall give a consistent account of 
 electrodynamic phenomena, i.e. to answer the question as to the possibility 
 of obtaining a complete parallel to the processes in any electromagnetic field 
 from those observed in some imaginary system of masses moving according to 
 the ordinary laws of mechanics. To reply completely to such a question it is 
 not necessary to make any definite assumptions as to the mechanism under- 
 lying the phenomena ; all we have to do is to show that they can be described 
 by means of the general equations of mechanics. 
 
 The most general dynamical principle which determines the motion of 
 every material system is the law of Least Action, expressible in the usual* 
 form 
 
 8 \(T - W) dt = oA 
 
 wherein T denotes the kinetic energy and W the potential energy of the 
 system in any configuration and formulated in terms of any coordinates that 
 are sufficient to specify the configuration and motion in accordance with its 
 known properties and connections; and where the variation refers to a 
 fixed time of passage of the system from the initial to the final configuration 
 considered. The power of this formula lies in the fact that once the energy 
 function is obtained in terms of any measurements of the system that are 
 convenient and sufficient for the purposes in view, the remainder of the 
 investigation involves only the exact processes of mathematical analysis. 
 
 643. Now we have in the first section succeeded in obtaining expressions 
 for the potential and kinetic energies associated with any electromagnetic 
 field and it thus only remains to interpret these functions in terms of suitable 
 coordinates, before applying the general laws of dynamics to determine the 
 sequence of events in any such system. But whatever view we may take as 
 to the constitution of the aether and the electrons it is quite obvious from 
 the whole of the preceding discussion that all electrical effects must be 
 explicable on the hypothesis of the aether with the electrons or discrete 
 atomic charges moving about in it freely or grouped into material atoms : 
 thus as far as we are at present concerned the only difference between the 
 
 * Cf. Lorentz, The Theory of Electrons, p. 140. 
 
 f Cf. Larmor, Aether and Matter, Ch. vi; Lorentz, La theorie electromagnetique, 55-61; 
 Helmholtz, Ann. Phys. Chem. XLVII. (1892), p. 1; Sommerfeld, Ann. d. Phys. XLVI. (1892), 
 p. 139; Reiff, Elastizitdt und Elektrizitdi (Leipzig, 1893). 
 
568 General electrodynamic theory [en. xiv 
 
 aether and any material medium must simply be due to the presence of 
 convection currents of electrons ; that is, wherever there is matter there are 
 these convection currents. Thus in a mechanical theory the electrodynamic 
 state of any system will be completely known if we can specify the positions 
 and motions of all the electrons in it together with the displacement, in 
 Maxwell's sense, in the aether, and herein we have sufficient data for our 
 present dynamical analysis. Of course for the purposes of electrodynamic 
 phenomena of material which we can only test by observation and experiment 
 on matter in bulk a complete atomic analysis of this kind is almost useless ; 
 for we are unable to take cognizance of the single electrons to which this 
 analysis has regard. The development of the theory which is to be in line 
 with experience ought instead to concern itself with an effective differential 
 element of volume, containing a crowd of molecules numerous enough to be 
 expressible continuously as regards their average relations, as a volume 
 density of matter. As regards .the actual distribution in the element of 
 volume of the really discrete electrons, all that we can usually take cognizance 
 jof is an excess of one kind, positive or negative, which constitutes a volume 
 density of electrification, or else an average polarisation in the arrangement 
 of the groups of electrons in the molecules which must be specified as a vector 
 by its intensity per unit volume : while the movements of the electrons, free 
 and paired, in such elements of volume must be combined into statistical 
 aggregates of translational fluxes and molecular whirls of electrification. 
 With anything else than mean aggregates of the various types that can be 
 thus separated out, each extended over the effective element of volume 
 mechanical science, which has for its object matter in bulk as it presents 
 itself to our observation and experiment, is not directly concerned. Never- 
 theless it is convenient on account of simplicity to formulate the problem in 
 terms of the separate electrons and to reserve the details of the process of 
 averaging, which must in reality be implied throughout, until the mechanical 
 relations of the system have been formulated in full. 
 
 644. Let us therefore proceed directly to the formulation of the 
 mechanical relations of a system of discrete electrons in a field of aether, the 
 potential or electrostatic energy of this system being expressed by the 
 integral 
 
 W = 6 ] - 
 
 O7T 
 
 extended throughout the entire volume of the electrodynamic field, whilst 
 the kinetic energy* is expressed by 
 
 * The principle of Least Action was employed in the manner here adopted by Prof. Larmor 
 but with the kinetic energy expressed in terms of the vector potential. The present deduction 
 was given by the author, Phil. Mag. xxxn. (1916), p. 195. 
 
043-646] The principle of least action 569 
 
 The complete Lagrangian function for the system is therefore 
 L = L +- ( - f(B 2 -E 2 )^, 
 
 O7T ./ 
 
 L being that part which does not depend on the aethereal configuration as 
 
 E 
 
 specified in the displacement , but which does depend essentially on the 
 
 size, constitution and motion of the electronic nucleus, as well as on the 
 forces, not of electric origin, exerted on it from the material atoms, or other- 
 wise, if such are presumed to exist. 
 
 645. We could now conduct the variation directly were it not for the 
 circumstance that our variables are not wholly independent: in fact the 
 variations of E and B are subject to the conditions as 
 
 I div E dv - TT^q = 0, 
 
 and (curl B -- ,- }dv -- - SOT = 0. 
 
 A c at) c 
 
 In these expressions E denotes a sum relative to all the electrons in the 
 volume considered, each with its proper charge q and velocity f, r being the 
 position vector of the typical electron. The seconu relation is a vector one 
 and is therefore equivalent to three independent equations. 
 
 Hence we must now introduce into the yariational equation four 
 Lagrangian undetermined functions of position (j>, A x , A y , A z , the last 
 three of which may be regarded as the rectangular components of a vector 
 A. It is thus the variation of 
 
 Ldt + 
 
 dt 
 
 \ I - U div E - (A, curl B - - S)l - S$0 + S 2 (Ar)l 
 
 I 4:77 I A ^ ("t / J C 
 
 that is to be made zero; afterwards determining the form of 4> and A to 
 satisfy the restrictions which necessitated their introduction. In conducting 
 the variation we can now treat the electric force, magnetic induction and 
 the position coordinates of the electrons as all independent. 
 
 646. As regards the electrons q the variation gives 
 
 \dt JSL - ^q (SrV) 6 + S | (r, (Sr, V) A) + S | (Sr, A)J- 
 
 where A must now be regarded as assuming the succession of values it takes 
 as the point whose position is defined by the vector r moves through the 
 aether, not the succession of values it takes at a fixed point. Integrating 
 by parts we get that this part of the variation is equal to terms at the time 
 limits together with 
 
 dt JSL - T,g (SrV)</> + S ^ (r, (SrV) A) - S \ 
 
570 General electrodynamic theory [OH. xiv 
 
 SA 
 where the symbol -j- is used to denote the time rate of variation of A 
 
 relative to the moving electron : thus 
 
 Now (r, (SrV) A) - (Sr (f V) A) = ([r curl A] Sr) 
 
 so that the main part of the variation of the integral due to the electrons is 
 
 Idt J8 + S? (Sr, - V</> - }~ + \ [f , curl A]) j . 
 
 J I \ f \ 
 
 As regards the variation of the state of the free aether defined by the 
 vectors E and B we have the terms 
 
 dt dv (BSB) - (ESE) + <j> div SE - (A curl SB) + A 
 Air. 
 
 On integrating the last three terms by parts we get that this part of the 
 variation is equal to terms at the time limits together with 
 
 f(B - curl A, SB) - (B+ V< + * 8 * , SE 
 
 wherein the last integral is taken over the infinite bounding surf ace of the field. 
 As usual in such problems we are not concerned with the terms at the 
 time limits because they may be as a rule suitably chosen. Also if we impose 
 the natural condition on <f> that it should be continuous everywhere and vanish 
 at the infinitely distant boundary, the surface integrals introduced also 
 vanish and we are left with the complete variation of our generalised 
 Lagrangian function in the form 
 
 |S/, - S f dtq (- d ~ - - [P curl A] + grad </>, 8r) 
 .' t \c ut c / 
 
 - | dt I Q A + ^ E + grad <f>, ^\ dr + [*' dt \dv (B - curl A, SB). 
 
 647. Now the variation Sr which determines the virtual displacement of 
 the electron q and the variations SE and SB which specify the electric 
 displacement of a point in the free aether, can now be considered as in- 
 dependent and arbitrary : hence the coefficient of each must vanish 
 separately in the dynamical variational equation. We thus obtain three 
 sets of equations of types 
 
 B - curl A = 
 and 
 
646-648] The complete electrodynamic force 571 
 
 where for simplicity L has been assumed to depend only on the coordinates 
 and velocities of the electronic charges. These equations are the same as 
 
 E = A grad</>, 
 
 c 
 
 B = curl A, 
 
 8Z 1 .: 
 
 w -5 A 
 
 These are the differentia] equations which determine the sequence of 
 events in the system. The first two show that the functions <f> and A, 
 introduced as undetermined multipliers, are respectively the scalar and 
 vector potentials of the theory. The third, expressed in 'the ordinary 
 language of electrodynamics which avails itself of the conception of force, 
 shows that 
 
 - ld * - grad,/, + - [rB] = E + - [rB] 
 c dt c c 
 
 is the electric force which tends to accelerate the motion of the electrons 
 q, each electron being presumed to have a constitution which enables it to 
 offer a kinetic reaction of 1 an electric nature to the action of this force. We 
 here speak of the electric force acting on the single electron which in strictness 
 is really more than our analysis gives us. The equation thus interpreted 
 should really have a S, a sign of summation in front of it to show that it is 
 an aggregate equation for all the electrons in the volume element, with which 
 we are in reality dealing. Certain considerations will however be offered 
 which point to the conclusion that the result is correct if interpreted for the 
 single electron separately, and we shall therefore often make use of the result 
 in this form. 
 
 648. Thus the whole mechanics of the electromagnetic system is summed 
 up in terms of these forces of ordinary type acting in the aggregate on the 
 individual electrons. Therefore for a complete specification of such a system 
 it is merely necessary to know in addition to the ordinary dynamical relations 
 of the masses moving in it, also the aggregate of the ' applied ' forces acting 
 on the individual electrons which they contain ; the force of electrodynamic 
 origin acting on the matter in bulk is the aggregate of the forces acting on 
 its electrons, and it is only in these impressed forces that the electrical 
 conditions manifest themselves. The total force of electrodynamic origin on 
 any body is thus* 
 
 S ? (E + - [uB]), 
 
 C 
 
 * The occurrence of the magnetic induction instead of the magnetic force in this expression 
 is important and must be emphasised. It points once again to the conclusion that the induction 
 is the fundamental vector of the theory, as in fact is obvious from our previous discussions of 
 the energy relations of the magnetic field; in fact from one point of view the only essential 
 point where our treatment of these relations differs from that usually given, lies in the choice of 
 
572 General electrodynamic theory [CH. xiv 
 
 which makes up in all a static part S^E and a kinetic part 
 
 S<? \ [nB]. 
 
 We shall return to a more detailed examination of these forces in a later 
 paragraph, but it is perhaps worth while considering at the present stage the 
 results of an experiment made by H. A. Wilson to distinguish between the 
 electromotive force acting on the electrons and the electric force in the aether, 
 by examining the effect on a dielectric body of motion through the aether. 
 
 Wilson rotated a hollow dielectric cylinder in a uniform magnetic field 
 parallel to its axis, thus bringing in a force of electrodynamic origin which 
 for all the electrons in the dielectric acts radially outwards from the axis 
 of the cylinder, i.e. perpendicular to the direction of their motion and to the 
 lines of force of the magnetic field and of amount at any point equal to 
 
 v being the velocity. This force gives rise to an electric displacement across 
 the shell of dielectric from the inner to the outer surface and in consequence 
 there will be a difference of potential established between these surfaces; 
 by coating them metallically and connecting by sliding contacts with the 
 quadrants of a galvanometer a measurement of the potential difference 
 was easily made. 
 
 The force producing the displacement being merely of kinetic origin there 
 will be no aethereal part in the displacement which will therefore be of 
 intensity per unit volume equal to 
 
 being the dielectric constant. Wilson verified that the potential difference 
 between the surfaces was proportional to 
 
 with sufficient exactness to justify the basis of the explanation here offered. 
 The result of this experiment also has another important bearing which will 
 be mentioned later. 
 
 649. The Hall effect. In 1879 Hall* found that the lines of flow of an 
 electric current through a metallic conductor are distorted when the conductor 
 is placed in a magnetic field, the distortion being of the character of that 
 produced by a slight additional electromotive force directed at right angles 
 
 the magnetic induction instead of the more usual magnetic force, as the 'aethereal vector.' 
 Again the conclusion that the induction is the true magnetomotive force would appear to in- 
 validate the argument of Kempken (Ann. d. Phys. xx. (1906)) whereby he derives the fact that 
 I In is constant in permanent magnets, instead of /, which is usually regarded as remaining 
 constant in these cases. Cf. also Gans, Ann. d. Phys. (4), xvi. (1905), p. 178. 
 * Phil. Mag. ix. (1880), pp. 225, 301. 
 
648-650] The Hall effect 573 
 
 to the current and to the direction of the magnetic force. Thus, if a metal 
 bar, along which a current is flowing, be placed in a magnetic field perpen- 
 dicular to the direction of the current, the current will tend to be deflected 
 from its course and to move towards the one or the other of the edges of the 
 bar. This cross flux of electricity will not however be permanent since it 
 will give rise to a slight accumulation of charge on the outer edges of the 
 bar, the additional electric field thus set up tending to oppose the transverse 
 current. A final steady state of equilibrium will be attained in which there 
 is no cross flux of electricity, the additional electric field being just sufficient 
 to balance the transverse electromotive force arising from the magnetic 
 field. But in this state of equilibrium there will be a difference of potential 
 between the opposite edges of the bar which may be measured by connecting 
 them up to the opposite terminals of an electrometer. This is the measure- 
 ment made by Hall. 
 
 The simple explanation of this phenomenon is almost obvious if we adopt 
 our former view that the current in the metal consists simply of a diffusion 
 of negative electrons through the spaces between the atoms : the action of 
 the electric driving field is then, to impose on the swarm of free electrons 
 thus imagined a finite average velocity in the direction opposite to that of 
 the electric force. If we suppose for the moment that this average drift 
 velocity is v then the average action of the magnetic field of force of intensity 
 
 H may be represented as a force on each electron equal to - [vB], if B 
 
 c 
 
 is the induction due to the force H. The statistical effect of all these forces 
 on all the electrons comprising the current is exactly the same as that of an 
 applied electromotive force in the same direction. This is the gist of the 
 explanation which has been offered by Riecke*, Drudef and Thomson J, 
 but on account of the great importance of the matter it seems worth while 
 at least indicating the more exact analysis , if only as an example of the 
 application of previous principles to such problems. 
 
 650. The general basis of the present explanation involves the usual 
 assumption that the whole of the electrical properties of the metals arises 
 entirely in the average motion impressed by the external circumstances on 
 the swarm of (negative) electrons which are otherwise moving about quite 
 irregularly and freely in the spaces between the atoms or atom-complexes. 
 In the absence of actions from any external agency the electrons are pre- 
 sumed to be moving about in such a manner that the distribution of Velocities 
 among them at any instant is precisely that specified by Maxwell's law so 
 that if N is the total number of free electrons per unit volume the number 
 
 * Wied. Ann. XLVI (1898). t Ann. d. Phys. i. (1900), p. 566; ni. (1900), p. 369. 
 
 J Rapp. Congr. Phys. in. p. 143 (Paris, 1900). 
 
 This was first given by Gans, Ann. d. Phys. xx. (1906). The present mode of treatment 
 is due to the author (Phil. Mag. xxx. (1915), p. 526). Cf. also N. Bohr, I.e. p. 313. 
 
574 General electrodynamic theory [OH. xiv 
 
 in the same volume with their velocity components between (. 17, ) and 
 (+ d$, 17 + drj, +, d) is given as usual by the formula 
 
 wherein u 2 = | 2 + -rf + 2 and q is a constant connected with the mean 
 square of u, viz. u m 2 , for all the 2V electrons by the relation 
 
 3 
 
 q ~2u m 2 ' 
 
 When however external electric and magnetic fields are in action through- 
 out the interior of the metal all this alters because then the velocity of each 
 electron while on its free path is modified by the forces in the external fields. 
 The main problem is now to determine the new steady velocity distribution. 
 If we follow the assumptions of our previous investigations and regard the 
 electrons and atoms as perfectly elastic spheres, the latter being of such 
 comparatively large mass that their energy and motion are unaffected by 
 the collision with them of the electrons, the problem is comparatively simple, 
 for if the new velocity distribution is one in which 
 
 SN-ffa-rj, I x, y, z, t)dg drift, 
 
 then we know from our previous investigation that the function / must 
 satisfy the differential equation 
 
 p a/ ?/ df df df df / / 
 
 F *8? + % + F *a 4 *te + ^ ~dy + tfo + ^ = r m > 
 where / is used, as there, for 
 
 Here (F,,., Y y , TF Z ) are the components of the applied forces on the typical 
 electron resulting from the action of the field ; and all that is now necessary 
 is to find these forces. 
 
 651. For the sake of simplicity we shall assume that the magnetic in- 
 duction is directed along the z-axis of the coordinates chosen and is of 
 uniform intensity B throughout the metal. The electric field is of uniform 
 intensity E but for the sake of generality we need not specify its direction 
 beyond stating that its components in the three principal directions are 
 E,,., E,,, E z . The motion of a typical electron while traversing a free path is 
 thus given by the following equations 
 
 & 
 
 m, - = PK 
 
 dt ***' 
 
 d-n eB y 
 
 m -^ eE, 
 
650 T 652] The Hall effect 575 
 
 The last two equations may be written in the form 
 
 fEeB 
 ' 
 
 z ~ m 2 c 2 m me 
 
 <M> = _^r- e ^v 
 dt 2 m 2 c 2 m 'me 
 
 eE 
 
 so that it we use n , 
 
 me 
 
 we shall have 
 
 'h = f + ( " + ) sin w *l + ( - ?? - COS 
 
 Vmw / \mw '/ 
 
 Y , f eE ~ \ f eE V , A / 
 
 4! = + - 1 sin <! - - + 4 (1 - cos ntj), 
 
 \mn '] \mn J 
 
 where (^, f) l , ^) denote the values of (, 17, ) at a time t l after the instant 
 ^ at which the latter are measured. Now in any real case n is always small 
 and since the formulae are to be applied with ^ of the order of magnitude 
 of the time of description of a free path, the product nt t will always be very 
 small so that we may approximate to these formulae and use 
 
 eE y , ,\ . 
 
 - + n MI 
 
 m V 
 
 y . , /eE 2 \ 
 
 ii = t + I - - nij 1 ij 
 
 \ m J 
 
 , . . eE z n 
 
 \- + n MI + - - 
 
 \ m V 2m 
 
 eE 
 
 - - - 
 
 2m 
 
 The accelerations of the typical electron may therefore be expressed as 
 functions of the time by 
 
 (F., F v ,F,) = f eE -', * + <+*?, !5._^- e A 
 V m m m m m 
 
 and the above differential formula is to be used with these values. 
 
 652. We shall now suppose that the physical conditions are the same at 
 all points of the metal so that the form of the velocity distribution function 
 / will not vary from point to point : in this simple steady case the equation 
 for / becomes 
 
 F 8 /4-F ^4-P _i_/. -A 
 *9| +JJ ^ + **a + T m r m ' 
 
 of which the solution just as previously may be written in the form 
 - dr t>=t F ^ -1- F dfo 4- F dfo \ dt 
 
 ^ 4 ^^^' 1 ' 
 
 the index 1 denoting that the values are to be properly expressed as functions 
 of the auxiliary time variable t^ used to express the integration. 
 
576 General electrodynamic theory [en. xiv 
 
 Now F^+F^+F,!^ -29(^ + ^ + ^/0, 
 so that to the same order of approximation we may write 
 
 - f" (F^ + F^ + F 2 C)) dt, 
 
 T m t,=t-r /i 
 
 and on inserting the values of (X , Y, Z), and (, 77, ), this is easily found to be 
 
 and again we may write 
 
 Vm 
 
 Tm ~ u 
 
 where l m is the length of the mean free path, which can be taken- to be inde- 
 pendent of the velocity : thus the expression for the general law of distribution 
 of velocities which is correct to the order of approximation adopted is obtained 
 with the function 
 
 or if we adopt the notations of vector analysis we can write this in the more 
 concise form 
 
 /= A 7 J q ~ e~^ jl + 2qelm (uE) - - m \ (B . [uE]) 
 V 7T 3 ( mu mcu 2 v 
 
 irrN \/i 
 
 The various constituents of the complete flux of electricity are now 
 easily calculated on the same lines as adopted above in the simpler example 
 of these same principles ; they are obtained as the components of the vector 
 
 Z 
 " 7 .. / 
 
 el m Virq 
 
 2mc 
 
 3 
 . / I 
 
 3mq 
 is the ordinary conductivity this may be written in the form 
 
 or since * ' ^ 
 
 653. Now in Hall's experiments if the current was sent along the metal 
 in the direction of the y-axis and the magnetic field was directed along the 
 -axis as above the induced potential gradient in the direction of the 0-axis 
 (i.e. perpendicular to the lines of the magnetic field and the direction of the 
 average electronic flow) necessary to maintain the current in its undisturbed 
 path was measured. The conditions parallel to the 2-axis are therefore such 
 that there is no flow along that axis : this implies that 
 
 el m V7rq 
 
 Az + ~K "~ B&y ~ V. 
 
 2mc 
 
652-654] The coefficient of the Hall effect 577 
 
 If J y denotes the current in the direction of the y-axis then we know that to 
 
 a first approximation 
 
 J y = crEj,, 
 o 
 
 so that E z = - BJ,,, 
 
 which is exactly the law usually adopted to express empirically the mag- 
 nitude of this effect. It shows that the potential increases in the positive 
 or negative direction of the 2-axis according as e is positive. Now since all 
 the other evidence points to the fact that it is only the negative electrons 
 that are freely moveable we must conclude that the observed potential 
 difference will always be in the direction of decreasing z and proportional 
 both to the strength of the current and to the magnetic induction. In non- 
 magnetic media the magnetic induction is equal to the force so that the 
 effect in such media would be proportional to the force, but in the ferro- 
 magnetic media it would practically be proportional to the polarisation. 
 
 Although these theoretical deductions are satisfactorily verified in a large 
 number of cases there are very big discrepancies in many other, and equally 
 prominent cases ; for instance the effects observed in iron and many other 
 metals are exactly in the opposite direction to that predicted by theory. 
 These discrepancies are probably however due to secondary constitutional 
 characteristics of the metals concerned and are in no way detrimental to the 
 general theory just developed. 
 
 654. There are other actions analogous to that discovered by Hall, which 
 we have however not thought it necessary to explain in detail, although they 
 are theoretically important as tending to confirm the general characteristic 
 basis of the present form of theory. We assumed above in discussing the 
 phenomenon of electrical and thermal conduction, that the thermal conduction 
 current was produced by the convection of the same electrons as give rise 
 to the electric current. Now suppose the ends of a metal bar are kept at 
 different temperatures so that there is a flow of heat along the bar : there 
 will on the average be no drift of the electrons from one end of the bar to the 
 other; but the electrons which are travelling from the hot end to the cold 
 end possess a greater kinetic energy and greater velocity than those which 
 are travelling in the opposite direction. And since the force on an electron 
 travelling in a magnetic field tending to deflect it at right angles to its motion 
 and to the direction of the field is proportional to the velocity of the electron, 
 the force tending to deflect the electrons travelling from the hot end to the 
 cold will be greater than that tending to deflect (in the opposite direction) 
 those travelling from the cold end to the hot. The electrons will thus tend 
 to move on the whole in the direction in which those moving from the hot 
 end to the cold are deflected by the magnetic field. This tendency to cross 
 flow of electricity will, in the steady state, be balanced by an initially 
 
 L. 37 
 
578 General electrodynamic theory [CH. xiv 
 
 established potential gradient which can be measured as in Hall's experi- 
 ments. This potential difference which is an essential consequence of the 
 present explanation of these phenomena has in fact been observed, first by 
 Nernst and von Ettinghausen in 1886. 
 
 It is not only in metals that the current of conduction is carried by freely 
 moving ions, but also in liquid electrolytes and in gases. The uses to which 
 Thomson has put the phenomenon analogous to the Hall effect in gaseous 
 conductors has already been dealt with in detail : it merely remains to add 
 that the same phenomenon has been often observed and measured in 
 electrolytes with results in full agreement with the theoretical predictions 
 as far as it is possible to follow them. 
 
 655. The Faraday and Zeeman effects. The next phenomenon which is 
 fundamental in this theory is concerned with an optical application of the 
 same principle. In 1845* Faraday discovered that on passing a plane polarised 
 ray of light through a piece of glass in the direction of the lines of force of 
 an imposed magnetic field, the plane of polarisation was rotated by an amount 
 proportional to the thickness of glass traversed and the strength of the field : 
 this discovery remained for a long time the only instance of an optical effect 
 brought about by a magnetic field, and the connexion between optical and 
 electromagnetic phenomena which is suggested by it could not be further 
 substantiated. In 1877, however, Kerrf showed that the state of polarisation 
 of the rays reflected by an iron mirror is altered by a magnetisation of the 
 metal; but it was not until 1896 % when Zeeman discovered the well-known 
 phenomenon of the magnetic separation of spectral lines which now bears 
 his name that a clue to the connection was obtained although previously 
 it had been formulated theoretically by Lorentz. 
 
 656. Now we have already explained how the emission and absorption 
 of light may be regarded as due to the vibratory motions of electric charges 
 or electrons contained in the atoms of ponderable bodies. The distribution 
 of these charges and their vibrations may be very complicated, but if we 
 wish only to explain the production of a single spectral line, we may be 
 content with the hypothesis already introduced that there is a single vibrating 
 electron in each molecule with the necessary positive charge. If this electron 
 executes simple harmonic vibrations it may be regarded as executing these 
 about a definite position of equilibrium to which it is bound by a force which 
 
 * On the magnetisation of light and the illumination of magnetic lines of force, Phil. Trans. 
 1846, i. p. 1; Experimental Researches, 1855, in. p. 1. 
 
 f On rotation of the plane of polarisation by reflection from the pole of a magnet, Phil. 
 Mag. (5), m. (1877), p. 321 ; (5), v. (1878), p. 161. 
 
 % Phil. Mag. (5), XLIII. (1897), p. 226. Zeeman has published a connected account of his 
 epoch-making investigations in book form in Magneto-Optics (1914, Macmillan), and the reader 
 is referred to this work for further details of the subject. 
 
654-657] The Zeeman effect 579 
 
 is proportional to its displacement from that position. The equations of its 
 simple vibratory motion will then be 
 
 mx = kx, my ky, mz = kz, 
 of which the general solution is 
 
 (x, y, z) = (X Q , y , z ) cos (pt + e ), 
 
 where p 2 = m' 
 
 so that the frequency of the free oscillation is . 
 
 Po 
 
 657. Let us next consider the influence of an external magnetic field H 
 giving rise to a magnetic induction B ; this will modify the motion of the 
 electron by introducing a new force 
 
 where e denotes the charge of the electron and v its velocity. If the magnetic 
 force H is parallel to the axis of z, the components of this force are 
 
 d&z dy eB z dx 
 
 c dt' c dt' 
 
 Hence the equations of motion of the typical electron become* 
 
 d 2 x eE z dy 
 
 m-^ = -kx-\ -- *--j- 
 
 at 2 c dt 
 
 d z y eB, dx 
 
 m w 2 = - k y- dt' 
 
 d 2 z 
 
 m -j-g = - kz. 
 dt 2 
 
 The last equation shows that the component vibrations in the direction 
 of the a;-axis are not affected by the magnetic field. The particular solution 
 given therefore still holds for this component of the vibration, which is there- 
 
 fore still simply periodic with period - . The first two equations are 
 
 Po 
 equivalent to the two equations 
 
 d 2 (x iy] . eB, d . 
 
 *-wr -^i^^-^^i*'/), 
 
 of which the solutions are 
 
 x + iy = A-^e 1 ^, 
 
 and x iy = A^e iP:it , 
 
 * This theory is due to Lorentz, Ann. Phys. Chem. LXIH. (1897), p. 278. Of. also Larmor, 
 Phil. Mag. (5), XLIV. (1897), p. 503. 
 
 372 
 
580 General electrodynamic theory [OH. xiv 
 
 eB 
 
 wherein ?i 2 + - f- Pi = 
 
 7} 2 
 
 P* 
 
 The corresponding real solutions are of the type 
 
 (i) x = % cos fat + j), y= + cti sin (p^t + e^, 
 (ii) a; = 2 cos (^ 2 ^ + c 2 ), y = - 2 sin (lM + e z)> 
 respectively, the constants a x , a 2 , e 1? e 2 being arbitrary. 
 
 These last two solutions represent circular vibration in a plane perpen- 
 dicular to the magnetic field and taking place in opposite directions. The 
 frequency of one is higher and that of the other lower than the original 
 frequency p . In all real cases the difference between the frequencies is 
 found to be very small compared with the frequency itself so that we may 
 
 put 
 
 eB z eB z 
 
 V^Vo-Zn*' 
 and therefore p* p = 
 
 me 
 
 658. We have now to consider the nature of the light emitted by the 
 vibrating electron. The total radiation is made up of several parts, corre- 
 sponding to the particular solutions we have obtained. 
 
 Our previous discussion of the radiation from a moving electron shows 
 that 1 , if such a particle has a vibration about a point .0 the vibration curve 
 of the electric force in the field due to it at a distant point P is similar to 
 the projection of the hodograph of the orbit of the electron on. the wave front 
 surface through P. It follows that the radiation emitted by the molecule 
 in a direction parallel to the lines of force in the magnetic field will be 
 composed of two circularly polarised constituent rays, one vibrating in a right- 
 handed direction and the other in a left-handed direction, our conventions 
 being such that the former has a period p and a period p 2 . 
 
 In a direction perpendicular to the lines of the field the radiation will 
 be composed of three linearly polarised constituents of frequencies p lt p , p 2 ', 
 the polarisation of the inner component (p Q ) being perpendicular to that of 
 the other two. 
 
 These theoretical predictions have been fully verified by Zeeman's 
 observations, who separated the various constituents in the radiation emitted 
 in any direction by passing it through a prism or grating, from which also 
 he was able to draw two very remarkable conclusions. 
 
 In the first place, it was found that, for light emitted in a direction 
 coinciding with that of the magnetic force the polarisation of the component 
 
657-659] The Faraday effect 581 
 
 of the doublet for which the frequency is lowest is right-handed. This 
 proves that the frequency p t is smaller than p% or that e is negative : this 
 agrees with the general result of other lines of research that the vibrations 
 are those of negative electrons, these having much greater mobility than the 
 positive charges. 
 
 The other result relates to the ratio between the numerical values of 
 the electric charge and mass of the electron. This ratio can be calculated 
 as soon as the distance between the components, from which we can find 
 Pi ~ Pz> an d the strength of the magnetic field have been measured. The 
 number deduced by Zeeman from the distance between the components of 
 of the D lines of sodium, or rather from the broadening of these lines, whose 
 components partly overlap, was one of the first values of e/m that have 
 been published and agrees remarkably well, considering the errors to which 
 its determination is subject, with the numbers that have been found for 
 the negative electrons of the cathode rays. 
 
 Unfortunately the satisfaction caused by this success of the theory of 
 electrons in explaining new phenomena, could not last long. It was soon 
 found that many spectral lines are split up into more than three com- 
 ponents; up to the present no very satisfactory explanation of these com- 
 plications have been offered*. 
 
 659. The magnetic rotation of the plane of polarisation of light trans- 
 mitted through a dispersive medium parallel to the lines of force of an imposed 
 field is closely connected with this phenomenon of the Zeeman effect and is 
 in fact due to a similar action of the magnetic field on the resonance vibrations 
 of the contained electrons which are effective in modifying the propagation 
 of light. This will be obvious when we realise that if in any propagation 
 the relation between the inducing field intensity and the dielectric displace- 
 ment current in the medium is of the form 
 
 f* 
 
 then the propagation is with the velocity --,-. Now let us consider the 
 
 circumstances in a simple case where the magnetic force is parallel to the 
 z-axis of coordinates and the propagation is that of a homogeneous wave 
 train also in this direction, so that the vectors of the theory depend only on 
 the time by the factor e ivt . If the material dielectric has molecules containing 
 
 * Cf . however W. Voigt, Magneto and elektro-optik, Ch. iv. In this work will also be found 
 full references to all the theoretical and experimental work bearing on this subject up to 1908. 
 Reference may also be made to the article "Theorie der magneto-optischen phanomene,' by 
 H. A. Lorentz, in Encyklop. d. math. Wiasensch. Bd. v. 
 
582 
 
 General electrodynamic theory 
 
 [CH. xiv 
 
 a system of vibrating electrons of the type we have already assumed the 
 equations of motion of the typical electron will be of the form 
 
 eB 
 
 mx + kx- i y = e (E x + aP x ), 
 c 
 
 my + kx -\ -- - x = e (E v + oP v ), 
 c 
 
 mz + kz = e (E z + aP e ), 
 
 where the notation is exactly as previously. In dealing with imposed 
 vibrations of period 2rr the first two of the above equations reduce to 
 
 (k-mp*)x- 
 
 aP x ), 
 
 (k - mp*) y + z x = e (E, + aP,), 
 
 C 
 
 whence we see that the two equations 
 - mp 2 T J -^- 1 (x iy) = 
 must be satisfied. , 
 
 660. Also since Sea; = P^, 
 
 we have 
 
 v ) + ae(P x iP y ) 
 
 iP, = 
 
 
 
 me 
 
 where the sums S are taken per unit volume over all the optically excitable 
 
 n 
 
 electrons and ft 2 = . 
 m 
 
 Now if also 
 we have 
 
 D E (e - 1) E 
 
 477 477 4r7 
 
 so that the two equations are equivalent to 
 
 Se 2 / 
 
 (E. *E.) 
 
 - 1 - 
 
 me 
 
 1 - 
 
 -^ 
 
 me J 
 
 0. 
 
 These can be satisfied in either of two ways. 
 
659-661] The ^Faraday effect '583 
 
 Either by* E x + iE v = 0, 
 
 v e z /m 
 
 v-y+* 
 
 1 1 , mc 
 and e = 1 H T. 
 
 or by E x - iE y = 0, 
 
 eB z 
 Po P ~ - 
 
 i i , mc 
 
 and e = 1 + . 
 
 2 -n 2 
 
 ro P 
 
 me 
 
 661. Now let us examine what this means : let e x and e 2 be the respective 
 values of the constants e determined by these equations and let us consider 
 the propagation of a beam of light which starts with the electric force polarised 
 in the plane of the axis of y and with an amplitude 
 
 E x = Ee ivt , E y = 0. 
 
 The medium effectively splits this beam into two oppositely circularly 
 polarised beams, in the one of which 
 
 initially, and in the other 
 
 E v , = 
 
 The first of these beams, in which E^ + iE Vi is propagated through 
 
 /i 
 
 the medium with a velocity = ; while the second in which E x iE v = is 
 
 
 
 propagated with a velocity equal to -7=- The result is that at a depth 
 
 z into the medium from the place of entry of this beam the amplitudes 
 in the respective components are 
 
 and ' (.,KJ 
 
 * These forms were first given by the author [Phil. Mag. 26 (1913), p. 362]. Those usually 
 given have a=0 but seem to be inconsistent with certain experimental results on the con- 
 stitutive character of the phenomena concerned. 
 
584 General electrodynamic theory [CH. xiv 
 
 so that in all 
 
 E 
 
 -ip - - ip 
 
 1,1,: E,, .e c -\- e 
 
 and therefore 
 
 . \e-, 
 
 
 
 - -- j= - 
 
 E,,, . Ve 2 z . 
 
 - in - ^p 
 
 e c e 
 
 Thus the beam of light is still plane polarised even at the depth z, but the 
 plane of polarisation has been turned through the angle 
 
 Vc* - Ve, 
 
 * z -v- 
 
 and this is the result determined by Faraday. 
 
 This simple explanation, although probably fundamentally .correct in its 
 essence, has nevertheless proved to be inadequate to describe many of the 
 observed complications of the phenomenon ; this is of course hardly surprising 
 when we consider the result of the explanation based on the same principles 
 of the more obvious Zeeman phenomenon described above. The discussion 
 of the necessary modifications would however take us beyond the scope of 
 the present work : reference may be made to Voigt, Magneto and Electro 
 Optik, for full details of the theoretical side of the question and to Zeeman' s 
 book already mentioned for a complete account of the experimental details 
 and results. 
 
 662. The phenomena just examined, which depend on the modifications 
 of existing electronic orbits by the magnetic field, are closely connected with 
 the phenomena of magnetic induction in diamagnetic bodies, for it is pre- 
 cisely these same modifications which give rise to the observed diamagnetic 
 polarity under similar circumstances. 
 
 Let us* examine the motions of the electrons in any small element of a 
 continuous piece of matter which give rise to its magnetic polarity. If r 
 denote the vectorial displacement of the typical electron, it is a condition 
 involved in the general permanency of the motions, that 
 
 are all independent of the time. If the distribution of electrons in the 
 
 * This analysis is due to Langevin, Ann. de chim. et phys. viu. t. 5, p. 70 (1905); but the 
 physical basis is due to Larmor and Lorentz. 
 
661, 662] The theory of diamagnetism 585 
 
 element is symmetrical the first three of these sums will be equal and the last 
 three zero. We assume that this is the case and write \<a for the general 
 value of the first three. 
 
 The motion of the typical electron is determined by an equation of the 
 type 
 
 mi = 
 
 where F e represents generally the resultant internal force of reaction on the 
 displaced electron and the suffix e is used to denote the values of functions 
 at the position of this electron. 
 
 It follows then since 
 
 that 
 
 where S is now used to denote a sum per unit volume. If the element 
 possesses no resultant magnetisation before the application of the field, then 
 
 2 - [rFJ = 
 m L 
 
 the position of the electron not being appreciably altered. Again we may 
 write 
 
 E e = E + (rV) E 
 
 B 6 = B+(rV)B 
 
 where the values without suffices denote those appropriate to the mean 
 position of the electrons. 
 
 Substituting these values in the expression for the rate of change of I 
 and using the conditions involved in the magnetic distribution we find that 
 
 dl e [ , B da 
 
 a curl E - -- Sl 
 
 dt 4mc L c at 
 
 IdB 
 
 But since curl E = - -j- 
 
 c dt 
 
 this is the same as 
 
 dl _ e 2 d 
 dt~ ~ (C 
 
 The aggregate change of I on the establishment of a field of strength B is 
 therefore 
 
 B 
 
586 General electrodynamic theory [CH. xiv 
 
 so that the susceptibility of diamagnetic polarisation is practically 
 
 , _ ae 2 
 4:mc' 
 
 The fact that this constant is dependent entirely on the conditions in the 
 atom is consistent with our experience of these things, as the diamagnetic 
 property is unaffected by the physical circumstance under which the mole- 
 cules exist*. 
 
 663. The transmission of force in the electromagnetic field. We now 
 
 turn to the discussion of the mode of transmission of force in the electro- 
 magnetic field. The discussion here is far more complicated than that of 
 the first paragraph; energy is a scalar thing whereas force is a vector 
 quantity and requires much more precise specification. We know from 
 experience that forces are exerted by one body on another as a result of 
 the existence of an electromagnetic field in the space surrounding them, or 
 even as the result of an interchange of radiation and we want to get a theory 
 of the matter. According to the ideas which we have developed the 
 mechanical actions on the parts of the material system resulting from the 
 established electromagnetic field are to be regarded merely as the terminal 
 aspects of a state of stress in the medium (the aether) between the 
 bodies, the action of one body on another being transmitted through and 
 by this medium. We should therefore be able to represent these forces as 
 an imposed geometrical stress system applied in the medium between the 
 bodies. The forces acting on the part of the system enclosed in any surface 
 drawn in the field would then be expressed as statically equivalent to a 
 system of tractions over the surface (statically equivalent meaning that the 
 resultants are the same as if we imagined the forces to be applied to rigid 
 systems). 
 
 The problem of determining this stress admits of an infinite number of 
 solutions owing to our indefinite knowledge of the actual properties of the 
 aether, which is the ultimate seat of the strain condition. A rough mechanical 
 analogy would be obtained by the consideration of two oppositely electrified 
 bodies set in an insulating jelly; they will tend to come together and will 
 thus create a state of stress in the jelly around them, and this stress will 
 balance their attractions. This balancing stress reversed would thus com- 
 pletely represent the actions between the bodies. But different kinds of 
 jelly would give different stress-representations, and the problem of deter- 
 mining this representation is indefinite until the jelly is specified. This 
 indefiniteness does not however trouble us much at the present stage. We 
 only want a physical solution of the problem which can be expressed in general 
 terms independent of the particular nature of the problem, and this can be 
 
 * Cf . however A. E. Oxley ' On the influence of molecular constitution and temperature on 
 magnetic susceptibility,' Phil. Trans. A, 214, pp. 109-146. 
 
662-665] The ponder omotive forces 587 
 
 obtained with certain limitations. The present discussion however leads 
 to one of the points where -electric theory has not yet been probed right to 
 the bottom. The solution obtained is useful and suggestive but it cannot 
 yet be linked with our general physical theories. 
 
 We must here emphasise that it is the mechanical forces on the material 
 bodies that we are going to deal with. If there is no matter in the small 
 volume any system of tractions over its surface must balance among them- 
 selves. 
 
 664. We examine quite generally the forcive on the portion of any elec- 
 trical system enclosed by an arbitrarily chosen surface /, assuming that it is 
 ultimately the same as the resultant of the forces on the separate electrons 
 associated with the matter of the system and constituting in their average 
 relations its free charge and dielectric and magnetic polarisation. In 
 estimating these forces account must however be taken of all the electrons 
 properly associated with the matter, even if they are displaced across to the 
 outside of the surface /, but not of those electrons temporarily inside / but 
 really belonging to the matter outside. The force exerted by the field on 
 any 'bound' electron is in reality applied to the matter at the point of it 
 to which the electron is bound and not at the point where the electron may 
 be found. For the purposes of the calculation we may therefore conveniently 
 regard the portion of the material system under consideration as abruptly 
 terminated by the surface / and therefore isolated from any portion outside 
 this surface. In all other respects it will be assumed to be perfectly con- 
 tinuous throughout. 
 
 665. If v denote the vector velocity of the typical electron with charge 
 e moving in a field at a point where the electric force and magnetic induction 
 vectors are E, B respectively, the force on it is 
 
 + '[V] 
 
 c 
 The force on the element Sv of the matter inside the surface / is therefore 
 
 1 / | 
 
 c J | 
 
 where p denotes the average charge density in the element and C' the average 
 current density of the electric flux. ' 
 
 The average charge on a small element Bv inside the surface is 
 
 (p divP)Si> 
 
 if p is the density of the free charge and P the intensity of the polarisation at 
 the point. In addition to this distribution there is however a surface charge 
 of density P n at the abrupt outer boundary of the portion of the system 
 under consideration, that is the surface / itself. 
 
588 General electrodynamic theory [OH. xiv 
 
 Again the average current density in the interior of the medium is 
 
 dP 
 Ci + pu + -r +c curl Ij 
 
 where C 1 is the true current of conduction; u the average velocity of the 
 matter at the point, and 
 
 I^I + ; [PU] 
 o 
 
 is the magnetic polarisation intensity, including the quasi -magnetism arising 
 from the convection of electrically polarised molecules. 
 
 This current distribution has also to be adjusted at the boundary of/ by 
 the inclusion of a current sheet on that surface of density 
 
 c [I^] 
 where n x is the unit normal vector at the point. 
 
 666. The electric part of the total forcive on the enclosed system will 
 therefore be given as regards its linear component by 
 
 f( P -di 
 
 divPEdv 
 
 the volume integral being taken throughout the space inside /and the surface 
 integral over this surface itself. A reduction of the latter integral by Green's 
 lemma shows that this forcive is the same as 
 
 i{ P E + (PV) E} dv. 
 
 Remembering now that the surface / was arbitrarily chosen we may interpret 
 this as implying that there is a forcive per unit volume on the system of 
 intensity 
 
 F e =pE+ (PV) E. 
 
 This is the whole of the average linear electric force on the medium. In 
 addition there is a torque per unit volume of intensity 
 
 C e = [PB] 
 as is easily seen on analogy with the statical case. 
 
 667. The force on the portion of the medium under review due to the 
 magnetic field is similarly 
 
 0, + P u + + c curl I 15 B dv + [ftnj B] df. 
 
 The second integral transforms similarly by Green's lemma to the volume 
 integral of 
 
 - [curl l!, B] + grad (IjB) - I div B 
 
 where the differential operator in the second term affects only the B function. 
 
665-669] The complete magnetic force 589 
 
 Thus since 
 
 divB= 
 
 we may take this part of the forcive as distributed throughout the medium 
 with intensity 
 
 per unit volume at any place. We shall henceforth use Cj to include the 
 convection and polarisation currents as well as the conduction currents, so 
 that the expression for this part of the forcive reduces to 
 
 where the restriction is still implied in the operator in the last term. 
 
 This is the complete expression for the magnetic part of the total forcive 
 per unit volume on the medium. 
 
 668. But of the magnetic induction B at any place a part, viz. H the 
 magnetic force arises from the system in general, and the remainder 47rl is 
 the expression of the local influence arising in the element of volume itself. 
 The question then arises whether the latter part is to be rejected in effecting 
 the summation over the element of volume, as being compensated by reaction 
 exerted by the elements of charge under consideration on the magnetism 
 existing in the same element. If, as at present, it is a question of finding 
 the mechanical force acting on the complete element of volume this com- 
 pensation will certainly subsist ; the action of the magnetism in the element 
 on the charges will just cancel the action of the charges on the magnetism. 
 In other conceivable cases this compensation may not occur. Thus in a 
 mechanical theory which considers only elements of volume the magnetic 
 part of the total force must be replaced by* 
 
 i [CJB] + grad (I X H). 
 
 C 
 
 669. Following our previous theory we now try and express these 
 forces on the electrical system or any part of it by surface integrals over 
 the boundary of the volume containing it. To do this we must first express 
 them in such a form that 
 
 dT xu 
 
 -\ -- o -- r 
 
 dx dy dz ' 
 
 * We have retained the induction vector in the first part of this expression because the 
 complete form is required in the subsequent analysis: the difference is a purely local term 
 representing forces between the magnetism in an element and the free electrons there and its 
 presence is a matter of indifference in a mechanical theory. It is however only fair to add that 
 if the induction vector is retained throughout a reduction of the forces to a representation by 
 a stress system is still possible but it is slightly different to that given in the text ; its uniform 
 pressure constituent contains the additional term 2n-/ 1 2 . 
 
590 General electrodynamic theory [OH. xiv 
 
 taken over the surface of the volume. This determines the T's as the com- 
 ponents of the stress system in the usual way. 
 
 670. We consider the electric and magnetic forces separately. As regards 
 the electric part we follow Maxwell's hint developed in our previous dis- 
 cussions* and try as before 
 
 F <* = 4, ( E ' D - - 1 E2 ) + 4 < E * D '> + <**> . ; ' 
 
 In this case however the relationship does not hold : there is an outstanding 
 term which cannot be included in the differentials. To see this easily we 
 notice on differentiating this expression out that it is 
 
 which differs from the above value by 
 
 dE. x D ?EJ _ _i/ E 8E,, SEA ais^ _ aE, 
 
 1 /_ dE x 3E,\ 1 
 which is I E v -jr~ =-= 
 
 4n \ oy ox J 4n 
 
 or in vector notation ^ (E 
 
 4rr 
 
 but curl E = =- 
 
 c at 
 
 so that this is 
 
 47TC 
 
 The difference between this case and that discussed in chapter iv is 
 obvious. In the statical case we were able to put 
 
 dEx _ dEy 8E 2 _ 8Ea. 
 dy dx ' dx dz ' 
 
 so that there was no discrepancy. These equalities are however no longer 
 satisfied because they imply the existence of an electric potential. 
 
 671. Similar results apply of course to the other components of the stress 
 system. Thus if we leave out for the present this outstanding part of the 
 forcive, we see that the main part of the electric force acting on the matter 
 is expressible as a stress system which can be specified by the matrix 
 
 * See page 181. 
 
669-672] 
 
 The representative stresses 
 
 591 
 
 En 
 x'-'y) 
 
 E 2 
 
 STT^' 
 
 E 
 
 
 
 E D 
 
 J v* J z 
 
 E D,,, . E? Jjy j E z D~ E 
 
 This is identical with that obtained in the statical theory and is therefore 
 amenable to the simpler specification there developed. It can in fact be 
 dissected into parts. 
 
 (i) A simple hydrostatic pressure ~- E 2 throughout the medium. The 
 negative sign shows that it is a pressure. 
 
 (ii) A tension along the internal bisector of the angle between D and E 
 equal to 
 
 E . D cos 2 ED. 
 (iii) A pressure along the external bisector equal to 
 
 E . D sin 2 ED. 
 
 (iv) And the torque per unit volume 
 
 [E.D]. 
 
 This couple or torque is represented as part of the stress system and so 
 the specification is complete, there is no discrepancy. 
 
 This is the general result ; if the medium is isotropic it reduces to a pull 
 along the lines of force equal to (ED) with a hydrostatic pressure 5- E 2 
 
 O7T 
 
 and this is Maxwell's system. 
 
 672. A similar argument applies to the magnetic part of the forcive on 
 the element. The x-component of this force is given by 
 
 Now the usual analysis shows that 
 
 ^ ( l sr) = Ix \ EA ~ I R2 1 + 1/ (B " Ha) + L (E * H * } + [B ' curl H ]< 
 
 \ VJU / UJU \ A UU U& 
 
 and since curl H = - ( 4^0, + -j- 
 
 c c \ at t 
 
 the last term in this expression is equal to 
 
 whence 
 
 = [BC]= [CB]= [ Cl B] - 
 
592 General electrodynamic theory [CH. xiv 
 
 and similar results apply to the other components. The main part of the 
 forcive can thus be specified as a stress system whose components are given 
 in the matrix 
 
 B a H 2 , H 2 B^ , H z E z | H 2 
 
 but this leaves out a part 
 
 which cannot be expressed by a surface integral. 
 
 Thus of the whole electrodynamic forcive per unit volume on the medium 
 there is a total outstanding part 
 
 E _ _ __ B 
 
 4:7TC [_' ' dt \ 4:7TC \_dt 
 
 which cannot be included in the stress specification. 
 
 673. The result is that if we have a lot of bodies in an electromagnetic 
 field attracting one another, the resultant of the electrodynamic forces acting 
 on the matter inside any given closed surface drawn in the field may in the 
 main be expressed as the resultant of the attractions across the surface as 
 specified above and which are to be treated accoiding to \he ordinary pro- 
 cedure in the theory of elasticity. In addition to this part there is however 
 a forcive per unit volume of amount 
 
 which cannot be so reduced. 
 
 What is the meaning of this outstanding term ? It is a complete differential 
 with respect to the time and thus following an ordinary dynamical analogy 
 there is a very strong temptation to say that it represents a rate of change 
 of some kind of momentum. This implies a distribution of 'electromagnetic 
 momentum ' throughout the field with a density at each point equal to 
 
 Such a tentative hypothesis would provide a convenient representation for 
 many purposes, but there are difficulties involved in it. In any case it is 
 a pure assumption, the only justification for it lying in the fact that it is 
 a complete differential with respect to the time. 
 
 From this, the modern, point of view the actual force distribution on the 
 matter enclosed in any surface would be expressible partly as a static stress 
 distribution over its surface and partly as a kinetic distribution throughout 
 
672-675] Electromagnetic momentum 593 
 
 the interior. Or to reverse the argument the actual forces between the 
 bodies partly bear a statical stress and are partly used up in a change of 
 momentum of some kind ; the part which does not appear as a stress being 
 capable of consideration as the kinetic reaction to a rate of change of 
 momentum. The forces of this latter type are of the nature of ' motional 
 forces,' to use Kelvin's phrase. 
 
 Part at least of this distribution of momentum would have to be ascribed 
 to the aether, as it would exist if there were no matter present. Thus even 
 if there is no matter inside the closed surface above considered there would 
 still be no balance between the stresses over the surface, which would act 
 to change the 'aethereal' electromagnetic momentum inside. 
 
 Even if we presume the existence of this momentum, it appears that it 
 is certainly not the main part of the momentum in the field ; it is for instance 
 not that part of the momentum whiqh is involved primarily in the propagation 
 of electromagnetic or light waves. The actual working forces which effect 
 the propagation of radiation depend on the first power of the field vectors 
 concerned whereas the momentum here discussed depends on the second 
 power of these vectors. It is therefore a second order effect and does not 
 change sign with the reversal of the field. This means that it would be too 
 difficult to detect even if it existed. 
 
 There is however after all no substantial reason for adopting this point 
 of view of the matter as anything more than a convenient mode of expression. 
 There is as yet no physical explanation to show why the forces act in this 
 way. 
 
 674. If we consider an electromagnetic field like tha in the propagation 
 of electromagnetic radiation of any sort the field is oscillating and so the 
 time-average value of 
 
 is zero. Thus in all cases with alternating fields of this kind we can neglect 
 this part when determining average forces, because it occurs as much positively 
 as negatively in any time average. 
 
 675. The mechanical pressure of radiation. An important application of 
 the foregoing principles is to the explanation of the pressure of radiation on 
 absorbing and reflecting bodies. In this case the average of the momentum 
 teim in the complete expression of the forcive is zero and the simpler 
 Maxwellian specification is applicable. 
 
 Consider the case of a train of plane polarised waves advancing through 
 an isotropic medium in the direction of the axis of x, so that all the quantities 
 are functions of x only, the electric force being (0, E VJ 0) and the magnetic 
 
 L 38 
 
594 
 
 General electrodynamic theory 
 
 [CH. xiv 
 
 force (0, 0, H z )*. We assume for simplicity that the permeability is unity. 
 The equations of propagation 
 
 4rr n _ dH z dEy = _ 
 
 dx 
 
 wherein 
 
 are satisfied by 
 
 dx ' 
 
 Ej, = A. e~ ax cos (nt bx), 
 
 C A t 
 
 H z = -- Va 2 + 6 2 e~ ax cos (nt - bx + 6), 
 
 wherein en 2 + i^nna = c 2 (a + ib) 2 , 
 
 or c 2 (a 2 6 2 ) en 2 and 
 
 The mechanical forcive per unit volume is as before given by its single 
 component t 
 
 1 dl 
 
 Now 
 
 C^Hj, dx = 
 
 4?r 
 
 c 
 
 Q- 
 
 STT 
 
 H, 2 
 
 x, 4?r 
 
 - i 'l 
 
 so long as H z is continuous between the limits of integration. Also 
 
 Z Jj 1 *' 9 Jj ' Jl J 
 
 at n 2 J Xi at max 
 
 since for the harmonic oscillatory motion assumed 
 
 dt* 
 
 = c 
 
 dtdx' 
 
 and thus this integral is 
 
 dt 
 
 r 
 provided -^ v is continuous throughout the range of integration as is always 
 
 the case, though e may change gradually or abruptly. We have thus 
 
 ' 
 
 which gives the aggregate mechanical forcive on the stretch of the medium 
 
 * Throughout this section we have used the magnetic force instead of the magnetic induction, 
 in order to conform to the usual practice. The two are identical in the present case of non- 
 magnetic media. 
 
 f We have followed Larmor in deducing the expression for the forcive directly from first 
 principles. The same result can however be readily obtained by an application of Maxwell's 
 stress formulae, but the present procedure deduces it without resort to these formulae, the 
 validity of which may be doubted. 
 
675, 676] The pressure of radiation 595 
 
 between a^ and x z in the form of pressures on its ends. Thus for the simple 
 forms of E v and H 2 assumed the time average of the pressure on either end is 
 
 L (A 2 i A 2\ 
 ic \ e i a -m h 
 j D7T 
 
 A r and A m representing the amplitudes of the magnetic and electric vibrations. 
 This is however the sum of the mean kinetic and potential energies per unit 
 volume of the radiation, less that involved in the electric polarisation in 
 the molecules; hence on any portion of the medium there is a mechanical 
 force, directed along the waves equal per unit cross section to the difference 
 of these densities of energy at its ends. 
 
 676. In a transparent medium 
 
 AflBiA 2 = c 2 ( 
 \ dt J '~ e \ 
 
 so that the above internal pressure may be expressed in the form 
 
 flBiA 2 = c 2 (dE y \ 2 = c 2 /dH, 
 dt J '~ e \dx) '~ e ( dt 
 
 STT 
 
 If there is in the medium a directly incident wave whose vibration at the 
 interface is A mf cos nt and also a reflected wave A m cos (nt e) and also 
 a refracted wave, this result may be applied to a layer of the medium con- 
 taining the interface ; thus there will be a mechanical traction on the interface 
 represented by a difference of pressure on its two sides, that on the incident 
 
 side being 
 
 If 1 1 _ ~| 
 
 g- {A m . cos nt + A mf cos nt - e} 2 + - {A m . sin nt + A mf sin nt e} 2 . 
 
 In air or vacuum this is 
 
 (A m * + A mf 2 + 2A m .A mf cos e), 
 
 where A m is the amplitude of the resultant magnetic vibration on that side. 
 
 When the radiation is directly incident on an opaque medium H z and 
 
 d"E 
 
 j~ are null in its interior ; so that, when the surrounding medium is air or 
 
 vacuum, its surface sustains in all a mechanical inward normal traction of 
 intensity %A m 2 , that is, equal to the mean energy per unit volume of the 
 radiation just outside it, in agreement with Maxwell's original statement*. 
 
 This is the famous pressure of radiation which has now been experimentally 
 examined and the theoretical conclusions as to its intensity verified to within 
 
 * Treatise n. 792. 
 
 382 
 
596 
 
 General electrodynamic theory 
 
 [CH. XIV 
 
 one per cent, by the measurements of Nicholls and Hull*. We can in a case 
 of this kind thus say that radiation exerts a force just as if it carried momentum. 
 This leaves open the question of the actual existence of the momentum 
 spoken of. 
 
 677. To illustrate these matters f further and to bring out another aspect 
 of the subject let us consider the opposite action, viz. the reaction or back 
 pressure exerted by the radiation emitted by a perfectly black body into 
 free space. Similar reasoning to the above will show that the back pressure 
 is of a similar amount to that there calculated. To exhibit the argument 
 in a simpler manner let us examine the component of the radiation emitted 
 from a small patch of the surface as a plane beam travelling out into space. 
 
 ISg. 97 
 
 SurrcJund the patch by a small closed surface cutting across the ray on the 
 one side. Our previous general theory then shows that the static resultant 
 of the forces on everything inside this surface is represented by a stress 
 system over the single patch of the surface where the ray cuts through it, 
 this being the only part on the surface where the field in this ray is different 
 from zero. If the ray cuts through this part of the surface normally it appears 
 that the normal stress at that point is a normal pull of amount 
 
 * Phys. Rev. xm. (1901), p. 293. Cf. also Lebedew, Arch, des Sciences Phys. et Nat. (4), vm. 
 (1899), p. 184; Poynting, Phil. Mag. ix. (1905), pp. 169, 475. 
 
 f The treatment here followed is given by Larmor in his lectures : certain aspects of it are 
 dealt with in his address on 'The dynamics of radiation' to the Mathematical Congress in 1912. 
 (Cf. Proceedings, vol. i. p. 206.) 
 
676-678] The pressure of radiation 597 
 
 and thus there is a total pull normally on the element of the emitting surface 
 equal to 
 
 n x being the normal vector direction to the elementary patch of area df^. 
 
 As it does not matter how big we draw this bounding sphere this stress 
 is the representative of a real force on the patch of the surface emitting the 
 beam of radiation ; and thus this beam exerts a back pressure on the body 
 emitting it which is equal per unit area to the density of the energy in the 
 field just outside it. 
 
 This back pressure is equal to the forward pressure on the body at the 
 other end receiving and absorbing the radiation. The action and reaction 
 are equal and opposite at the two ends of the beam, so that on this view of 
 things the ray behaves as if it were a carrier of momentum. 
 
 678. The question naturally arises as to what happens before the ray 
 reaches the second body. Where is then the corresponding reaction to the 
 
 \ 
 \ 
 
 Fig. 98 
 
 back pressure on the body giving out the radiation ? To answer this question 
 completely we must return to our original scheme. According to the general 
 theory the stresses acting over any geometrical surface drawn in the field 
 which does not contain any matter are balanced by the kinetic reaction to 
 the rate of change of the quasi-momentum in the aether inside the surface. 
 Now consider a small parallel beam of light advancing into free space. The 
 plane perpendicular front of the beam is advancing with the velocity c of 
 radiation. Draw a surface in the field much as that shown in the figure. 
 The stresses over the boundary of this surface are represented by the pressure 
 of radiation over the small patch of the surface where the beam cuts through 
 it : and this pressure must account for and just balance the rate of change 
 of the quasi-momentum of the aether included inside the surface. Now 
 where does this change of momentum come in? The propagation by waves 
 
598 General electrodynamic theory [CH. xiv 
 
 is an -alternating affair and so on the average the momentum in any part 
 of the beam remains constant; the beam is however getting longer; a new 
 region is being added in which there is momentum and so the change due to 
 this added momentum per unit time must be equal to the pressure. The 
 quasi-momentum per unit volume in the electromagnetic field in the general 
 case is of amount 
 
 and is directed perpendicular to both vectors. In our case of plane pro- 
 pagation of wave motion in the aether 'this is 
 
 and the general equations give 
 
 IL = E 
 
 **e 
 
 so that the quasi-momentum per unit volume is 
 
 1 
 
 47TC 
 
 H 2 
 ** ? 
 
 or in the mean it is - H 2 . 
 
 07TC 
 
 Thus on the average the momentum added in a time St is 
 
 reckoned per unit area of cross section in the beam. The rate of change 
 of this is equal to 
 
 But the energy per unit volume in our wave is on the average equal to 
 
 16rr~ ' 167r~ STT " 
 
 That is the average momentum per unit length in the beam is equal 
 to the energy transmitted per unit time across any cross section and this 
 shows that it is balanced exactly by the pressure of radiation on the patch 
 of the surface aforesaid. 
 
 679. The whole theory is thus consistent. It must however be noticed 
 that the stresses and momentum involved in this discussion, about whose actual 
 existence there may still be some doubts, are excessively small. The stress 
 for example which gives rise to the phenomena of radiation pressure depends 
 on the square of the vectors defining the field and is therefore nearly always 
 smothered by the stress which propagates the wave and depends only on 
 
678-680] 
 
 The back-pressure of radiation 
 
 599 
 
 the first power of the vectors. A rough analogy is provided by the attraction 
 of small objects by a vibrator like a tuning fork in air. Very near such 
 a vibrator in air the atmospheric pressure is less than that at a distance 
 and so any object placed near the vibrator would have a greater pressure 
 on its surface farthest from the vibrator and would therefore be impelled 
 towards that body. But this resultant pressure depends on the square of 
 the average pressure of the air whereas the sound propagation depends on 
 the first order things. Thus in a body emitting light the reactions of the 
 pressure of radiation would hardly ever be appreciable, being almost entirely 
 swamped by the reaction to the setting up of the vibrations. This fact 
 renders it almost impossible experimentally to test the existence of the 
 ' momentum ' force in free aether by testing for the reaction on a radiator, 
 before the radiation from it has reached an absorber, 
 
 In these considerations the radiator has been considered as at rest, we 
 must now calculate the effects due to motion. 
 
 680. In order to investigate whether the back pressure depends on the 
 motion we proceed as before ; and examine the reaction of an oblique beam 
 emitted by a plane radiator travelling normally to itself with a velocity v. We 
 
 Fig. 99 
 
 consider any boundary drawn in the field as shown. The force acting on the 
 path of the radiating surface which is inside this volume would then be 
 balanced by the radiation pressure on the single patch of the geometrical 
 surface were it not for the fact that the average ' momentum ' of the field 
 inside the surface is changing, owing to the fact that the beam is becoming 
 
600 General electrodynamic theory [CH. xiv 
 
 shorter at a rate v cos 6, 6 being the angle between the beam and normal 
 to the surface. This rate of change of momentum together with the pressure 
 along the ray from the radiator are balanced by the radiation pressure on 
 the patch of the geometrical surface. Thus if E' is the energy density in 
 the beam the back push or radiation pressure is easily seen to be 
 
 / vcos6\ 
 p = 1 -) . 
 
 * ^ / 
 
 681. But now we want E' '. The question is whether the energy per unit 
 volume in the radiation from a moving body is different from that of the 
 same body at rest or does the nature of the radiation from a 'perfect radiator 
 depend on its velocity ? 
 
 A simple thermodynamic argument can be adopted* to prove that the 
 periods and amplitudes of the motions of the molecules or electrons in a 
 body moving with uniform velocity do not depend on the velocity, so that 
 the amplitude of the oscillation in the emitted wave train is the same but 
 the wave length is necessarily altered by the motion according to the Doppler 
 principle being shortened by the factor 
 
 v cos 
 
 because if the radiator moves forward the waves are crowded up or shortened. 
 The period of the wave is therefore shortened by a similar factor and thus 
 the average energy per unit length is increased by the factor 
 
 v cos 0\~ 2 
 
 Thus if we use E for the energy of the statical radiation we have 
 
 *-('-'-?-')- 
 
 /.. V COS d}- 1 
 
 and thus *=(!- E. 
 
 \ c J 
 
 Similarly for a perfect black body absorbing radiation and moving with 
 a velocity v' cos 6' in the same direction we should have the pressure of 
 radiation on its surface equal per unit area to 
 
 682. The thermodynamics of radiation f. According to the views de- 
 veloped in detail in the previous chapter all radiation whether of thermal, 
 
 * Of. Larmor, Phil. Trans. A, 185 (1894), p. 781. 
 
 f The discussion of the following paragraph follows closely that given by Larmor in the 
 article 'Radiation,' in the Encyclopedia Britannica. Reference may also be made to Planck, 
 'Die Theorie der Wannestrahlung.' 
 
680-683] The thermodynamics of radiation 601 
 
 * 
 
 optical or electrical type consists essentially in vibrational waves of funda- 
 mentally identical types in the aether of space. A molecule or in fact any 
 piece of matter is to be regarded as a kinetic system compounded of simpler 
 systems so that its energy may be classified into constitutive energy essential 
 to its constitution and vibratory energy which it can receive from or radiate 
 away into the aether. A piece of matter isolated in free space would in time 
 lose all energy of the latter type by radiation : but the former will remain so 
 long as the matter persists, along with the energy of uniform translatory 
 motion to which it is ultimately reduced. Thus all matter is in continual 
 exchange of vibratory energy with the aether and it is with the laws of this 
 exchange of energy that the general theory of the present title deals. 
 
 683. The foundation of this subject is the principle arrived at inde- 
 pendently by Balfour Stewart and Kirchhoff * about the year 1858 that the 
 constitution of the radiation which pervades an enclosure surrounded by bodies 
 in a steady thermal state must be a function of the temperature of these 
 bodies, and of nothing else. Their reasoning rests on the dynamical principle 
 that by no process of ordinary reflexion or transmission can the period and 
 therefore the wave-length of any harmonic constituent of the radiation be 
 changed ; each constituent remains of the same wave-length from the time it 
 is emitted until the time it is absorbed again. If we imagine a field of 
 radiation to be enclosed within perfectly reflecting walls, then, provided there 
 is no material substance in the field which can radiate and absorb, the con- 
 stitution of the radiation in it may be any whatever and it will remain 
 permanent. It is only the presence of material bodies that can transform the 
 surrounding radiation towards the unique constitution which corresponds to 
 their temperature. We can define the temperature of an isolated field of 
 radiation, of this definite ultimate constitution, to be the same as that of 
 the material bodies with which it would thus be in equilibrium. Further the 
 mutual independence of the various constituents of any field of radiation 
 enclosed by perfect reflectors allows us to assign a temperature to each 
 constituent, such as the part involving wave lengths lying between A and 
 A + SA, that will be the temperature of a material system with which this 
 constituent by itself is in equilibrium of emission and absorption. But to 
 reason about the temperature in this way we must be sure that it completely 
 pervades the space and has no special direction; this is ensured by the 
 continual reflections from the walls of the enclosure. The temperature of 
 each constituent in a region of undirected radiation is thus a function of its 
 wave length and its intensity alone. 
 
 It is the fundamental principle of thermodynamics that temperatures 
 tend to become uniform. In the present case of a field of radiation, this 
 equalisation cannot take place directly between the various constituents of 
 * Ann. Phys. Chem. 109 (1860), p. 275; Oes. Abhdl (1882), p. 578. 
 
602 General electrodynamic theory [OH. xiv 
 
 i 
 
 the radiation that occupy the same space, but only through the intervention 
 of the emission and absorption of material bodies ; the constituent radiations 
 are virtually partitioned off adiabatically from direct interchange. Thus in 
 discussing the transformations of temperatures of the constituent elements 
 of radiation, we are really reasoning about the activity of material bodies 
 that are in thermal equilibrium with those constituents ; and the theoretical 
 basis of the idea of temperature as depending on the fortuitous residue of 
 the energy of molecular motions is preserved. 
 
 684. Let us consider a spherical enclosure filled with radiation and 
 having walls of ideal perfectly reflecting quality, so that none of the radiation 
 can escape. If there is no material body inside it, any arbitrarily assigned 
 constitution of this radiation will be permanent. Let us suppose the radius 
 a of the enclosure is shrinking with extremely small velocity v. A ray inside 
 it incident at an angle i, will always be incident on the walls in its successive 
 reflexions at the same angle, except as regards a negligible change due to 
 the motion of the reflector; and the length of its path between successive 
 reflexions is 2<z cos i. Each undulation in this ray will thus undergo reflexion 
 
 at intervals of time equal to - - , where c is the velocity of light and it 
 
 c 
 
 s'1) f* O ^ *? 
 
 is easily verified that on each reflexion it is shortened by the fraction - 
 
 of itself : thus in the very long time T required to complete the shrinkage 
 
 
 
 it is shortened by the fraction vT/a, which is , where Sa is the total shrinkage 
 
 a 
 
 in the radius, and is independent of i. The wave length of each undulation 
 in the radiation inside the enclosure is therefore reduced in the same ratio 
 as the radius. Now suppose the constitution of the enclosed radiation corre- 
 spond initially to a definite temperature. During the shrinkage thermal 
 equilibrium must be maintained among its constituents ; otherwise there 
 would be a running down of their energies towards uniformity of temperature, 
 if material radiating bodies were present, which would be superposed on the 
 mechanical operations belonging to the shrinkage and the process could not 
 be reversible. Such a state of affairs is not possible for it would land us in 
 processes of the following type. Expand the enclosure gaining the mechanical 
 work of the radiant pressure against its walls, whatever that may be. Then 
 equalise the intensities of the constituent radiations to those corresponding 
 to a common temperature, by taking advantage of the absorptions of material 
 bodies at the actual temperatures of these radiations : when this is done, 
 as it may actually be to some extent by aid of sifting produced by partitions 
 which transmit some kinds of radiation more rapidly than others, a further 
 gain of work can be obtained at the expense of the radiant energy. Now 
 contract the remaining radiant energy to its previous volume, which requires 
 
683-686] Stefan's law 603 
 
 an expenditure of less work on the walls of the enclosure than the expansion 
 of the greater amount of radiation originally afforded ; and finally gain still 
 more work by equalising the temperature of its constituents. The energy 
 now remaining being of smaller amount and under similar conditions must 
 have a temperature lower than the initial one. This process might be 
 repeated indefinitely and would constitute an engine without an extraneous 
 refrigerator, violating Carnot's principle by deriving an unlimited supply of 
 mechanical work for thermal sources at a uniform temperature. Thus 
 independently of any knowledge of the intensity of the mechanical pressure 
 of radiation, or indeed of whether such a pressure exists at all, it is established 
 that the shrinkage of the enclosure must directly transform the contained 
 radiation to the constitution which corresponds to some definite new tempera- 
 ture. 
 
 685. Let us next consider the enclosure filled with radiation of energy 
 density E at volume F, of any constitution but devoid of special direction 
 and let it be shrunk to volume F 8 F against its own pressure ; if the 
 density thereby becomes (E 8E) the conservation of energy requires that 
 
 EV - (E - 8E)(V - 8V), 
 
 is equal to the work done against the pressure, viz. p8V. To obtain the 
 pressure p we have to average the direct pressures for the different values of 
 the angle of incidence 6, there being no sideway pressure : the foreshortening 
 of the area pressed gives a factor cos 6, and resolving of the pressure normally 
 gives another cos 6, so that the resultant pressure on the interface is equal 
 to one-third of the total density of radiant energy in the enclosure 
 
 and thus EV + %E8V = (E - 8E) (V - 8V), 
 
 so that %E8V + V8E = 0, 
 
 or E~V~*. 
 
 686. Again, but now with the restriction to radiation with its energy 
 distributed as regards wave length so as to be of uniform temperature, the 
 performance of the mechanical work ^E8 V has changed the energy of radiation 
 EV from the state that is in equilibrium of absorption and emission with 
 a thermal source at temperature ^ to the state in equilibrium with an absorber 
 of some other temperature ^ SSr and that in a reversible manner, thus by 
 Carnot's principle 
 
 1 E8V _ _&* 
 3 = 
 
 so that S- varies as F~^, or inversely as the linear dimensions when the 
 enclosure is shrunk. 
 
604 General electrodynamic theory [CH. xiv 
 
 Combining these results it appears that E varies as ^ 4 ; this is Stefan's* 
 empirical law for the complete radiation corresponding to the temperature, 
 first established on these lines by Boltzmann|. Starting from the principle 
 that this radiation must be a function of the temperature alone, this adiabatic 
 process has in fact given us the form of the function. These results cannot 
 however be extended without modification to each separate constituent of 
 the complete radiation, because the shrinkage of the enclosure alters its 
 wave length and so transforms it into a different constituent. The effect 
 of compressing the complete radiation is thus to change it to the constitution 
 belonging to a certain higher temperature, by shortening all its wave lengths 
 by the proportion of one-third of the compression of volume, the temperature 
 being in fact raised by the same proportion at the same time increasing in 
 a uniform ratio the amounts corresponding to each interval 8 A, so as to get 
 the correct total amount of energy for the new temperature. In the com- 
 pression each constituent alters so that ^A remains constant and the energy 
 .E^SA in the range SA in other respects changes as a function of T alone. 
 Hence generally E^\ must be of the form 
 
 r 
 But for each temperature E^dX is equal to E and so varies as ^ 4 , by 
 
 J o 
 
 Stefan's law; that is 
 
 %-i-F () f / (*A) d (*A) ~ ^ 4 , 
 
 J o 
 
 so that *t-*F () ~ . 
 
 Thus finally j A 8A is of the form 
 
 or AX~ 5 <I> (*A) 8 A, 
 
 which is Wien's J general result known as the displacement law : it states that 
 the distribution of the energy among the various wave lengths are, at any 
 two temperatures homologous, in the sense that the intensity curves after 
 the wave lengths in one of them have been reduced in a ratio depending 
 definitely on the two temperatures differ only in the absolute scale of mag- 
 nitude of the ordinates. 
 
 * Wiener Bericht. 79 (1879), p. 39. This proof of Stefan's law is apparently all right, but 
 there are assumptions involved in it which ought not really to be allowed to stand without further 
 examination. An amended form of the proof is given by Larmor (Aether and Matter, p. 137), 
 but it can be shown to lead to the additional result that the energy density varies as the tem- 
 perature. A critical examination of the subject would require too much space and cannot be 
 attempted in the present work, but it may be stated that there is ample experimental evidence 
 of the truth of the law as stated. 
 
 f Ann. Phys. Chem. 22 (1884), p. 29. 
 
 J Berlin. Ber. (1893), p. 55. Ann. Phys. Chem. 52 (1884), p. 132. 
 
686-688] Wieris law 605 
 
 687. It is of interest to follow out this adiabatic process for each con- 
 stituent of the radiation as a verification and also to ascertain whether anything 
 new is thereby gained. To this end let E (X, ^) 8A represent the intensity 
 of the radiation between A and A + SA which corresponds to the temperature 
 S-. The pressure of this radiation when it is without special direction, is in 
 intensity one-third of this ; thus the application of Carnot's principle shows, 
 as before, that in adiabatic compression ^ ~ V ~ * so that a small linear 
 shrinkage in the ratio 1 # raises T in the ratio 1 + x. We have still to 
 express the equation of energy. The vibratory energy E (A, ^) SAF in 
 volume F, together with the mechanical work \E (A, S-) SA . 3xF yields the 
 vibratory energy 
 
 E (A (1 - x), ^ (1 + x)) SA (1 - x) V (1 - 3z), 
 thus writing E for E (A, ^) we have, neglecting x 2 
 
 r)f clF 
 
 E(l + x) = E-xX~ + x*~(l- 4z), 
 so that 5E + A ^- - ^ 5- == 0, 
 
 U\ O^f 
 
 a partial differential equation of which the integral is 
 
 E = AX~ 5 <j> (&A), 
 the same formula as was before obtained. 
 
 This method treating each constituent of the radiation separately, has 
 in one respect some advantage, in that it is necessary only to postulate an 
 enclosure which totally reflects that constituent, this being a more restricted 
 hypothesis than an absolutely complete reflector. 
 
 688. To determine theoretically the form of the function^* we must have 
 some means of transforming one type of radiation into another, different 
 in essence from the adiabatic compression already utilised. The condition 
 that the entropy of the independent radiations in an enclosure is a maximum 
 when they are all transformed to the same temperature with total energy 
 unaltered is already implicitly fulfilled; it would thus appear that any 
 further advance must involve the dynamics of the radiation and absorption 
 of material bodies. 
 
 We cannot here discuss all the attempts* that have been made to obtain 
 an insight into the form of this fundamental function ; but it is of interest 
 to follow the reasoning employed by Lorentz f to deduce it in a particular 
 
 * Of. Wien's article, ' Theorie der Strahlung ' in Encyklop. der math. Wiss. Bd. v. 23, where 
 a short account of the subject is given with full references to the more important original work 
 bearing on the subject. 
 
 f Amsterdam Proc. (1903). Cf. also Theory of Electrons, ch. n. The theory has been subse- 
 quently generalised along the lines suggested for the theory of metallic conduction in ch. vn. 
 
606 General electrodynamic theory [OH. xiv 
 
 case, as it depends merely on an application of principles already expounded 
 in previous chapters. 
 
 In the electron theory of metals as expounded in various places above 
 we deal with substances in which the circumstances of the emission and 
 absorption of radiation are completely known, and it would appear then 
 that if a calculation of these quantities can be made in a particular case 
 some knowledge of the density of the radiation which would be in equilibrium 
 with the metal at a given temperature could be obtained. 
 
 689. We consider, with Lorentz, a thin metallic plate in which a large 
 number of free electrons a*e moving about in a perfectly irregular manner, 
 consistent with the general laws of the conservation of their total energy and 
 momentum. We know that an electron can be the centre of an emission of 
 energy when its velocity is changing, consequently, as a result principally of 
 the numerous collisions of the electrons with the atoms, which produce 
 alterations of the directions and magnitudes of the velocities of the electrons, 
 a part of the heat energy of the irregular motion of the electrons will be 
 radiated away from the metal. This radiant energy, which is subsequently 
 to be the subject of a detailed examination is, however, presumed to be so 
 small compared with the energy of motion of the electrons that it can be 
 neglected in any dynamical considerations respecting those motions extended 
 over a finite time. To this extent the analysis offered is only a first ap- 
 proximation to the actual state of affairs. 
 
 690. Now let/ and/' be two infinitely small parallel surface elements,/ 
 being on the plate itself and /' at a distance r outside it on the normal to the 
 plate through the centre of /. Then of the whole radiation emitted by the 
 metal plate, a certain portion will travel outwards through / and /'. Suppose 
 we decompose this radiation into rays of different wave lengths and each ray 
 again into its plane polarised constituents in two planes at right angles through 
 the chosen normal to the plate (these two planes and the plane of the plate 
 being parallel to a system of properly chosen rectangular coordinate planes 
 in which z = is the plane of the plate). Now consider in particular those 
 of the rays in this beam whose wave length lies between the two infinitely 
 near limits A and A + d\ and which are polarised in the plane y = ; the 
 amount of energy emitted by the plate per unit time through both elements 
 / and /' so far as it belongs to these rays, must be directly proportional to 
 /, /' and dX and inversely proportional to r 2 , so that it can be represented by 
 
 an expression of the form 
 
 The coefficient E is called the emissivity of the plate and is a function not 
 only of the positions of/,/' and A but also of the conditions and type of the 
 metal composing the plate. 
 
688-691] Lorentz's theory 607 
 
 We have already seen how, as a result of the collisions between the 
 electrons and atoms, part at least of any regular or organised energy acquired 
 by the electrons during their free motion between the atoms can be dissipated 
 into heat energy of the irregular motion of the same electrons. In this way 
 it is possible for a metal to absorb a portion of the. energy from an incident 
 beam of radiation, because the electric force in the electromagnetic field 
 associated with the radiation will pull the electrons about during their other- 
 wise free motion between collisions, imparting kinetic energy to them, which 
 is then dissipated by collision at the end of each path into irregular heat 
 motion. Suppose then that a plane polarised beam such as that specified 
 above is incident through the small surface /', on the patch / of the plate : 
 then we know that a certain portion of the energy of this beam will be 
 absorbed in the metal and converted into heat energy, instead of being 
 reemitted as a portion of the reflected or transmitted beams. The fraction 
 expressing the proportion of the energy absorbed is called the coefficient of 
 absorption of the plate under the conditions specified and may be denoted 
 by A. 
 
 691. If however the metal plate is in exchange of steady thermal 
 radiation either with itself or other bodies at the same uniform temperature 
 in a perfectly reflecting enclosure the amount of radiant energy of a particular 
 type absorbed by it must be equal to the amount of the same type which is 
 emitted by it : this means that the. ratio 
 
 E 
 
 A 
 
 determines the density of the energy in the particular constituent of the 
 radiation under consideration and this from the principles discussed above 
 must be dependent only on the temperature and wave-length of this con- 
 stituent. 
 
 But in the present case both E and A are directly calculated by known 
 principles. If we consider that the thickness A of the metallic plate is so 
 small that the absorption may be considered as proportional to it, we shall 
 find by an obvious calculation 
 
 a 
 
 A = - A*, 
 
 c 
 
 c being the usual radiation constant and a the conductivity of the metal. 
 Now in all applications involving steady or only slowly varying currents the 
 conductivity a is given by 
 
 Ne*L, 
 
 where N denotes the number of free electrons per unit volume in the metal, 
 each of mass m and charge e, moving with velocities the average square of 
 
 * See Ex. 334 in the Appendix. 
 
 
 
608 General electrodynamic theory [CH. xiv 
 
 which is u m z ; l^ is the length of the mean free path; but in applications 
 involving more rapid alternations in the current it is necessary to use the 
 more complete formula obtained in the previous chapter, viz. 
 
 / 
 
 V 
 
 mu 
 
 ze~ z dz 
 
 The coefficient of absorption of the metal plate is thus completely determined. 
 
 692. Now let us consider the radiation from the plate. We need only 
 consider the radiation normally from the small volume /A of the plate, as 
 this is the only part of all the radiation through / from the whole plate that 
 gets to /'. Now according to a formula established above, a single electron 
 moving with a velocity v in the part of the plate under consideration will 
 produce at the position of /' an electromagnetic field in which the x-com- 
 ponent of the electric force is given by 
 
 e dv x 
 
 c 2 r dt ' 
 
 if we take the value of the differential quotient at the proper instant. But 
 on account of the assumption as to the thickness of the plate, this instant 
 
 may be represented for all the electrons in the portion /A by (t - - j , if t is 
 
 V c/ 
 
 the time for which we wish to determine the state of things at the distant 
 surface/'. We may therefore with the same notation as previously employed 
 write for the x-component of the electric force in the total field at /' 
 
 and then the flow of energy through /' per unit of time will be 
 
 47T 
 as far as this one component is concerned. 
 
 Since however the motion of the electrons between the metallic atoms is 
 highly irregular and of such a nature .that it is impossible to follow it in 
 detail, we must rather content ourselves with mean values of the variable 
 quantities calculated for a sufficiently long interval of time. We shall 
 therefore always consider only the mean values of our quantities taken over 
 the large time between the instants t = and t = ^. For example, the 
 flow of energy. through /' is, on the average, equal to 
 
 4:77 ^ J 477- 
 
 say. 
 
691-694] Lorentz's theory 609 
 
 693. Now whatever be the way in which E,,. changes from one instant 
 to the next we can always expand it in a series by the formula 
 
 sirt 
 
 x = a s sin 
 =i ^* 
 
 where s is a positive integer and 
 
 2 P . STTt- 
 
 a s = - sin ^- E x dt. 
 ;* ./ 5 
 
 The frequency of the sth term of this series is ^ , so that the wave length of 
 
 *J 
 
 the vibration represented on it is 
 
 v 2cS- 
 A = -- . 
 S 
 
 If ^ is very large the small part of the spectrum corresponding to the small 
 interval of length d\ between the wave lengths A and A + dX will contain 
 
 a large number -^- dX of spectral lines represented by terms of this series. 
 
 If now we substitute the Fourier series for E,,. into the expression for the 
 mean energy of flux through /', we shall find in the usual manner that it is 
 equal to 
 
 n TT 8=1 
 
 the product terms when averaged up giving each separately zero. To obtain 
 the portion of this flux corresponding to wave-lengths between A and A + dX 
 
 we have only to observe that the -s^-dA spectral lines lying within that 
 
 interval, may be considered to have equal intensities. In other words the 
 value of a s may be regarded as equal for each of them, so that they contribute 
 to the sum S in the last equation an amount 
 
 Consequently the energy flux through /' belonging to the interval of 
 wave length dX is given by 
 
 477A 2 
 
 and we now want to find a a . 
 
 694. From the value of E x given above we find that 
 
 2 
 
 where the square bracket round v x serves to indicate the value of this quantity 
 at the time t r/c . 
 
 L. 39 
 
610 General electrodynamic theory [CH. xiv 
 
 The sign S now refers to a sum taken over all the electrons in the part 
 /A of the plate. 
 
 On integration by parts we find 
 
 ^TTSe ^ [ * r -, STft 
 
 or what is the same thing 
 
 Now the integral on the right is made up of two parts, arising respectively 
 from the intervals between the consecutive impacts of the electrons and from 
 the intervals during these impacts. If we can suppose, as we shall do, that 
 the duration of an encounter of an electron with an atom is much smaller 
 than the time between two successive encounters of the same electron, we 
 may neglect altogether the part that corresponds to the collisions and confine 
 ourselves entirely to the part corresponding to the free paths between the 
 collisions. But while an electron travels over one of these free paths, its 
 velocity v x is constant. Thus the part of the integral in a s which corresponds 
 to one electron and to the time during which it traverses one of its free paths 
 is therefore 
 
 t+T 
 
 cos 
 
 STT 
 
 (* + 1} At, 
 
 where t is now the instant at which this free path is commenced and r the 
 duration of the journey along it; but this is equal to 
 
 2^-V,,, . STTT STT ( r r 
 ' + - 
 
 STT 2^r ^ \ c 2 
 
 We can now fix our attention on all the paths described by all the electrons 
 under consideration during the time ^, and we use the symbol S to denote 
 a sum relating to all these paths. We have then 
 
 2-n-se _ 2^V,,. . STTT STT ( r T\ 
 
 a s = cv2 2 S ~~ sln ^T cos ^ ( ^ H o ) 
 
 .j c T STT Zj rJ \ c Z/ 
 
 695. We now want to determine the square of the sum S. This may be 
 done rather easily because the product of two terms of the sum whether they 
 correspond to different free paths of one and the same electron, or to two 
 paths described by different electrons, will give if all taken together. 
 Indeed the velocities of two electrons are wholly independent of one another, 
 and the same may be said of the velocities of one definite electron at two 
 instants separated by at least one encounter. Therefore positive and negative 
 values of Va- being distributed quite indiscriminately between the terms of 
 the series S, positive and negative signs will be equally probable for the 
 
694-696] 
 
 Lorentz's theory 
 
 611 
 
 product of two terms. We have therefore only to calculate the sum of the 
 squares of the terms in S or simply 
 
 S 
 
 sin 1 ' 
 
 cos 
 
 2 S7T 
 
 Now since the irregular motion of the electrons takes place with the same 
 intensity in all directions, we may replace v x 2 by |v 2 . Also in the immense 
 number of terms included in the sum the quantities T and v are very different, 
 and in order to effect the summation we may begin by considering only those 
 
 terms for which the product ( v sin-^-j has a certain value. In these terms 
 
 \ 3 / 
 
 s,* T / V T\ 
 
 which are still very numerous, the angle ( t -\ \- - ] has values that are 
 
 j \ c 2/ 
 
 distributed at random over an interval ranging from to STT. The square 
 of the cosine may therefore be replaced by its mean value \, so that 
 
 or introducing after Lorentz, the length of the path I instead of the time 
 T in it, this may be written 
 
 2 
 
 O /I V \ 
 
 sin 
 
 sf 
 
 696. The metallic atoms being considered as practically immovable, the 
 velocity of an electron will not be altered by a collision. Let us, therefore,- 
 now fix our attention on a certain group of electrons moving along their 
 zig-zag lines with the definite velocity u. During the time ^, one of these 
 particles describes a large number of free paths, this number being given by 
 
 if l m is the mean length of the paths; the number of these paths whose 
 length lies between I and I + dl, is 
 
 w <x _ I 
 
 U/rJ j 77 
 
 e l m dl, 
 
 so that 
 
 (*irl m I 
 in . 
 
 sm 
 
 dl 
 
 is the part of the sum contributed by these paths. On integration of this 
 
 392 
 
612 General electrodynamic theory [CH. xiv 
 
 expression from I = to I = oo we find the part of the sum due to one of the 
 typical electrons, which is therefore 
 
 , 
 
 "*" 
 
 Now the total number of electrons in the part of the metallic plate under 
 consideration is Nfb, and by Maxwell's law, among these 
 
 have velocities between u and u + du; the constant q is related to the mean 
 velocity u m already mentioned above by the formula 
 
 3 
 
 y ~~ 9<>/ 2* 
 
 ^ W TO 
 
 Thus the total value of the sum S in a s z is given by 
 
 Jo 
 
 27TC STT , 
 or, using z = <?w 2 and -- = , oy 
 
 1 + 
 'o 
 
 Thus we have 
 
 32 /2 
 
 a = TV 3^ 
 
 , 
 o 
 
 or introducing the expression for the conductivity 
 
 and the expression for the partial energy flux through the element/' thus 
 
 and thus the einissivity of the plate under the conditions specified is 
 
 3A 4 
 
 E 
 
 so that 
 
696, 697] Lorentz's theory 613 
 
 697. Now as we have already mentioned the mean kinetic energy of an 
 electron for which we may write \ mu m * is presumed to be equal to the mean 
 kinetic energy of a gaseous molecule at the same temperature; and the 
 latter is proportional to the absolute temperature 6 so that we may put 
 
 |mw w 2 = ad, 
 
 E STrad 
 
 and then -7 = -~^y 4 , 
 
 A. oA 
 
 so that KirchhofFs principle which requires this ratio to be absolutely 
 independent of the nature of the metal is exactly verified : its form as a 
 function of A and 6 is also consistent with the general principles laid down 
 above and appears in fact to be perfectly consistent with the experimental 
 facts. 
 
 It must however be remembered that the deduction here given restricts 
 the present law for application only to long-wave radiation as it is only in 
 that case that the condition implied in the duration of a collision is satis- 
 factorily verified. The rigorous deduction of a law which shall apply to all 
 wave lengths has not yet been accomplished although one or two successful 
 formulae have been arrived at by distinctly artificial means : a proper 
 discussion of these would however take us too far beyond the purpose which 
 we have at present in view. 
 
CHAPTER XV 
 
 THE ELECTRODYNAMICS OF MOVING MEDIA 
 
 698. The general equations of electrodynamic theory. We have up to 
 the present confined our considerations mainly to the electromagnetic and 
 electrodynamic phenomena of systems in which the ponderable matter is either 
 actually at rest or is at least in a state of such slow motion that it may at 
 any instant be regarded as at rest relative to the instantaneous electromagnetic 
 field. We must now discuss certain aspects of the more general case of 
 rapidly moving electromagnetic systems : such a discussion appears to be 
 necessary not only because of its intrinsic theoretical interest but because 
 all electrodynamic phenomena are concerned with the more or less rapid 
 motion of electrically charged bodies. Absolute rest is of course unknown 
 by the human intelligence, and for instance all electrostatic fields created on 
 this earth necessarily partake of the motion of the earth through space, so 
 that they are of a type more general than that discussed in the earlier chapters 
 of this book. 
 
 We shall begin by formulating the general equations of electrodynamic 
 theory ; these have already been set out in full on a previous occasion, but 
 with a view to emphasising the point we may here briefly indicate their 
 deduction on a purely dynamical basis. 
 
 699. It has been shown in the previous chapter that on the tentative 
 assumption of appropriate forms for the potential and kinetic energies of an 
 electrical system presumed to comprise merely a group of electrons or 
 electrically charged particles in motion in the aether, the complete circum- 
 stances of the configuration and motion of the system can be described by 
 means of the ordinary equations of dynamical theory : in such a mode of 
 formulation of the theory the only effect of the interaction between the 
 electromagnetic condition of the aether and the charge on the moving 
 electron is completely specified as a force of ordinary mechanical type on 
 the typical electron of vector amount 
 
 where e is the charge on the electron and v its velocity ; E is the aethereal 
 
698-700] The general equations 615 
 
 electric force, defined in terms of the vector and scalar potentials by the 
 relation 
 
 IdA 
 
 B is the magnetic induction vector and 
 
 B - curl A. 
 On such a theory the total effective current is 
 
 1 dE dP 
 
 1 dE 
 
 where C x is the true current of conduction; - the fictitious current of 
 
 4?r at 
 
 aethereal displacement ; -=-- the true current of material polarisation ; c curl Ij 
 
 at 
 
 the current which in its magnetic aspects is the effective equivalent of the 
 distribution of magnetic polarisation, including both the true magnetisation 
 I and the quasi-magnetisation due to the convection of the polarised medium 
 with velocity v 
 
 finally vp is the current due to the convection of the material medium charged 
 to density p at any point. 
 
 700. All of these relations can be regarded either in the light of 
 definitions or as relations of a purely dynamical nature. It follows from 
 them that 
 
 curl F = curl E + - curl [vBl 
 c 
 
 IdB 1 
 - c -S + c curl 
 
 Now on reference to the general theorem established in the introduction, 
 19, we see that the right-hand side of this equation when multiplied 
 by c and integrated as regards its normal component over any surface, 
 expresses the time rate of change of the magnetic induction through the 
 surface regarded as moving at each point with the charge system with velocity 
 v. Our equation is thus the analytical expression of Faraday's circuital 
 relation which states that the line integral of the electromotive force F round 
 any circuit which is carried along with the matter is equal to the time rate 
 of diminution of the magnetic induction through it multiplied by 1/c. 
 
616 The electrodynamics of moving media [CH. xv 
 
 701. We have also, of course, 
 
 or if it is preferred not to include the magnetism as molecular current whirls 
 so that the total current is only 
 
 C - 477-cl!, 
 this relation becomes 
 
 curl H = curl (B - 477 IJ 
 
 - (total current), 
 
 C 
 
 where the magnetic force vector H is defined by the relation 
 
 H= B-47TI!. 
 
 This is the expression of Ampere's circuital relation that .the line integral 
 of the magnetic force round any circuit, fixed or moving*, is at each instant 
 equal to the flow of the Maxwellian total current through it multiplied by 
 the factor 4rr/c. It is important to notice that as the magnetic force 
 is here introduced into the theory it is a subsidiary quantity defined in 
 terms of B and the magnetisation. 
 
 702. When the material medium, however heterogeneous, is at rest 
 in the aether, these electrodynamic equations reduce precisely to Maxwell's 
 original scheme 
 
 in ld * 
 curlE- -- -=-, 
 
 c at 
 
 ITI 
 curl H = 
 
 _ 1 dD 
 
 C=Cl + 4^' 
 C x being the true conduction current. 
 
 When the material medium is in motion these equations are modified in 
 the following respects; there is a term arising from convection of electric 
 polarisation added to the magnetism, which changes I to Ij and there is 
 the current arising from the convection of electric charge which supplies the 
 term pv, a term which Maxwell in some connections temporarily overlooked 
 but which has been fully restored by Fitzgerald and others. 
 
 703. The existence of a magnetic field due to the convection of 
 electrically charged bodies and of polarised dielectrics has been experimentally 
 verified by Roentgen f and Rowland J; doubts were subsequently thrown 
 on the interpretation of their results by Cremieu but the experiments have 
 
 * Time differentials are not involved. 
 
 f Berlin. Ber. (1885), p. 198. $ American Jmir. of Sc. (3) 15 (1878), p. 30. 
 
 Paris C. B. 130 (1900), p. 1544, 131 (1900), pp 575, 797. 
 
701-703] 
 
 Eichenwald' s experiment 
 
 617 
 
 been repeated with much greater precision by Eichenwald* , with results 
 which completely substantiate the verification of the theoretical predictions. 
 
 The arrangement finally adopted by Eichenwald consisted mainly of a 
 parallel circular plate condenser with a uniform dielectric slab. The rapid 
 motion was produced by rotating the whole condenser round an axis of 
 
 Fig. 100 
 
 symmetry perpendicular to the plates. If the charge on the plates of the 
 condenser is of density a at any point there will be a convection current due 
 to its being dragged on with the system which will be of amount 
 
 (TV, 
 
 v denoting the velocity of the system at that point. 
 
 In addition the dielectric medium will be polarised to intensity P and the 
 convection of this will also be equivalent to a current of intensity curl [vP]. 
 If we neglect the irregularity of the edges this field between the plates will 
 be uniform right across and thus P will be constant throughout the interior 
 of the slab and there will be no volume distribution of current of this latter 
 type ; but it exists as a surface distribution on the abrupt interfaces of the 
 dielectric where the density is 
 
 [% [Pv]], 
 
 if n x is the unit normal vector whose direction is in the positive direction of 
 P, i.e. straight across between the plates in the present instance. This 
 current thus appears as of magnitude P . v and is directed parallel to the 
 direction of v at each place, but in the opposite sense. 
 
 * Ann. d. Phys. 11 (1904), p. 421. 
 
618 The electrodynamics of moving media [CH. xv 
 
 The total effective current in this arrangement is a surface current on 
 the plates of the condenser and of density 
 
 But if D is the total electric displacement across the dielectric 
 
 D E p 
 
 ~ 47T ~ 477 + 
 
 and thus the surface current density is simply 
 
 jEv| 
 
 47T 
 
 in the direction of v, i.e. directed in circles rfcund the axes of rotation. 
 
 The important point to notice is that this current and therefore also the 
 magnetic field associated with it does not in any way depend on the dielectric 
 material, but only on the potential difference between the plates : and this 
 was exactly verified by Eichenwald. 
 
 704. The importance of this experiment is the confirmation which it 
 provides for the fundamental hypothesis on which the present theory is 
 .based. The modern theory of electromagnetism is built on the idea of an 
 aether permeated by a large number of electrons or electric point charges, 
 either free or grouped together in material atoms, and it is with these charges 
 and their general- configuration and motion that we are alone concerned. 
 The motion of a material medium is thus effectively accounted for in the 
 motion of its constituent electrons. But what about the aether? Can this 
 medium move also; and is it dragged along with the matter which is in 
 motion through it? We have in our discussions tacitly neglected the 
 possibility of any such motion of the fundamental medium and this course 
 appears not only the simplest one but it is found to be more consistent with 
 experimental facts. This is the original view of Fresnel, Lorentz and Larmor ; 
 but the opposite view has been strongly advocated by Stokes in optical theory 
 and Hertz in electrical theory. According to their views the -motion of a 
 piece of matter through the aether necessarily produces by a sort of me- 
 chanical dragging action a convective motion of the aether itself in the 
 neighbourhood of the piece of moving matter. That such a view is in- 
 consistent with the result of Eichenwald's experiment is however easily seen, 
 for according to it no distinction need be made between the separate parts of 
 the total displacement current, and the whole effect summed up in the term 
 D is presumed to be convected with the matter : the Roentgen current 
 would in such a theory have, as is easily seen, a density 
 
 - curl [Dv], 
 
 477 
 
 and this adopted into the theory of Eichenwald's experiment would lead to 
 
703-705] A rotating conductor in a magnetic field 619 
 
 the result that there should be no resultant current at all, the current due 
 to the convection of the polarised dielectric and aether just balancing that 
 due to the convection of the charges on the plates. 
 
 The experiment carried out by H. A. Wilson and described above*, p. 572, 
 also in some respects affords another result in favour of the theory of a 
 stationary aether. It was there shown that the effects of the rotation of 
 a dielectric substance in a magnetic field can be fully and accurately ex- 
 plained on the assumption that it is merely the electrons in the dielectric 
 atoms that are convected with that substance, the aether itself remaining 
 absolutely at rest. 
 
 Thus it cannot but be admitted that the course adopted in the above 
 exposition of the theory is at least perfectly consistent with our experience. 
 We shall refer to this point later and mention further and perhaps more 
 exact evidence in its favour, and also some difficulties in the way of its 
 acceptance. We may perhaps here mention a direct attempt made by Lodge 
 to detect an aethereal drag accompanying the mass of a very large rapidly 
 rotating flywheel, but with negative results. 
 
 705. On the rotation of a conductor in a magnetic field f. As a first ex- 
 ample of the general principles formulated in the previous paragraph we may 
 consider the practically important problem of a conductor in motion in a 
 steady magnetic field. If we consider any closed circuit in the conductor, it 
 is clear that the electromotive force round it depends only on the change 
 produced by its motion in the number of tubes of force that it encloses, and is 
 therefore quite independent of whether the relative motion of the conductor 
 and the field be ascribed to the conductor or to the magnetic field, or to both 
 conjointly. Therefore the currents induced in the body are derived from the 
 same equations whether the axes are fixed or moving in any manner, uniform 
 or not. But in the case of an unclosed circuit there is a difference introduced 
 in the value of the electrostatic potential. In fact such an open line which 
 is at rest relatively to the moving axes is displaced across the field owing to 
 the motion of the body : if we suppose the ends of the line in a former 
 position (1) and in a near position (2) at a very short distance from it, to 
 be connected so as to form a closed circuit, the number of tubes of force on 
 the positive side of the line will be diminished by the number which pass 
 through this closed circuit supposed circulating in an anti-clockwise direction. 
 The diminution is therefore equal to the flux, of the vector potential 
 along (2) minus its flux along (1), together with its fluxes along the two 
 lines of motion of the ends of the open circuit. Thus when the line has 
 moved from (1) fco (2) we must suppose the potential at each end diminished 
 
 * Phil. Trans. A, 204 (1904), p. 121 
 
 f Cf. Larmor, 'Electromagnetic Induction in Conducting Sheets and Solid Bodies,' Phih 
 Mag. (1884), Maxwell, Treatise n. p 275. 
 
620 The electrodynamics of moving media [OH. xv 
 
 by the flux of the vector potential along the line of motion of that end 
 divided in each case by the usual constant c. Therefore in the equation for 
 the electromotive force we must include terms for the change of the rate of 
 variation of this flux 'as we pass from point to point of the conductor; that 
 
 Fig. 101 . 
 
 is instead of the true electrostatic potential <f>, we shall get from our 
 equations </> +</>', where (f> r is the scalar product of vector potential and 
 velocity of the point supposed connected with the moving system of axes 
 and is therefore 
 
 This method of statement brings out clearly what it is on which the term <' 
 really depends. 
 
 706. The same result here deduced from first principles also follows 
 immediately from the analytical relations between the functions involved. 
 The electromotive force in the field referred to a system of axes at rest is 
 given quite generally by the expression 
 
 1 9A 1 r -m 
 
 P = - - V< + - [uB] 
 
 c dt c 
 
 where u is the absolute velocity of the moving charge. 
 Now if v is any other velocity 
 
 [vB] - [v . curl A] - V (vA) - (vV) A 
 so that 
 
 Thus we see that if the system is referred to a set of axes moving with a 
 velocity v the expression for the electromotive force is of exactly the same 
 form as that given above except that the scalar potential cf> is altered to 
 
705-708] The charge induced by the rotation 621 
 
 as there explained. This alteration does not of course affect the aggregate 
 force in a closed circuit. 
 
 We conclude then that when a constant electromagnetic system is moving 
 through the aether the effect produced by the relative motion is an electro- 
 static charge of the system of such character that its potential is cf>'. This 
 static charge, however, itself exerts a magnetic effect by virtue of its motion ; 
 
 (y\ 2 
 - j and is therefore very minute 
 
 in a real case. 
 
 707. In the case of a conductor rotating steadily in a magnetic field 
 a steady distribution of currents in space will ensue when the conductor is 
 symmetrical about the axis of rotation ; and the electromotive force along 
 any line will be given by the number of tubes of force of this steady field 
 that are cut through by the line per second. Now when the magnetic field is 
 symmetrical round the axis of rotation the number of tubes enclosed in any 
 closed moving circuit in the conductor will not alter at all, so that there will 
 be no current round any circuit, and therefore no induced currents whatever : 
 the electric force along each open line will accumulate a statical electric 
 charge at one end of it, so that the conductor will become electrified until 
 the induced electromotive force is exactly neutralised by the statical difference 
 of potential. This conclusion holds whatever be the shape of the body. 
 
 The state being steady the electromotive force F given by 
 
 1 dA. , . 1 r -n-i 
 
 - [vB], 
 
 . dA. . 
 
 where of course is now zero. 
 dt 
 
 Thus, since F is necessarily derived from a potential <X>, 
 
 so that <t> - O = - |([v . B] . da). 
 
 cj 
 
 708. If the rotation is with steady angular velocity a> about a fixed 
 line which may be chosen as the axis of z in a rectangular frame of reference 
 v x = coy, v y = + a>x, 
 
 and thus <j> - <D = - (E x x + B tf y) dz - E z (xdx + ydy), 
 
 c J 
 
 and from this the electrification of the conductor may be determined. For 
 example let us take a uniform field of intensity H parallel to the axis of 
 
 rotation 
 
 B, = B v = 0, B 2 = H, 
 
622 The electrodynamics of moving media [OH. xv 
 
 we have then 
 
 9 2 c r " 
 
 so that V 2 (< - O) = - . 
 
 Inside the conductor the electromotive potential O is constant because 
 electromotive force would induce a compensating charge : thus at internal 
 points 
 
 thus the electrification there is that belonging to the static potential </> and 
 
 TT 
 
 involves a volume density - - as well as a surface density. 
 
 ZiTTC 
 
 In outside space the circuitality of the aethereal displacement, the force 
 producing which is naw identical with the electromotive force, requires that 
 
 V 2 c = 0, 
 
 and (f> must be itself continuous across the surface because there cannot be 
 discontinuity in the aethereal strain produced in the manner specified ; but 
 it is O or 
 
 </> + \- c H (x* + y*} - C; 
 
 and not $ itself that is constant over the surface of the conductor. 
 
 If the conductor is a sphere the outside potential which corresponds to 
 the given value at the surface of the internal potential is 
 
 where v is the radial line. Thus the surface density determined by the 
 difference of the normal gradients of the internal and external electric 
 potentials is 
 
 coHa/4 5 (x z + 
 
 ~ 
 
 The arbitrary constant C allows us to superpose any free distribution. 
 If the charge on the body is normally zero we may give it such a value that 
 the charge shall remain zero ; but if the axis of rotation is uninsulated the 
 condition is that (7 = 0. 
 
 709. The case when the axis of rotation is perpendicular to the direction 
 of the field has also a certain amount of practical interest. We consider only 
 the one simple case of a linear conductor consisting of a rigid plane circuit 
 enclosing an area F and rotating with angular velocity co in a uniform magnetic 
 field of intensity H. If the axis of rotation is the z-axis of the rectangular 
 
708-710] Uni-polar induction 623 
 
 coordinate system and the lines of force in the field are parallel to the y-axis 
 then 
 
 B-= B = B = H 
 x ** ^> **i/ ** 
 
 and thus 
 
 to f__ , coH f 7 
 - O = - Hyefe = ydz 
 cj c j y 
 
 i ^H r , 
 
 <P = <p - - ydz. 
 
 c J y 
 
 In the whole circuit there is therefore an additional potential available for 
 driving a current of amount 
 
 ~ cos 6 
 
 c 
 
 where 6 is the angle between the normal to the plane of the current and the 
 direction of the field, 
 
 = Itadt. 
 
 If the motion is with uniform angular velocity 
 
 6 = cot + a ' 
 so that the electromotive force in the circuit induced by its motion is 
 
 Hw.Fcos (cut + a) 
 
 c 
 and is simply periodic. 
 
 710. For the case of a flat disc rotating about an axis perpendicular to 
 its plane in any uniform field, we may divide the force into two components, 
 one parallel to the plane of the disc "which produces no induction, on account 
 of the thinness of the sheet, and the other perpendicular to it, whose effect 
 has just been estimated. By connecting one terminal of a wire to the axis 
 and making the other terminal rub along the circumference in Faraday's 
 manner, we utilise the difference of potential to produce a current in the 
 external circuit*. 
 
 The well-known phenomenon of uni-polar induction, in which a current 
 is induced when a magnet revolves round its axes of symmetry through its 
 own field, is also explained in a similar manner. We may infer that for a 
 solid magnet of any form, in motion of any type, the induced electromotive 
 
 force is derived from a potential - - (Av) where v is the velocity through 
 
 G 
 
 the aether of the elements of the magnet considered ; so that it can at each 
 instant be compensated by the static force due to a minute induced electrifica- 
 tion, or it may be used to drive a current in an external circuit. 
 
 * Cf. Exp. Ees. i. 81 (1831). 
 
624 The electrodynamics of moving media [CH. xv 
 
 The conclusions thus drawn from the theory of the previous paragraph, 
 which have been fully verified by experiments made to substantiate 
 them, have an important bearing on the general theory, as they essen- 
 tially involve the consideration of electrification as made up of discrete 
 elements surrounded and influenced only by the aether which is the real 
 distinct seat of the electromagnetic field : it is only in such a case that the 
 motion of a conductor independently of the aether in reality involves the 
 transference of electric charges through the electromagnetic field in that 
 aether. 
 
 711. Apart however from these theoretical considerations the problem of 
 the rotation of an uncharged conductor in a magnetic field is of practical 
 importance in applications in technology where rotating conducting masses 
 frequently occur*. In these applications however it is the magnetic effects 
 with which we are concerned, so that cases in which the circumstances are 
 as simple as those just discussed are of little importance. It is with the 
 more general case when the steady circumstances involve a distribution of 
 currents in the conductor that we have to deal in actual practice. Confining 
 ourselves entirely to motion in steady fields we see as above that the 
 electromotive force round any circuit depends only on the change produced 
 by its motion in the number of tubes of force that it encloses so that the 
 effects are independent of whether the relative motion of the conductor and 
 the field be ascribed to the conductor or to the field. We can therefore 
 simplify the equations which give the electric currents when the conductor 
 is in motion as we can reduce it to rest and solve the corresponding relative 
 problem, where the motion across the lines of force is replaced by a variation 
 of the field itself. The general considerations of the analogous problems 
 solved in chapter X above will then apply. 
 
 712. Let us first consider the phenomenon associated with Arago's disc*. 
 Arago discovered that a magnet placed near a rotating metallic disc ex- 
 periences a force tending to make it follow the motion of the disc, although 
 when the disc is at rest there is no action between it and the magnet. This 
 action is due to the currents induced in the elements of the plate by their 
 motion across the lines of force in the magnetic field. The distribution of 
 these currents being independent of whether the relative motion is due to 
 the motion of the conductor or magnet, will be the same as that induced in 
 the disc at rest by the same magnetic system to which a rotation about the 
 same axis is imparted in the opposite direction : and this fact, taken in 
 conjunction with the results deduced in the previous problem, shows that 
 
 * Cases have been worked out by Larmor, I.e. p. 389; Jochmann, Crelles Journ. 63 (1863); 
 Hertz, Dissertation (Berlin, 1880); Ges. Werke, I. p. 37; Riecke, Gott. Abhdl. 21 (1876), p. 1; 
 Gans, Zeitschr. /. Math. u. Phys. 48 ( 1902), p. 1 ; cf. also S. Valentiner, Die electromagnet. Rotationen 
 und die Uni-polarinduction (Karlsruhe, 1904). 
 
 t Arago, Ann. de chim. et phys. 27 (1824), p. 363, 28 (1825), p. 325. 
 
710-713] Arago's disc 625 
 
 the magnetic action of the currents in a disc supposed very large so that 
 the irregularities at the edges may be neglected is equivalent to that of a 
 trail of images of the magnetic system in the form of a helix. 
 
 If the magnetic system consists of a single magnetic pole of strength 
 unity the helix will lie on the cylinder whose axis is that of the disc and which 
 passes through the magnetic pole. The image will begin at the position of 
 the optical image of the pole in the disc. The distance parallel to the axis 
 between consecutive coils of the helix will be l/aa>. The magnetic effect of 
 the trail will be the same as if this helix had been magnetised everywhere in 
 the direction of a tangent to the cylinder perpendicular to its axis, with an 
 intensity such that the magnetic effect of any small portion is numerically 
 equal to the length of its projection on the disc. The calculation of the 
 force on the magnetic pole would be complicated but it is easy to see 
 that it will consist of (1) a dragging force, parallel to the direction of motion 
 of the disc, (2) a repulsive force acting from the disc, (3) a force towards the 
 axes of the disc. All these were observed by Arago. 
 
 713. Let us next consider the case of a thin spherical shell rotating with 
 the angular velocity a> in a uniform field. The conditions being steady the 
 displacement currents in the free aether are non-existent so that the internal 
 and external magnetic fields are derivable from potentials 0j and 2 
 respectively, which in the most general case satisfy at the surface of the 
 shell the relation 
 
 2 
 
 dt\ dr dt\ dr 2 
 
 dr\ dr 
 
 * 
 
 If the sphere is held at rest and the uniform magnetic field rotated so as 
 to produce the same relative motion then the operator d/dt is equivalent to 
 the operator o (d/d(f)), (f> denoting the azimuth round the axis of rotation 
 through the centre of the sphere : the condition is therefore 
 
 / SV dr~2dr dr 
 The inducing field has in the neighbourhood of the sphere the potential 
 
 iff = Hr sin Qe^, 
 
 or at least the real part of this expression; and we therefore try solutions 
 for the internal and external potentials 
 
 0! = - Hr sin de^ + =^ sin Be 1 *, 
 
 2 = _ Hr sin de** + A 2 r sin 6e i4> , 
 which satisfy all the conditions if 
 
 iw (- H + A 2 ) = iaj (- H - 2AJ = ak (A - A 2 ), 
 
 * k = [ ~) where a is radius, k conductivity and 6 thickness of the shell. 
 \ ac 2 / 
 
 L. 40 
 
626 The electrodynamics of moving media [CH. xv 
 
 so that if we write a = -^ , 
 
 H 1 + ia ,. 
 
 we have A 2 = 1 r- = T -^H. 
 
 I la 1 + a 2 
 
 Taking real parts only, we find, if 
 
 /f = ifj Hr sin 6 cos </>, 
 then the external field is determined by its potential 
 
 ^2 = *Ao ~~ ^ r g i n ^ cos </* ~~ ^*" g i n ^ cos X cos (</> ~ x) 
 
 y \ 
 
 = j/r fl^r sin sin ^ cos (</> + j ~ Xi > 
 where tan x = . 
 
 (X 
 - x)j while its 
 
 intensity is reduced in the ratio sin^ : 1. 
 
 We have also 
 
 H 
 
 2(1 
 
 so that the shell has the same outside effect as a simple magnet of moment 
 
 Ha 3 
 
 , whose axis is inclined to the direction of the original field at an 
 
 2-V/l + a 
 angle tan" 1 a. 
 
 The opposing couple experienced by the rotating shell will therefore be 
 the same as for this magnet, i.e. it will be 
 
 F*a 3 a 
 "2(1+ a 2 )' 
 
 and the rate of expenditure of power required to keep up the rotation will 
 be Go). 
 
 The case of the solid sphere can be worked out on similar lines to that 
 adopted above and the results are analogous. 
 
 714. The steady linear translation of an electrostatic material system*. 
 
 The general equations of the first paragraph enable us to treat in detail the 
 electrodynamic relations of an electrical system in steady uniform motion 
 through the aether. In order that a steady electric state may be possible 
 without permanent currents of conduction, it is necessary that the configura- 
 tion of the matter shall be permanent and that its motion shall be the same 
 at all times relative to this configuration and to the aether, and also to the 
 
 * Cf. Larmor, Phil. Trans. A. 190 (1897) p. 226. The theory is due originally to J. J. Thom- 
 son, Phil. Mag. (5), 11 (1881), p. 229; Phil. Mag. (6), 28 (1889), p. 1 ; Phil. Mag. (5), 31 (1891), 
 p. 149; Recent Researches, p. 16. Cf. also Heaviside, Phil. Mag. (5), 27 (1889), p. 324; 
 G. F. C. Searle, Phil. Trans. 187 A. (1896), p. 675; Phil. Mag (5), 44 (1897), p. 329. 
 
713-715] The uniform translation 627 
 
 extraneous magnetic field, if there is one : this confines it to uniform spiral 
 motion on a definite axis fixed in the aether. We shall here confine our 
 attention to the case when the motion is one of uniform translation and in 
 which there is no extraneous field, electric or magnetic. Under these circum- 
 stances the magnetic induction through any circuit moving with the system 
 being constant, the electromotive force F is derived from a potential O 
 
 F = grad <, 
 
 because its line integral round such a circuit vanishes. Inside a conductor 
 the electromotive force F must vanish, otherwise electric separation would 
 be going on ; therefore $ must be a constant over and inside any conductor 
 in the system. 
 
 <I> is called after Searle the convection potential of the field of the moving 
 system*. 
 
 715. If we refer the field to axes fixed in and moving with the material 
 system and use v as the vector velocity of the system then the total current 
 density at any point in the system is 
 
 1 8E , 
 
 * = to& + l"' 
 
 p being the density of the charge distribution at the point; it is presumed 
 that the system consists entirely of conductors and free aether, no dielectrics 
 being present. But on account of the steadiness of the motion 
 
 so that the current density may be written in the form 
 
 C = - -^ (vV) E + pv, 
 kn 
 
 or since 4^rp = div E, 
 
 = _ J_{(vV)E- vdivE} 
 
 the velocity v being uniform throughout the system. We have therefore 
 from Ampere's relation 
 
 curl H = = - - curl [E . v], 
 c c 
 
 so that curl JH + - [E . vll = 0, 
 
 I c J j 
 
 * Schwarzschild calls it the electrokinetic potential. Cf. Qott. Nachr. (math. phys. Kl.) (1903), 
 p. 125. 
 
 402 
 
628 The electrodynamics of moving media [CH. xv 
 
 which implies that the vector 
 
 H - - [v . E] 
 c 
 
 is the gradient of a potential function : we write 
 
 H - - [vE] = - grad /, 
 
 C 
 
 i/r is an undetermined function which will be continuous as to itself and its 
 gradient except at the surfaces of transition. The most general value of 
 H consistent with the circuital relation is thus 
 
 H = - [vE] + grad 0, 
 
 
 
 the part of it depending on would include the extraneous magnetic field, 
 if there were one, and also the field due to magnets, if any, that belong to 
 the material system itself. 
 
 If there is no external applied magnetic field and the moving system itself 
 contains no magnetic matter the magnetic field of the moving charges will 
 be sufficiently defined by the magnetic vector potential A so that 
 
 the function being not now necessary, there being no external circumstances 
 to be allowed for. 
 
 716. Combining the relation between E and H (= B in free space)* with 
 the direct dynamical relation 
 
 F = E + - [vB] = E + - [v . H], 
 
 C C 
 
 we get F = E - 1 [v . V] + i [[E . v] . v] 
 
 wherein as above F = grad >, 
 
 and (vF) = (vE). 
 
 Again since the total current is always a stream we have 
 
 div E = 47T/>, 
 
 so that div (F - ^ (vF)) = (l - ^)divE, 
 
 or V 2 0> = ~ (vV) 2 <D - 4-rrp (l - J) , 
 
 * Throughout the remainder of this chapter we have conformed to the usual practice of using 
 the magnetic force vector instead of the magnetic induction vector to define the conditions in 
 the free aether. For the present purposes the two vectors are always identical. 
 
715-718] The characteristic equation 629 
 
 where now iff has disappeared. This is the characteristic equation from which 
 the single independent variable O of the problem is to be determined, subject 
 to the condition that it is to be constant over each conductor. 
 
 For the interior of a conductor O is constant and the electromotive force 
 
 F vanishes ; but the aethereal displacement - E does not vanish in the 
 
 4:77 
 
 conductors, being now given by 
 
 which makes it circuital so that there is no volume distribution of electrifica- 
 tion. 
 
 717. In an investigation in detail of the field produced by the motion, 
 it will conduce to brevity if we take v to be a velocity parallel to one of 
 the axes of coordinates, say the se-axis. We shall also use the notation 
 
 The characteristic equation for the convection potential <E> is then 
 
 and this has to be solved subject to the condition that O is constant over 
 each conductor of the system : as the change in the form of the equation 
 arising from the motion depends on jS 2 , the differences thereby introduced 
 will all be of the second order of small quantities. 
 
 We can restore the above characteristic equation for O, the potential of 
 the electromotive force, to an isotropic form by a geometrical strain of the 
 system and the surrounding space represented by 
 
 (*o *A Zo) - (K-^X, y, z), 
 where of course K Z = 1 /3 2 .* 
 
 718. Now let us compare our moving system which we may generally 
 describe as S, with the correlative system S obtained by this transfor- 
 mation and supposed at rest : we shall assume that the density p of the charge 
 distribution in S is reduced from the corresponding value in 'S in the ratio 
 K : 1 so that corresponding elements of volume contain the same total 
 charges : then if </> is the electrostatic potential of these charges on S 
 
 is the characteristic equation satisfied by </> in this system. The general 
 type of solution of this equation is, as we had it before, 
 
 * This transformation was suggested by Thomson and Heaviside. 
 
630 The electrodynamics of moving media [CH. xv 
 
 the integral being taken over the entire field and r denoting the distance 
 of the element dv from the point in the field at which the function is calculated. 
 
 Now the potential O in the moving system satisfies the equation 
 
 [p dv 
 
 so that <P = K<f> = K I- - , 
 
 J r o 
 
 and since pdv = p dv Q 
 
 and r 2 = ^ (a; - X P )* + (y - y P )* + (z - Z P ) Z , 
 
 (x f , y p , Z P ) denoting the coordinates of the point at which <D is calculated, 
 and (x, y, z) the coordinates of the position of dv, we may write 
 
 and this is the general type of solution for the convection potential in any 
 moving system of the type under consideration. 
 
 719. Again comparing the components of the electrostatic force 
 
 ET7JL 
 O = - V<PQ , 
 
 in S with the corresponding components of the electromotive force 
 
 F = - VO, 
 
 in S we see that 
 
 x dx dx n x ' 
 
 = K[B 0if , E J. 
 
 Thus the forces on corresponding elements of charge in the two systems 
 are equal as regards their components in the direction of motion, but the 
 components in any direction at right angles to this direction are smaller in 
 the moving, system in the ratio /c : 1. 
 
 Thus if we have solved the electrostatic problem for any system S at 
 rest, i.e. if we have determined the equilibrium distribution of electricity on 
 the conductors in the system under the influence of the rigid charge distribu- 
 tion, then we can immediately deduce the solution for the equilibrium 
 distribution of electricity on the conductors in a uniformly moving system 
 S obtained from S by a uniform contraction in the direction of the motion 
 in the ratio K : 1. 
 
 720. Suppose, for example, that the system S is represented by a uniform 
 distribution of electricity of total amount Q Q throughout the thin homoeoidal 
 ellipsoidal shell between two similar and similarly situated concentric ellipsoids 
 
718-720] The two special solutions 631 
 
 and that there are no other bodies in the system. If a , a , a are the 
 
 123 
 
 axes of the mean ellipsoid on which this shell lies then we know from the 
 investigation of the second chapter that the appropriate form of the electro- 
 static potential <f> has the constant value 
 Q r dt 
 
 2 ' o V(a 0] 2 + (a 02 2 + t) K 3 2 + t) 
 throughout the interior of the ellipsoid, whilst at external points the value is 
 
 <2_o f ' dt 
 
 2 -^V(a 0l 2 + K 2 2 + K 3 2 + *)' 
 
 where Q is the total charge on the ellipsoid and in the last expression A is 
 the positive root of the cubic equation 
 
 o j 7 I " A , , ""r" o i _* * 
 
 Moreover we have seen also that, since the potential <f> is constant throughout 
 the interior of the ellipsoid, the distribution of charge thus specified is identical 
 in the limit with the surface distribution of charge of the same total amount 
 on the same ellipsoid when composed of conducting material. 
 
 Now by uniformly contracting this ellipsoid and its space in the ratio 
 K parallel to any definite line we obtain another ellipsoid with semi-axes 
 (], 2 , a 3 ). If this new ellipsoid is moved parallel to the chosen line with 
 the velocity appropriate to the ratio K the original statical system and its 
 field will exactly correspond in the manner just defined to the system it 
 defines ; on it the convection potential will therefore take the constant value 
 
 2 .1 o V(a 0l 2 + (a 02 2 + (a 0s 2 + t) 
 
 whilst at external points its value is 
 
 K Q f dt 
 
 ~2~1 /'~ a 2 ' 
 
 A being the positive root of the equation 
 
 ftp 2 S/o 2 Zo 2 = -, 
 
 The equilibrium distribution of electricity on a moving conductor is 
 characterised by the fact that the electromotive force in its interior vanishes, 
 i.e. the convection potential O is constant there. Thus the distribution on 
 the moving ellipsoid obtained by contraction of the static distribution on 
 the conducting ellipsoid in S is identical with the distribution which would 
 hold if the moving ellipsoid were conducting. But when it is remembered 
 that the electric distribution in S Q is the limit of a uniform distribution 
 between two concentric, similar and similarly situated ellipsoids ancLthat in 
 the process of uniform contraction these ellipsoids remain concentric similar 
 
632 The electrodynamics of moving media [CH. xv 
 
 and similarly situated, it follows that the new distribution of charge on the 
 moving ellipsoid (a lf a z , a s ) would be exactly the same as if it were in 
 equilibrium. Thus the distribution of charge on a conducting ellipsoid is 
 not disturbed by imparting a uniform translatory motion to it*. 
 
 Two particular cases of this general theorem have assumed special import- 
 ance on account of the applications which have been made of them to 
 illustrate the properties of a moving electron, which is nothing more nor less 
 than a charged particle. 
 
 721. In the first case the conductor in motion is assumed to be spherical 
 in form, say of radius af. The conductor in the correlative static system will 
 
 then be a prolate spheroid with axes (-, a, a), if the motion is along the 
 
 direction of the z-axis. The appropriate form of the convection potential 
 can then be written in the form 
 
 g_*Qr- dt 
 
 where A is the positive root of the quadratic 
 
 on the surface of the sphere. 
 
 The integrals in these cases can be directly evaluated by the substitution 
 
 T'-i + V. 
 
 K* 
 
 so that it becomes 
 
 K' 
 
 a 2 + XK 2 + aVl -/c 2 
 A/^-aVT^P' 
 
 * Mr H. S. Jones has suggested to me a modification of this proof. If it is assumed that the 
 distribution on the conducting ellipsoid in motion which gives zero force inside it is the equilibrium 
 one, we can argue exactly as in the statical case that the surface density varies as the central 
 perpendicular on the tangent plane at the point, since we have shown that the electric force 
 due to any moving point charge is radial and, for any given direction, varies inversely as the radius 
 squared. Thus since a ccp and the total charge is unaltered, the distribution must remain 
 unaffected by the motion. 
 
 f M. Abraham, Ann. d. PJiys. (iv.) x. p. 105 (1903). 
 
720-722] Lorentz's electron 633 
 
 which reduces to the value 
 
 K 2 Q 1 + Vl ~K Z 
 
 <D = -- ;?- ---- log 
 
 2a VT- /c 2 1 - Vl - /c 2 
 
 on the surface of the sphere. The full details of the field can now be deter- 
 mined : the charge on the sphere is uniformly distributed over the surface. 
 
 722. In the second case* the moving surface is an oblate spheroid 
 of axes (a, K , a, a], . 
 
 the first being in the direction of motion. The surface in the correlative 
 static system which has axes 
 
 /a/c \ 
 
 [ i a, ), 
 
 \K 
 
 is therefore a sphere of radius a. In this case the electrostatic potential <f> is 
 
 so that the convection potential O is 
 
 _ == 
 
 Vx 2 + /c 2 (^/ 2 + z 2 ) 
 which reduces to the constant value 
 
 *,&, 
 
 a 
 on the surface of the moving conductor of which the equation is 
 
 z 2 + K 2 (y 2 + z 2 ) - a z K z . 
 
 The field in the moving system is of a much simpler character than that 
 of the previous example and we may therefore examine it in more detail 
 with a view to illustrating some of the general features of these convection 
 fields. The electromotive force F at any point in the field has components 
 
 F F F = _ (1 1 *\ 
 
 \dx' dy' dz) 
 
 {x* + /c 2 (y* + 2 2 )} f 
 
 the electric force at the same point, which in the most general case of transla- 
 tion along the axis Ox has components 
 
 . / F F \ 
 
 (E x , E tf , E z ) = [P., =i -.), 
 
 V /c 2 /c 2 / 
 
 is therefore given by 
 
 f (^ 2/- ) 
 
 * Cf. Lorentz, The Theory of Electrons, p. 210. This is also the solution for a point charge. 
 
634 
 
 The electrodynamics of moving media [CH. xv 
 
 The magnetic force in the field, given generally by the relation 
 
 has therefore the components 
 Hj,, H z ) = 
 
 (x 2 + K* (y z + 
 
 (0, - z, y). 
 
 723. The formulae thus obtained for the electric and magnetic force 
 vectors in the field are considerably simplified by a simple transformation in 
 which account is taken of the fact that the conditions in the field at any point 
 have really originated from the motion of the sphere not at the instant at 
 which they are examined but at a previous effective instant and that they 
 have been propagated out with the velocity c from the position of the sphere 
 at this instant. We now choose as the origin of a spherical polar coordinate 
 system the centre (see figure) of the sphere in its effective position for the 
 
 Pig. 102 
 
 special point P of the field at which the conditions are investigated : the 
 polar axis is taken along the direction of the motion : then if OP = r and 
 if A is the actual position of the centre of the sphere at the instant t (and 
 this is the origin of the rectangular coordinate system), then 
 
 OA = pr, . 
 and thus x z = (r cos 6 f3r) z , 
 
 Thus 
 
 x* + /c 2 (y 2 + z 2 ) = r 2 - 2r cos + 2 r 2 cos 2 
 
722-724] The dynamics of moving systems 635 
 
 and thus the electric force has the components 
 
 K zn 
 (E X} E y , E z ) - r2(1 _0 cog6))3 (cos 6 - p, sin 6 cos </>, sin 6 sin </>}, 
 
 whilst the components of the magnetic force are 
 
 - K*BQ sin d 
 rMl-^os0)3(< Sm ^- COS ^ 
 
 The electric force at any point has therefore a radial component from the 
 effective centre of the sphere of amount 
 
 Q i- 2 
 
 r 2 (l -j8cos0) 3 ' 
 
 the remaining part of it being a single component in the meridian plane of 
 amount 
 
 Q (I - jS) jS sin 6 
 
 r 2 ~(1 -jScosfl) 3 
 parallel to the line of motion. 
 
 The magnetic force is in circles round the direction of the motion and at 
 any position (r, 6) its intensity is 
 
 r 2 (1 - j8 cos~0j 8 ' 
 which is equal to the latter component of the electric force*. 
 
 724. The dynamics of moving electrified systems. In the previous para- 
 graph we have discussed the character of the field in the neighbourhood of 
 an electrical system moving with uniform velocity along a straight line. It is 
 of course assumed that the system thus discussed has been in motion in the 
 manner specified for a sufficiently long time previous to the instant at which 
 it is examined : the conditions implied in this restriction will be fairly obvious 
 when reference is made to the discussions of chapter xn where the mode 
 of establishment of such steady fields is reviewed in detail. It was there 
 seen that in the process of setting up the uniform motion, say from rest, 
 a shell of disturbance is sent out into the surrounding electrostatic field : 
 this shell travels out and away from the system with the velocity c of radiation 
 leaving behind it the new steady field associated with the charges in motion 
 and in which the field vectors are expressed by functions of position decreasing 
 rapidly (like 1/r 2 ) as the point is taken more and more distant from the charges 
 themselves. Thus if the shell of disturbance has got to such a distance from 
 the system that the field vectors in the uniform field are negligibly small we 
 may regard the effective conditions for the uniform motion as practically 
 established, and we have then no further concern with the expanding 
 radiation field in which the energy has attained a constant value. 
 
 * These formulae are due to Heaviside, Phil. Mag. 27 (1889), p. 332; Electrician, Dec. 7, 
 1888, p. 148. 
 
636 The electrodynamics of moving media [OH. xv 
 
 The practical aspect of the results thus obtained lies, of course, in their 
 application to the explanation of the observed mechanical relations of such 
 moving systems of charges and it is with these explanations that we shall 
 now concern ourselves. The considerations of the present paragraph will be 
 confined mainly to the translatory motion of such rigid electrical systems as 
 have been examined in the previous paragraph. 
 
 725. In the previous chapter we showed quite generally that the resultant 
 force of electrodynamic origin on any material system contained within any 
 closed surface / can be separated into two parts, the first of which can be 
 represented simply as a stress across the surface / which per unit area is of 
 intensity proportional to the square of the field vectors, whilst the second 
 turns out to be expressible as the result of a distribution of bodily forces 
 throughout the interior field of intensity ' (vectorial) per unit volume 
 
 This conclusion takes a remarkable form if the system under consideration 
 is of finite dimensions and if the surface /is then extended indefinitely. For 
 then the field vectors at points of the surface are so small that the first part 
 of the total forcive just specified may be neglected. We conclude then that 
 the total force on the whole system in this case can be represented as a 
 wrench which when reduced to the origin of the rectangular coordinates as 
 base point is specified by the linear and angular components F and G 
 respectively, where 
 
 F = - 7 - f 4 CEH] dv 
 4-n-c J dt 
 
 _d^ 
 dt ' 
 
 , 8 --M r -s lw ]*' 
 
 -- + P..V], 
 
 wherein M e = -: [EH] dv 
 
 and N e = -. - f[r . [E . H]] dv, 
 
 47TC J 
 
 all integrals being extended throughout the entire field : r denotes the vector 
 coordinate of position of a point -whose components are (z, y, z). 
 
 The forcive exerted by and through the aether on any electrical system 
 without induced or intrinsic magnetisation is therefore equal and opposite 
 to the change per unit time of the quantity which we have tentatively defined 
 above as the electromagnetic momentum in the aethereal field of the system. 
 
724-727] Electromagnetic momentum 637 
 
 726. If the motion of the charges in the system is one of uniform trans- 
 lation in a fixed direction the field will, as we have just seen, be carried 
 on with the system in a steady configuration. The total electromagnetic 
 momentum in it will therefore be constant. The linear component of the 
 reacting electromagnetic force in the system would then be zero but there 
 will be a couple unless the field is symmetrical about the direction of motion. 
 
 In the more general case however whenever the motion of the system is 
 accelerated there will arise a reaction on account of its charges specified as 
 a wrench with linear and angular components 
 
 ar- -ar 
 
 There will of course in the general case be a reaction force to the acceleration 
 of the motion of the system on account of its material momentum and this 
 will be a wrench with linear and angular components 
 
 dMp _ dH 
 dt > dt "* 
 
 Thus to maintain the accelerated motion external action of some kind is 
 necessary ; if the external force system reduces to the same base as a wrench 
 with components p Q. 
 
 then we shall have 
 
 w , ^ M F _ 
 
 FO+F '" F - 
 
 - ~dT 
 
 
 
 Go + G = d ~ + [v . N ], G = I (N + N.) + [v . N + NJ. 
 
 From this point of view it would appear that the idea of an electromagnetic 
 momentum is just as legitimate a conception as that of ordinary material 
 momentum ; in any case, of course, the conception, legitimate or not, provides 
 a convenient mode of expressing a definite fact of theory. 
 
 727. In the practical application of these principles it is first necessary 
 to determine the vectors M e and N/for the system under consideration, and 
 this requires a knowledge of the complete electromagnetic field of the moving 
 charges. Now we have already examined the details of the mode of generation 
 and propagation of such electromagnetic fields : we saw that the conditions 
 at any point of the field at a definite instant were made up by superposition 
 of disturbances from all the separate electrons in the system, emitted at the 
 appropriate effective previous time and place for each of them and transmitted 
 thence with the velocity of radiation. The definition of the field at any 
 instant in the most genera] case is therefore tremendously complicated and 
 it is only in a very few restricted cases that it has been accomplished. But 
 for the majority of the applications we shall have to make of these principles, 
 such generality of procedure, however desirable it might be from the theoretical 
 
638 The electrodynamics of moving media [OH. xv 
 
 point of view, is not at all necessary. In fact it will be seen by reference 
 back to the analysis of the previous chapter that the effective conditions in 
 the field for the determination of the force on the system of electrodynamic 
 origin are in reality merely the conditions which hold in the immediate 
 neighbourhood of the charges in the system. Thus if the system of charges 
 is of finite extent and the changes in its motion are not too rapid the conditions 
 in the field throughout the extent of the field covering the charge distribution 
 will at any instant be practically the same as those which exist in a system 
 moving uniformly with the instantaneous configuration and velocity of the 
 given system. Now the new conditions in the field are smoothed out with 
 the velocity c of radiation so that the condition here implied is that the 
 relative configuration and motion of the system must not change appreciably 
 in the time taken by radiation to cross the system, so that the effective field 
 is at each instant smoothed out to the type appropriate to the motions of 
 the charges before these are appreciably altered. We shall of course in our 
 calculations assume that the uniform field extends to an indefinite distance 
 beyond the system so that we can avail ourselves of the separation of the 
 forcive mentioned above. This procedure is legitimate because, since the 
 force is in reality defined by the conditions of the field in the immediate 
 neighbourhood of the system, the type of field to which this local field 
 continues is irrelevant and may for the purposes of the detailed calculations 
 be taken to be the simple uniform field of the instantaneous motion. 
 
 This.is the essence of an equilibrium theory and it restricts our analysis 
 to application only in the cases of so-called quasi-stationary motion. The 
 importance in a dynamical theory of the restriction thus implied lies^in the 
 fact that the effective conditions in the field are then determined solely by 
 the instantaneous configuration and motion of the charge system giving rise 
 to it, so that instead of having to include in the analysis, in addition to the' 
 finite number of the coordinates of the charge system, an infinite number of 
 coordinates to specify the conditions in the aethereal medium surrounding 
 it, we have only to reckon with the former by themselves. 
 
 There is a mechanical analogy in the theory of the motion of a solid 
 body through an ordinary elastic medium such as the air. If the motion 
 of the solid is slow enough the conditions of the elastic medium adjust 
 themselves at each instant to the state of motion then existing, and an 
 equilibrium theory applies just as if the medium were incompressible. But 
 in the other extreme case of rapid motion with large accelerations the whole 
 of the circumstances in the surrounding medium are complicated by the 
 continual generation of waves of compression starting out from the solid. 
 In this case the resistance to the motion of the solid is practically all due 
 to the elastic resistance of the surrounding medium to the setting up of waves 
 in it. 
 
727, 728] The kinetic potential 639 
 
 728. Thus in the calculation of the quantities involved we may assume 
 that at each instant the field is of the steady type appropriate to that in- 
 stantaneous motion and position investigated in the previous paragraph; 
 but in this case the motive forces of electromagnetic origin on the elements 
 of charge in the system have a potential <I>, just as the ordinary static forces 
 in electrostatics. We may therefore conclude as in the electrostatic theory 
 that the mechanical forces of electrodynamic origin acting on the material 
 bodies of the system carrying charges have a potential which in the general 
 case is expressed by the integral 
 
 taken throughout the field, but which in simpler cases will reduce to the 
 forms already discussed at length in the electrostatic theory. From another 
 aspect this quantity may be regarded as potential energy stored up in th'e 
 system ; but it is to be noticed that it also includes a part, due to the magnetic 
 field which we have previously classed as kinetic energy and which therefore 
 when reckoned as potential energy must have its sign changed : it is easy to 
 verify this in the particular case of a charged conductor moving uniformly 
 in a straight line such as discussed in the previous paragraph : for in that 
 case the magnetic or kinetic energy of the system is 
 
 T = 
 
 STT ' 
 
 and H = - [vE], 
 
 c 
 
 so that if the motion is along the #-axis 
 
 T = j^(E 2/ 2 + E z 2 ): 
 
 whilst the ordinary potential or electric energy of elastic strain in the aether 
 is 
 
 so that 
 
 W = U -T=- 
 
 IO \ X "* 
 STT l 
 
 - x 
 
 OTT 
 
 1 ' 2p 2 
 
 * V 
 
 - # If 2 (I*) 2 + 2 + (w)l dv ' 
 
640 The electrodynamics of moving media [OH. XY 
 
 and this is easily transformed in the usual manner by Green's theorem and 
 gives 
 
 2 2 < 2 
 
 dv, 
 
 , t - 
 
 87TK 2 J T \ ox z dy 
 
 as above*. 
 
 It must of course be remembered that this is not necessarily all the energy 
 that is in the field, but it represents the variable part in the present case. 
 The motion itself when under examination is of course of the quasi-stationary 
 type so that the radiation field and the energies associated with it are 
 negligible : but the motion must have been started by a non-stationary 
 impulse in some remote past time and in this process a circular shell of 
 disturbance of the wave motion type is sent out into the field and the energy 
 in this part which however tends to a constant value is not necessarily 
 negligible. But the more distant the time of generation, the farther is 
 this wave shell away from the system and the nearer does the field inside 
 it approximate to the actual field belonging to the quasi-stationary motion 
 which the analysis maps out. 
 
 729. The above deduction of the form of W shows that the function 
 
 L= - W 
 
 also serves as a sort of Lagrangian function. The fact that the force function 
 and Lagrangian function with the sign changed agree in the cases of quasi- 
 stationary motion is not necessarily confined to the present problem : in 
 fact the general equations of motion of any system are of the type 
 
 d_ fdL\ _ dL d_W 
 
 dt\de) dd ~ W ' 
 
 and if the changes in the motion only take place very slowly, if the accelera- 
 tions, that is, are small this equation practically reduces to 
 
 _dL dW_ 
 
 36 ~ dd ' 
 
 so that W L, 
 
 except perhaps for a constant. 
 
 This interpretation of the force function as a Lagrangian function has 
 another significance : in fact we see from the form given for L, viz. 
 
 that 
 
 dL 
 
 dv dv -L 2 F , 
 
 y ~ 
 
 * Cf. Schwarzschild, Zwei Formen des Prinzips der kleinsten Aktion in der Elektronentheorie, 
 Gott. Nachr. (math. phys. KL), 1903. p. 125. 
 
728-730] Electromagnetic mass 641 
 
 Now the former of the integral? on the right is 
 
 "dv ,, 
 
 the second is on the other hand equal to zero for it is 
 
 If /^ff\ C^T3^ C^/T\ ^TH ClfT\ ^t^ \ 
 
 I fO^V C/J2j r t/U* (yJcj,, C/VP UKAy \ 
 
 I I _t . " I * 1 /7/yi 
 
 4 I I ^ * ^ f~\ I f\ * *"\ /" I ^ *\ t-\ I lv I/ 
 
 477jV8cc 8,8 dy 8)8 82 3j8 / 
 
 and this transforms in the usual way by Green's theorem and becomes 
 
 1 
 
 f 4T7.T3p' 
 
 It follows therefore that ~ cM e , 
 
 dp 
 
 I v 
 
 or since 8 = - 
 
 c 
 
 8 | v | c 8 
 
 and this relation again exhibits the analogy between the so-called electro- 
 magnetic momentum of the system and an ordinary momentum in dynamical 
 theory : it also provides us with the simplest means of calculating M e . 
 
 It is important to notice either as a deduction from this last equation or 
 as a consequence of the particular type of field determined that the vector 
 of linear momentum of a rigid system of charges moving in a straight line 
 is directed entirely along that line : there is no tendency to motion across the 
 direction of translation. 
 
 730. Next let us turn to another fundamental aspect of these matters. 
 In ordinary dynamical theories the existence of momentum implies the 
 presence of a material mass moving with a velocity. Now we have seen that 
 for instance an electrostatic system in uniform motion in a straight line would 
 possess something akin to momentum, viz. what we have called electro- 
 magnetic momentum, in the direction of its motion, even if its material 
 mass were negligibly small. Thus if we are prepared to adopt the analogy 
 between electromagnetic and material momentum it appears as, at least 
 convenient to extend the analogy still farther and to say that the existence 
 of electromagnetic momentum implies also an electromagnetic mass moving 
 with a velocity. All that it is intended to imply in such a statement is that 
 an electrical system possesses a certain amount of inertia on account of the 
 field and charges in it, just as if it had an additional mass of ordinary type : 
 the inertia offered by this additional mass of the system to accelerated motion 
 
 L. 41 
 
642 
 
 The electrodynamics of moving media [CH. xv 
 
 exactly accounts for the reaction which according to our theories arises from 
 the electromagnetic field surrounding the system. 
 
 We can put this point in another way. The equations of linear translation 
 of an electrical sstem were obtained in the form 
 
 F being the applied force vector : now suppose that the motion is of the 
 quasi-stationary type so that both M and M e are functions of geometrical 
 configuration and velocities only, and are both parallel to the direction of 
 the velocity v of the system. If then we use R for the radius of curvature 
 of the path of the system, and Rj a unit vector along R then 
 
 P = 
 
 d 
 
 d 
 
 M + M 
 
 d \ v 
 dt 
 
 RT 
 
 and now we recognise the ordinary definition of inertia mass in dynamics 
 on the basis of Newton's second law of motion. Thus even if M , the material 
 mass of the body is zero there will still be an apparent mass of electrodynamic 
 origin which for accelerations in the direction of the line of motion is of 
 amount 
 
 d VL 
 
 Id 
 
 while it is 
 
 d v 
 
 Me 
 
 V 
 
 ?2 ' 
 
 M, 
 
 for accelerations perpendicular to the direction of motion. 
 
 Thus the quasi-stationary motion of an electromagnetic system is effectively 
 modified on account of the charges on it just as if it possessed additional 
 mass of an amount however depending on the relative direction of its main 
 velocity and its acceleration. 
 
 731. These results may be further illustrated by application to the two 
 special cases examined in the previous paragraph. When the moving system 
 consists solely of a uniformly charged sphere the convection potential assumes 
 the constant value 
 
 *~2a tog l-' 
 
 on the surface of the sphere : the force function or, if we prefer it, the 
 Lagrangian function of the motion with the sign changed is therefore 
 
 
 
 = L = 
 
 4a 
 
 log 
 
 Thus we have 
 
 IdL 
 
 c ~d3 
 
 
 L 
 
730-733] The mass of the electrons 643 
 
 and then the longitudinal and transverse masses turn out to be* 
 
 Q 2 1 ( 2 1,1 + 
 
 If 1 + )3 
 
 These masses, functions of the velocity, increase indefinitely as ft = - 
 
 G 
 
 is increased and become infinite for j8 = 1, i.e. when the velocity of the electron 
 attains the velocity of light. This means of course that such a velocity 
 would never be attained. They have the common limit 
 
 2D2 
 u/ 
 
 when ft is small. 
 
 732. In the second case the moving system is of variable dimensions, 
 having the form of an oblate spheroid with axes (/ca, a, a) when the motion 
 is with velocity v, where 
 
 v 2 
 
 O 1 -| /")o 
 
 K 2 = 1 2 = 1 p 2 . 
 
 In this case the convection potential assumes the constant value 
 
 l^.tt^WST 
 
 a a ' 
 
 on the surface of the body where the charge is confined : and thus now 
 
 follows therefore, just as above, that 
 __ IdL 
 
 so that f m e = 
 
 c dft 2ac ' Vl - j3 2 ' 
 Q 2 'I 
 
 which again become infinite as the velocity approaches that of light, starting 
 from the same limit 
 
 m = 2oc 2 = 3 m ' 
 
 733. The great interest in these results lies in their application to the 
 explanation of the properties of the electron. It is impossible to charge an 
 ordinary piece of matter with sufficient electricity to produce an electro- 
 magnetic inertia at all comparable with its ordinary inertia ; but an electron 
 
 * Abraham, Ann. d. Phys. 10 (1903), p. 105, Die Theorie der Eleklrizitat, n. p. 181. 
 f Lorentz, Theory of Electrons, p. 210. 
 
 412 
 
644 The electrodynamics of moving media [CH. xv 
 
 carries a charge which is so enormous, compared with its mass, that its 
 electromagnetic inertia is far greater than its material inertia if it has any. 
 Again pieces of matter even so small as the individual molecules are much 
 too cumbersome to get up a speed comparable with that of light, whereas the 
 electrons are often found under natural circumstances travelling with such 
 speeds, so that they should exhibit the additional characteristic of the electro- 
 magnetic mass in increasing with the speed. Some time after Thomson's 
 determination of the ratio of the charge to the mass, Kaufmann* proceeded 
 by the same method to determine the functional form of this ratio in terms 
 of the velocity, with a view to testing the effectiveness of the theoretical 
 formulae. He worked with the negative electrons which are thrown off as 
 the/3-rays from radium with velocities ranging up to '95 c. Now it was found 
 that while the velocity increases from about '5 c. to the higher value the 
 corresponding value of e/m considerably diminishes, and exactly in the way 
 that we might expect if the charge remains constant and the mass increases 
 according to the theoretical formulae. The first experiments were unable to 
 distinguish between the two types of formulae given, the first by Abraham 
 and the second by Lorentz, but later and more exact methods tend to the 
 view that the simpler functional forms given by Lorentz' s method are the 
 more exact of the two. 
 
 The conclusion to be drawn from these facts is that at all events the 
 electromagnetic mass of 'an electron has an appreciable influence; but the 
 experiments went farther, the type of function obtained pointed to the 
 conclusion that the electromagnetic mass greatly preponderates over any 
 material mass that the electrons may have. Indeed Kaufmann's numbers 
 show no trace of an influence of the material mass at all, his ratio of the 
 effective masses for two different velocities agreeing within the limits of 
 experimental error with the theoretical ratio of the electromagnetic masses 
 themselves. 
 
 734. If the material mass of an electron is inappreciable under all 
 circumstances it may be treated as absolutely zero. It is no use talking of 
 something that cannot be traced. On such a view the electron consists merely 
 of an element of negative electricity. 
 
 Of course by the negation of a material mass, the electron loses much of 
 its substantiality because our theory has been interpreted entirely in terms 
 of charged matter and the ordinary mechanical relations of such matter. 
 In speaking therefore of the electron as having no material mass, we must 
 nevertheless leave it sufficient substantiality to enable us to speak of forces 
 acting on it. In terms of a pure aether theory we might say that the electron, 
 which is necessarily a centre of an electric field of strain in the aether, is 
 
 * Gott. Nachr. 1 (1901), 5 (1902), 3 (1903); Phys. Zeitschr. 1902, p. 55; Ann. d. Phys. 19 
 (1906), p. 487. 
 
733-735] Optical phenomena in moving media 645 
 
 nothing more nor less than the centre or nucleus of the strain thus depicted, 
 a knot, so to speak, which can however move freely from one point of the 
 medium to another. This point of view has been advocated by Larmor and 
 it has many points in its favour on account of the simplicity it introduces 
 into the subject. 
 
 It is natural to enquire whether there is similarly no material mass for 
 the elements of positive electricity. Unfortunately no decisive answer is yet 
 possible to this question, but the view is now generally held that not only is 
 there no material mass in the ordinary sense of the word in the positive 
 particles, but that the material mass of any ordinary piece of matter is nothing 
 but the inertia mass of the electric charges involved in it*. This is the ' electro- 
 magnetic theory of matter ' and although it is not possible for us to enter 
 here into any further details, it must be admitted that the evidence of both 
 the experimental and theoretical researches of the last 10 years is gradually 
 tending to confirm the substantiality of the view. 
 
 735. Optical phenomena in moving media. We now leave these in- 
 teresting questions concerning moving charges to turn to another aspect of 
 electrodynamic theory of moving systems in general. We have so far merely 
 treated of steady electromagnetic fields convected with a system of bodies 
 having a motion of translation, but it is equally important to consider the 
 other extreme case of the propagation of very rapid electromagnetic dis- 
 turbances across the field between such moving systems. The practical 
 aspect of these considerations lies in their application to the explanation of 
 the optical phenomena in systems having a motion of translation, as for 
 instance all terrestrial bodies have by the annual motion of the earth. 
 
 We consider in the first place the propagation of electric waves across a 
 dielectric medium, which is moving with uniform velocity v parallel to a 
 definite direction which we shall take to be the x-axis of a system of rectangular 
 coordinates. According to the general scheme of equations formulated 
 above we have the electromotive force expressed in terms of the vector and 
 scalar potentials and the magnetic induction by the relation 
 
 1 dA 1 r 
 
 where B = curl A, 
 
 and if C is the total current of Maxwell's theory 
 
 4rrC 
 
 - = curl H, 
 
 and B = H + - 
 
 * Cf. Larmor, Phil. Trans. A, 186 (1895), p. 697. Cf. also Aether and Matter. 
 
646 The electrodynamics of moving media [OH. xv 
 
 if there are no magnetic substances about ; P is the dielectric polarisation : 
 thus* 
 
 - 4?r cur] [Pv] . 
 
 c 
 
 These equations are satisfied by the propagation of a train of transverse 
 waves along the z-axis, in which P z , H x , 0*, A x are all null, while ^ is a 
 function of y, z. 
 
 Thus for such a wave train 
 
 [ V B] = [v [VA]] 
 
 = V (Av) - (vV) A 
 9A 
 
 __. n\ 
 
 dx 
 
 and thus 
 
 c \dt ~* dxj ' dz ' 
 Now since div A = any theory that makes 
 
 div F = 0, 
 
 will also make V 2 </> = 0, that is, will make <f> merely the static potential of 
 the electric charges in the field. ' We shall then have 
 
 and the remainder of the analysis depends on the relation between C and F. 
 
 736. Now we have seen that the total current C is composed of two 
 parts : the first, an aethereal part, of density 
 
 JL5? 
 
 47T dt 9 
 
 which is not convected, presumably because the aether does not participate 
 in the motion of the matter; the second part depends on the material 
 polarisation and is measured by its time rate of change 
 
 8P 
 dt' 
 
 Again curl 0Pv] = (vV) P - v (VP) 
 
 and as far as concerns the two components under consideration this is the 
 same as 
 
 v d ~ 
 dx 
 
 whilst P = e'F, 
 
 * We are using Maxwell's instantaneous vector potential. 
 
735, 736] FresneVs convection coefficient 647 
 
 where e = 1 + 47re' is the dielectric constant of the medium*. Thus 
 
 _ 1 0"E - I /9 , 
 
 C + c curl [_Pv] = -. j \- - [ - + v 
 477 at 4rr \ot 
 
 Now putting (f) null for purely transverse waves, which will be found to cause 
 no discrepancy, we have 
 
 F - _ 1 
 
 c dt ' 
 
 and thus keeping A as th more convenient independent variable 
 
 If the period of the waves propagated is all the functions may be taken 
 to depend on the time and position coordinates by the factor 
 
 gip (c't art 
 
 where c' is the velocity of propagation ; substituting this form we find that 
 
 C 2 = c'2 + ( - 1) ( C ' - V) 2 , 
 
 whence since v/c' is very small we have approximately 
 
 c / / IN v 
 
 Ve\ \ *J c' 
 
 c 
 ~V~e 
 
 Thus the velocity of propagation of radiation in the moving medium is 
 increased from its value in the same medium at rest by the amount 
 
 This result was formulated by Fresnelf long before the electromagnetic theory 
 had been formulated, and it has been experimentally verified first by Fizeau 
 but much more accurately by Michelson and Morley J in connection with the 
 propagation of light in flowing water. In the experiments water was made 
 to flow in opposite directions through two parallel tubes placed side by side 
 and closed at both ends by glass plates : the two interfering beams of light 
 were passed through the tubes in such a manner that throughout their course 
 one went with the water and the other against it. 
 
 These experiments provide the most decisive, as well as the most accurate 
 test of Fresnel's hypothesis of a stagnant aether, on which the above theory 
 has been based, and they definitely exclude the more complicated hypothesis 
 
 * It is to be understood that the constant here involved is that appropriate to rapidly 
 alternating fields and is a function of the frequency of alternation, 
 t Paris C. S. 33 (1851), p. 349; Ann. d. Phys. (1853), p. 377. 
 j Amer. Jwr. of Sc. 31 (1885). 
 
648 The electrodynamics of moving media [OH. xv 
 
 advanced by Stokes. This is easily seen because if the motion of a piece of 
 matter through the aether drags that medium with it, all that is contained 
 in the flowing column of water in the experiments will share the translatory 
 motion of flow : the propagation of light in the water will then go on in its 
 interior in exactly the same' manner as if it were at rest ; so that its actual 
 
 velocity in space will be ( + v ) . 
 
 We / 
 
 737. A further important point is also emphasised by the result obtained 
 above. It essentially involves not only the distinction between aethereal 
 and material polarisation currents but also the distinction between the forces 
 producing these currents, which is brought out in the general dynamical 
 theory of the previous chapter. It was there shown that the force of electro- 
 dynamic origin which acts on the electrons and produces the true current of 
 material displacement is the electromotive force F, whereas the force which 
 strains the aether and produces the aethereal displacement current is the 
 electric force, differing from the electromotive force F by the dynamical 
 
 part - [vB]. 
 
 The fundamental hypotheses on which the theory is based are therefore 
 the only ones which are consistent with the results of experiments made 
 especially with a view to testing them : we must now follow the theory 
 through into some of its more important consequences in order to see whether 
 it is entirely sufficient or whether it still requires modification and amplifica- 
 tion. Before proceeding at once to the optical theory it will however be 
 convenient to elaborate at the present stage a few further details of a question 
 already dealt with at some length on a previous occasion. 
 
 738. The ponderomotive force which an electromagnetic field exerts on 
 any interface separating a perfectly conducting or absorbing medium from 
 the surrounding free space is determined mainly in the form of a traction 
 per unit area 
 
 T = 1 (2EE W + 2HH W - n x (E + H 2 )}, 
 
 n x being the unit vector along the normal to the surface. 
 
 The resultant force exerted by the field on any such body is obtained 
 by integration of this surface force over the outer surface of the body. 
 
 Now let us find how this surface force is altered when the body moves in 
 a vacuum : the above expression is then increased for momentum is taken 
 up as a result of the motion. If v is the velocity of the typical element df 
 of the surface, the momentum picked up by the body on account of the 
 motion of df is then 
 
 ZZL TEH] df. 
 - 
 
736-739] Radiation pressure 
 
 It follows that the pressure per unit area on the moving surface is of the 
 form 
 
 T' = 1 \EE n + HH n - I n (W + H*) + ^ [EHjl. 
 
 We can transform this into a more convenient form : to do this we start 
 from the identity 
 
 v n [EH] + E w [H . v] + H n [ V E] = n x (v . [EH]), 
 so that we may write 
 
 877-T' = 2E'E W + 2H'H n - n JE 2 + H 2 - 2 (v . [EH])1, 
 
 where we have written E' and H' for the vectors 
 
 E' = E + - [vH], H' = H + - [E . v]. 
 We may notice also that 
 
 (BE') = E 2 - - (v . [EH]), (HH') = H 2 - - (v . [EH]), 
 
 C 
 
 so that r = -i E'E W + H'H K - ? ((EE') + 
 
 is the general expression for the electromagnetic surface force on a moving 
 surface. 
 
 739. If the problem concerns the pressure of radiation on a perfectly 
 absorbing surface in motion, then E, H are simply the vectors of the field of 
 the incident radiation, there being no reflexion. But with a moving mirror 
 things are different ; here the reflexion takes place so that at all points of 
 the surface the tangential components of the electromotive force (E') and 
 the normal components of magnetic induction (H and therefore also of the 
 vector H') must vanish*. Thus at the surface 
 
 H w = 0, [nE'] - 0, 
 
 so that = [E . [E'n]] = E'E n - n (EE'), 
 
 so that we have now 
 
 r = i a (EE' - HH') 
 
 O7T 
 
 Each of these two last formulae determine the surface forces exerted by 
 the electromagnetic field on the moving mirror. Since HJ is a unit vector 
 parallel to the normal to the surface T' is always perpendicular to this surface. 
 The radiation pressure gives no tangential forces on the perfectly reflecting 
 surface. 
 
 * Provided of course the substance of the mirror is perfectly reflecting, so that there is 
 no penetration. 
 
650 The electrodynamics of moving media [CH. xv 
 
 740. Now let us return to our purely optical considerations in moving 
 media. 
 
 The fundamental conception of optical science is the relation between the 
 direction of the ray and that in which the radiant waves are travelling. 
 When the material medium is at rest in the aether, the ray, or the path of 
 the radiant energy, is the same relative to the matter as to the aether. In 
 the circumstances of moving matter there are however two kinds of rays to 
 be distinguished, one of them being the paths of the radiant energy with 
 respect to the particles of the moving matter, the other the absolute path of 
 the radiant energy in the stagnant aether, or as we may say in space. As 
 radiation is revealed to us wholly by its action on matter it is the former 
 type of rays that is of objective importance. 
 
 Let us suppose that a ray from a fixed source passes through a small 
 opening in a screen at 0* and after traversing the distance OP in the free 
 aether arrives at P. An observer situated at P and the screen at may 
 be supposed to have a common velocity v. The opening has then arrived 
 
 *p 
 
 Fig. 103 
 
 at the new position 0' by the time that the radiation which passed through 
 it at has reached the observer at P and tnis observer, supposed ignorant 
 of the motion, would regard O'P as the direction of the ray : this is therefore 
 the direction of the relative ray and it is parallel to the direction of the relative 
 velocity of the actual light and the observer 
 
 c' = c - v. , 
 
 Bradley explained by this simple construction the aberration of the 
 starlight which arises from the yearly motion of the earth : the periodic 
 motion of the earth round the sun gives rise to a periodic change in the 
 direction of the relative ray of light from a star and therefore also to yearly 
 periodic change in the apparent position of the star. 
 
 741. This simple definition of the relative ray can also be obtained by 
 purely electromagnetic reasoning without introducing the conceptions of 
 geometrical optics. 
 
 * For example the object glass of a telescope. 
 
740, 741] The radiation vector 651 
 
 The absolute ray of light is determined by the Poynting vector 
 
 and it gives the flux of energy per unit area through a small surface placed 
 perpendicular to its direction. If this surface be regarded as perfectly 
 black all the energy which is thus transferred is absorbed by the surface. 
 
 The relative flux of energy per unit area through a small surface perpen- 
 dicular to the direction denoted by n and moving with the velocity v is 
 
 S n - ^ (E 2 + H 2 ), 
 
 the second term arising from the fact that the energy provided in the flux 
 S is partly used up in establishing the energy in the ray which gets longer 
 and longer on account of the motion of the surface. Suppose now that the 
 moving surface is perfectly absorbing; then we may regard the component 
 of the relative ray in the direction perpendicular to this surface as defined 
 by the amount of energy absorbed per unit time by the surface and trans- 
 formed into heat. Now only part of the energy which reaches the surface 
 is thus transformed,, the remainder being used up in doing mechanical work 
 as a radiation pressure on the surface : moreover the rate of working of this 
 latter force per unit area is 
 
 - 2E W (vE) + 2H W (v . H) - v n (E 2 + H 2 ) + (v . [EH]), 
 
 so that the component of the relative ray perpendicular to the direction of 
 the surface is determined by 
 
 S n + E n (vE) + H w (vH) - v w (E 2 + H 2 ) + -* (v . [EH]) , 
 
 *T ( ) 
 
 which may be regarded as the component in the same direction of the vector 
 S' = S + -i- IE (vE) + H (vH) - v (E 2 + H 2 ) + - <v . [EH])j, 
 
 47T ( C ) 
 
 which determines the relative ray vector. If we write 
 
 E' = E + i [v . H], H' = H - - [vE], 
 c c 
 
 then we see that 
 
 [E'H'] = [EH] - - [E . [vE]] - - [H . [v . H]] + \ [[vE] . [vH]], 
 c c c 
 
 and [E . [v . E]] - vE 2 - E (vE), 
 
 [H [vH]] = vH 2 - H (vH), 
 [[vE] . [vH]] - v ([vE], H) - H ([vE] . v) 
 = v(v.[EH]), 
 
 so that finally S' = [E', H'], 
 
652 The electrodynamics of moving media [CH. xv 
 
 which is the simplest form of the general expression of the relative ray vector : 
 its component in any direction determines the amount of heat developed per 
 unit time per unit area in a perfectly absorbing surface moving with the 
 appropriate velocity and placed perpendicular to this direction. 
 
 742. It remains to show that for plane waves this definition of the relative 
 ray direction agrees with the elementary rule given in the earlier part of the 
 paragraph. 
 
 For a ]5lane polarised wave train the absolute vector velocity c of propaga- 
 tion, and the vectors of electric and magnetic force intensities are mutually 
 perpendicular to one another : and since the intensities of these forces are 
 equal to one another we have 
 
 Thus E' = - [H . c'], H' = - [c'E], 
 
 so that S' = ^ . i [[He'] . [c'E]] 
 
 so that S' is parallel to the relative velocity of the light to the receiving 
 surface : the elementary rule for the construction of the path is therefore 
 completely in accordance with the electromagnetic theory of these things. 
 
 743. These considerations are confined to cases where the source of light 
 is at rest in the aether and they therefore only apply to such problems as the 
 aberration of light from fixed stars. We have next to enquire whether the 
 motion of a system has any influence on optical phenomena taking place 
 inside it. Such a. question is of practical importance in considerations re- 
 specting ^the effect of the motion of the earth on optical phenomena taking 
 place on its surface. Can we determine the motion of the earth by optical 
 measurements in a laboratory? Let us imagine that at the time t = a 
 light signal is emitted from the point 0. At the instant t it will have arrived 
 at P, the absolute direction of its path being OP, which we can denote by 
 the vector r. In the interval thus occupied by the actual radiation in 
 travelling from to P the source of light will have moved to 0' with the 
 velocity v of the system to which it and the observer at P belong. The 
 radius O'P in the vector sense is given by 
 
 r' - r - v, 
 and of course r = ct, 
 
 so that r' = (c - v) - = - c', r = r I 
 
 c c 
 
741-744] The absolute and relative light paths 653 
 
 Thus the observed direction of the ray is identical with the actual direction 
 of the line joining the instantaneous positions of source and observer. Thus 
 in a uniformly moving system the source of light is seen just where it happens 
 ta be at the instant. The common motion of source and observer cannot 
 therefore be detected by an observation of the direction of the relative ray. 
 We might however still hope to detect this motion by a measurement of the 
 
 Fig. 104 
 
 length of the path of the ray, for the surface through P of constant absolute 
 light distances is a sphere of centre and the point 0' from which the relative 
 ray starts lies eccentric to this sphere so that given lengths r of the absolute 
 light path would correspond to different lengths r' of the relative light path 
 O'P depending on its direction. Now we see from the figure as given 
 
 V 
 
 where ft is used for - -' just as before so that again using /c 2 = 1 /3 2 
 c 
 
 cos 
 
 / 
 
 K 
 
 If now we choose moving axes with origin at P and the x-axis in the direction 
 of motion of the observer and source we can write 
 
 " = v 
 
 where (x 1 ' , y' , z'} are the coordinates of 0' : this relation determines the 
 absolute length of the path in terms of the relative coordinates of source and 
 observer. 
 
 744. Now let us compare with this moving system a stationary system 
 obtained from it by a uniform expansion in the ratio 1 : K parallel to the 
 
The electrodynamics of moving media [OH. xv 
 
 direction of motion, i.e. the #-axis. The analytical transformation is ex- 
 pressed by the relations 
 
 giving the coordinates of the point P ' corresponding to P'. If we use 
 
 then K r = r f + j8& '. 
 
 The coordinates of the absolute effective position of the source in the moving 
 system relative to P are therefore connected with the coordinates of the 
 relative position in the stationary system by the relations 
 
 KX O ' = x- fir, y ' = y, z f = z, 
 and we have also 
 
 KT O ' = r - fix, 
 
 KX = X ' + p^', y = y, z = z '. 
 
 745. Now let us examine the question as to whether a measurement of 
 the length of the relative light path by an observer moving with the system 
 will enable a determination of the motion of the system to be made. The 
 method is to send two parts of the same beam along different paths and then 
 unite them and determine by an interference method whether the one has 
 had to traverse a longer path. Suppose that light is sent along the relative 
 path O'P' and then reflected back by a mirror to 0' '. The length of the 
 absolute path of the incident beam is 
 
 and that of the reflected beam is 
 
 r if-^-r ' Rif-^-^r ' 
 
 1 2 K / pK J/ , 
 
 the sum of the two being 
 
 Now draw a sphere round 0' as centre and of radius r'. For any point P 
 on this sphere the path O'PO' would be the same if the system were at rest, 
 but when it is in motion this is by no means the case. In this case the 
 absolute path is determined not by the relative radius in S but by the corre- 
 sponding length r ' in the expanded system S , 
 
 r '2 
 '2 _i_ '2 _i_ '2 
 
 T = ~K~ 2 J ' ' 
 
 whilst r' 2 = x' 2 + y" 2 + z' 2 . 
 
 Thus a given total relative path corresponds to different lengths of the 
 
 absolute path. If the relative path is parallel to the direction of motion 
 
 2f' 
 
 so that r + r = - 
 
 AC 2 
 
744746] The Fitzgerald-Lorentz contraction 655 
 
 whilst if the relative path is perpendicular to the direction of motion 
 
 r - r, 
 
 2r' 
 
 and r 1 + r.> = , 
 
 K 
 
 Thus with equal relative paths the ratio of the absolute light paths in the two 
 cases is 1 : K greater in the first than in the second. The difference of the 
 light paths is then A/ when 
 
 if we neglect higher powers of j8. 
 
 746. This difference in length of the absolute light paths for a beam 
 propagated over a given length in a moving system parallel and perpen- 
 dicular to the motion was first noticed by Maxwell. It formed the object 
 of investigation in a now famous experiment conducted by Michelson and 
 Morley* with a view to determining the motion of the earth. Two pencils 
 of light coming from the same source are brought to inference after traversing 
 perpendicular paths of equal length, the one perpendicular to the direction 
 of motion of the earth and the other parallel to it. Although the precision 
 attained in the measurements was such that a difference equal to ^ of that 
 expected would have been observed and measured no difference whatever 
 was observed at all which was not within the very close limits of experi- 
 mental error. The conclusion must therefore be drawn that no effect of the 
 kind mentioned can be experimentally detected. 
 
 Thus if we retain the view tacitly assumed throughout that the dimensions 
 of a body are unaltered by imparting a uniform motion to it, the theory as 
 we have expounded it is inconsistent with experience. In order however to 
 surmount this difficulty it is suggested by Fitzgerald f and Lorentz J that we 
 make the further not unnatural assumption that the dimensions of a body 
 in motion are altered in such a way that its diameter parallel to the direction 
 of motion are contracted in the corresponding ratio 
 
 1 or (l - L^j-Y : 1. 
 
 This of course disposes of the difficulty of the Michelson-Morley experiment 
 for it throws the points in the stationary system S which correspond to 
 loci of constant relative paths round 0' on to a sphere and constant relative 
 paths then correspond to constant absolute paths. 
 
 * Phil. Mag. Dec. 1887. Cf. also Morley and Miller, Phil. Mag. 9 (1905). 
 t Cf. Lodge, Phil. Trans. 184 A (1893), p. 727. 
 
 | 'Versuch einer Theorie der elektrischen u. optischen Erscheinungen in bewegters Korpen,' 
 Leiden (1895). 
 
656 The electrodynamics of moving media [CH. xv 
 
 747. However extraordinary this hypothesis may appear at first sight, 
 it must be admitted that it is by no means gratuitous, if we assume that the 
 intermolecular forces act through the mediation of the aether in a manner 
 similar to that which we know to be the case in regard to electric and magnetic 
 forces. If that is so, the translation of the matter will most likely alter the 
 action between two molecules or atoms in a manner similar to that in which 
 it alters the attraction or repulsion between electrically charged particles. 
 As then the form and dimensions of a solid body are determined in the last 
 resort solely by the intensity of the molecular forces, an alteration of the 
 
 dimensions cannot well be left out of consideration. 
 
 i 
 
 In its theoretical aspect there is thus nothing to be urged against the 
 hypothesis. As regards its experimental aspect we at once notice that the 
 elongation or contraction which it implies is extraordinarily minute. It 
 would involve a shortening in the diameter of the earth of about 6| centi- 
 metres. The only experimental arrangements in which it could come into 
 evidence would be just of the type of this one of Michelson's which first 
 suggested it. 
 
 748. Relativity. Before finally closing this work reference must be 
 made to a fundamental property of the electromagnetic equations which is 
 of great importance in the discussions relating to the conditions in moving 
 electrical systems and which co-ordinates in one principle many of the pro- 
 perties of these systems treated separately above. We shall simplify our 
 discussion by assuming that the whole electrodynamic properties of matter 
 can be explained on the basis of a stationary aether and electrons, and we 
 shall so far substantiate these latter elements as to assume that they are 
 extremely small but finite entities consisting essentially and solely of electricity 
 distributed with a finite volume density throughout their small extent, 
 where the fundamental equations are also presumed to hold. In this case 
 the only constituents of the complete current of the theory are its purely 
 aethereal part 
 
 JL-rfE 
 
 4?r ~dt 
 
 and a part due to the convection of the electrons which, at any place, is of 
 density pu> 
 
 p being the density of the charge in the point of the electron passing over it. 
 
 The equations of the'theory referred to a frame of reference fixed in the 
 aether thus assume the form 
 
 . _ IdB 
 
 CUrl H = - -=- + 47TOU 
 
 c at 
 
 IdH 
 curl E = - 
 
 c at 
 
 with div E = 4wp, div H = 0. 
 
747-749] Relativity 657 
 
 It is now at least of philosophical interest to determine how the electrical 
 phenomena in such a system would behave to an observer moving uniformly 
 relative to the fixed frame of reference with the velocity v, which we may, for 
 simplicity, take to be parallel to 'the z-axis of the fixed co-ordinate system. 
 The first step naturally will be to take a set of axes moving with the observer, 
 but parallel to the old ones. When this is done the equations lose their 
 original simplicity; but it is found that the original form can be recovered 
 by a slight modification of the variables*. 
 
 v 2 
 
 749. We write /c 2 = 1 and then put 
 
 c 2 
 
 x' = - (x vt], y' = y, z' = z 
 
 and then if </> is any function of (x, y, z, t) 
 
 _ _ 
 
 ~ dx' ~ c 2 dt' ' 
 
 and ttf 
 
 Thus in terms of the new variables the fundamental equations of Ampere 
 assume the form 
 
 /9E ? _ dE x \ , , 
 
 \dt' ~ V dx' J^ c ~ dy' "" dz' 
 
 _ _ 3BL, 
 
 c A9? " ' ~~ = 
 
 z_ 8E Z \ TT P U Z _ fdH y v 8H V \ 8H,, 
 c \dt' ~ dx'j ^ c K \ dx' c* dt' ) dy f 
 
 The second and third of these can be written in the form 
 
 c dt'~ c dz' dx' 
 
 an; 
 
 c dt c dx' 
 
 where 
 
 H; ** H, + B, , H; - ^ H, - B, 
 
 * Cf . Lorentz, ' Versuch einer Theorie der elektrischen u. optischen Erscheinungen, ' etc. ; 
 Lannor, Aether and Matter, chaps, x and xi; Lorentz, Amsterdam Proc. (1904), p. 809; Einstein, 
 Annalen d. Phys. 17 (1905). 
 
 L. 42 
 
658 The electrodynamics of moving media [OH. xv 
 
 These last equations are equivalent to 
 
 /-I- It' 
 
 If now we substitute these values in the first equation and use also 
 
 E X ' = E Z , H^H* 
 it becomes 
 
 K-I ME.' /3E.' , 3E,' 8E.'\ ) 4pn. _ 85.' _ 9H,/ 
 
 T t?? " H 8^ + a/ + W-)\ 
 
 while the equation di v E = 
 
 becomes under similar circumstances 
 
 Multiplying this by - and adding it to the first equation and remembering 
 
 -.O 
 
 that K z i _ 
 
 &> 
 
 i9E' 4^ _ 
 
 ' ( x ' 
 
 8E,' , _ 
 
 and also 8^ " " "^ "" 82' * c 
 
 750. On treating Faraday's equations in the same way we find that they 
 are equivalent to 3 R 
 
 _ 
 
 c dt 1 ~ d' ' dz' 
 
 8E' 
 
 c dt' ~ dx' dy' 
 together with the conditional equation 
 
 SH; 8H V ' 8H; _ 
 
 -5 r T a_/ "or/ u - 
 
 OX a?/ (72 
 
 Thus finally if we write 
 
 Ua. V _ MX 
 
 ^ ^ i ^L* ~ dt ' 
 
 L ~~" 
 
 c 
 
 ' - KU * -^ 
 " \ E *' 
 
 X 
 
 c 
 
 ' = KU ^ = ^ 
 Uz " vu x dt' 
 
 c 
 
749-752] The theorem of relativity 659 
 
 j / _ _! (-i vu x\ _ dt' 
 
 the original form of the equations is completely restored. 
 
 Moreover since the determinant of the transformation is unity and the 
 transformation itself is an orthogonal one we must have 
 
 dx'dy'dz'dt' = dx dy dz dt 
 so that also p > dx > dy > dz > = pdxdy dz 
 
 or the charges in corresponding elements of volume in the two systems are 
 the same. 
 
 Thus if we transform any electrical system S in the manner specified above 
 into another system S' in such a way that the charges in corresponding 
 volume elements in the two systems are the same, the electromagnetic equa- 
 tions defining the sequence of events in the system remain unaltered, so long 
 as the vectors E', H' in S' correspond respectively to the vectors E, H in S. 
 This mathematical law is called by Minkowski the 'Theorem of Relativity.'* 
 
 751. According to the transformation as a mere geometrical correspon- 
 dence, the length of a line in the direction of the axis of x, measured in the 
 co-ordinates (x', y' , z'}, is greater than its length measured in the co-ordinates 
 (x, y, z) in the ratio 1 : K. Thus the Fitzgerald-Lorentz hypothesis of the 
 reduction in the dimensions of a body when it moves relatively to an observer 
 is reduced by this geometrical transformation to the assumption that in the 
 variables associated with it, the shape of the body is unaltered. 
 
 A similar result holds as regards the time. Suppose we have a clock 
 moving with the origin of the (x 1 ' , y' , z'} system of co-ordinates : then x' 
 and so x = vt. Thus if t^', t 2 ' are the times of two consecutive events as 
 recorded on the moving clock and t lt t 2 the times for the same events as 
 registered in the fixed system, then 
 
 K-*(t 2 ' -ti) = t 2 -t v 
 
 We may for instance take t z ', t^ to represent the two consecutive hour strokes 
 of the clock. It is then clear from the equation above that the moving 
 clock as observed from the fixed system will appear to have its periodic time 
 increased in the ratio 1 : K. The frequency will be decreased in the inverse 
 ratio. 
 
 752. Let us| now imagine an observer whom we shall call A and to 
 whom we shall assign a fixed position in the aether, to be engaged in the study 
 of the phenomena in the stationary electromagnetic system S. We shall 
 suppose him to be provided with a measuring rod and a clock. By these 
 
 * Cf . H. Minkowski, ' Zwei Abhandlungen iiber die Grundgleichungen der Elektrodynamik ' 
 (Leipzig, 1910); the first appeared in Oottinger Nachr. (maih.-phys. Kl.}, 1907. 
 f Cf. Lorentz, Theory of Electrons, ch. v. 
 
 422 
 
660 The electrodynamics of moving media [CH. xv 
 
 means he will be able to determine the co-ordinates (x, y, z) of any point 
 and the time t for any instant, and by studying the electromagnetic field 
 as it manifests itself to him at different places he will be led to the intro- 
 duction of the vectors E and H and the fundamental equations connecting 
 them. * 
 
 Let A' be a second observer, whose task it is to examine the phenomena 
 in the system S', and who himself also moves through the aether with the 
 velocity v, without being aware of his motion or that of the system S r . Let 
 this observer also be provided with a similar measuring rod, which will how- 
 ever be contracted on account of its possessing the motion v, and a clock 
 working properly to his time. If this observer study the electromagnetic 
 phenomena in the system S' he will introduce the vectors E', H' and find 
 that they are connected by exactly the same equations as were the vectors 
 used by the observer A in the stationary system. 
 
 Thus if both A and A' were to keep a record of their observations and the 
 conclusions they draw from them, these records would, on comparison, be 
 found to be identical. In other words neither observer can detect, by an 
 examination of circumstances in his own system, whether he is in motion or 
 at rest. 
 
 753. Thus far it has been the task of each observer to examine the con- 
 ditions in his own system. Let us now suppose that each observer is able 
 to see the system to which the other belongs and to study the phenomena 
 in it. The observer A, in studying the electromagnetic field in S' will be 
 led to introduce the new variables x',y',z',t'\ E', H', etc., and so will 
 establish our relative equations which are exactly those deduced directly 
 by A'. On the other hand A', in studying the field of S, would introduce 
 new variables (x, y, z, t) defined by the reversed relations (the v to him is 
 now negative) and eventually he would deduce the original equations found 
 by A. 
 
 Thus the impressions received by the two observers in examining their 
 own or any other electromagnetic system would be exactly identical, so that 
 in reality nsi ther could be sure of his own or the other's state of motion : all 
 that they know is that their relativity velocity is v. We conclude that if 
 the transformation mathematically defined above has any truth in fact 
 then no measurements of electromagnetic conditions will ever enable anyone 
 to determine whether he is in motion or not, except relatively to other bodies. 
 The idea of absolute motion, frequently employed in our discussions, thus 
 proves to be immaterial to our physical theories. 
 
 754. The conclusion just stated can hardly be avoided on purely philo- 
 sophical grounds, and if it is granted as intuitive then a sufficient scientific 
 reason is provided for the general validity of the transformation on which 
 
752-754] Einstein's principle 661 
 
 it was based. In fact Einstein*, starting from the philosophical principle 
 that absolute motion is physically indeterminate, derives the transformation 
 by purely mathematical considerations. As a consequence of the principle 
 he assumes that the velocity of propagation of light in any direction referred 
 to any axes must be the same and then proceeds to determine what amount 
 of arbitrariness in the space and time variables is consistent with this assump- 
 tion. If the changes between the variables from the fixed to a moving system 
 is linear and such that a point at rest in the fixed system corresponds to a 
 point moving with a uniform velocity v parallel to the direction of the z-axis 
 in the other, then we are limited to transformations of the type 
 
 x' = k (x vt), y' = ly, z' = Iz 
 
 t' = ax + fiy + yz -f 8t. 
 The analytical criterion is that the equation 
 
 dx z + dy* + dz 2 - c 2 dt 2 = 
 must lead to the equation 
 
 dx' 2 + dy' 2 + dz' 2 - cW 2 - 
 
 and it is easy to verify that the only forms of the above transformation are 
 those in which 
 
 a- -3 
 
 C 2 K 
 
 The transformation thus only differs from that adopted above by the presence 
 of the arbitrary constant factor I; this can however easily be reduced to 
 unity by a proper choice of units. 
 
 A full discussion of the developments of this theory is quite beyond the 
 scope of the present work. There are however several treatises f specially 
 devoted to the subject and the reader who desires to pursue the subject is 
 referred to these for further information. He will find in them much elegant 
 mathematical analysis and several important applications of the fundamental 
 principle to the dynamics of moving electrical systems ; to radiation problems, 
 and finally also to the fundamental problem of gravitation. 
 
 * Ann. d. Phys. 17 (1905). 
 
 f Cf. M. Lane, Das Selativitdtsprinzip (Brunswick, 1913); E. Cunningham. The Principle 
 of Relativity (Cambridge, 1914); L. Silberstein. The Theory of Relativity (London, 1914). 
 
I. ELECTROSTATICS 
 
 1. Charges 4q, - q are placed at the points A lt A 2 and B is the point of equilibrium. 
 Prove that the line of force which passes through B meets A 1 A 2 at an angle of 60 at A 
 and at right angles at B. Find the angle at A 1 between A V A Z and the line of force which 
 leaves A 2 at right angles to A-^A^, 
 
 2. Two positive charges q 1 and q 2 are placed at the points A 1 and A z respectively. 
 Show that the tangent at infinity to the line of force which starts from q l making an angle 
 a with A 2 A 1 produced, makes an angle 
 
 2 sin- 1 . 
 
 ?i + ? 2 
 
 with A z A lt and passes through B in A^A^ such that 
 
 A^B : BA Z = q z : q 1 . 
 
 3. Point charges + q, - q are placed at the points A lt A 2 . The line of force which 
 leaves A l making an angle a with A 1 A 2 meets the plane which bisects AA Z at right angles 
 in P. Show that 
 
 .a fa . 
 
 in- = V2si 
 
 sin 
 
 4. The potential is given at four points near each other and not all in one plane. 
 Obtain an approximate construction for the direction of the field in their neighbourhood. 
 
 5. The potentials a_the four corners of a small tetrahedron A^A Z A 3 A^ are <p lt <p 2 , $ 3 , 
 4 respectively. G is the centre of gravity of masses m lt m z m 3 , m 4 at A lt A z , A 3 , A 
 respectively. Show that the potential at O is 
 
 6. Charges 3q, - q, - q are placed at A lt A 2 , A 3 respectively, where A z is the middle 
 point of A 1 A 3 . Draw a rough diagram of the lines of force: shew that a line of force 
 which starts from A 1 making an angle a with A t A 2 > cos~ 1 (- ^) will not reach A z or 
 A 3 , and show that the asymptote of the line of force for which a = cos" 1 (- f ) is at right 
 angles to A t A 3 . 
 
 7. If there are three electrified points A l , A 2 , A 3 in a straight line, such that A t A 3 = f, 
 A Z A 3 = -j, and the charges are q, - -~ and <a respectively, show that there is always 
 a spherical equipotential surface, and discuss the position of the points of equilibrium on 
 
 the line A^A 3 when < = g;/ + a and when = q "" 
 
 - 
 
 . 
 a ) (J + a ) 
 
Electrostatics 663 
 
 8. A negative point charge - q z lies between two positive point charges q l and q 3 
 on the line joining them and at distances a t , a 2 from them respectively. Show that, if 
 the magnitudes of the charges are given by 
 
 <h = 9, = J^_ and a K x , 
 
 a 2 <*! a 1 + a 2 
 
 there is a circle at every point of which the force vanishes. Determine the general form 
 of the equi-potential surface on which this circle lies. 
 
 9. Charges of electricity q v , -q 2 ,q 3 (?a > ?i) are placed in a straight line, the negative 
 charge being midway between the other two. Show that, if 4q z lie between (q^ - q^fi 
 and (q$ + qfy 3 , the number of unit tubes of force that pass from q^ to q 2 is 
 
 i (?1 +'.fc - ft) + P| (<?3 f - ?!*) (if - 2V + ft 1 )*- 
 
 10. Three infinite parallel wires cut a plane perpendicular to them in the angular 
 points A lt A 2 , A 3 of an equilateral triangle, and have charges q, q, - q' per unit length 
 respectively. Prove that the extreme lines of force which pass from A to A 3 make at 
 
 starting angles ^ * IT and rr with A^A^, provided that q' ^> 2q. 
 
 11. Draw the equi-potential surfaces when the field is that due to two small conductors 
 with charges q and 4q placed at a distance 3a apart, and find the capacity of a conductor 
 in the form of the equi-potential surface, whose potential is A<?/a, where X< 3. 
 
 12. If the algebraic sum of the charges on a system of conductors be positive, then 
 on one at least the surface density is everywhere positive. 
 
 13. Two conductors carrying respectively a positive charge q l and a negative charge 
 q 2 are introduced into the interior of a larger conductor maintained at potential $; show 
 that if <h > q 2 the potential of the first conductor is greater than 0. 
 
 14. There are a number of insulated conductors in given fixed positions. The capacities 
 of any two of them in their given positions are C and C 2 , and their mutual coefficient 
 of induction is B. Prove that if these conductors be joined by a thin wire, the capacity 
 of the combined conductor is Q + fj + ^B 
 
 15. A system of insulated conductors having been charged in any manner, charges 
 are transferred from one conductor to another till they are all brought to the same potential 
 <. Show that = Qfa + 2s 2 ), 
 
 where *, , s 2 are the algebraic sums of the coefficients of capacity and induction respectively 
 and Q is the sum of the charges. 
 
 16. Two Leyden jars of capacities C, C' are placed at such a distance apart that their 
 mutual induction is negligible. The outer coating of the former is connected to a distant 
 charged conductor of vary large capacity which maintains its potential at a constant 
 value 0, whilst that of the latter is earthed. A charge Q is given to the inner coating of 
 the former and the knobs are then connected by a wire. Show that the total dissipation 
 
 of energy due to the discharge is 
 
 C' (Q + C<j>)* 
 ~2C (C + C')' 
 
 17. Prove that the effect of the operation described in the last question is a decrease 
 of the electrostatic energy equal to what would be the energy of the system if each of the 
 original potentials were diminished by 0. 
 
664 Appendix 
 
 18. Two equal similar condensers, each consisting of two spherical shells, radii a, b, 
 are insulated and placed at a great distance r apart. Charges q-^ and q 2 are given to the 
 inner shells. If the outer surfaces are now joined by a wire, show that the loss of energy 
 is approximately 
 
 19. Four equal uncharged conductors are placed symmetrically at the corners of 
 a regular tetrahedron, and are touched in turn by a moving spherical conductor at the 
 points nearest to the centre of the tetrahedron, receiving charges q lt q 2 , q 3 , q t . Show 
 that the charges are in geometrical progression. 
 
 Replace the word 'tetrahedron' by 'square' and prove that 
 
 (?i - ?s) 
 
 20. Show that if the distance x between two conductors is so great compared with 
 the linear dimensions of either, that the square of the ratio of these linear dimensions to 
 
 CC" 
 x may be neglected, then the coefficient of induction between them is - , where C, C' 
 
 are the capacities of the conductors when isolated. 
 
 21. Prove (i) that if a conductor, insulated in free space and raised to unit potential, 
 produce at any external point P a potential denoted by (P), then a unit charge placed 
 at P in the presence of this conductor uninsulated will induce on it a charge - (P) ; 
 
 (ii) that if the potential at a point Q due to the induced charge be denoted by (P, Q), 
 then (P, Q) is a symmetrical function of the positions of P and Q. 
 
 22. Two small uninsulated spheres are placed near together between two large parallel 
 planes, one of which is charged, and the other connected to earth. Show by figures the 
 nature of the disturbance so produced in the uniform field, when the line of centres is 
 (i) perpendicular, (ii) parallel to the planes. 
 
 23. A portion P of a conductor, the capacity of which is C, can be separated from 
 the conductor. The capacity of this portion, when at a long distance from other bodies, 
 is c. The conductor is insulated, and the part P when at a considerable distance from the 
 remainder is charged with a quantity q and allowed to move under the mutual attraction 
 upon it ; describe and explain the changes which take place in the electrical energy of the 
 system. 
 
 24. A hollow conductor A is at zero potential, and contains in its cavity two other 
 insulated conductors, B and C, which are mutually external: B has a positive charge, 
 and C is uncharged. Analyse the different types of lines of force within the cavity which 
 are possible, classifying with respect to the conductor from which the line starts, and the 
 conductor at which it ends, and proving the impossibility of the geometrically possible 
 types which are rejected. 
 
 Hence prove that B and 'C are at positive potentials, the potential of C being less than 
 that of B. 
 
 25. A conductor having a charge Q l is surrounded by a second conductor with charge 
 Q t . The inner is connected by a wire to a very distant uncharged conductor. It is then 
 disconnected, and the outer conductor connected. Show that the charges Q, Q z ' are now 
 
 , _ mQ 1 - nQ 2 . , = (m 
 
 1 ' 
 
 n) 
 
 m + n + mn' m + n 
 
 where C, C ( I + m) are the coefficients of capacity of the near conductors, and Cn is the 
 capacity of the distant one. 
 
Electrostatics 665 
 
 26. The inner sphere of a spherical condenser (radii a, b) has a constant charge Q, 
 and the outer conductor is at zero potential. Under the internal forces the outer conductor 
 contracts from radius b to radius b 1 . Prove that the work done by the electric forces is 
 
 1 Q2 b ~ b l 
 
 661 ' 
 
 27. Two equal and similar conductors A and B are charged and placed symmetrically 
 with regard to each other; a third moveable conductor C is carried so as to occupy 
 successively two positions, one practically wholly within A, the other within B, the positions 
 being similar and such that the coefficients of potential of C in either position are p, q, r 
 in ascending order of magnitude. In each position C is in turn connected with the con- 
 ductor surrounding it, put to earth, and then insulated. Determine the charges on the 
 conductors after any number of cycles of such operations, and show that they ultimately 
 lead to the ratios 
 
 1 : - j8 : /3 2 - 1, 
 where /3 is the positive root of 
 
 * rx 2 - -qx + p - r 0. 
 
 28. Three conductors A lt A 2 , A 3 are such that A 3 is practically inside A z . A l is 
 alternately connected* with A z and A 3 by means of a fine wire, the first contact being with 
 A 3 . AI has a charge Q initially, A z , A 3 being uncharged. Prove that the charge on 
 A after it has been connected n times with A is 
 
 f a(y-0)/a + 0\- 1 ) 
 | ^ (a + y) V + y) } 
 
 a + 
 where a, /3, y stand for p u - p V2 , p 2Z - p 12 and p 3S - p 12 respectively. 
 
 29. Two equal and similar conductors are placed symmetrically with regard to each 
 other, one of them being uncharged. Another insulated conductor is made to touch them 
 alternately in a symmetrical manner, beginning with the one which has a charge. If 
 <7i 1z be their charges when it has touched each once, show that their charges, when it has 
 touched each r times, are respectively 
 
 and 
 
 ft 1 
 
 - ? 2 
 
 30. Two closed equi-potentials <f> lf <po are such that (f) 1 contains $o, an d <pp is the 
 potential at any point P between them. If now?a charge Q be put at P, and both equi- 
 potentials be replaced by conducting shells and earth connected, then the charges Q v Q 
 induced on the two surfaces are given by 
 
 Q, Qo Q 
 
 31. A condenser is formed of two thin spherical shells of radii a, b. A small hole 
 exists in the outer sheet through which an insulated wire passes connecting the inner sheet 
 with a third conductor of capacity c, at a great distance r from the condenser. The outer 
 sheet of the condenser is put to earth, and the charge on the two connected conductors is 
 Q. Prove that approximately the force on the third conductor is 
 
 ab 
 
 r^b 
 
 32. Two insulated fixed condensers are at given potentials when alone in the electric 
 field and charged with quantities Q lt Q 2 of electricity. Their coefficients of potential are 
 
666 Appendix 
 
 Pu> PiZ'-Pzz- But if tnev are surrounded by a spherical conductor of very large radius 
 R at potential zero with its centre near them, the two conductors require charges Q^, Q 2 ' 
 
 to produce the given potentials. Prove, neglecting -, that 
 
 Ql ~ Ql _ P22 - Pl2 
 
 Qz - Qz PU - PIZ 
 
 33. Show that the locus of the positions, in which a unit charge will induce a given 
 charge on a given uninsulated conductor is an equi-potential surface of that conductor 
 supposed freely charged. 
 
 34. A condenser consists of a spherical shell of internal radius b, surrounding a con- 
 centric sphere of radius a. If the thickness t of the shell is such that a, b, b + t are in 
 harmonic progression, and if it is kept at a constant potential while the sphere is connected 
 to earth, show that the presence of a small distant sphere of radius r, also connected to 
 
 rt 
 earth, will increase the charge of the shell by -=^ f itself, R being the distance between the 
 
 centres. 
 
 35. With the usual notation, prove that 
 
 Pll + P23>Pl2 + Pl3, 
 PllP23 
 
 36. A, B, C are three condensers at large distances from each other, which can be 
 connected by wires. Their charges are originally q l , q z , q 3 . A and B are then connected, 
 and after they are disconnected, B is connected with C. If after this the charges on A 
 and C are q^ and q 3 , prove that the capacity of B : to the capacity of C 
 
 37. Show that if #., p n , p sg be three coefficients before the introduction of a new 
 conductor, and p' rr , p' rg , p' 8s the same coefficients afterwards, then 
 
 (P-rrPts - P'rrP'ss) < (Prs ~ P'rs) Z - 
 
 38. A, B, C, D are four conductors, of which B surrounds A and D surrounds C. 
 Given the coefficients of capacity and induction 
 
 (i) of A and B when C and D are removed, 
 (ii) of C and D when A and B are removed, 
 (iii) of B and D when A and C are removed, 
 determine those for the complete system of four conductors. 
 
 39. A system of conductors consists of three, C lt C 2 , C 3 ; and C : completely surrounds 
 C 2 . If C 2 were annihilated the coefficients of potential of C^ and C 3 would be P u , P 13 , P 33 ; 
 and if C 3 were annihilated the coefficients of potential of C 1 and C 2 would be or 11( O7 12 , w zz . 
 Show that the actual coefficient of potential of C 2 on itself is 
 
 p n ~ &11 + 22> 
 and write dow,ii the remaining coefficients of potential of the system. 
 
 40. If one conductor contains all the others, and there are (n + 1) in all, shew that 
 there are (n + 1) relations between either the coefficients of potential or the coefficients 
 of induction, and if the potential of the largest be $ , and that of the others (^ <f> 2 , ... $ n> 
 then the most general expression for the energy is W(f) z increased by a quadratic function 
 of 0j - <f) , <f) 2 -0 , ... n - <; where C is a definite constant for all positions of the 
 inner conductors. 
 
Electrostatics 667 
 
 41. Two conductors are of capacities C t and C 2 when each is alone in the field. They 
 are both in the field at potentials fa and fa respectively, at a great distance r apart. Prove 
 that the repulsion between the conductors is 
 
 G-iC* (n/h - C 2 fa) (rfa - C^fa) 
 
 As far as what power of - is this result accurate? 
 
 42. A system of conductors consists of one large conductor and n conductors whose 
 dimensions and distances from the large conductor and from each other are small compared 
 with the dimensions of the large conductor. Prove that if fa is the potential of the large 
 conductor, fa, fa, ... tf) n the potentials of the other conductors, the potential energy of 
 the system is approximately 
 
 Discuss the significance of this result relatively to the theory of electrostatic measurements. 
 
 43. An electrometer consists of two fixed similar conductors A and B, together with 
 a conductor C which can turn round an axis placed symmetrically with respect to A and 
 B. The position of C is specified by the angle B between its axis of symmetry and a line 
 which is symmetrically situated with respect to A and B; in any position, C is acted on 
 by a restoring couple <B towards the symmetrical position (6 = 0). From the following 
 data, valid for positive values of 6, find the position of equilibrium of C when A, B and C 
 are insulated and at potentials fa, fa and fa respectively: 
 
 A and B earthed, C at unit potential, charge on 
 
 C = a- ad 2 , 
 
 A and B earthed, C at unit potential, charge on 
 
 A = - b - ftd, 
 B and G earthed, A at unit potential, charge on 
 
 A = C + &e and on B = - d - yd 2 . 
 
 Show further that for given values of fa and fa, if these are small compared with fa, the 
 maximum displacement of C occurs when 
 
 fa = \/K/a. 
 
 44. Two small pith balls, each of mass m, are connected by a light insulating rod. 
 The rod is supported by parallel threads, and hangs in a horizontal position in front of 
 an infinite vertical plane at potential zero. If the balls when charged with q units of 
 electricity are at a distance a from the plate equal to half the length of the rod, show that 
 the inclination d of the strings to the vertical is given by 
 
 1-K-M. 
 
 45. An uninsulated conducting sphere is under the influence of an external electric 
 charge; find the ratio in which the induced charge is divided between the parts of its 
 surface in direct view of the external charge and the remaining part. 
 
 46. A conducting surface consists of two infinite planes which meet at right angles, 
 and a quarter of a sphere of radius a fitted into the right angle. If the conductor is at 
 zero potential, and a point charge q is symmetrically placed with regard to the planes and - 
 the spherical surface at a great distance / from the centre, show that the charge induced 
 on the spherical portion is approximately - 5qa 3 /irf 3 . 
 
668 Appendix 
 
 47. . If two infinite plane uninsulated conductors meet at an angle of 60, and there 
 is a charge q at a point equi-distant from each, and distant r from the line of inter- 
 section, find the electrification at any point of the planes. Show that at a point in a 
 principal plane through the charged point at a distance r \/3 from the line of intersection, 
 the surface density is 
 
 49. An infinite conducting plane at zero potential is under the influence of a charge 
 of electricity at a point 0. Show that the charge on any area of the plane is proportional 
 to the angle it subtends at O. 
 
 49. A charged particle is placed in the space between two uninsulated planes which 
 intersect at right angles. Sketch the sections of the equi-potentials made by an imaginary 
 plane through the charged particle, at right angles to the planes. 
 
 Supposing the particle to have a charge q and be equi-distant from the planes, show 
 that the total charge on a strip, of which % one edge is the line of intersection of the planes, 
 and of which the width is equal to the distance of the particle from this line of intersection, is 
 
 -}<? 
 
 50. Two equal parallel plates of area A are connected by a wire, and an equal thin 
 plate with a charge Q is placed between them at distances a, b from them. Prove that 
 it is repelled from the nearer one with a force 
 
 a - b\ 
 
 51. Within a spherical hollow in a conductor connected to earth, equal point charges 
 q are placed at equal distances / from the centre, on the same diameter. Shew that each 
 is acted on by a force equal to 
 
 4a"/ 
 
 _ 
 
 4f 
 
 52. An uncharged insulated conductor formed of two equal spheres of radius a cutting 
 one another at right angles, is placed in a uniform field of force of intensity E, with the 
 line joining the centres parallel to the lines of force. Prove that the charges induced on 
 the two spheres are JgkEa 2 and - ^-Ea 2 . 
 
 53. A point charge q is brought near to a spherical conductor of radius a having a 
 charge Q. Show that the particle will be repelled by the sphere, unless its distance from 
 
 the nearest point of its surface is less than A/ft' approximately. 
 
 * v V 
 
 54. A hollow conductor has the form of a quarter of a sphere bounded by two per- 
 pendicular diametral planes. Find the image of a charge placed at any point inside. 
 
 55. Show that the image of a doublet of moment m in a sphere is a doublet of strength 
 
 3- at the inverse po t int together with a distinct charge y- at the same point, where 
 c c 
 
 m x denotes the component of m 1 along the direction of the diameter through the point. 
 Interpret these results when the initial doublet is at a great distance from the sphere. 
 
 56. An infinite plate with a hemispherical boss of radius a is at zero potential under 
 the influence of a point charge q on*the axis of the boss distant / from the plate. Find 
 the surface density at any point of the plate, and show that the charge is attracted towards 
 the plate with a force 
 
 q z 4? 2 a 3 f 3 
 
Electrostatics 669 
 
 57. A conductor is formed by the outer surfaces of two equal spheres, the angle between 
 their radii at a point of intersection being 27T/3. Show that the capacity of the conductor 
 so formed is 
 
 5 V3 - 4 
 
 7IZ Gtj 
 
 2v'3 
 where a is the radius of either sphere. 
 
 58. A conducting plane has a hemispherical boss of radius a, and at a distance / from 
 the centre of the boss and along its axis there is a point charge q. If the plane and the 
 boss be kept at zero potential, prove that the charge induced on the boss is 
 
 f f 2 - a 2 ) 
 
 - q \\ - J [ . 
 
 59. A conductor is bounded by the larger portions of two equal spheres of radius 
 a cutting at an angle ^TT, and of a third sphere of radius c cutting the two former ortho- 
 gonally. Show that the capacity 6f the conductor is 
 
 c + a (f - f V3) - ac {2 (a 2 + c 2 ) ~i - 2 (a 2 + 3c 2 ) ~* + (a 2 + 4c 2 ) ~*} . 
 
 60. A conductor formed by the outer portions of two equal spheres cutting at right 
 angles is placed in a uniform field of force of strength E parallel to the line of centres. 
 Show that the external field of the charge induced on the spheres is the same as that of 
 
 the field due to doublets of strength Ea 3 at the centres and 3- midway between them 
 
 Ea? Ea 3 
 
 together with charges ---- at one centre and --- at the other. 
 c . c 
 
 Work out the corresponding results when the spheres are unequal. 
 
 61. A conducting spherical shell of radius a is placed, insulated and without charge, 
 in a uniform field of force of intensity E. Show that if the sphere be cut into two hemi- 
 spheres by a plane perpendicular to the field, these hemispheres tend to separate and 
 require forces equal to fya z E z to keep them together. 
 
 62. A conducting sphere of radius a is electrified to potential (f). If the sphere consists 
 of two separate hemispheres, shew that the force between them is $ 2 ; and if the whole 
 be surrounded by an uninsulated concentric spherical conductor of internal radius b and 
 the potential of the solid sphere is still (f>, prove that the force between the hemispheres is 
 
 8 (b - a) 2 ' 
 
 63. If a non-conducting portion of an electrostatic field suddenly becomes conducting, 
 discuss the effect upon the electrostatic energy. If there was previously only one con- 
 ductor how is its capacity affected by the change ? 
 
 64 . Two spherical insulated conducting shells of radius a are placed with their centres 
 at a distance h apart, and carry equal charges <?;, within one of them there is situated a 
 concentric spherical conductor of radius b carrying a charge q'. Show that, if a/h is small, 
 the joining of the equal shells by a wire of negligible capacity will result in a loss of electro- 
 static energy whose measure is approximately 
 
670 Appendix 
 
 65. If a particle charged with a quantity q of electricity be placed at the middle 
 point of the line joining the centres of two equal spherical conductors kept at zero potential, 
 show that the charge induced on each sphere is 
 
 - 2qm (1 - m + ra 2 - 3ra 3 + 4m 4 ), 
 
 neglecting higher powers of m, which is the ratio of the radius to the distance between 
 the centres of the spheres. 
 
 66. Two insulating conducting spheres of radii a, b, the distance c of whose centres 
 is large compared with a and b, have charges Q l , Q 2 respectively. Show that the potential 
 energy is approximately 
 
 67. Show that the force between two insulated spherical conductors of radius a placed 
 in an electric field of uniform intensity E perpendicular to their line of centres is 
 
 c being the distance between their centres. 
 
 68. Two uncharged insulated spheres, radii a, b, are placed in a uniform field of force 
 so that their line of centres is parallel to the lines of force, the distance c between their 
 .centres being great compared with a and b. Prove that the surface density at the point 
 at which the line of centres cuts the first sphere (a) is approximately 
 
 E^ ( 6ft 3 i^jjs 28a 2 6 3 57a 3 6 3 
 
 47T f + c 3 "* c 4 c 5 ~J~ 
 
 69. Assuming that the force on an element df of a conductor is 2Tro- 2 df, show that 
 if a conductor with a charge Q is slightly deformed so that the outward normal displacement 
 of the element df is ATI, the alteration in the capacity C is given by 
 
 1 Q 2 ( 
 
 2%2 8C = I 27TO- 2 Att . df. 
 
 A conductor has the form of a circular cylinder with hemispherical ends; show that 
 if the length / is small the capacity is a + |7, where a is the radius. 
 
 70. A prolate spheroid, semi-axes a, b, has a charge Q of electricity. Show that 
 the repulsion between the two halves into which it is divided by its diametral plane is 
 
 4 (a 2 - 6 2 ) 6 6' 
 
 71. One face of a condenser is a circular plate of radius a: the other is a segment 
 of a sphere of radius R, R being so large that the plate is almost flat. Show that the 
 capacity is %R log (tjtg), where t lt t are the thickness of the dielectric at the middle and 
 edge of the condenser. Determine also the distribution of the charge. 
 
 72. Prove that if v is a solution of Laplace's equation the capacity of the condenser 
 formed by the conducting surfaces v = a, v = /3 separated by a heterogeneous dielectric 
 whose specific inductive capacity is / (v) is 
 
 where the integral in the numerator is an area integral over the surface v = a and 8n is 
 an element of the normal measured into the dielectric. 
 
Electrostatics 671 
 
 Prove that the function v which is determined in terms of Cartesian coordinates by the 
 relation 
 
 tanh v = 2ax/(x 2 + y 2 + a 2 ) 
 
 satisfies Laplace's equation, and ascertain what surfaces are represented by the equation 
 v = const. 
 
 Complete the integrations in this case when / (v) is e v . 
 
 73. A condenser is formed of the two prolate spheroids 
 
 *-* 
 
 z 2 r 2 
 
 and -= + j- = 1 - p., 
 
 a 2 b 2 
 
 where /* is very small. Show that its capacity is 
 
 29z>2 1 02 1 I a oZ 
 
 tQ i ~- & _ 1-rC C 
 
 where 
 
 74. The surface of a conductor being one of revolution whose equation is 
 
 4 1. _ 1_ 
 
 r + r'~ 12' 
 
 where r, r' are the distances of any two fixed points at distance 8 apart, find the electric 
 density at either vertex when the conductor has a given charge. 
 
 75. The curve 
 
 1 9af a + x a - x \ _ 1 
 
 when rotated round the axis of a, generates a single closed surface, which is made the 
 bounding surface of a conductor. Show that its capacity will be a, and that the surface 
 density at the end of the axis will be q/3ira 2 , where q is the total charge. 
 
 76. An uninsulated conductor consisting of two equal spheres in contact is under 
 the influence of a charge q at a point in the tangent plane at the point of contact. Prove 
 
 (702 
 
 that the charge is attracted with a force --, where C is the capacity of the conductor 
 and / is the distance of the charge from the point of contact of the spheres. 
 
 77. A conducting sphere of radius a is in contact with an infinite conducting plane. 
 Show that if a unit charge be placed beyond the sphere and on the diameter through the 
 point of contact at distance c from that point, the charges induced on the plane and sphere 
 are 
 
 no, ira , ira . net , 
 
 - cot and cot 1. 
 
 c c c c 
 
 78. Show that the capacity of a spherical conductor of radius a with its centre at a 
 distance c from an infinite conducting plane is 
 
 a sinh a 2 cosech na, 
 i 
 
 where c = a cosh a. 
 
672 Appendix 
 
 79. . An insulated conducting sphere of radius a is placed midway between two parallel 
 
 /a\ 2 
 infinite uninsulated planes at a great distance 2c apart. Neglecting ( - ) , show that the 
 
 . \ c / 
 
 capacity of the gphere is approximately 
 
 ail + -log 2J. 
 ( c ) 
 
 80. Two spheres of radii a, b touch each other, and their capacities in this position 
 are c, d. Show that 
 
 where / = j . 
 
 a + b 
 
 81. If the centres of the two shells of a spherical condenser be separated by a small 
 distance d, prove that the capacity 'is approximately 
 
 ab f, abd 2 
 
 b - a { (b - a) (b 3 - a 3 )] 
 
 82. A condenser is formed of two spherical conducting sheets, one of radius 6 
 surrounding the other of radius a. The distance between the centres is c, this being so 
 small that (c/a) 2 may be neglected. The surface densities on the inner conductor at the 
 extremities of the axis of symmetry of the instrument are a- 1 , <r 2 and the mean surface 
 density over the inner conductor is a. Prove that 
 
 er 2 ~ f\ 
 
 83. The equation of the surface of a conductor is r = a (1 + aP n ), where a is very 
 small, and the conductor is placed in a uniform field of force E parallel to the axis of 
 harmonics. Show that the surface density of the induced charge, at any point is greater 
 than it would be if the surface were perfectly spherical, by the amount 
 
 + l)JVn + (n-2)P n _ 1 }. 
 
 84. The'conductor of the last question is uninsulated. Show that the charge induced 
 on it by a unit charge at a distance / from the origin and of angular coordinate 6, is 
 approximately 
 
 * 85. An uninsulated conducting sphere is placed in a field whose potential before the 
 introduction of the sphere was 
 
 <j> = 2A n r"P n . 
 
 o 
 
 Determine the charge distribution induced on the sphere and prove that there is a force 
 urging it in the positive direction of the polar axis of amount 
 
 86. A circular disc of radius a is under the influence of a charge q at a point in its 
 plane at distance 6 from the centre of the disc. Show that the density of the induced 
 distribution at a point on the disc is 
 
 q /b 2 - a 2 
 
 2n 2 R 2 V a 2 - r 2 ' 
 where r, R are the distances of the point from the centre of the disc and the charge. 
 
Electrostatics 673 
 
 87. An ellipsoidal conductor differs but little from a sphere. Its volume is equal 
 to that of a sphere of radius a, its axes are 2a (1 + aj), 2a (1 + a 2 ), 2a (1 + a 3 ). Show 
 that, neglecting cubes of a lt a 2 , a 3 , its capacity is 
 
 88. Electricity is induced on an uninsulated spherical conductor of radius a, by a 
 uniform distribution, density a-, over an external concentric non-conducting spherical 
 segment of radius c. Prove that the surface density at the point A of the conductor at 
 the nearer end of the axis of the segment is 
 
 c(c + a)( AB\ 
 
 -^- I -' 
 
 where B is the point of the segment on its axis, and D is any point on its edge. 
 
 89. Two conducting discs of radii a, a' are fixed at right angles to the line which 
 joins their centres, the length of this line being r, large compared with a. If the first 
 has potential (f> and the second is uninsulated, prove that the charge on the first is 
 
 2arrr z (J) 
 
 90. Two equal spheres of radius a are in contact. Show that the capacity of the 
 conductor so formed is 2a log,, 2. 
 
 91. Two spheres of radii a, b are in contact, a being large compared with b. Show 
 that if the conductor so formed is raised to potential <, the charges on the two spheres are 
 
 , /, 7T 2 & 2 \ , , 7T 2 & 2 
 
 Aft 1 1 : rr5 and d>a - r^. 
 
 \ cr (a + b) z j a- (a + o) 2 
 
 92. Prove that if the centres of two equal uninsulated spherical conductors of radius 
 a be at a distance 2c apart, the charge induced on each by a unit charge at a point midway 
 between t, em is 
 
 00 
 
 2 ( - l) n sech na, 
 i 
 
 where c = a cosh a. 
 
 93. A spherical shell of radius a with a little hole in it is freely electrified to potential 
 
 <. Prove that. the charge on its inner surface is less than ~ , where s is the area of the 
 
 oirCl 
 
 whole. 
 
 94. A thin spherical conducting shell from which any portions have been removed 
 is freely electrified. Prove that the difference of densities inside and outside at any point 
 is constant. 
 
 95. Prove that the capacity of a hemispherical shell of radius a is 
 
 Prove that the capacity of an elliptic plate of small eccentricity e and area A is approxi- 
 mately 
 
 /Z2/, e* #\ 
 V TT TT \ 64 647' 
 
 i. 43 
 
674 Appendix 
 
 96. A thin circular disc of radius a is electrified with charge Q and surrounded by 
 a spheroidal conductor with a charge $1 placed so that the edge of the disc is the locus 
 of the focus S of the generating ellipse. Show that the energy of the system is 
 
 2 a 2 a 
 
 B being an extremity of the polar axis of the spheroid, and C the centre. 
 
 97. If the two surfaces of a condenser are concentric and coaxial oblate spheroids 
 of small ellipticities f and (' and polar axes 2c and 2c', prove that the capacity is 
 
 cc' (c f - c)~ 2 {c' - c + f (ec' - e'c)}, 
 
 neglecting squares of the ellipticities ; and find the distribution of electricity on each surface 
 to the same order of approximation. 
 
 98. A thin spherical bowl is formed by the portion of the sphere x z + y 2 + z 2 = cz 
 
 2-2 y2 2 2 
 
 bounded by and lying within the cone + j- = and is put in connection with the 
 
 O> C 
 
 earth by a fine wire. is the origin, and C, diametrically opposite to 0, is the vertex 
 of the bowl; Q is any point on the rim, and P is any point on the great circle arc CQ. 
 Show that the surface density induced at P by a charge Q placed at O is 
 
 Qc CQ 
 
 ,...*==-. tie 
 
 where 
 
 = f 
 
 J o (a? si 
 
 (a 2 sin 2 6 + b 2 cos 2 6 
 
 99. A flat piece of corrugated metal (y = a sin mx) is charged with electricity. Find 
 the surface density at any point, and show that it exceeds the average density approxi- 
 mately in the ratio my : 1. 
 
 100. A long hollow cylindrical conductor is divided into two parts by a plane through 
 the axis and the parts are separated by a small interval. If the two parts are kept at 
 potentials (/> x and </> 2 , the potential at any point within the cylinder is 
 
 <fti + < 2 , 0i ~ 02 fnn _, 2ar cos 6 
 
 tan r , 
 
 TT a 2 - r 2 
 
 where r is the distance from the axis, and 6 is the angle between the plane joining the 
 point to the axis and the plane through the axis normal to the plane of separation. 
 
 101. Show that the capacity per unit length of a telegraph wire of radius a at height 
 h above the surface of the earth is 
 
 f, A - aT 1 
 
 4tanh~ x */ T 
 
 V h + a J 
 
 102. A cylindrical conductor of infinite length, whose cross section is the outer 
 boundary of three orthogonal circles of radius a, has a charge q per unit length. Prove 
 that the electric density at distance r from the axis is 
 
 q (3r 2 + a 2 ) (3r 2 - a 2 - */6ar) (3r 2 - a 2 + VW) 
 6^a ~ r 2 (9r* - 3a 2 r 2 + a 4 ) 
 
 103. If the cylinder r a + b cos 6 be freely charged, show that in free space the 
 resultant force varies as 
 
 r-ir 2 + 2rccosd + c 2 )~i, 
 
Electrostatics % 675 
 
 and makes with the line 6 an angle 
 
 / r sin 6 \ 
 6 + itan" 1 
 
 \c + r cos 6) 
 
 where a 2 - 6 2 = 2bc. 
 
 104. If (J> + fy f(x + iy), and the curves for which < = const, be closed, show 
 that the capacity C of a condenser with boundary surfaces <f> = <f> lt </> = </> is 
 
 47T (0! - < ) 
 
 per unit length, where [^] is the increment of \^ on passing once round a <-curve. 
 
 105. Using the transformation 
 
 x + iy c cot | (<j) + ii^), 
 
 show that the capacity per unit length of a condenser formed by two right circular cylinders 
 (radii a, b) one inside the other, with parallel axes at a distance d apart, is 
 
 a* + b* - d 2 
 2 cosh 1 , 
 
 2ab 
 
 106. A plane infinite grating fe made up of equal and equi-distant parallel thin metal 
 plates, the distance between their successive central lines being TT, and the breadth of 
 
 each plate 2 sin" 1 I -=. J. Show that when the grating is electrified to constant potential, 
 the potential and charge functions <f>, -^ on the surrounding space are given by the equation 
 
 sin (</> + ty) = K sin (x + iy). 
 
 Deduce that when the grating is to earth and is placed in a uniform field of force of 
 unit intensity at right angles to its plane, the charge and potential functions of the portion 
 of the field which penetrates through the grating are expressed by 
 
 + ty - (x + iy), 
 and expand the potential in the latter problem in a Fourier series. 
 
 107. Two insulated uncharged circular cylinders outside each other, given by 77 = a 
 and TJ - ft, where x + iy = c tan ( + irj), are placed in a uniform field of force of 
 potential < - Ex. Show that the potential due to the distribution on the cylinders is 
 
 i smh n (a + ft) 
 
 the summation being taken for all odd positive integral values of n. 
 
 108. Two circular cylinders outside each other, given by rj = a and rj = - ft, where 
 
 x + iy = c tan | ( + *?)> 
 
 are put to earth under the influence of a line-charge Q on the line x = 0, y 0. Show 
 that the potential of the induced charge outside the cylinders is 
 
 . n _ 1 e~ wa sinh n (rj + ft) + e~ nfi sinh n (a - 17) 
 - 4$ 2 - cos | + const., 
 
 n smh n(a + ft) 
 
 the summation being taken for all odd positive integral values of n. 
 
 109. What problems are solved by the transformation 
 
 d , c(t 2 - 1)* a + t 
 
 dt (x + ^ = ^a^T' ^* + '*>= 1< *^ri 
 
 where a > 1 ? 
 
 432 
 
676 , Appendix 
 
 110. What problem in electrostatics is solved by the transformation 
 
 x + iy = en ((f> + ty), 
 where ^ is taken as the potential function, <f> being the function conjugate to it ? 
 
 111. Verify that, if r, s be real positive constants, z = x + iy, a= pe 1 ?, - = - + -, 
 
 the field of force outside the conductors x 2 + y z .+ 2sx = 0, x 2 + y 2 - 2rx = due to a 
 doublet at the point z a, outside both the circles, of strength p. and inclination a to the 
 axis, is given by putting 
 
 A + ^ = e - cot CT ---- -- cot e - 
 
 P 2 I . \ z / \ 2 
 
 where z = a is the inverse point to z = a with regard to either of the circles. 
 
 112. A semi-infinite conducting plane is at zero potential under the influence of an 
 electric charge q at a point Q outside it. Show that the potential at any point P is given by 
 
 r, , ., i , /cosh An + cos A (6 - #,) 
 
 =\ {cosh 77 - cos (6 - 0J} -i tan" 1 */ - 
 
 V cosh 9 - cos (<9 - <9 X ) 
 
 _ l /COsh in + COS i (0 + #]) 1 
 
 - {cosh r; - cos (6 + 6 l )} * tan- 1 ,*/ - -f-^ - ^ , 
 
 V cosh \r\ - cos \ (6 + 6i) J 
 
 where r, 0, z are the cylindrical coordinates of the point P, (r 1 , 6 lt 0) of the point Q, 6 = 
 is the equation of conducting plane, and 
 
 2rr x cosh n = r 2 + r^ + z 2 . 
 
 Hence obtain the potential at any point due to a spherical bowl at constant potential 
 and shew that the capacity of the bowl is 
 
 sin a 
 
 where a is the radius of the aperture, and a is the angle subtended by this radius at the 
 centre of the sphere of which the bowl is a part. 
 
 113. Show that the potential at any point P of a circular bowl, electrified to potential 
 
 d> ( . AB OA . , /OP AB 
 
 ? | Sm APVPB + OB Sm (OA ' AP^ 
 where O is the centre of the bowl, and A, B are the points in which a plane through P 
 and the axis of the bowl cuts the circular rim. 
 
 Find the density of electricity at a point on either side of the bowl and shew that the 
 capacity is 
 
 a . 
 
 - (a + sm a), 
 
 7T 
 
 where a is the radius of the sphere, and 2a is the angle subtended at the centre. 
 
 114. Two spheres are charged to potentials </>j and < 2 . The ratio of the distances 
 of any point from the two limiting points of the spheres being denoted by e 1 * and the angle 
 between them by , prove that the potential at the point , r; is 
 
 where n = a, n = - /3 are the equations of the spheres. Hence find the charge on either 
 sphere. 
 
Electrostatics 677 
 
 115. Two large parallel conducting plates are maintained at potentials fa and <p z 
 and the space between them is filled up by slabs of dielectric whose inductive capacities 
 are f l and f 2 , whose thicknesses are d l and t/ 2 and whose common face is parallel to the 
 plates. Find the potential at any point between the plates and shew that it is everywhere 
 the same as if the dielectrics were replaced by an insulated conducting sheet along their 
 common face, and the charge on this plate per unit of area were 
 
 ( f i - c a) (<fri ~ ftz) 
 4?r (d^z + ^2 6 i) 
 
 116. Three thin conducting sheets are in the form of concentric spheres of radii 
 a + d, a, a - c respectively. The dielectric between the outer and middle sheet is of 
 specific inductive capacity e, and that between the middle and inner sheet is air. At 
 first the outer sphere is uninsulated, the inner sheet is uncharged and insulated and the 
 middle coating is charged to potential < and insulated. The inner sheet is now uninsulated 
 without connection with the middle sheet. Prove that the potential of the middle sheet 
 falls to 
 
 e0c (a + d) 
 {ec (a + d) + d (a - c)}' 
 
 117. A conductor has a charge Q and < 1( 2 are the potentials of two equi-potential 
 surfaces completely surrounding it (0 X > < 2 ). The space between these two surfaces is 
 now filled with a dielectric of inductive capacity e. Show that the change in the energy 
 of the system is 
 
 W <-'*) f -^ 
 
 118. The surfaces of an air-condenser are concentric spheres. If half the space 
 between the spheres be filled with solid dielectric of specific inductive capacity e, the 
 dividing surface between the solid and the air being a plane through the centre of the 
 spheres, show that the capacity will be the same as though the whole dielectric were of 
 uniform specific inductive capacity (1 + e). 
 
 119. The radii of the inner and outer shells of two equal spherical condensers, remote 
 from each other and immersed in an infinite dielectric of inductive capacity e, are respec- 
 tively a and b, and the inductive capacities of the dielectric inside the condensers are 
 e 1( e 2 . Both surfaces of the first condenser are insulated and charged, the second being 
 uncharged. The inner surface of the second condenser is now connected to earth, and 
 the outer surface is connected to the outer surface of the first condenser by a wire of 
 negligible capacity. Show that the loss of energy is 
 
 Q 2 {2 (b - a) e + ae 2 } 
 2fb {(b - a)f + ae 2 ] ' 
 
 where Q is the quantity of electricity which flows along the wire. . 
 
 120. The outer coating of a long cylindrical condenser is a thin shell of radius a, 
 and the dielectric between the cylinders has inductive capacity e on one side of a plane, 
 through the axis, and e' on the other side. Show that when the inner cylinder is connected 
 to earth, and the outer has a charge q per unit length, the resultant force on the outer 
 cylinder is 
 
 4 <? 2 Jl-JL) 
 ira (e + c') 
 
 per unit length. 
 
678 Appendix 
 
 121. A heterogeneous dielectric is formed of n concentric spherical layers of specific 
 inductive capacities f lt e 2 , *3> f n> starting from the innermost dielectric, which forms 
 a solid sphere ; also the outermost dielectric extends te infinity. The radii of the spherical 
 boundary surfaces are a lf a 2 , ... a w _j respectively. Prove that the potential due to a 
 quantity Q of electricity at the centre of the spheres at a point distant r from the centre 
 in the dielectric e. is 
 
 122. A condenser is formed by two rectangular parallel conducting plates of breadth 
 b and area A at distance d from each other. Also a parallel slab of a dielectric of thickness 
 t and of the same area is between the plates. This slab is pulled along its length from 
 between the plates, so that only a length x is between the plates. Prove that the electric 
 force sucking the slab back again to its original position is 
 
 2irQ 2 dbt f (d - t') 
 {A(d - t') + xbt'}*' 
 
 where t' = t, e is the specific inductive capacity of the slab, Q is the charge, and 
 the disturbances produced by the edges are neglected. 
 
 123. Three closed surfaces 1, 2, 3 are equi- potentials in an electric field. If the space 
 between 1 and 2 is filled with a dielectric , and that between 2 and 3 is filled with a dielectric 
 e', show that the capacity of a condenser having 1 and 3 for faces is C, given by 
 
 1 - _L J_ 
 
 C~ A* + BT" 
 
 where A, B are the capacities of air-condensers having as faces the surfaces 1, 2 and 2, 3 
 respectively. 
 
 124. The surface separating two dielectrics (e x , f 2 ) has an actual charge a- per unit 
 area. The electric forces on the two sides of the boundary are F { , F 2 at angles c a , c 2 with 
 the common normal. Show how to determine F 2 , and prove that 
 
 /, 47TO- \ 
 
 e, cot C 2 = e, cot c, 1 - - . 
 
 1 V ^i cos cj 
 
 125. The space between two concentric spheres radii a, b which are kept at potentials 
 A, B is filled with a heterogeneous dielectric of which the inductive capacity varies as the 
 nth power of the distance from their common centre. Show that the potential at any 
 point between the surfaces is 
 
 Aa n + 1 - Bb n + 1 a n + 1 b n + 1 A - B 
 
 - b n 
 
 126. An infinitely long elliptic cylinder of inductive capacity e, given by = a, where 
 x + iy = c cosh ( + irj), is in a uniform field E parallel to the major axis of any section. 
 Show that the potential at any point inside the cylinder is 
 
 I + coth a 
 f + coth a ' 
 
 127. An infinite slab of a dielectric of constant e and thickness t has air on either side 
 of it. A point charge q is situated at a point A in the air on one side of the slab. Prove 
 that the potential at any point on the other side is the same as if the slab were removed, 
 
Electrostatics 679 
 
 4/7f 
 
 the charge at A altered to ^ and point charges placed at points situated at distances 
 
 2t, 4t, 6t, ... from A, the charge at the nth point being 
 
 (e ~ I) 2 " 
 9f '( f + !) + 
 
 128. A dielectric hemisphere of radius a and inductive capacity is laid flat against 
 an unlimited plane conductor charged to surface density a-; prove that the disturbance 
 which its presence produces in the field of force is derived from a potential 
 
 e - 1 a 3 x 
 
 47TCT p. 5-. 
 
 e + 2 r 3 
 where x is the distance from the plane and r the distance from the centre of the hemisphere. 
 
 129. A condenser is formed of two very long circular and coaxal cylindrical conductors 
 If a portion of the intermediate space, bounded by planes through the axes inclined to one 
 another at an angle a, be filled with a dielectric of specific inductive capacity e, show that 
 the capacity per unit length is increased in the ratio 
 
 130. The plates of a condanser are vertical and at a difference of potential equal to 
 (j> ; show that if a slab of dielectric (constant e) of thickness t is suspended partly inside 
 the condenser and partly outside, it will be sucked in by a force 
 
 fc^ 2 /_! 
 
 STT Ui 
 
 where b is the horizontal breadth of the slab, a the distance between the plates and 
 
 a,i=[a - t + t/e. 
 
 131. A point charge is placed in front of an infinite slab of dielectric, bounded by 
 a plane face. The angle between a line of force in the dielectric and the normal to the 
 face of the slab is a ; the angle between the same two lines in the immediate neighbourhood 
 of the charge is /3. Prove that 
 
 s _/3 
 
 132. An electrified particle is placed in front of an infinitely thick plate of dielectric. 
 Show that the particle is urged towards the plate by a force 
 
 e - 1 _q*_ 
 e + 1 ' 4d 2 ' 
 
 where d is the distance of the point from the plate. 
 
 133. Two dielectrics of inductive capacities ej and c 2 are separated by an infinite 
 plane face. Charges q lt q z are placed at points on a line at right angles to the plane, each 
 at a distance a from the plane. Find the forces on the two charges, and explain why they 
 are unequal. 
 
 134. A conducting sphere of radius a is placed in air, with its centre at a distance 
 c from the plane face of an infinite dielectric. Show that its capacity is 
 
 fe - IV" 1 
 a sinh a 2 I - ) cosech na, 
 
 i V + V 
 where a = c/a. 
 
680 Appendix 
 
 135. Two conductors of capacities q and q z in air are on the same normal to the plane 
 boundary between the dielectrics f lf e 2 at great distances a, b from the boundary. They 
 are connected by a thin wire and charged. Prove that the charge is distributed between 
 them approximately in the ratio 
 
 JI *1~ f 2 2*2 _ I . , j I, 1~ *2 2 fl _ 1 
 
 :i \C 2 26 ( fl + e,) (e x + e,) (a + &)/ ' ' K 2a (e, + e,) ( ei + *,) (a + 5)J ' 
 
 136. A conducting sphere of radius a is embedded in a dielectric (e) whose outer 
 boundary is a concentric sphere of radius 2a. Show that if the system be placed in a 
 uniform field of force E equal quantities of positive and negative electricity are separated 
 of amount 
 
 57+"?' 
 
 137. A sphere of glass of radius a is held in air with its centre at a distance c from a 
 point at which there is a charge q. Prove that the resultant attraction is 
 
 where /3 = r-. 
 
 e + 1 
 
 138. A spherical conductor of radius a is surrounded by a uniform dielectric e, which 
 is bounded by a sphere of radius b having its centre at a small distance y from the centre 
 of the conductor. Prove that, if the potential of the conductor is <f>, and there are no other 
 conductors in the field, the surface density at a point where the radius makes an angle 
 B with the line of centres is 
 
 6 (e - 1) ya 2 cos 6 
 
 lira {(e - 1) a + b} \_ 2 (e - 1) a 3 + ( e + 2) P 
 
 139. A sphere of s.i.c. e is placed in air in a field of force due to a potential X n (before 
 the introduction of the sphere) referred to rectangular axes through the centre of the sphere, 
 where X n is a solid harmonic of order n. Prove that the potential inside the sphere is 
 
 --n' 
 
 n + 1 + en 
 
 140. A charge q is placed at a distance c from the centre of a sphere of s.i.c. e and 
 outside the sphere. Prove that the potential at any point inside the sphere at a distance 
 r from the centre is 
 
 q 2n + 1 r 
 
 ~ I * 
 
 c o en + n + 1 \c 
 
 141. Find the potential at any point when a sphere of specific inductive capacity 
 e is placed in air in a field of uniform force. 
 
 A circle has its centre on the line of force which passes through the centre of the sphere 
 and its plane perpendicular to this line of force. Prove that if the plane of the circle 
 does not cut the sphere, the presence of the sphere increases the induction through the 
 circle in the ratio 
 
 1 i 2 6 ~ 1 .in*a-l 
 
 e + 2 
 
 where 2a is the angle of the enveloping cone from any point on the circumference of the 
 circle to the sphere. 
 
Electrostatics 681 
 
 142. A shell of glass of inductive capacity e, which is bounded by concentric spherical 
 surfaces of radii a, b (a < b), surrounds an electrified particle with charge Q which is at a 
 point P' at a small distance c from 0, the centre of the spheres. Show that the potential 
 at a point P outside the shell at a distance r from P' is approximately 
 
 Q 2Qc (b 3 - a 3 ) ( - I) 2 cos 6 
 
 < r 2a 3 (e - I)" 2 - b 3 (e + 2) (2e + 1) r 2 ' 
 where 6 is the angle PP' makes with OP' produced. 
 
 143. A conductor at potential rf) whose surface is of the form r = a ( 1 + eP n ) is 
 surrounded by a dielectric (e) whose boundary is the surface r = b (1 + e'P n ), and outside 
 this the dielectric is air. Show that the potential in the air at a distance r from the origin is 
 
 (2ro + 1) f a"6 2TO+1 + (c - 1) t'b n \nb* u + 1 + (n + 1) a 2 "* 1 } P n 
 
 (e - 1) a + 6 _r (1 + TO + we) 6 2n + 1 + (e - 
 
 where squares and higher powers of e and e' are neglected. 
 
 144. A dielectric sphere is surrounded by a thin circular wire of large radius b carrying 
 a charge Q. Prove that the potential within the sphere is 
 
 Q ( 1 + 4n 1 . 3 . 5 ... 2n - 1 r 
 
 ' 
 
 b l + 2nl + -" 2.4.6...2n \b 
 
 145. An electrified line with charge Q per unit length is parallel to a circular cylinder, 
 of radius a and inductive capacity e, the distance of the wire from the centre of the cylinder 
 being c. Show that the force on the wire per unit length is 
 
 e - 1 4a 2 g 2 
 * + 1 ' c (c 2 - a 2 ) ' 
 
 146. The space between two concentric conducting spheres is filled on one side of 
 a diametral plane with dielectric of specific inductive capacity e, and on the other side with 
 dielectric of specific capacity e'. The inner sphere is of radius a and has a charge Q. 
 Prove that the force on it perpendicular to this diametral plane is 
 
 (e + ') 2 ' 2a 2 ' 
 
 147. A dielectric hemisphere of radius a and inductive capacity e is placed with its 
 base in contact with the plane boundary of an otherwise . unlimited conductor. Prove 
 that the potential at any point of the field outside both the conductor and the dielectric is 
 
 <f> = - 47TO- cos d(r- i-j-g . 
 
 where the origin is at the centre of the hemisphere and a- is the surface density of the 
 charge on the plane conductor at a great distance from the hemisphere. 
 
 148. A point charge q is within a sphere of homogeneous dielectric (e) of radius a and 
 is a short distance c from the centre. Show that the force on the point is approximately 
 
 2 (e - 1) q*c 
 
 2 (e + 2) a 3 ' 
 
 149. A condenser is formed of two parallel plates, distant h apart, one of which is at 
 zero potential. The space between the plates is filled with a dielectric whose inductive 
 capacity increases uniformly from one plate to the other. Show that the capacity per unit 
 area is 
 
 e 2 ~ e i 
 4:7rh log fa/ej' 
 
 where ej and e 2 are the values of the inductive capacity at the surfaces of the plate. The 
 inequalities of distribution at the edges of the plates are neglected. 
 
682 Appendix 
 
 150. A spherical conductor of radius a is surrounded by a concentric spherical con- 
 ducting shell whose internal radius is b, and the intervening space is occupied by a dielectric 
 
 whose specific inductive capacity at a distance r from the centre is . If the inner 
 sphere is insulated and has a charge Q, the shell being connected with the earth, prove that 
 
 the potential in the dielectric at a distance r from the centre is log- , . 
 
 c & r(c + b) 
 
 151. A spherical conductor of radius a is surrounded by a concentric spherical shell 
 of radius b, and the space between them is filled with a dielectric of which the inductive 
 
 capacity at distance r from the centre is ne~ p * p~ 3 , where p= -. Prove that the capacity 
 
 
 
 of the condenser so formed is 
 
 62 
 
 152. If the specific inductive capacity varies as e~, where r is the distance from a 
 fixed point in the medium, verify that a solution of the differential equation satisfied by 
 the potential is 
 
 a \ 
 
 and hence determine the potential at any point of a sphere, whose inductive capacity is 
 the above function of the distance from the centre, when placed in a uniform field of force. 
 
 153. Show that the capacity of a condenser consisting of the conducting spheres 
 r = a, r = b and a heterogeneous dielectric of inductive capacity e = f(6, <) is 
 
 154. If the electricity in the field is confined to a given system of conductors at given 
 potentials, and the inductive capacity of the dielectric is slightly altered according to any 
 law such that at no point is it diminished, and such that the differential coefficients of the 
 increment are also small at all points, prove that the energy of the field is increased. 
 
 155. A slab of dielectric of inductive capacity e and of thickness x is placed inside 
 a parallel plate condenser so as to be parallel to the plates. Show that the surface of the 
 slab experiences a tension 
 
 2n . 2 /i _ 1 _ L /T\l 
 
 r I e dfeWJ' 
 
 156. For a gas e = 1 + dp, where p is the density and B is small. A conductor is 
 immersed in the gas : show that if 6 2 is neglected the mechanical force on the conductor 
 is 2ira 2 per unit area. 
 
 157. A metallic shell is surrounded by a thin concentric conducting shell formed by 
 two hemispheres with their rims in contact, the space between the sphere and shell being 
 filled with a dielectric of specific inductive capacity e. If charges Q, Q' be given to the 
 shell and the sphere, show that if the halves of the shell remain in contact the charges must 
 
 be of opposite signs and the ratio of their magnitudes must lie between the limits 1 -p . 
 
 Ve 
 
 158. If a spherical conductor, of radius a, with no other conductor in the neighbourhood 
 is coated with a uniform thickness d of shellac of which e is the specific inductive capacity, 
 shew that the capacity of the conductor is increased in the ratio e (a + d) : ta + d. If 
 the spherical conductor consist of two separate hemispherical portions, what would be the 
 force tending to separate one hemisphere from the other? 
 
Magnetostatics 683 
 
 II. MAGNETOSTATICS 
 
 159. A small magnet ACB, free to turn about its centre C, is acted on by a small 
 fixed magnet PQ. Prove that in equilibrium the axis ACB lies in the plane PQC, and that 
 tan 6 = - % tan 6', where 6, 6' are the angles which the two magnets make with the line 
 joining them. 
 
 160. The axis of a small magnet makes an angle (j> with the normal to a plane. Prove 
 that the line from the magnet to the point in the plane where the number of lines crossing 
 it per unit area is a maximum makes an angle 6 with the axis of the magnet, such that 
 
 2 tan d = 3 tan 2 (< - 0). 
 
 161. Two small magnets lie in the same plane, and make angles 6, 6' with the line 
 joining their centres. Show that the line of action of the resultant force between them 
 divides the line of centres in the ratio 
 
 tan 6' + 2 tan 6 : tan 6 + 2 tan 6'. 
 
 162. Two small magnets having their centres at distance r apart, make angles 6, 6' 
 with the line joining them and an angle e with each other. Show that the force on the 
 first magnet in its own direction is 
 
 j (5 cos 2 6 cos 6' - cos 6' - 2 cos e cos 6). 
 r 
 
 Show also that the couple about the line joining them which the magnets exert on one 
 another is 
 
 mm' , 
 
 j- d sin t , 
 r* 
 
 where d is the shortest distance between their axes. 
 
 163. Two magnetic needles of moments M, M' are soldered together so that their 
 directions include an angle a. Show that when they are suspended so as to swing freely 
 in a uniform horizontal field, their directions will make angles 6, 6' with the lines of force, 
 given by 
 
 sin 6 _ sin 6' _ sin a 
 
 M' M (jlf /2 + M 2 + 2MM' cos a)*' 
 
 164. Two small magnets at distance r are in one plane and are inclined at angles 
 \TT and |TT to the line joining their centres. Prove that the action between them reduces 
 to a single force parallel to this line and at a distance l/9r from it. 
 
 165. Show that the action between two small magnets with centres A, A' reduces to 
 a single force if the image of each magnet in the plane perpendicularly bisecting A A' is 
 perpendicular to the other magnet. Show also that if the axis of the magnets meet the 
 plane in the points B, B' and if AO = 20B and A'O' = 2O'B', then 00' is the line of action 
 of the force. 
 
 166. A sphere of hard steel is magnetised uniformly in a constant direction and a 
 magnetic particle is held at an external point with the axis of the particle parallel to the 
 direction of magnetisation of the sphere. Find the couples acting on the sphere and on 
 the particle. 
 
 167. Two magnetic particles of equal moment are fixed with their axes parallel to 
 the axis of z, and in the same direction, and with their centres at the points ( a, 0, 0). 
 
684 Appendix 
 
 Shew that if another magnetic molecule is free to turn about its centre, which is fixed at 
 the point (0, y, z), its axis will rest in the plane x = and will make with the axis of z the 
 angle 
 
 tan TT ~ a o 
 2z 2 - a 2 - y 2 
 
 Examine which of the two positions of equilibrium is stable; 
 
 168. Two small equal magnets have their centres fixed and can turn about them in 
 a magnetic field of uniform intensity H, whose direction is perpendicular to the line r 
 joining the centres. Show that the position in which the magnets both point in the direction 
 of the lines of force of the uniform field is stable only if 
 
 169. Obtain the equation 
 
 T = 27r*/7/mH 
 
 for the time of a small oscillation of a magnet of moment m swung about its centre of 
 inertia in a horizontal plane in a uniform field of horizontal intensity H, the moment of 
 inertia of the magnet being I. 
 
 170. Assuming the earth to be a sphere uniformly magnetised parallel to the axis 
 of rotation with intensity M, show that the time of a small horizontal oscillation in latitude 
 (f> is 
 
 \3Iir/mM cos 0. 
 
 171. A small magnet of moment m is held fixed at the origin of coordinates, with its 
 axis in the direction (I, m, n) ; another small magnet of moment m' has its centre fixed at 
 the point (x, 0, z), and is free to turn so that its axis moves in a plane parallel to the plane 
 z = 0. Find the position of stable equilibrium of m' and show that the period of its free 
 oscillations about this position is 
 
 'm'~(x 2 + z 2 )* [{I (x 2 + z 2 ) - 3x (Ix + nz)} 2 + m 2 (x 2 + z 2 ) 2 ] ~ , 
 where / is the moment of inertia of m'. 
 
 172. Four small equal magnets are placed at the corners of a square and oscillate 
 under the actions they exert on each other. Prove that the times of vibration of the 
 principal oscillations are 
 
 I*, 
 2J 
 
 m3 (2 + |V2) (m 2 (3 - 
 
 where m is the magnetic moment, / the moment of inertia of a magnet and d is a side of 
 the square. 
 
 173. If a small cylindrical cavity be made within a magnetised body with its axis 
 parallel to the direction of magnetisation at the point, prove that the magnetic force within 
 the cavity is simply H if the length of the cavity is large compared with its radius but is 
 B if the radius is large compared with the length. 
 
 174. A uniformly magnetised substance has an ellipsoidal cavity in it with a principal 
 axis in the direction of magnetisation of the substance ; prove that the ratio of the magnetic 
 force in the cavity to the magnetic force at a great distance from the cavity is 
 
 where 
 
 {-(HP 
 
 I - , 
 
 J (a 2 + u)* (b 2 + ) (c 2 + ) 
 
Magnetostatics 685 
 
 the direction of the principal axis a being that of magnetisation of the substance, and show 
 that when the cavity is an oblate spheroid with its least axis in the direction of magnetisa- 
 tion the ratio tends to p as the least axis diminishes. 
 
 175. A steel shell, bounded internally and externally by non-concentric spherical 
 surfaces, is magnetised uniformly. Prove that there is no magnetic field in the hollow, 
 and that the external field is the same as would be due to two magnetic particles at the 
 centres of the spheres, whose moments are proportional to the respective volumes of the 
 spheres. 
 
 176. A small magnet of moment m is held in the presence of a very large fixed mass 
 of soft iron of permeability p with a very large plane face: the magnet is at a distance 
 a from the plane face and makes an angle 6 with the shortest distance from it to the plane. 
 Show that a certain force, and a couple 
 
 (p - 1) m 2 sin 6 cos 6 
 8a 3 (p + 1) 
 
 are required to keep the magnet in position. 
 
 177. A sphere of soft iron of radius a is placed in a field of uniform magnetic force 
 parallel to the axis of z. Show that the lines of force external to the sphere lie on surfaces 
 of revolution, the equation of which is of the form 
 
 r being the distance from the centre of the sphere. 
 
 178. A sphere of soft iron of permeability p is introduced into a field of force in which 
 the potential is a homogeneous polynomial of degree n in (x, y, z). Show that the potential 
 inside the sphere is reduced to 
 
 2n + I 
 up + n + 1 
 of its original value. 
 
 179. If a shell of radii a, b is introduced in place of the sphere in the last question, 
 show that the force inside the cavity is altered in the ratio 
 
 (2n + I) 2 p : (np + n + 1) (np + n + p.) - n (n + 1) (p. - I) 2 f ^ 
 
 180. If the magnetic field within a body of permeability p be uniform, shew that 
 any spherical portion can be removed and the cavity filled up with a concentric spherical 
 nucleus of permeability p^ and a concentric shell of permeability p 2 without affecting the 
 external field, provided p lies between p t and p 2 and the ratio of the volume of the nucleus 
 to that of the shell is properly chosen. Prove also that the field inside the nucleus is 
 uniform and that its intensity is greater or less than that outside according as p is greater 
 or less than p 1 . 
 
 181. A sphere of radius a has at any point (x, y, z) components of permanent magnetisa- 
 tion (ir - I,, - ) , the origin of coordinates being at its centre. It is surrounded by a 
 
 \ x a' v a ) 
 
 spherical shell of uniform permeability p, the bounding radii being a and 6. Determine 
 the complete circumstances of the field. 
 
 182. An infinitely long hollow iron cylinder of permeability p, the cross section being 
 concentric circles of radii a, b, is placed in a uniform field of magnetic force the direction 
 
686 Appendix 
 
 of which is perpendicular to the generators of the cylinder. Show that the number of lines 
 of induction through the space occupied by the cylinder is changed by inserting the cylinder 
 in the field, in the ratio 
 
 b z (n + I) 2 - a 2 (n - I) 2 : 2/x {b 2 (/*+!)- a 2 (p. - 1)}. 
 
 183. An infinite cylinder of soft iron is placed in a uniform field of potential 
 
 f - H x x - H y y, 
 
 2*2 ^/2 
 
 the equation of the cylinder being .+ V = 1. Show that the potential of the induced 
 
 a o 
 
 magnetism at any internal point is 
 
 + , H,y 
 
 ya 
 
 r x 
 pb 
 
 184. A circular wire of radius a is concentric with a spherical shell of soft iron of radii 
 b and c. Show that, if a steady unit current flow round the wire, the presence of the iron 
 increases the number of lines of induction through the wire by 
 
 27r 2 a 4 (c 3 - & 3 ) f/i -l)(/i + 2) 
 6{(2/i + 1) (fi + 2) c 3 - 2 (p - I) 2 6 3 }' 
 
 185. A solid elliptic cylinder of iron whose equation is = a given by 
 
 x + iy = c cosh ( + irj) 
 
 is placed in a field of magnetic force whose potential is A (x z - y 2 ). Show th,at in the 
 space external to the cylinder the potential of the induced magnetism is 
 
 - J Ac 2 cosech 2 (a + j8) sin 4ae 2(a "~^~ cos 2rj, 
 where coth 2/3 is the permeability. 
 
 186. An infinite elliptic cylinder of permeability p is placed in a uniform magnetic 
 field of strength H so that the direction of the force is perpendicular to the axis of the 
 cylinder. Show that there is a couple on the cylinder tending to sot the major axis of a 
 principal section in the direction of the force; the moment of the couple per unit length 
 being 
 
 ab(a 2 - fe 2 )( M - l 
 
 8 (a + fib) (b + pa) 
 where 6 is the angle between the major axis of a section and the direction of the force. 
 
 187. A unit magnetic pole is placed on the axis of 2 at a distance / from the centre of 
 a sphere of soft iron of radius a. Show that the potential of the induced magnetism at any 
 external point is 
 
 ( tl 
 f +l dtd6 
 / . a 2 t\ 2 ' 
 
 J ^ + &07COS0 - -jj 
 
 where 2, tsr are the cylindrical coordinates of the point. Find also the potential at an external 
 point. 
 
 188. A magnetic pole of strength m is placed in front of an iron plate of permeability 
 p. and thickness c. If this pole be the origin of rectangular coordinates x, y and if x be 
 perpendicular and y parallel to the plate, show that the potential behind the plate is given by 
 
 z~ xi J (yt) dt 
 
 where _ 
 
 + 1 
 
Electrokinetics 687 
 
 TIL ELECTROKINETICS 
 
 189. A network is formed of uniform wire in the shape of a rectangle of sides 2a, 3a, 
 with parallel wires arranged so as to divide the internal space into six squares of sides 
 a, the contact at the points of intersection being perfect. Show that if a current enter the 
 framework by one corner and leave it by the opposite, the resistance is equivalent to that 
 of a length 12 la/69 of the wire. 
 
 190. A fault of given earth resistance developes in a telegraph line. Prove that the 
 current at the receiving end, generated by an assigned battery at the signalling end, is 
 least when the fault is at the middle of the line. 
 
 191. The resistances of three wires BC, CA, AR, of the same uniform section and 
 material, are a, b, c respectively. Another wire from A of constant resistance d can make 
 a sliding contact with BC. If a current enter at A and leave at the point of contact with 
 BC, show that the maximum resistance of the network is 
 
 (a + b + c) d 
 a + b + c + 4d' 
 and determine the least resistance. 
 
 192. The resistances in the arms AC, CB, BD, DA of a Wheatstone's bridge are 
 P, R, 8, Q respectively. The points A and C are also connected through a condenser of 
 capacity K in parallel with the resistance P ; and the arm BD includes a coil of inductance 
 L. A and B are connected to the battery and C and D to the galvanometer. If there is 
 no flow, instantaneous or permanent, through the galvanometer on making the battery 
 circuit, show that L = PSK. 
 
 193. A quadrilateral is formed of wire and A, B, C, D are its corners taken in order. 
 The resistances of the wires AB, BC, CD, DA are p, q, r, s respectively. The excess of the 
 potential of A over that of B when unit current enters the quadrilateral at C and leaves it 
 at D by wires applied at these points is denoted by [AB, CD]. The resistance of the quadri- 
 lateral when A and B are the electrodes is denoted by [^4-B]. The other symbols have 
 corresponding meanings. Calculate [AB, CD] in terms of p, g, r, s and show that 
 
 [AD] + [BC] - [AC] - [BD] = Z[AB . CD], 
 [AB . CD] + [BC . AD] + [CA . BD] = 0. 
 
 194. Two planes A, B are connected by a telegraph of which the end at A is connected 
 to one terminal of a battery, and the end at B to one terminal of a receiver, the other 
 terminals of the battery and receiver being connected to earth. At a point C of the line 
 a fault is developed, of which the resistance is r. If the resistances of AC, CB be p, q 
 respectively, show that the current in the receiver is diminished in the ratio 
 
 r (p + q) : qr + rp + pq, 
 the resistance of the battery, receiver and earth circuit being neglected. 
 
 195. Two cells of electromotive forces e 1 , e 2 and resistances r lt r 2 are connected in 
 parallel to the ends of a wire of resistance R. Show that the current in the wire is 
 
 e^ + e^ 
 r : R + r 2 R + r-^r^ 
 and find the ratio at which the cells are working. 
 
 196. A network of conductors is in the form of a tetrahedron PQRS ; there is a battery 
 of electromotive force E in PQ, arid the resistance of PQ, including the battery, is R. If 
 the resistances in QR, RP are each equal to r, and the resistances in PS, RS are each equal 
 to \r, and that in Q8 = f r, find the currents in each branch. 
 
688 Appendix 
 
 197. A cell of resistance r is connected to the ends of a wire AB. The cell is then 
 replaced by two different cells, of resistances R, R' arranged in parallel, producing the 
 same current in AB and having the combined resistance r when in parallel. Show that the 
 total heat production is greater in the second case than in the first, by the amount which 
 would be produced in the circuit of the two cells if the wire AB were broken. 
 
 198. An electric circuit contains a galvanometer and a battery of constant electro- 
 motive force (j). The resistance of the galvanometer is G and that of the rest of the circuit 
 including the battery S. Show that on shunting the galvanometer with resistance S the 
 current through the galvanometer is decreased by 
 
 ___ <j>RG 
 
 (R + G)(RS + RG + SG)' 
 
 199. A circuit contains two lamps, each of resistance R, in parallel on leads each of 
 resistance S. The resistance of the rest of the circuit, including the battery of constant 
 voltage <j), is r. Show that if one lamp is broken, the heat emitted in unit time by the other 
 is increased by 
 
 tfR (3r 2 + 2r ( R + S)} 
 (r + R + S) 2 (2r + R + S) 2 ' 
 
 200. A, B, C, D are the four junctions of a Wheatstone's bridge, and the resistances 
 c, /3, b, y on AB, BD, AC, CD respectively are such that the battery sends no current 
 through the galvanometer in BC. If now a new battery of electromotive force E be intro- 
 duced into the galvanometer circuit, and so raise the total resistance in that circuit to a, 
 find the current that will flow through the galvanometer. 
 
 201. A wire is interpolated in a circuit of given resistance and electromotive force. 
 Find the resistance of the interpolated wire in order that the rate of generation of heat 
 may be a maximum. 
 
 202. The resistances of the opposite sides of a Wheatstone's bridge are a, a' and b, b' 
 respectively. Show that when the two diagonals which contain the battery and galvano- 
 meter are interchanged, 
 
 E _ E _ (a - a') (b - b') (G - R) 
 C~ C''~ oaf - W ' ' 
 
 where C and C" are the currents through the galvanometer in the two cases, G and R are 
 the resistances of the galvanometer and battery conductors, and E the electromotive force 
 of the battery. 
 
 203. A current C is introduced into a network of linear conductors at A, and taken 
 out at B, the heat generated being H . If the network be closed by joining A, B by a 
 resistance r in which an electromotive force E is inserted, the heat generated is H 2 . Prove 
 that 
 
 204. A number JV of incandescent lamps, each of resistance r, are fed by a machine 
 of resistance R (including the leads). If the light emitted by any lamp is proportional to 
 the square of the heat produced, prove that the most economical way of arranging the 
 lamps is to place them in parallel arc, each arc containing n lamps, where n is the integer 
 nearest to *jNR/r. 
 
 205. A system of 30 conductors of equal resistances are connected in the same way 
 as the edges of a dodecahedron. Show that the resistance of the network between a pair 
 of opposite corners is ^ of the resistance of a single conductor. 
 
Electrokinetics (389 
 
 206. Four points A, B, C, D are connected, in order, by a battery of electromotive 
 force E and resistance b, and three resistances c, d, a. A and C are connected by a resistance 
 X, and B, D by a galvanometer of resistance O. Find the current through the galvanometer, 
 and prove that it is independent of X if ac = bd. 
 
 207. Six wires of equal length and resistance are arranged so as to form the sides 
 and the lines joining the middle points of the opposite sides of a square; prove that the 
 resistance of the network between diagonally opposite corners is f that of each wire. 
 
 208. An octahedron is formed of twelve bars of equal length and thickness and of the 
 same material; a current enters the system at one end of a bar and leaves at the other 
 end of the same bar ; show that the resistance of the octahedron is -f% of that of a single bar. 
 
 209. Two conducting circuits OPQ, OP'Q' are connected from P to P' and Q to Q' by 
 wires of resistances r and r'. A current enters the circuit at and leaves it at O'. Show 
 that, if the resistances of OP, PQ, QO are A, B, C and of O'P', P'Q', Q'O' are a, b, c 
 respectively, the currents in PP' and QQ' are in the ratio 
 
 BC . be AB ab 
 
 + r : -r ^ ~ + = + r. 
 
 A + B + C a + b + c ' A + B + C a + b + c 
 
 210. A wire forms a regular hexagon and the angular points are joined to the centre 
 by wires of equal resistance. Show that it is possible to adjust this resistance so that the 
 resistance to a current entering at one angular point of the hexagon and leaving by the 
 opposite is equal to that of a side of the hexagon. 
 
 211. A battery of mn equal cells is such that when it is arranged in m parallel sets of 
 n cells in series the maximum current C is produced for a given external circuit. Show 
 that when the cells are arranged in n parallel sets of m cells the current is 
 
 2mnC/(m* + n 2 ). 
 
 212. If there be n points A and n points B such that the resistance between two 
 A points and two B points is r and that between an A and a B point R, then the resistance 
 to a current entering at an A point and leaving at a B point is 
 
 R (R + 2n - Ir) 
 R + r 
 
 213. A telegraph wire joining two places A, B drops from one of its supports at a 
 place C and rests on another wire which is earthed at both ends. If X is the ratio of the 
 current strength at A to that at B when the current in AB is sent from A, and /z is the 
 ratio when the current is sent from B, show that C divides AB in the ratio 
 
 ( M -i-l):(X- 1). 
 
 214. An assemblage of n points, of which A, B, C, D, P are any five, has each point 
 connected to every other point by wires of resistance r. A current enters at any point 
 X in the wire AB and leaves from any point Y in the wire CD. Prove (i) that the sum 
 of the currents in AP, BP is l/n of the whole current ; (ii) that the mean of the currents in 
 AC, BD or in AD, BC is also l/n of the whole current; and (iii) that the currents in AP, 
 BP are inversely proportional to the resistances of AX, BX. 
 
 Show also that the whole resistance of the network lies between r (n + 2)/2n and 2r/n. 
 
 215. A, B, C, D are four points in succession at equal distances along a wire; and 
 A, C and B, D are also joined by two other wires of the same length as the distances between 
 those pairs of points measured along the original wire. A current enters the network 
 thus formed at A and leaves at D : show that J of it passes along BC. 
 
 44 
 
690 Appendix 
 
 216. A battery of electromotive force E and of resistance B is connected with the two 
 terminals of two wires arranged in parallel. The first wire includes a voltameter which 
 contains discontinuities of potential such that a unit current passing through it for a unit 
 time does p units of work. The resistance of the first wire, including the voltameter, is 
 R : that of the second is r. Show that if E is greater than p (B + r)/r, the current through 
 the battery is 
 
 E (R + r) - pr 
 Rr + B(R + r)' 
 
 217. In a network PA, PB, PC, PD, AB, BC, CD, DA, the resistances are a, ft y, 8, 
 y + 8, 8 + a, a + ft /3 + y respectively. Show that if AD contains a battery of electro- 
 motive force E, the current in BC is 
 
 P (aft + yd) E 
 2P 2 Q + (/3d - ay) 2 ' 
 where P = a + /3 + y + 8, Q = &y + ya + a/3 + a8 + (ft + y8. 
 
 218. A wire forms a regular hexagon and the angular points are joined to the centre 
 by wires each of which has a resistance 1/n of the resistance of a side of the hexagon. Show 
 that the resistance to a current entering at one angular point of the hexagon and leaving 
 
 it by the opposite point is 
 
 2 (n + 3) 
 (n +!)( + 4) 
 times the resistance of a side of the hexagon. 
 
 219. Two long equal parallel wires AB, A'B' of length I have their ends B, B' joined 
 by a wire of negligible resistance, while A, A' are joined to the poles of a cell whose 
 resistance is equal to that of a length r of the wire. A similar cell is placed as a bridge 
 across the wire at a distance x from A, A'. Show that the effect of the second cell is to 
 increase the current in BB' in the ratio 
 
 2 (21 + r) (x + r) 
 r(4J+ r) + 2x(2l - r) - 4x 2 ' 
 
 220. There are n points 1,2 ... n joined in pairs by linear conductors. On introducing 
 a current C at electrode 1 and taking it out at 2, the potentials of these are < 1; < 2 , $n- 
 If x lz is the actual current in the direction 12, and x' 12 any other that merely satisfies the 
 conditions of introduction at 1 and abstraction at 2, show that 
 
 2 r lz x lz x, z = (</>! - (f> 2 ) C = 2 (r 12 x lz 2 ), 
 
 and interpret the result physically. If x typify the actual current when the current enters 
 at 1 and leaves at 2, and y typify the actual current when the current enters at 3 and leaves 
 at 4, shew that 
 
 2 (r ia * 18 y 12 ) - (X 3 ~ *) C = (Y, - Y 2 ) C, 
 
 where the X's are potentials corresponding to the currents x, and the T's are potentials 
 corresponding to the currents y. 
 
 221. In a simple network of conductors joining n points, the currents C lt C 2 , ... C n _ 1 
 are supplied at (n - 1) of them and taken out at the nth point. There are also electro- 
 motive forces in the conductors. Show that the heat function 2-R M C' pg 2 can be expressed 
 as the sum of two quadratic functions H c and H e of the entering amounts and the electro- 
 motive forces respectively and that the current in any conductor pq is given by 
 
 r -L (^ - ^ 
 
 Develope from this point of view the theorems of minimum heat dissipation and the 
 reciprocal relations between the currents and potentials in the circuits. 
 
Electrokinetics 691 
 
 222. A, B, C are three stations on the same telegraph wire. An operator at A knows 
 that there is a fault between A and B, and observes that the current at A when he uses 
 a given battery is i, i' or i", according as B is insulated and C to earth, B to earth, or B 
 and C both insulated. Show that the distance of the fault from A is 
 
 {ka - k'b + (b - afi (ka - k'bfi] /k - k', 
 where AB = a, BC = b - a, k = : 7-,, 
 
 223. Six conductors join four points A, B, C, D in pairs, and have resistances a, a, 
 b, /3, c, y, where a, a refer to BC, AD respectively, and so on. If this network be used as 
 a resistance coil, with A, B as electrodes, show that the resistance cannot lie outside the 
 limits 
 
 11 in- 1 
 
 - + 7 + 
 
 c a + b 
 
 
 
 [1 + {(i + I)- 1 + (i + i)" 1 }" 1 ]". 
 
 and 
 
 224. Two equal straight pieces of wire A A n , B B n are each divided into n equal 
 parts at the points A lt A 2 , ... A n _ l and B lt B z , ... B n _ l respectively, the resistance of 
 each part and that of A n B n being R. The corresponding points of each wire from 1 to n 
 inclusive are joined by cross wires, and a battery is placed in A^B . Show that, if the 
 current through each cross wire is the same, the resistance of the cross wire A g B s is 
 
 {( - s) 2 + (n - s) + 1}R. 
 
 225. A network is formed of a system of conductors joining every pair of a set of 
 n points, the resistances of the conductors being all equal, and there is an electromotive 
 force in the conductor joining the points A lt A 2 . Show that there is no current in any 
 conductor except those which pass through A or A 2 and find the current in these con- 
 ductors. 
 
 226. Each member of the series of n points A lt A 2 , A 3 , ... A n is united to its successor 
 by a wire of resistance p, and similarly for the series of n points B lt B z , ... B n . Each pair 
 of points corresponding in the two series, such as A r and B r , is united by a wire of resistance 
 R. A steady current i enters the network at AI and leaves it at B n . Show that the current 
 at A! divides itself between A^A Z and ' A^B^ in the ratio 
 
 sinh a + sinh (n - 1) a + sinh (n - 2) a : sinh a + sinh (n - I) a - sinh (n - 2) a, 
 
 R+ p 
 
 where cosh a = s . 
 
 K 
 
 227. An underground cable of length a is badly insulated so that it has faults throughout 
 its length indefinitely near to^one another and uniformly distributed. The conductivity of 
 the faults is l/p' per unit length of cable, and the resistance of the cable is p per unit length. 
 One pole of a battery is connected to one end of the cable and the other pole is earthed. 
 Prove that the current at the farther end is the same as if the cable were free from faults 
 and of total resistance 
 
 Vpp'tanh [a \/ ,} 
 
 \ v pj 
 
 \ 
 
 228. Two parallel conducting wires at unit distance are connected by (n + 1) cross 
 pieces of the same wire, so as to form n squares. A current enters by an outer corner 
 
 442 
 
692 Appendix 
 
 of the first square, and leaves by the diagonally opposite corner of the last. Show that, 
 if the resistance is that of a length \n + a n of the wire, 
 
 a n + 1 = ~~ ~T~n 
 
 a n + 2 
 
 229. A, B are the ends of a long telegraph wire with a number of faults, and C is an 
 intermediate point on the wire. The resistance to a current sent from A is R when C is 
 earth-connected, but if C is not earth-connected the resistance is S or T according as the 
 end B is to earth or insulated. If R', S', T' denote the resistances under similar circum- 
 stances when a current is sent from B towards A, show that 
 
 T' (R - S) = R' (R - T). 
 
 230. The inner plates of two condensers of capacities C, C' are joined by wires of 
 resistances R, R' to a point P, and their outer plates by wires of negligible resistance to 
 a point Q. If the inner plates be also connected through a galvanometer, show that the 
 needle will suffer no sudden deflection on joining P, Q to the poles of a battery, if CR = C'R'. 
 
 231. An infinite cable of capacity and resistance K and R per unit length is at zero 
 potential. At the instant t = one end is suddenly connected to a battery for an 
 infinitesimal interval of time and then insulated. Show that, except for very small values 
 of t, the potential at any instant at a distance x from this end of the cable will be pro- 
 portional to 
 
 232. A condenser is formed of two large parallel plates of area A, separated by a 
 thickness d of a medium with dielectric constant e and resistance k. If it is charged, show 
 
 1 Afk 
 
 that the time in which the charge will sink to - of its original amount is log n. 
 
 233. The sectional area of a wire of uniform material of specific resistance o- is equal 
 to A + Bx 2 , where x is the distance from one end. Determine the resistance of a length 
 of the wire ; and find the position of a point at which the electric potential is the mean 
 of those at the ends, when a steady current is flowing through it. 
 
 234. The ends of a rectangular conducting lamina of breadth c, length a, and uniform 
 thickness T, are maintained at different potentials. If p = / (x, y) be the specific resistance 
 of a point whose distances from an end and a side are x, y, prove that the resistance of the 
 lamina cannot be less than 
 
 a f dx 
 
 -775' ; 
 
 oJ -/op 
 or greater than 
 
 235. Two large vessels filled with mercury are connected by a capillary tube of uniform 
 bore. Find superior and inferior limits to the conductivity. 
 
 236. A cylindrical cable consists of a conducting core of copper surrounded by a thin 
 insulating sheath of material of given specific resistance. Show that if the sectional areas 
 of the core and sheath are given, the resistance to lateral leakage is greatest when the 
 surfaces of the two materials are coaxal right circular cylinders. 
 
Electrokinetics 693 
 
 237. Prove that the product of the resistance to leakage per unit length between 
 two practically infinitely long parallel wires insulated by a uniform dielectric and at different 
 
 potentials, and the capacity per unit length, is -~ , where e is the inductive capacity and 
 
 p the specific resistance of the dielectric. Prove also that the time that elapses before 
 the potential difference sinks to a given fraction of its original value is independent of the 
 sectional dimensions and relative positions of the wires. 
 
 238. If the right sections of the wires in the last question are semi-circles described 
 on opposite sides of a square as diameters, and outside the square, while the cylindrical 
 space whose section is the semi-circles similarly described on the other two sides of the 
 square is filled up with a dielectric of infinite specific resistance, and all the neighbouring 
 space is filled up with a dielectric of resistance p, prove that the leakage per unit length 
 in unit time is 2F/p, where F is the potential difference. 
 
 239. If 4> + ty = f (x + iy), and the curves <p const, be closed curves, show that the 
 insulation resistance between lengths / of the surfaces <p = (f> , (p = < t is 
 
 where [>//] is the increment of \^ on passing once round a </>-curve, and p is the specific 
 resistance of the dielectric. 
 
 240. Current enters and leaves a uniform circular disc through two circular wires of 
 small radius e whose central lines pass through the edge of the disc at the extremities of 
 a chord of length d. Show that the total resistance of the sheet is 
 
 (2r/7r) log (dje). 
 
 241. Using the transformation 
 
 log a: + iy = + iy, 
 
 prove that the resistance of an infinite strip of uniform breadth IT between two electrodes 
 distant 2a apart, situated on the middle line of the strip and having equal radii 8, is 
 
 "i / 2 . \ 
 log I j: tanh a 1 . 
 
 242. Show that the transformation 
 
 , TT (x + iy) 
 x + iy = cosh 
 
 enables us to obtain the potential due to any distribution of electrodes upon a thin conductor 
 in the form of the semi-infinite strip bounded by y = 0, y = a, and x = 0. 
 
 If the margin be uninsulated, find the potential and flow due to a source at the point 
 
 x = c, y = *. Show that if the flows across the three edges are equal, then 
 
 TTC = a cosh" 1 2. 
 
 243. Equal and opposite electrodes are placed at the extremities of the base of an 
 isosceles triangular lamina, the length of one of the equal sides being a, and the vertical 
 
 an gl e JL m Show that the lines of flow and equal potentials are given by 
 o 
 
 U S W 1 /Q ^ ~*~ Cn% 
 
 2 1 - cnu' 
 
 where 3* r (T) ua = r (|) r () (*'- a 
 
 VB/ \V VV \ 
 
 and the modulus of en u is sin 75, the origin being* at the vertex. 
 
694 Appendix 
 
 244. A circular sheet of copper, of specific resistance o^ per unit area, is inserted in 
 a very large sheet of tinfoil (o- ) and currents flow in the composite sheet, entering and 
 leaving at the electrodes. Prove that the current function in the tinfoil corresponding 
 to an electrode at which a current e enters the tinfoil is the coefficient of i in the imaginary 
 part of 
 
 or e |~, (TO - ar 1 cz 
 
 - log (z - c) + - log , 
 2ir L o- + o-j 6 C2 - a 2 J 
 
 where a is the radius of the copper sheet, z is a complex variable with its origin at the centre 
 of the sheet, and c is the distance of the electrode from the origin, the real axis passing 
 through the electrode. 
 
 Investigate the corresponding expressions determining the lines of flow in the copper. 
 
 245. A uniform conducting sheet has the form of the catenary of revolution 
 
 OC 
 
 y z + z 2 = c 2 cosh 2 - . 
 c 
 
 Prove that the potential at any point due to an electrode at (x , y , z ) introducing a 
 current C is 
 
 c<r. ( x - x 
 
 const. - log 4 cosh 
 
 47T C 
 
 246. Electric currents are introduced into an infinite plane sheet at a series of electrodes. 
 Show that the asymptotes of the lines of flow all meet in the mean centre (centre of gravity) 
 of the electrodes. 
 
 247. The molecules of a metal are assumed to be fixed centres of force repelling the 
 electrons in collision with a force n/r n . If it is still assumed that the average duration of 
 a collision is small compared with the time in an average path between two collisions, 
 show that the velocity distribution function is given by an equation of the type 
 
 where the notation is as in the text except that now 
 
 and l m is a constant depending on /*. 
 
 Hence develope a general theory of conductivity on this basis and shew that the funda- 
 mental laws are still satisfied. 
 
 248. Prove that the velocity distribution function determined in the last question 
 may be determined directly from the fact that the electrons at any instant started the free 
 paths which they are then pursuing with the velocities assigned to them by Maxwell's law. 
 
 249. Prove by direct calculation that the energy dissipated per unit volume of a metal 
 subject to an applied field of strength E, but in which the conditions are otherwise 
 uniform, is 
 
 where a- is the ordinary expression for the conductivity. 
 
 250. Prove that a theoretically sufficient explanation of the Volta, Peltier and 
 Thomson effects in metals can be obtained on the assumption that a part of the force 
 exerted by an atom on an electron on collision varies with the conditions and type of the 
 metal molecules. 
 
 How far does this explanation differ from that given by Lorentz which is reproduced 
 in the text above ? 
 
Electrodynamics 695 
 
 IV. ELECTRODYNAMICS. 
 
 251. A current J flows in a very long straight wire. Find the forces and couples it 
 exerts upon a small magnet. 
 
 Show that if the centre of the small magnet be fixed at a distance c from the wire, it 
 has two free small oscillations about its position of equilibrium, of equal period 
 
 /2mJ 
 
 V'~Jc 
 
 where / is the moment of inertia, and m the magnetic moment of the magnet. 
 
 252. Two parallel infinite wires convey equal currents of strength J in opposite 
 directions, their distance apart being 2a. A magnetic particle of strength p. and moment 
 of inertia / is free to turn about a pivot at its centre, distant c from each of the wires. 
 Show that the time of a small oscillation is that of a pendulum of length I given by 
 
 4Jaml gc z l. 
 
 253. Regarding the earth as a uniformly and rigidly magnetised sphere of radius a, 
 and denoting 'the intensity of the magnetic field on the equator by H, show that a wire 
 surrounding the earth along the parallel of south latitude X, and carrying a current J from 
 west to east, would experience a resultant force towards the south pole of the heavens of 
 
 amount 
 
 QnaJH sin X cos 2 X. 
 
 254. A current J flows in a circuit in the shape of an ellipse of area A and length /. 
 Show that the force at the centre is irJl/A. 
 
 255. Show that at any point along a line of force the vector potential due to a current 
 in a circle is inversely proportional to the distance from the centre of the circle to the foot 
 of the perpendicular from the point on to the plane of the circle. Hence trace the lines of 
 constant vector potential. 
 
 256. A current J flows round a circle of radius a, and a current J' flows in a very long 
 straight wire in the same plane. Show that the mutual attraction is ^irJJ' (sec a - 1), 
 where a is the angle subtended by the circle at the nearest point of the wire. 
 
 If the circle is placed perpendicular to the straight wire with its centre at a distance 
 c from it, shew that there is a couple tending to set the two wires in the same plane, of 
 moment 2irJJ'a 2 /c or 2irJJ'c, according as c > or <a. 
 
 257. A current J flows round a circular wire which can turn about a fixed diameter ; 
 and a current J' passes through a long straight wire parallel to this diameter and so placed 
 that the plane through the wire and the diameter is perpendicular to the plane of the 
 circle. Show that there is a couple on this circular wire tending to set the two wires in the 
 
 (c \ 
 1 ) , where c is the distance of the 
 
 v a 2 + c 2 / 
 
 wire from the plane of the circle and a the radius of the circle. 
 
 258. A long straight current intersects at right angles a diameter of a circular current, 
 and the plane of the circle makes an acute angle a with the plane through this diameter 
 and the straight current. Show that the coefficient of mutual induction is 
 
 4n- {c sec a - (c 2 sec 2 a - a 2 )*} or 4?rc tan (^ - ^ J , 
 
 according as the straight current passes within or without the circle, a being the radius of 
 the circle, and c the distance of the straight current from its centre. 
 
696 Appendix 
 
 259. Prove that the coefficient of mutual induction between a pair of infinitely long 
 straight wires and a circular one of radius a in the same plane, and with its centre at a 
 distance (b ~>a) from each of the straight wires, is 
 
 STT (6 - V& 2 - a 2 ). 
 
 260. A circuit contains a straight wire of length 2a conveying a current. A second 
 straight wire, infinite in both directions, makes an angle a with the first, and their common 
 perpendicular is of length c and meets the first wire in its middle point. Prove that the 
 additional electromagnetic forces on the first straight wire, due to the presence of a current 
 in the second wire, are equivalent to a wrench of pitch 
 
 / . a sin a\ / . a sin a 
 
 2 a sm a - c tan" 1 / sin 2a tan" 1 . 
 
 V e // c 
 
 261. Two circular wires of radii a, b have a common centre, and are free to turn on 
 an insulating axis which is a diameter of both. Show that when the wires carry currents 
 J, J' a couple of magnitude 
 
 9 & 2 \ 
 " 16 V 
 
 is required to hold them with their planes at right angles, it being assumed that b/a is so 
 small that its fifth power may be neglected. 
 
 262. Two circular circuits are in planes at right angles to the line joining their centres. 
 Show that the coefficient of induction 
 
 C0s2ddd 
 
 o V a 2 sin 2 6 + C 2 cps 2 
 
 where a, c are the longest and shortest lines which can be drawn from one circuit to the 
 other. Find the force between the circuits when currents are flowing in them. 
 
 263. Two currents J, J' flow round two squares each of side a, placed with their edges 
 parallel to one another and at right angles to the distance c between their centres. Show 
 that they attract with a force 
 
 a 2 + 
 
 l _ 
 
 264. A current J flows in a rectangular circuit whose sides are of lengths 2a, 2b and 
 the circuit is free to rotate about an axis through its centre parallel to the sides of length 
 2a. Another current J' flows in a long straight wire parallel to the axis and at a distance 
 d from it. Prove that the couple required to keep the plane of the rectangle inclined at 
 an angle <f> to the plane through its centre and the straight current is 
 
 8.7 J' abd (6 2 + d 2 ) sin ft 
 b* + d* - 26 2 d 2 cos2ft ' 
 
 265. Two circular wires lie with their planes parallel on the same sphere, and carry 
 opposite currents inversely proportional to the areas of the circuits. A small magnet has 
 its centre fixed at the centre of the sphere and moves freely about it. Show that it will 
 be in equilibrium when its axis is either at right angles to the planes of the circuits, or makes 
 an angle tan" 1 4 with them. 
 
 266. An infinite long straight wire conveys a current and lies in front of and parallel 
 to an infinite block of soft iron bounded by a plane face. Find the magnetic potential at 
 all points, and the force which tends to displace the wire. 
 
Electrodynamics 697 
 
 267. A small sphere of radius b is placed in the neighbourhood of a circuit, which 
 when carrying a current of unit strength would produce a magnetic force H at the point 
 where the centre of the sphere is placed. Show that, if K is the coefficient of induced 
 magnetisation for the sphere, the presence of the sphere increases the coefficient of self- 
 induction of the wire by an amount approximately equal to 
 
 (3 + 47TK) 2 
 
 268. A circular wire of radius a is concentric with a spherical shell of soft iron of radii 
 b and c. If a steady unit current flow round the wire, show that the presence of the iron 
 increases the number of lines of induction through the wire by 
 
 27r 2 a 4 (c 3 - 6 3 )Q* - l)(/i + 2) 
 b 3 {(2fi + 1) (p + 2) c 3 - 2 (p- I) 2 6 3 } ' 
 
 approximately, where a is small compared with b and c. 
 
 269. A right circular cylindrical cavity is made in an infinite mass of iron of perme- 
 ability p. In this cavity a wire runs parallel to the axis of the cylinder carrying a steady 
 current of strength I. Prove that the wire is attracted towards the nearest part of the 
 surface of the cavity with a force per unit length equal to 
 
 2(/i- I)/ 2 
 (/*+ l)d ' 
 where d is the distance of the wire from its electrostatic image in the cylinder. 
 
 270. A wire is wound in a spiral of angle a on the surface of an insulating cylinder 
 of radius a so that it makes n complete turns on the cylinder. A current J flows through 
 the wire. Prove that the resultant magnetic force at the centre of the cylinder is 
 
 2irJn 
 
 a (1 -t- 7r 2 % 2 tan 2 a) 
 along the axis. 
 
 271. Coils of wire in the form of circles of latitude are wound upon a sphere and 
 produce a magnetic potential Ar n P n at internal points when a current is sent through them. 
 Find the mode of winding and the potential at external points. 
 
 272. A current of strength J flows along an infinitely long straight wire, and returns 
 in a parallel wire. These wires are insulated and touch along generators the surface of an 
 infinite uniform circular cylinder of material whose coefficient of induction is k. Prove that 
 the cylinder becomes magnetised as a lamellar magnet whose strength is 
 
 2irkJ/(l 
 
 273. A fine wire covered with insulating material is wound in the form of a circular 
 disc, the ends being at the centre and circumference. A current is sent through the wire 
 such that / is the quantity of electricity that flows per unit time across unit length of any 
 radius in the disc. Show that the magnetic force at any point on the axis of the disc is 
 
 27r/ {cosh" 1 (sec a) - sin a}, 
 where a is the angle subtended at the point by any radius of the disc. 
 
 274. A given current sent through a tangent galvanometer deflects the magnet through 
 an angle 6. The plane of the coil is slowly rotated round the vertical axis through the 
 centre of the magnet. Prove that if 6 > \ IT the magnet will describe complete revolutions, 
 but if 6 < \ir, the magnet will oscillate through an angle sin" 1 tan 6 on each side of the 
 meridian. 
 
698 Appendix 
 
 275. . Prove that if a slight error is made in reading the angle of deflection of a tangent 
 galvanometer, the percentage error in the deduced value of the current is a minimum if 
 the angle of deflection is JTT. 
 
 276. A tangent galvanometer is incorrectly fixed, so that equal and opposite currents 
 give angular readings a and /3 measured in the same sense. Show that the plane of the 
 coil, supposed vertical, makes an angle f with its proper position such that 
 
 2 tan e = tan a + tan /3. 
 
 Hence show that the real value of the current is the harmonic mean of its apparent magni- 
 tudes when sent in opposite directions round the galvanometer circuit. 
 
 277. In a tangent galvanometer, the sensibility is measured by the ratio of the incre- 
 ment of deflection to the increment of current, estimated per unit current. Show that the 
 galvanometer will be most sensitive when the deflection is jr/4, and that in measuring the 
 current given by a generator whose electromotive force is E, and internal resistance R, the 
 galvanometer will be most sensitive if there be placed across the terminals a shunt of 
 resistance 
 
 HRr 
 
 E - 
 
 where r is the resistance of the galvanometer and H is the constant of the galvanometer. 
 What is the meaning of this result if the denominator vanishes or is negative ? 
 
 278. A galvanometer coil of n turns is in the form of an anchor ring described by the 
 revolution of a circle of radius b about an axis in its plane distant a from its centre. Show 
 that the constant of the galvanometer 
 
 I cifiudn^udu (k = b/a) 
 
 a> Jo 
 
 = (8w/3A; 2 a) [(1 + F) E - (1 - P) JT|. 
 
 279. A coil is rotated with constant angular velocity about an axis in its plane in 
 a uniform field of force perpendicular to the axis of rotation. Find the current in the coil 
 at any time, and show that it is greatest when the plane of the coil makes an angle 
 
 tan " 1 (^) 
 with the lines of magnetic force. 
 
 280. The ends B, D of a wire (R, L) are connected with the plates of a condenser of 
 capacity C. The wire rotates about BD which is vertical with angular velocity o>, the area 
 between the wire and BD being A. If H is the horizontal component of the earth's 
 magnetism, show that the average rate at which work must be done to maintain the rotation 
 
 281. The resistance and self- induction of a coil are E and L, and its ends A and B 
 are connected with the electrodes of a condenser of capacity C by wires of negligible resist- 
 ance. There is a current J cos pt in a circuit connecting A and B, and the charge of the 
 condenser is in the same phase as this current. Show that the charge at any time is 
 
 LJ cos pt 
 ~R ' 
 
 and that C (R z + p 2 L 2 ) = L. 
 
 Obtain also the current in the coil. 
 
Electrodynamics 699 
 
 282. A closed solenoid consists of a large number N of circular coils of wire, each 
 of radius a, wound uniformly upon a circular cylinder of height 2h. At the centre of the 
 cylinder is a small magnet whose axis coincides with that of the cylinder, and whose 
 moment is a periodic quantity p sin pt. Show that a current flows in the solenoid whose 
 intensity is approximately 
 
 2irpNp 
 
 sm (pt + a), 
 
 {(a 2 + h*)(R* + V)P 
 
 p 
 
 where R, L are the resistance and self-induction of the solenoid, and tan a = y- . 
 
 Li 
 
 283. A circular coil of n turns, of radius a and resistance B, spins with angular velocity 
 w round a vertical diameter in the earth's horizontal magnetic field H: show that the 
 average electromagnetic damping couple which resists its motion is ^H 2 n 2 TT 2 a 2 a>R. 
 
 284. A condenser, capacity C, is discharged through a circuit, resistance R, induction 
 L, containing a periodic electromotive force Esinnt. Show that the 'forced' current 
 in the circuit is 
 
 E sin (nt - 6) \ R 2 + (nl - gr-) | , 
 
 , . n*CL - 
 
 where tan 6 = 
 
 nCR 
 
 285. Two electric circuits of negligible resistances have self -inductances L it L z and 
 are tuned separately to the same natural free period by varying the capacities C lt C 2 of 
 the condensers in the two circuits. Find the periods of free oscillation in the system con- 
 sisting of the two circuits in the presence of one another, showing how they are affected by 
 changes in the value of the mutual inductance, M . 
 
 The condenser of the first circuit is initially charged to a difference of potential <f) lt 
 while the second condenser is uncharged and there is no initial current in either circuit; 
 
 show that if M <Vi 1 L 2 the potential difference of the second condenser cannot exceed 
 
 286. Two equal long straight coils are placed end to end so that the direction of the 
 winding is the same in both and they are acted on in series by an electromotive force 
 E cos pt. Show that if, without change in position, they are placed in parallel (the current 
 being led in at the middle and out at the two ends) the maximum current in either coil 
 is increased and the lag diminished. 
 
 Discuss also the case where the coils are wound in opposite directions. 
 
 287. Two circuits, resistances R and R 2 , coefficients of induction L, M, N, lie near 
 each other, and an electromotive force E is switched into one of them. Show that the 
 total quantity of electricity that traverses the other is 
 
 288. A current is induced in a coil B by a current J sin pt in a coil A. Show that the 
 mean force tending to increase any coordinate of position 6 is 
 
 _ 1 J*p z LM 8M 
 2R* + L 2 p 2 80 ' 
 where L, M, N are the coefficients, of induction of the coils, and R is the resistance of B. 
 
 289. A plane circuit, area S, rotates with uniform velocity o> about the axis of 2, 
 which lies in its plane at a distance h from the centre of gravity of the area. A magnetic 
 
00 Appendix 
 
 molecule Of strength m is fixed in the axis of a; at a great distance a from the origin pointing 
 in the direction Ox. Prove that the current at time t is approximately 
 
 28o>m 
 
 , COS (cot - e) + - = COS (2cot - T]), 
 
 a 3 (R 2 + L 2 o> 2 )* a 4 (R 2 + 4Z, 2 a> 2 )* 
 
 where e, i) are determinate constants. 
 
 290. Two points A, B are joined by a wire of resistance R without self-induction ; 
 B is joined to a third point C by two wires each of resistance R, of which one is without 
 self- induction, and the other has a coefficient of induction L. If the ends A, C are kept 
 at a potential difference E cos pt, prove that the difference of potentials at B and C will be 
 
 E' cos (pt - y), 
 
 291. A condenser, capacity C, charge Q, is discharged through a circuit of resistance 
 R, there being another circuit of resistance 8 in the field. If LN M 2 , show that there 
 will be initial currents - NQ/C (UN + SL) and MQ/C (UN + SL), and find the currents 
 at any time. 
 
 292. Two insulated wires A, B of. the same resistance have the same coefficient of 
 self-induction L, while that of mutual induction is slightly less than L. The ends of B 
 are connected by a wire of small resistance, and those of A by a battery of small resistance, 
 and at the end of a time t a current J is passing through A. Prove that except when t is 
 very small 
 
 J=l(Jo + J'), 
 
 approximately, where J is the permanent current in A, and J' is the current in each after 
 a time t, when the ends of both are connected in multiple arc by the battery. 
 
 293. The ends of a coil forming a long straight uniform solenoid of m turns per unit 
 length are connected with a short solenoidal coil of n turns and cross section A, situated 
 inside the solenoid, so that the whole forms a single complete circuit. The latter coil can 
 rotate freely about an axis at right angles to the length of the solenoid. Show that in free 
 motion without any external field, the current J and the angle 6 between the cross sections 
 of the coils are determined by the equations 
 
 RJ = - -r (L! J + L 2 J + SirmnAJ cos 6), 
 dt 
 
 d z d 
 
 I -y-r- + 
 dt? 
 
 where LI , L 2 are the coefficients of self-induction of the two coils, I is the moment of inertia 
 of the rotating coil, R is the resistance of the whole circuit, and the effect of the ends of 
 the long solenoid is neglected. 
 
 294. Two electrified conductors whose coefficients of electrostatic capacity are y^ , y z , T 
 are connected through a coil of resistance R and large inductance L. Verify that the 
 frequency of the electric oscillations thus established is 
 
 JL f 2r + ?i' + ya I 
 T 2 ' L 
 
Electrodynamics 701 
 
 295. An electric circuit contains an impressed electromotive force which alternates 
 in an arbitrary manner and also an inductance. Is it possible, by connecting the ex- 
 tremities of the inductance to the poles of a condenser, to arrange so that the current in 
 the circuit shall always be in step with the electromotive force and proportional to it ? 
 
 296. Two coils (resistances R, 8; coefficients of induction L, M, N) are arranged in 
 parallel in such positions that when a steady current is divided between the two, the 
 resultant magnetic force vanishes at a certain suspended galvanometer needle. Prove 
 that if the currents are suddenly started by completing a circuit including the coils, then 
 the initial magnetic force on the needle will not in general vanish, but that there will be 
 a 'throw' of the needle, equal to that which would be produced by the steady (final) current 
 in the first wire flowing through that wire for a time interval 
 
 M - L _ M - N 
 ~~IT~ S ' 
 
 297. A condenser of capacity C is discharged through two circuits, one of resistance 
 R and self-induction L, and the other of resistance R' and containing a condenser of capacity 
 C'. Prove that if Q is the charge on the condenser at any time 
 
 L \d*Q (R^R V\dQ Q _ 
 
 c + c f+ H ) dt\ + \c ' c r + c)~dt + cc' ~ 
 
 298. A condenser of capacity C is connected by leads of resistance r, so as to be in 
 parallel with a coil of self-induction L, the resistance of the coil and its leads being R. If 
 this arrangement forms part of a circuit in which there is an electromotive force of period 
 Zirlp, show that it can be replaced by a wire without self-induction if 
 
 (,R 2 - L/C) = p z LC (r* - L/C), 
 and that the resistance of this equivalent wire must be 
 
 (Rr + L/C)/(R + r). 
 
 299. Two coils, of which the coefficients of self and mutual induction are L lt L z , M, 
 and resistances R l , R 2 , carry steady currents C l , C 2 produced by constant electromotive 
 forces inserted in them. Show how to calculate the total extra currents produced in the 
 coils by inserting a given resistance in one of them, and thus also increasing its coefficients 
 of induction by given amounts. 
 
 In the primary coil, supposed open, there is an electromotive force which would produce 
 a steady current C, and in the secondary coil there is no electromotive force. Prove that 
 the current induced in the secondary by closing the primary is the same, as regards its 
 effects on a galvanometer and an electrodynamometer, and also with regard to the heat 
 produced by it, as a steady current of magnitude 
 
 -t z a ^ 
 
 lasting for a time ' * l , 
 
 % Jti K^ 
 
 while the current induced in the secondary by breaking the primary circuit may be repre- 
 sented in the same respects by a steady current of magnitude CM/2L 2 lasting for a time 
 2L 2 /R 2 . 
 
 300. Four points A, B, C, D are connected up as follows: A, B are joined through 
 a coil of self-induction L and resistance P; A, D through a resistance Q; B, C through a 
 resistance R; C, D through a resistance S and through a condenser of capacity K, the 
 
702 Appendix 
 
 resistance- and condenser being in parallel; B, D through a galvanometer; A, C through 
 a source of current of period 2ir/p. Show that if no current passes through the galvanometer 
 
 PS = QR and L = QRK. 
 [The resistances of the connecting wires may be neglected.] 
 
 301. Two conductors ABD, ACD are arranged in multiple arc. Their resistances are 
 R, 8 and their coefficients of self and mutual induction L, N, and M . Prove that when 
 placed in series with leads conveying a current of frequency p, the two circuits produce 
 the same effect as a single circuit whose coefficient of self-induction is 
 
 NR 2 + LS 2 + 2MRS + p 2 (LN - M*) (L + N - 2M) 
 (L + N - 2M) 2 p 2 + (R+ S) 2 
 
 and whose resistance is 
 
 MS (8 + R) + p 2 {R (N - M ) 2 + 8 (L - M) 2 } 
 (L + N - 2M) 2 p 2 + (R + S) 2 
 
 302. A condenser of capacity C containing a charge Q is discharged round a circuit 
 in the neighbourhood of a second circuit. The resistances of the circuits are R, S and 
 their coefficients of induction are L, M, N. Obtain equations to determine the currents at 
 any moment. 
 
 If x is the current in the primary and the disturbance be over in a time less than 
 T, prove that 
 
 and that 
 
 + + S 2 LR x 2 dt = {CS*L + CSNR + N 2 } 
 
 
 C \ G / ) J o 
 
 r 
 
 Examine how I x z dt varies with S. 
 Jo 
 
 303. Prove that the currents induced in a solid with an infinite plane face, owing to 
 magnetic changes near the face, circulate parallel to it, and may be regarded as due to the 
 diffusion into the solid of currents induced at each instant on the surface so as to screen 
 off the magnetic changes from the interior. 
 
 Show that for periodic changes the current penetrates to a depth proportional to the 
 square root of the period. 
 
 304. A magnetic system is moving towards an infinite plane conducting sheet with 
 velocity w. Show that the magnetic potential on the other side of the sheet is the same 
 as it would be if the sheet were away, and the strengths of all the elements of the magnetic 
 system were changed in the ratio R/(R + w), where 2-rrR is the specific resistance of the 
 sheet per unit area. Show that this result is unaltered if the system is moving away from 
 the sheet and examine the case when R = - w. 
 
 If the system is a magnetic particle of mass m and moment p, with its axis perpendicular 
 to the sheet, prove that if the particle has been projected at right angles to the sheet, then 
 when it is at a distance z from the sheet its velocity is given by 
 
 \m (z-R) 2 = C- . 
 
Electrodynamics 703 
 
 305. A small magnet horizontally magnetised is moving with a velocity u parallel to 
 a thin horizontal plate of metal. Show that the retarding force on the magnet due to the 
 
 currents induced in the plate is 
 
 * 2 uR 
 
 (&j* Q (Q + R) ' 
 
 where p is the moment of the magnet, c its distance above the plate, 2irR the resistance 
 of a unit area of the plate, and Q 2 = u 2 + R 2 . 
 
 306. A uniform line of magnetic poles is suddenly generated parallel to the axis of 
 a long cylindrical conducting shell (mean radius a) of small thickness 8. Prove that the 
 field inside the shell is equivalent to that of a line of opposite poles of the same strength 
 starting from the original position of the generated line and moving radially away from 
 
 the axis so that its distance from this axis at any subsequent time' is be a . 
 
 Show also that the external field is equivalent to that of two lines of poles of the same 
 uniform strength, a negative one along the axis of the cylinder and a positive one starting 
 from the position of the inverse of the given initial line and moving towards the axis so 
 
 _4ir<rSt 
 
 a?e a 
 
 that its distance at the time t is - 
 
 b 
 
 307. Examine the effect of the sudden generation of a line of magnetic poles inside 
 a thin cylindrical conducting shell and parallel to the axis, obtaining the complete image 
 system for both the internal and external fields. 
 
 308. If a linear current of strength J be suddenly generated parallel to the axis of a 
 conducting cylindrical shell of mean radius a and thickness 8 and at a distance b from the 
 axis (b >a), show that the image representing the induced currents for points inside the 
 
 47TO-? 
 
 sphere will be a current of strength - J in a symmetrical line at a distance be from 
 the axis. Find also the images representing the effect of these currents at external 
 points. 
 
 309. A thin spherical shell of radius a is introduced into an oscillating magnetic field 
 whose potential in the immediate neighbourhood of the sphere can be written in the form 
 
 / r \n 
 
 ^ = ^ + A cos pt ( - } Y n , 
 
 'where Y n is a surface harmonic of order n. Show that the field inside the shell has a 
 potential 
 
 / r \n 
 
 ^ = ^ + A cos x cos (pt + x ) ( - ) Y n , 
 
 (2n + 1) <r8 
 
 where tan Y = - , 
 
 irap 
 
 8 being the thickness of the shell and a- its specific inductive capacity. 
 Find also the external field. 
 
 310. A uniform solid conducting sphere of radius a and conductivity a- is introduced 
 into an oscillating magnetic field whose potential in its neighbourhood can be written in 
 the form + = + + A i* Y n cos pt. 
 
 Show that to a first approximation when the period of oscillation is not too small the 
 external field is derived from the potential 
 
 and examine the internal field. 
 
704 Appendix 
 
 311. -A magnet of moment m is suddenly generated at a distance b from the centre 
 of a thin spherical conducting shell of mean radius a, and thickness 8, with its axis directed 
 towards the centre. 
 
 Prove that at any time t after the generation of the magnet, the potential of the field 
 due to the induced currents is, at points inside the sphere, equivalent to that in the field 
 of a magnet of moment - me skt '' 2a at a distance from the centre be kt / a and along the 
 same axis. 
 
 Prove also that at points outside the sphere the potential is equivalent to that of a 
 
 magnet of moment - me~ ' rs at a distance from the centre -j-e~ ''. 
 
 A 
 
 312. A circular current is suddenly generated symmetrically outside a thin spherical 
 conducting shell. Examine the magnetic effect of the currents induced in the shell. 
 
 Examine also the mechanical reaction between the shell and the current. 
 
 313. Examine the effect of the sudden generation of a small symmetrical magnet at 
 a point inside a thin spherical shell and obtain the magnetic image system of the currents 
 induced in the shell. 
 
 314. A copper disc of radius a, with an axle hole, is rigidly and symmetrically attached 
 to a thin bar magnet of length 2h and moment m, so that the axis of the magnet is at 
 right angles to the plane of the disc, and the system is spun about the axis of the magnet 
 with angular velocity o>. A wire in a fixed position has one end in contact with one end of 
 the axis, of the magnet, and the other makes a sliding contact with the edge of the disc. 
 Prove that the electromotive force generating a current in the wire is 
 
 m<{h- 1 - (h* + a 2 )-*}. 
 
 315. A circular conducting ring is fixed in space. A fixed conducting wire joins the 
 centre of the circle to a fixed point of the ring and another straight wire rotates round the 
 centre making sliding contact at the centre with the fixed wire and with the ring at its 
 circumference. The whole arrangement is placed in a uniform magnetic field so that the 
 plane of the circle is perpendicular to the lines of force. Show that the induced electro- 
 motive force in the closed circuit formed by the wires is 
 
 2c ' 
 
 H being the strength of the field, co the angular velocity and a the length of the rotating 
 wire. 
 
 Determine the couple necessary to keep up the rotation. 
 
 316. Prove that the rotation with angular velocity co of a solid uniform spherical 
 conductor in a field whose magnetic potential is 
 
 ^ = >^ + A r n Y. n * cos s(j>, 
 
 where Y n & is a surface harmonic of order n and class s, generates a system of currents in 
 the conductor which gives a magnetic field outside the conductor derived from a potential 
 
 *(* 
 
 - Y n s sin (j 
 4:Tr<ra>sn 
 
 (n + l)(2n + l)(2n + 3)' 
 Examine also the internal field and current distribution and prove that the retarding 
 couple exerted by the field on the sphere is approximately for slow rotations 
 2 2(r> 
 
 (n - s)l ' (n + 1) (2n + 3) (2n + 1) ' 
 
Electrodynamics 705 
 
 317. A solid sphere can execute torsional vibrations about an axis of period 
 Prove that if a uniform magnetic field is imposed perpendicular to the axis of vibration 
 the oscillations will decay at a logarithmic rate which is approximately equal to 
 
 15/ ' 
 
 where H is the strength of the applied field, a- the conductivity of the material of which the 
 sphere is made, and / the moment of inertia of the sphere about its axis of vibration. 
 
 318. Investigate the field of a rigid electrified system moving with a uniform screwing 
 motion comprising a linear translation parallel to the axis of x with velocity (cjQ) and angular 
 rotations (a> x , <a v> u> z ) about the three axes. 
 
 Show that if the system is a uniformly charged sphere the force function of the electro- 
 magnetic reactions is of the general form 
 
 w - " log + 
 
 ~ 4^ /3 g 1 - + 6c 2 
 
 1 1-f ff 2 , 1 + 
 
 02 + 03 I0g 1 
 
 12c 2 V /3 2 2$ 
 where a is the radius of the sphere and e the charge on it. ' 
 
 319. Prove that the electric and magnetic energies in the field of a charged conducting 
 sphere in uniform motion in a straight line are 
 
 ^Sifi/r^rHr 1 
 
 +- 2 , 1 + /3 
 
 - lOg -T T- - 1 
 
 320. If W is the total energy of a rigid electrical system in quasi-stationary motion 
 show that the longitudinal electromagnetic mass of the system is 
 
 _L dw 
 
 v\ d\v\' 
 
 Show also that W v -.-. , - L. 
 
 a \v\ 
 
 Explain why the above formula for the electromagnetic mass is inapplicable to the 
 case of a system which experiences the Fitzgerald-Lorentz contraction on account of its 
 motion. 
 
 321. Show that in the general case of a system of charges moving through the aether 
 in any manner the effective forces of reaction of electrodynamic origin are derivable in the 
 usual way from the Lagrangian function 
 
 both integrals being taken throughout the volume of the field, p being the charge density 
 in the position dv and v the velocity of this element through the aether. 
 
 Does this form imply any restriction as to the vector potential function to be used? 
 
 322. Find the force exerted between two point charges moving in any manner and 
 show that the rectangular components of the force on the first electron are of the type 
 
 8L _ d dL 
 dx-t dt 8%i ' 
 
 T,. 45 
 
706 Appendix 
 
 (x, y, z) being the coordinates of the position of the electron and 
 
 the square brackets indicating that the functions are taken at the time t , r being the 
 
 c 
 
 distance apart of the electrons and v x , v 2 their velocities. 
 
 323. A perfectly conducting sphere is initially at rest and carries a charge Q. Show 
 that if a uniform acceleration * is imparted to the sphere the effective force of electro- 
 dynamic origin resisting the initial accelerated motion at the time t is 
 
 * - - 
 
 o 9 \l ~ e a COS -- - - F^ sm j f > 
 
 3 ac 2 ( 2a V3 2a J 
 
 a being the radius of the sphere, c the velocity of radiation and is small compared with 1. 
 
 c 
 
 324. Investigate the initial accelerated motion of a uniformly rigidly charged dielectric 
 sphere and shew that the force necessary to maintain the uniform acceleration s is of the 
 form 
 
 2f) / pa \ 
 
 " I" , *? r D a 
 o - (ii + 2- 2 B r e 
 o a \c Ui 
 
 where the B's are constants and the different values of X are the roots of the equation 
 
 being the dielectric constant of the material of the sphere. 
 
 325. Show that the small linear oscillations in a period (2n/n) of a perfectly conducting 
 charged sphere under the periodic force P may be obtained as a solution of the equation 
 
 mx + ki- = P, 
 
 2 Q 2 
 where m = ^ 
 
 3 ac 2 
 
 1 - 
 
 an 1 
 
 2 Q 2 T' n 
 
 k ~3M 2 7 a*n z \ 2 a 2 n 2 ' 
 
 \ ~ 5~ / ~^~ 2~ 
 
 \ C / 
 
 and the sphere is presumed to have no material mass of the ordinary kind. 
 
 326. Three linear Hertzian vibrators placed at a point in three directions mutually 
 at right angles have the same period of vibration but are not necessarily in the same phase ; 
 prove that the rate at which energy is radiated from them is the sum of the rates at which 
 energy is radiated from each separately. 
 
 Hence obtain the rate at which energy is radiated from any number of Hertzian vibrators 
 of the same period placed at a point. 
 
 327. Electrical oscillations take place on two parallel rectangular conducting plates 
 so that the wave surfaces in the dielectric disturbance between the plates are normal to 
 
Electrodynamics 707 
 
 these plates. Prove that if a, b are the lengths of the edges of the plates the possible 
 oscillations have wave lengths 
 
 2 
 
 b 2 
 m, n being any positive integers. 
 
 [It can be assumed that the current flux outwards at the edges of the plates is zero. 
 
 328. Electrical oscillations take place on two coaxal cylindrical conducting sheets of 
 length / and nearly equal radii a : if the electric force in the thin shell of field is everywhere 
 radial show that the wave lengths of the possible oscillations are of the type 
 
 27T 
 
 _1_ 
 
 a 2 
 
 m, n being positive integers. 
 
 329. Electric oscillations take place in the shell of dielectric between two perfectly 
 conducting spherical surfaces of radii a, b. Show that if the conditions in the shell are 
 symmetrical round the centre and the wave surfaces are concentric spheres the period 
 equation is 
 
 p(b - a) 
 
 tan p(b - a) = f- ~. 
 1 + p 2 ab 
 
 If the wave surfaces are perpendicular to the conducting surfaces shew that the period 
 equation for the first order oscillations is 
 
 p(b - a) (I + p 2 ab) 
 
 tan p (b - a) = 
 
 - 2 a 2 
 
 p 2 a 2 )(l - p 2 b z ) + p 2 ab' 
 
 330. Prove that the components of the electric force in any electromagnetic radiation 
 field referred to spherical polar coordinates r, 6, (f> can be written in the form 
 
 E = l - - i 1* - 
 
 r ' ~ 
 
 - - 
 8r 2 c 2 8t 2 ' ~ r 8r86 r sin 6 ' 
 
 18W 
 
 rsin<98r8$ r 86 ' 
 where W and W z satisfy the equation 
 
 8 2 W 1 8 / . fl 8W\ 1 8 2 W 1 8 2 W 
 
 ~8r 2 ' + r 2 sin<9 ' dd \ ~80 ) + r- sin 2 6 ~8^ 2 = c? ~W ' 
 
 and c is the velocity of radiation. 
 
 Find the corresponding expressions in terms of W t and W z for the components of the 
 magnetic induction in the same directions. 
 
 331. Develope a theory of the Hall effect when the metal molecules are regarded as 
 point centres of force repelling the electrons with a force which varies as the inverse nth 
 power of the distance between them, and prove that in this case the gradient of the cross 
 potential which gives rise to the effect is 
 
 - I 
 
 t/ 
 r ( 
 V 
 
 r 2+ 
 
 H being the strength of the field and J y that of the current directed in perpendicular 
 directions. 
 
 452 
 
708 Appendix 
 
 332. The electric current in Hall's experiment is replaced by a thermal current driven 
 down a uniform temperature gradient -=- parallel to the ?/-axis of a coordinate system. 
 
 The magnetic field is directed parallel to the axis of x : show that there is a potential gradient 
 parallel to the axis of z of amount 
 
 1 2_ 
 
 o -l 
 
 n-5 \ \2 n-lj Rl m q< 
 
 ^(n- })) / 2 \ me "dy' 
 
 1 I -r ~ ) 
 
 \ n - 1) 
 
 the notation being as before in the text and the questions above. 
 [The Nernst and von Ettinghausen effect.] 
 
 333. Electric waves are travelling along the plane interface separating a dielectric 
 medium from a moderately bad conducting medium. Show that the electric force is nearly 
 normal to the interface if the conduction is bad but that under other conditions it has a' 
 considerable slant in the direction of propagation. 
 
 Discuss the bearing of this result on the disposition of the aerial wires at a wireless 
 telegraph receiving station. 
 
 334. Electric waves are incident on a thin metallic plate. Show that the coefficient 
 of absorption (i.e. percentage of incident energy absorbed) is proportional to the product 
 
 , of the conductivity and the thickness of the plate. 
 
 335. Light is incident on a metallic or other densely opaque thin prism of small angle 
 a ; prove that when the angle of incidence of the light is small, the real part of the complex 
 index of refraction is given by 1 + 8/a, where 8 is the deviation of the light. 
 
 Prove further that if 8 X is the deviation for the same prism when the light is incident 
 at an angle y, the imaginary part of the refractive index is given by 
 
 K' 
 
 cos 2 7 
 
 336. Prove that in the passage of a beam of light normally through a very thin film 
 of metal the phase is accelerated by an amount approximately equal to 
 
 P(1- M 2 - K 2 ), 
 
 where 8 is the thickness of the film and p, and K are defined by the relation 
 
 (' 4-ira-i . . 
 
 = (p. - in)*, 
 e pe 
 
 e', a- referring to the metal and e to the surrounding dielectric and p is the frequency of the 
 light. 
 
 337. Waves of light are incident perpendicularly on a transparent plate of thickness 
 d ; prove that the ratio of the intensities of the reflected and incident light is 
 
 (1 - p 2 ) sin 2 
 (l~- p 2 ) sin 2 6 + 4 M 2 ' 
 
 where p- is the index of refraction of the plate and 2nd/ '6 the wave length of the light on 
 the plate. Find also the intensity of the transmitted light. 
 
 338. Show that the equations of the electromagnetic field in free aether can be satisfied 
 by functions of the type 
 
 d(a, ft) id.(a, ft) 
 
 Hx + i ' E ^s^Tz)- c^^'t)' 
 
 where a, /3 are arbitrary functions which satisfy the conditions implied in these forms. 
 
Electrodynamics 709 
 
 Show further that suitable pairs of functions a, # are 
 
 (i) a = x cos 6 + y sin 6 + iz. 
 
 P-X sin - / cos - c/. 
 
 339. Plane electromagnetic waves fall on the convex surface of an infinite paraboloid 
 of revolution x = a - r, whose surface is a perfect reflector. If the incident waves are 
 given by expressions of the type 
 
 17 -P ft x 8 ( a '0) i fl R\ 8(a ' 
 H x + lE x = f(a, ft) = -/(a, ft) ^ T} , 
 
 where a = y + iz, @= x + ct, prove that the boundary conditions at the surface of the 
 paraboloid may be satisfied by subtracting from the primary field a secondary field repre- 
 sented by expressions of a similar type with 
 
 w - iz 
 
 8 = a - r + ct. 
 
INDEX OF NAMES 
 
 Abraham, M. 16, 187, 290, 422, 433, 483, 547, 
 
 549, 632, 643 
 
 Ampere, A. 157, 343, 344, 360, 430 
 Arago, F. 624, 625 
 Arrhenius, S. 330 
 
 Baedeker, K. 318 
 
 Barrett, E. 263 
 
 Bateman, H. 486, 504, 530, 533 
 
 Becquerel, H. 43, 333 
 
 Bichat, E. 147 
 
 Bidwell, A. E. 234 
 
 Biot, J. B. 235, 409 
 
 Bjerknes, V. 485 
 
 Blondlot, R. 147 
 
 Boggio, T. 253, 257 
 
 Bohr, N. 45, 313, 573 
 
 Boltzmann, L. 604 
 
 Born, M. 511 
 
 Boussinesq, J. 48 
 
 Bragg, W. H. and W. L. 511 
 
 Bryan, G. H. 389 
 
 Burali-Forti, C. 1 
 
 Byerly, W. E. 78 
 
 Campbell, N. 42 
 
 Carnot, S. 57 
 
 Cauchy, A. 77, 104 
 
 Cavendish, Hon. H. 38, 152, 158, 282 
 
 Chasles, M. 73, 93 
 
 Christoffel, E. B. 110 
 
 Clairaut, A. 92 
 
 Classen, E. 422 
 
 Clausius, R. 58, 308 
 
 Colin, E. 186, 243, 270, 301, 422 
 
 Cole, A. D. 476 
 
 Coulomb, Ch. A. 38, 41, 85, 139, 235 
 
 Cremieu, V. 616 
 
 Crookes, Sir W. 42, 340 
 
 Crowther, J. A. 342 
 
 Gumming, W. 306 
 
 Cunningham, E. 661 
 
 Curie, Mme E. 44 
 
 Curie, J. 233 
 
 Curie, P. 233, 263, 271 
 
 Debye, P. 525, 533 
 Drude, P. 318, 573 
 Duhem, P. 486 
 
 Eichenwald, A. 617, 618 
 Einstein, A. 661 
 Ettinghausen, A. A. von 578 
 Eucken, P. 275 
 Ewald, P. P, 511 
 Ewing, Sir J. A. 260, 265 
 
 Faraday, M. 33, 41, 42, 87, 91, 157, 158, 159, 
 164, 169, 177, 178, 185, 190, 325, 343, 348, 
 350, 351, 362, 428, 578, 623 
 
 Feddersen, M. 434 
 
 Felici, R. 304 
 
 Fitzgerald, G. F. 616, 655, 659 
 
 Fizeau, H. L. 647 
 
 Fleming, J. A. 476 
 
 Foppl, A. 16 
 
 Fourier, J. B. 40 
 
 Franz, R. 318 
 
 Fresnel, A. 517, 618, 647 
 
 Gans, R. 75, 187, 243, 270, 533, 572, 573, 624 
 
 Garnett, M. 533 
 
 Gauss, C. F. 39, 46, 50, 61, 77, 87 93, 129, 
 
 250 
 
 Gibbs, J. W. 1, 328 
 Giuliani, C. 256 
 Goldstein, W. 340, 341 
 Green, G. 9, 46, 53, 76, 77, 117, 141 
 Greenhill, Sir G. 304 
 Grube, 0. 256 
 
 Hagen, E. 523 
 
 Hall, E. H. 572, 596 
 
 Hamilton, Sir W. 1, 436 
 
 Harms, F. 547 
 
 Heaviside, 0. 1, 148, 458, 486, 545, 559, 626, 
 
 629, 635 
 
 Heine, H. E. 78, 84, 527 
 Helmholtz, H. von 42, 53, 54, 58, 77, 135, 186, 
 
 270, 304, 327, 424, 567 
 Hermann, R. A. 257 
 Hertz, H. 158, 186, 350, 400, 434, 476, 477, 
 
 485, 498, 618, 624 
 Hicks, W. M. 304 
 Hittorf, W. 340 
 Hobson, E. W. 81, 85 
 Holder, 0. 51 
 Hondros, D. 547 
 Hopkinson, J. 263 
 Huyghens, Chr. 467 
 
 Ignatowsky, W. von 1, 525 
 
 Jeans, J. H. 187, 422 
 Jochmann 624 
 tones, H. S. 632 
 Joule, J. P. 624 
 
 Kammerlingh-Onnes, H. 271 
 
 Karman, T. von 511 
 
 Kaufmann, W. 644 
 
 Kelvin, Lord, see Sir W. Thomson 
 
 Kempken, E. 572 
 
 Kerr, J. 578 
 
 Kirchhoff, G. 23, 134, 186, 293, 300, 304, 601 
 
 Koldcek, F. 270, 483 
 
 Korteweg, D. J. 186 
 
 Kundt, A. 522, 525 
 
 Lagrange, J. 45, 49, 55, 436 
 Lamb, H. 84, 400 
 Lampa, A. 483 
 
Index of Names 711 
 
 Lange, Fr. 233 Poynting, J. H. 550, 596 
 
 Laplace, P. S. 39, 72, 77, 103, 154, 156 
 
 Larmor; Sir J. 42, 52, 156, 187, 190, 196, 203, Rayleigh, Lord 85, 196, 444, 483, 525 529 
 
 204, 207, 212, 221, 228, 230, 234, 242, 264, 531, 533, 541, 545, 559 
 
 270,271,308,347,349,358,389,400,403, Reiff, R. 157,567 
 
 416, 417, 424, 441, 467, 483, 513, 547, 550, Richardson, O. W. 42, 313, 422 
 
 562, 567, 568, 574, 579, 584, 594, 596, 600, Riecke, E. 234, 310, 573, 624 
 
 604, 618, 619, 624, 626, 645, 657 Riemann, B. 75, 457, 473, 483 
 
 Laue, M. 511, 661 Rive, L. de la 504 
 
 Lecher, E. 535, 540 Roentgen, W. C. 510, 616 
 
 Lee, Miss A. 485 Rowland, H. A. 616 
 
 Lejeune-Dirichlet, P. 77 Rubens, H. 523-525 
 
 Levi-Civita, T. 389 Rutherford, Sir E. 42, 44, 334 
 
 Lienard, A. 504 
 
 Lippraann, G. 234 Sano, S. 270 
 
 Livens, G. H. 187, 228, 313, 422, 524, 568, Sarasin, E. 500 
 
 573, 583 Savart, S. 409 
 
 Lodge, Sir 0. 655 Schaefer, C. 422, 525 
 
 Lorberg, H. 186 Schott, G. A. 357, 504 
 
 Lorentz, H. A. 1, 23, 42, 187, 221. 228, 232, Schwarz, H. A. 110 
 
 310, 357, 511, 514, 567, 578, 579, 584, 605, Schwarzschild, K. 504, 627, 640 
 
 606, 618, 633, 643, 644, 655, 657, 659 Searle, G. F. C. 626, 627 
 
 Lorenz, R. 228 Seebeck, Th. J. 305 
 
 Love, A. E. H. 178, 465, 485, 486, 487, 489, Seitz, W. 525 
 
 529, 564 Silberstein, L. 661 
 
 Somigliana, C. 253 
 
 MacCullagh, J. 47 Sommerfeld, A. 157, 547, 567 
 
 Macdonald, H. M. 357, 555 Stefan, J. 545, 604 
 
 Maclaurin, C. 72, 92 Stewart, Balfour 601 
 
 Marcolongo, R. 1 Stokes, Sir G. G. 13, 93, 510, 618 
 
 Maxwell, J. C. 33, 42, 97, 126, 134, 147, 157, Straubel, R. 233 
 158, 164, 174, 176, 178, 184, 197, 214, 237, 
 
 240, 241, 293, 348, 350, 356, 362, 366, 379, Tait, P. G. 1, 80, 117, 147 
 
 389, 417, 422, 435, 439, 444, 459, 473, 498, Thomson, Sir J. J. 42, 44, 134, 229; 310, 334, 
 
 591, 655 336, 338, 340, 342, 422, 444, 482, 483, 533, 
 
 Mazotto, D. 535 547, 573, 578, 626, 629 
 
 Michelson, A. A. 647, 655 Thomson, Sir W. (Lord Kelvin) 58, 80, 87, 93, 
 
 Mie, G. 533, 547 117, 134, 135, 146, 147, 148, 164, 176, 190, 
 
 Minkowski, H. 659 228, 232, 233, 260, 288, 291, 293, 307, 429 
 
 Morley, E. W. 647, 655 Townsend, J. S. 42 
 
 Morton, W. B. 547 Trouton, F. T. 500 
 
 Mossotti, 0. F. 189 
 
 Valentiner, S. 624 
 
 Nernst, W. 578 Voigt, W. 54, 232, 581, 584 
 
 Neumann, C. 50, 84, 104, 256, 257 Volta, A. 285, 289 
 
 Neumann, F. 384, 441 
 
 Newton, Sir I. 40, 157 Waals, A. van der 196 
 
 Nichols, E. F. 441 Warburg, E. 260, 265 
 
 Nicholson, J. W. 525, 533, 547 Weber, H. 75 
 
 Nielsen, N. 84 Weber, R. H. 457, 483 
 
 Niven, C. 400 Weber, Wm. 157, 343 
 
 Weingarten, J. 50 
 
 Oersted, H. Chr. 157, 343, 428 Weiss, P. 262, 275, 21f, 278 
 
 Ohm, G. S. 280, 282, 284, 285 Wheatstone, C. 293, 297 
 
 Ornstein, L. S. 511 Whetham, W. C. D. 324, 330 
 
 Oxley, A. E. 586 Whitehead, C. S. 400 
 
 Whittaker, E. T. 78, 79, 82, 527, 530 
 
 Pearson, K. 485 Wiechert, E. 340, 504 
 
 Peltier, J. C. 306 Wiedemann, G. 318 
 
 Perrier, A. 271 Wien, W. 342, 604, 605 
 
 Pidduck, B. 151 Wilson, C. T. R. 336 
 
 Planck, M. 55, 566, 600 Wilson, E. B. 1 
 
 Pluecker, A. 340 Wilson^ H. A. 338, 619 
 
 Pockels, F. 187 
 
 Pocklington, H. C. 547 Young, T. 196 
 
 Poincare, H. 27, 48 
 
 Poisson, S. D. 41, 47, 77, 88, 134, 164, 190, Zeeman, P. 578, 579, 580, 581 
 196, 237, 253, 255 
 
GENERAL INDEX 
 
 Absorption of energy by an electric current, 
 290, 558 ; of electric waves in metallic 
 substances, 464, 521, 544, 545 
 
 Action, medium or distance, 156-157; 
 question of velocity of transmission of, 
 157 ; the Faraday-Maxwell theory of the 
 transmission of electrostatic, 158; New- 
 ton's principle of, and reaction, 413 
 
 Aether, 157; displacement currents in, 224, 
 348 ; conception of the, as a carrier of 
 energy, 549; rigidity of, 618, 624. 645- 
 647 
 
 Ampere's electromagnetic theory, 343-354; 
 the circuital equation of, 347 ; mag- 
 netism in, 360; general dynamical for- 
 mulation of, 616 
 
 Anion, 324 
 
 Anode, 324, 335 
 
 a particles, 44; magnetic and electric de- 
 flection of, 342, 644; charge and nature 
 of, 342, 644 ; mass of, 342 ; dependence 
 of mass of, on velocity, 644 
 
 Availability of energy in electrostatic fields, 
 211; in magnetic fields, 269, 422-423; in 
 the general electromagnetic field, 552-557 
 
 Bessel functions, 82 
 
 Biot and Savart, law of, 409 
 
 Boundary conditions at a statically charged 
 surface, 7 ; at the surface of a dielectric, 
 163, 201, 516; at the surface of a con- 
 ductor, 163, 201, 391, 520; at the surface 
 of a magnetised body, 239, 253; at the 
 surface of two conductors, when a cur- 
 rent is flowing across, 299 ; at the surface 
 of a moving conductor, 629, 649; at an 
 advancing wave front, 487 
 
 * 
 
 Capacity, electrostatic, of ellipsoid, 75; of 
 . prolate spheroid, 75; of sphere, 75; of 
 oblate spheroid, 75 ; of a circular plate, 
 75; of a general type of condenser, 141, 
 144; of a spherical condenser, 142; of 
 a cylindrical condenser, 143 ; measure- 
 ment of, 151; effect of, on distribution 
 of alternating currents in linear con- 
 ductors, 443452 ; effect of, on the pro- 
 pagation of waves in a cable, 454, 539 
 
 Carnot's cycle, 57 
 
 Cathode, 324, 335 
 
 Cathode layer in discharge tube, 339 
 
 Cathode rays, 340; the charge and mass of 
 the particles in, 341 
 
 Charge of electricity, 33; method of pro- 
 ducing a, of any amount on a conductor, 
 36 ; absolute unit of, 39 ; dimensions of 
 a, 41 ; Faraday's conception of a, 42, 
 16.1; electronic constitution of, 41-44; 
 on an electron, 42, 341 ; the entity of 
 the positive, 43 ; volume distribution of 
 a, 51 ; surface and double sheet distribu- 
 tions of, 53 ; location of the static, on 
 a conductor, 100; on an ion in a gas, 
 336, 338; density of, dissipated by 
 conduction in the general field, 461 ; 
 moving in a magnetic field, force on, 
 588-589, 638-641; magnetic effect of a 
 moving, 616, 626-635; induced on a 
 conductor in motion in a magnetic field, 
 620-622 ; field of, in steady motion, 627 ; 
 momentum of a, in steady motion, 635- 
 641 ; mass of a, in quasi-stationary 
 motion, 642 
 
 Circuitality of electric displacement flux, 
 225 ; of total current flux, 347 ; of mag- 
 netic induction, 239 
 
 Condenser, 139; field of a general type of, 
 140; leyden jar, 141; the spherical, 142; 
 the parallel plate, 142 ; the cylindrical, 
 143; a general form of, 144; combina- 
 tions in parallel and series, 145; parallel 
 plate, with a dielectric slab, 169; dis- 
 charge of, through an induction, 430 
 
 Conduction of electricity by metals, 36, 283 ; 
 by gases and air under the action of 
 Roentgen rays, 334 ; by liquids and 
 electrolytes, 333 ; electron theory of, in 
 metals, 316-323 
 
 Conduction of heat by metals explained by 
 the electron theory, 317 
 
 Conductors; general properties of the static 
 field of a system of, 99; relations be- 
 tween the potentials and charges of, 
 126; coefficients of capacity and in- 
 duction for, 128; energy of a system -of, 
 134; ponderomotive forces acting on 
 and between, 137-139; induction of 
 currents in, by changing magnetic fields, 
 390403 ; propagation of electric waves 
 in, 459465 ; the reflection and diffraction 
 of electric waves by, 520525 ; charge 
 due to rotation of, in a magnetic field, 
 621 ; general relations of charged, in 
 motion, 626-641 
 
 Critical temperature for ferromagnetic ef 
 fects, 263, 276 
 
General Index 
 
 713 
 
 Crooke's dark space in discharge tube, 339 
 Curie's law connecting magnetisation and 
 
 temperature, 272 
 
 Currents of electricity, the origin of, 279 ; 
 general definition of, 281 ; special defi- 
 nition of, 281 ; resistance to flow of, 283; 
 hydrostatic analogy for, 284; energy 
 relations of, 291 ; distribution of steady, 
 in a network of conductors, 294; the 
 law of minimum dissipation for the flow 
 of, 292 ; in three dimensions, 298 ; law 
 of refraction of lines of flow of, 300 ; the 
 thermal relations of, 305; thermody- 
 namic discussion of these relations, 308 ; 
 electron theory of the mechanism under- 
 lying, 309 : in liquids, phenomena ac- 
 companying, 324; in solutions, due to 
 variations of concentration, 333; satura- 
 tion value of, in a gas, 335 ; magnetic 
 fields due to, 342 ; work done in taking 
 a magnetic pole round, 345; dimensions 
 of, 352 ; the complete expression for, on 
 Maxwell's theory, 362 ; due to con- 
 vection of charged and polarised media, 
 364-365; electronic aspects of the total 
 current, 366; circuital property of, 390; 
 induction of, by varying magnetic fields, 
 350 ; induced in a plane conducting sheet, 
 by varying magnetic field, 389-396 ; in- 
 duced in a spherical conducting shell by 
 a uniform oscillating field, 396 ; induced 
 in a solid sphere by a similar field, 400; 
 induction of, by moving electric charges, 
 403; mutual mechanical relations of 
 linear, 408-458 ; interaction between 
 magnets arid, 408-415; energy in the 
 field of linear, 417-423; in a discharging 
 condenser, 430; dynamical theory of the 
 interaction of linear, 435-452; distribu- 
 tion of oscillating, in a network of con- 
 ductors, 449 ; distribution of oscillating, 
 over the cross section of a conductor, 
 .").")!! : induced in a conductor by rotation 
 in a magnetic field, 623-626 
 
 D'Alembert's principle, significance of, 52 
 Damping in a condenser circuit, 432 ; of 
 waves along a cable, 456, 544-546 ; of 
 waves propagated into metallic con- 
 ductors, 464, 520; of waves, by cubical 
 expansion, 485, 489-490 
 Dark space, Faraday's, 339 ; Crooke's, 339 
 Diamagnetism, 259; electron theory of, 585 
 Dielectric constant, 158, 161, 162, 171, 200; 
 relation between, and density of medium, 
 226; constitutional expression of, for 
 steady fields, 232 ; for rapidly alternating 
 fields, 475; for gases, 476; for metals, 523 
 Dielectrics, on Faraday-Maxwell theory, 161 ; 
 the inverse electrostatic problem with, 
 165; the effect of, on the capacity of 
 condensers, 169; the Faraday-Maxwell 
 theory for the distribution of energy in, 
 1 72 ; the Faraday-Maxwell theory of the 
 transmission of mechanical force through. 
 177; the polar theory of, 188; mathe- 
 matical specification of the field of 
 polarised, 199; validity of expressions 
 
 for the field functions in the theory of, 
 193 ; Poisson's transformation of the 
 potential of polarised, 194; the com- 
 position of the displacement current in 
 the theory of polarised, 197; the cha- 
 racteristic properties of the field of 
 polarised, 198; the law of induction of 
 the polarisation in, 200 ; energy relations 
 of polarised, 201 ; intrinsic energy of 
 polarised, 207; mechanically available 
 energy of polarised, 207; energy, re- 
 strictions on the law of induction in, 
 207 ; complete expression for the energy 
 in the field of polarised, 209, 552; 
 mechanical relations of polarised. 212, 
 588-590; force and couple on the ele- 
 ments of, 213, 588 ; the stress in polarised, 
 216, 591 ; electron theory of the polarisa- 
 tion of, 228 
 
 Diffraction of electric waves by a grating, 
 499 ; by a dielectric cylinder, 526 ; by 
 a dielectric sphere, 529 
 
 Dimensions, doctrine of, 40; of electric 
 charge, 41, 352; of electric force, 352; 
 of electric displacement, 352 ; of electric 
 potential, 352; of electric current, 352; 
 of magnetic force, 352; of magnetic 
 induction, 352 
 
 Disc, Arago's rotating, in a magnetic field, 
 623 
 
 Discharge tube, phenomena in, 339 
 
 Electric displacement, 160, 197; character- 
 istic properties of, on Faraday-Max- 
 well theory, 164; circuital properties of, 
 197, 225; physical significance of, on 
 the theory of polarised dielectrics, 224- 
 226 ; currents of, 347 
 
 Electricity, constitution of, 41 ; two fluid 
 theory of, 41 ; atomic structure of, 42, 
 326; specific heat of, 308 
 
 Electrification, by friction, 33 ; by induction, 
 34; by conduction, 35; by induction, of 
 a sphere in a uniform field, 111; by in- 
 duction, of an ellipsoid in a uniform 
 field, 115: of a sphere, by induction from 
 a point charge, 121 ; of orthogonal planes 
 and spheres, by induction from point 
 charges, 119-124; of a conductor pro- 
 duced by rotating it in a magnetic field, 
 621 
 
 Electro-caloric effects, 233 
 
 Electrodynamics, general theory of, for 
 linear currents, 435 ; for the most general 
 system, 567, 614; of electrostatic systems 
 in quasi-stationary motion, 635-645 
 
 Elect rodynamic potential, 441 
 
 Electrolysis, 324; Faraday's laws of, 325; 
 electrolytic dissociation, 330; electro- 
 lytes, 324; velocity of ions in, 331 
 
 Electromagnetic field, Ampere's circuital 
 relation for, 343 ; Faraday s circuital re- 
 lation for, 350; the differential form of 
 these fundamental relations, 353 ; defini- 
 tion of, in terms of the scalar and vector 
 potentials, 356, 357, 359-360; Max- 
 well's general theory of, 362; general 
 dynamical theory of, 567-572, 614-617 
 
714 
 
 General Index 
 
 Electromagnetic momentum, 592 ; as a 
 motional force, 593 ; of moving electrical 
 systems, 634-641 
 
 Electromagnetic theory of light, 473 
 
 Electrometer, 146; trap-door, 147; quad- 
 rant, 148; cylindrical, 148 
 
 Electromotive force, complete expression for, 
 on moving electron, 571; in any circuit, 
 615 
 
 Electrons, 42 ; mass of, 42, 341 ; charge of, 
 42, 335-338; determination of velocity 
 of, 338-341 ; determination of mass of, 
 341; electromotive force on, 414; field 
 of moving, 506-507, 626-635; radiation 
 from accelerated, 508; radiation from 
 a group of, 513; momentum of, in quasi- 
 stationary motion, 635-641 ; mass of, 
 in quasi-stationary motion, 643 ; the 
 Abraham type of, 632, 642 ; the Lorentz 
 type of, 633, 643 
 
 Electron theory, 42 ; of material constitu- 
 tion, 43, 44, 509, 644-645; of dielectric 
 polarisation, 228 ; of metallic conduction 
 of heat and electricity, 310-323; of 
 Peltier effect, 321; of Thomson effect, 
 323 ; of magnetisation, 368 ; of optical 
 dispersion, 475; of Hall effect, 573; of 
 Zeeman effect, 579 ; of Faraday effect, 
 581 ; of diamagnetism, 585; of long wave 
 thermal radiation from a metal, 605; 
 of the optical properties of metals, 523 
 
 Electrostatic field, definition of, of point 
 charges, 45 ; definition of, of continuous 
 charge distributions, 48-53 ; the poten- 
 tial energy in, 57-59, 172-177, 201-212; 
 ponderomotive forces due to, 63, 177-187, 
 212-218; of line charge, 67; of spherical 
 shell, 68; of a flat disc, 69, 85; of a 
 homogeneous sphere, 70; of a homo- 
 geneous ellipsoid, 71; of an ellipsoidal 
 shell, 7.4; Green's analysis of, 76; cha- 
 racteristic properties of the general, 76, 
 91-94, 99, 103; of concentric spheres, 
 84 ; of the freely charged ellipsoid, 86 ; 
 mapping of, by lines of force and equi- 
 potential surfaces, 91-94 ; of point charge 
 systems, 94-99; of a system of con- 
 ductors, 99-103; in two dimensions, 103- 
 110 
 
 Electrostriction. 234 
 
 Energy, conservation of, 55, 415, 549; of 
 electrostatic system, 57-62 ; distribution 
 in the electrostatic field, 173-177; of 
 polarised dielectrics in the electrostatic 
 field, 202-209 ; separation of the, of an 
 electrostatic field into available and in- 
 trinsic energy, 207, 209, 211; of mag- 
 netisation, 264-268, 425; the thermal 
 and mechanical relations of magnetic, 
 265-271 ; separation of magnetic, into 
 available and intrinsic, 266-269 ; rela- 
 tions of an electric current, 291, 301 ; 
 chemical origin of, of an electric current, 
 292; relations of a voltaic cell, 329; 
 mutual potential, of a current and a 
 magnetic system, 408414; distinction 
 between kinetic and potential, 415; of 
 magnetic fields treated as kinetic, 415; 
 
 of the magnetic field of a system of 
 currents, 417-420; the separation of 
 this energy into its constituent parts, 
 422; radiated from an accelerated elec- 
 tron, 508, 566; general conception of, 
 549; the density of magnetic, on Poyn- 
 ting's theory, 554 ; the density of mag- 
 netic, on Macdonald's theory, 555; the 
 flux of, in its general aspects, 551 ; the 
 flux of, on Poynting's theory, 553 ; the 
 flux of, on Macdonald's theory, 555; 
 the flux of, along a beam of light, 561 ; 
 the flux of, from a Hertzian oscillator, 
 563 ; the flux of, from a charge oscillating 
 on a sphere, 565 ; the potential, of a 
 moving electrical system, 639-641 
 Equilibrium theory in dynamics, 426, 638 
 Equi-potential surfaces, properties of, 93 
 
 Faraday dark space, 339 
 
 Faraday effect, 578 ; theory of, 581 
 
 Faraday-Maxwell theory of the electrostatic 
 field, 156-187; of the electromagnetic 
 field, 362-370; general dynamical for- 
 mulation of, 614-616 
 
 Faraday's law of electromagnetic induction, 
 350; differential form of, 355; deduced 
 from the energy principle, 428; for 
 circuits in general, 440; general dynami- 
 cal formulation of, 615 
 
 Ferromagnetism, 259; critical temperature 
 for, 263, 277 ; Weiss' theory of, 275 
 
 Fizeau's experiment, 647 
 
 Flux of displacement, see Electric displace- 
 ment; of induction, see Magnetic in- 
 duction; of energy, see Energy 
 
 Force, line of, 91 ; tube of, 92; properties of 
 lines and tubes of, in electrostatic fields, 
 92, 99-102 
 
 Force, electromotive, definition of, 45 ; as 
 the gradient of potential, 46, 49, 55; in 
 field due to uniformly charged infinite 
 plates, 89; in field due to a circular 
 cylinder, 89; in field due to concentric 
 spheres, 90; definition of, independent 
 of distance action ideas, 159; relation 
 between, and displacement, 160, 175; 
 relation between and polarisation, 199, 
 208 ; drawing currents in a circuit, 286- 
 289; induced in linear circuits due to 
 changing magnetic fields, 350; induced 
 in a circuit by its motion through a 
 magnetic field, 620 
 
 Force, ponderomotive, on statically charged 
 bodies, 62, 136; on statically charged 
 body when charge is maintained con- 
 stant, 64, 137; on statically charged body 
 when the potential is maintained con- 
 stant, 65, 138; on a sphere and an ellip- 
 soid in a uniform field, 114-116; on di- 
 electrics in any field", 183-187, 212-221, 
 588 ; on magnetised media in any field, 
 270, 589 ; between a magnetic pole and 
 a current, 408 ; on a current element in 
 any field, 409, 413; in steady dynamical 
 systems, 416, 640; on a moving electron, 
 414, 517, 636-641; estimate of, on the 
 medium in any field, 571, 587; on the 
 
General Index 
 
 715 
 
 general static system in quasi-stationary 
 motion, 636-641 
 
 Galvanometer, tangent, 379; ballistic, 381 
 Gases, conduction of electricity in, 333; 
 discharge in rarefied, 339; ionic theory 
 of conduction in, 334 
 
 Gauss's normal induction theorem, 87; its 
 relation to Poisson's equation, 88; for 
 unclosed surfaces, 91 ; on the Faraday- 
 Maxwell theory, 161 
 
 Gauss's reciprocal energy theorem, 61, 129 
 Green's analysis, 18-23; of the electrostatic 
 
 field, 76 
 
 Green's equivalent strata, 117 
 Green's lemma, 8; for moving circuits, 17 
 Green's theorem, for external points, 18; for 
 internal points, 19 ; for discontinuous 
 fields, 19; for infinite fields, 21 ; its appli- 
 cation in the theory of electrostatics, 76 
 
 Hall effect, 572; electron theory of, 573; 
 anomalies of, 577 
 
 Harmonics, solid, 79; spherical, 81 
 
 Hertz's experiments, 498 
 
 Huyghen's principle, 467; Larmor's dis- 
 cussion of, 467-471 
 
 Hysteresis in magnetism, 260 
 
 Ignoration of coordinates, physical signifi- 
 cance of, in analytical dynamics, 416 
 
 Images, electromagnetic, of varying mag- 
 netic field in a plane conductor, 393-396; 
 of currents induced by moving charges, 
 403-407; of a steady magnetic field in 
 a moving conductor, 624-626 
 
 Images, electrostatic, general theory of, 120; 
 of a point charge in a sphere, 121 ; of a 
 point charge in orthogonal spheres, 123; 
 of a point charge in orthogonal planes, 
 125; method of, for two spheres, 132; 
 of a point charge in an infinite dielectric 
 with a plane face, 168 
 
 Index of refraction, 474; relation between, 
 and dielectric constant, 476 
 
 Induction, electromagnetic, Faraday's law 
 of, 351 ; coefficients of self and mutual, 
 for linear circuits, 383 ; in plane con- 
 ducting sheets, 391-396; in a spherical 
 shell, 397; in a solid sphere, 400; general 
 electrodynamic theory of, as regards 
 linear circuits, 408-456; Faraday's law 
 of, deduction from energy principle, 428 
 
 Induction, electrostatic, electrification by, 
 34; theory of electrification by, 110-117, 
 118-120; coefficients of self and mutual, 
 for a system of conductors, 129 
 
 Induction, magnetic, vector of, 239; cir- 
 cuital property of, 239, 351, 363; as 
 distinguished from the magnetic force, 
 412, 571, 589 
 
 Inverse square's law, 38, 156, 235; proof of, 
 152; validity of, 156 
 
 Kirchhoff's theorem, 23 
 
 Lagrange's equations, for current circuits, 
 438 
 
 Laplace's equation, 77; in cartesian coor- 
 dinates, 79-80 ; construction of solutions 
 of, 80; in spherical polar coordinates, 81 ; 
 in cylindrical polar coordinates, 82 
 
 Larmor-Lorentz transformation of the electro- 
 magnetic equations, 657 ; derived from 
 the principle of relativity, 661 
 
 Least Action, principle of, 439, 454, 567; 
 applied to the cable problem, 454; ap- 
 plied in electromagnetic theory, 569 
 
 Level surfaces, see Equi-potential surfaces 
 
 Leyden jar, 141 
 
 Light, electromagnetic theory of, 473 
 
 Lines of force in electric field, 91 ; refraction 
 of, at dielectric interface, 172 
 
 Local time, 659 
 
 Logarithmic potential, 104 
 
 Maclaurin's theorem, 73 
 
 Magnet, poles of a long, 235 ; axis of a, 235 ; 
 field of a finite, 237 ; field of a uniform, 
 238 
 
 Magnetic field, potential of, 237; Poisson's 
 transformation of the potential of, 237 ; 
 mathematical relations of, 239 ; vector 
 potential of, defined, 240; vector poten- 
 tial of, of any magnet, 241 ; vector poten- 
 tial of the, of a shell, 243 ; of permanent 
 magnets, 244 ; of a current, 343-345 ; of 
 given current distributions, 360; of a 
 straight current, 371 ; of parallel currents, 
 373; of a solenoid, 374; of a circular 
 current, 377 ; of a system of currents in 
 general, 381 ; inductive action of a vary 
 ing, 391-407; the energy of, 417-426, 
 553-557 ; of a system of charges in uni- 
 form motion, 628-629 
 
 Magnetic force, 237; derived as an average 
 vector, 424; derived by rejecting the 
 local part of the induction, 589 
 
 Magnetic induction, 239 ; circuital property 
 of, 240; force and, in Maxwell's theory, 
 363 ; contrasted with the magnetic force, 
 589 
 
 Magnetic particles, field of, 237, 247 ; mutual 
 relations of, 248 ; force on, in any field, 
 250; Gauss's arrangement of two, 251 
 
 Magnetic shell, vector potential of, 243; 
 equivalence of field of, and of a current, 
 345 
 
 Magnetic stress 270, 592 
 
 Magnetisation, 237 ; induced in a uniform 
 sphere, 253 ; induced in spherical shell, 
 255 ; induced in ellipsoid, 256 ; saturation 
 value of, 260, 277 ; of iron, pyrrhotite, 
 magnetite, 260; energy of, 264; loss of 
 energy of, in hysteresis cycle, 265 ; 
 thermal relations of energy, 271 ; treated 
 as a current distribution, 368, 424 
 
 Magnetism, 235; law of force in, 234; per- 
 manent and induced, 243; field of per- 
 manent, 244 ; induced, 251 ; law of in- 
 duction of, 251; field of induced, 252; 
 retentiveness of, 260; hysteretic quality 
 of, 260; energy of, 263-270; mechanical 
 relations of, 270, 591-592 ; Weiss's theory 
 of, 275-278; Ampere's theory of, 360; 
 electron theory of, 514; in radiation, 557 
 
716 
 
 General Index 
 
 Magneton, 275 
 
 Mass of cathode particle, 341 ; of electron, 
 42, 341; of Abraham's electron, 643; 
 of Lorentz'e electron, 643 ; electro- 
 magnetic, of a moving system, 641 ; 
 material, entirely of electromagnetic ori- 
 gin, 645 
 
 Maxwell's theory, of the electrostatic field, 
 156-187; of the electromagnetic field, 
 362-370; dynamical aspects of, 567-572, 
 614-617 
 
 Metallic glass, colours of, 533 
 
 Molecular theory, effectiveness of detailed, 
 51, 192-196, 201-205, 567 
 
 Momentum, electromagnetic, 592; of a 
 moving electrical system, 635-642 
 
 Moving media, 635-642 
 
 Mutual induction, Neumann's formula for, 
 384; of parallel currents, 385; of parallel 
 circular currents, 387 
 i 
 
 Negative glow in discharge tube, 339 
 Nernst and von Ettinghausen effect in metals, 
 
 578 
 Neumann's theorem on magnetic induction, 
 
 257 
 
 Ohm's law, 282; for circuits of different 
 
 media, 288 
 Oscillations, electric, in discharging c.on- 
 
 denser, 429 ; of charge on a spherical 
 
 conductor, 484, 488; damping of, 485; 
 
 detectors of, 482 
 Oscillator, electric, Hertz's, 477 ; mode of 
 
 establishment of the field of an, 491 
 
 Paramagnetism, 258; see also Magnetism 
 
 Peltier effect, 306; electron theory of, 321 
 
 Perpetual motion, negation of, involved in 
 the definition of potential, 55 
 
 Piezo-electricity, 232 
 
 Points of equilibrium in electrostatic fields, 
 99 
 
 Poisson's equation, 77; on Faraday-Max- 
 well theory, 162; in two dimensions, 103 
 
 Polar coordinates, definition of spherical 
 and cylindrical, 1 ; differential opera- 
 tions in, 11; solutions of Laplace's 
 equation in, 80-85; solutions of the 
 wave equation in, 478-485, 525-533, 
 545-548 
 
 Polarisation, dielectric, see Dielectrics; mag- 
 netic, see Magnetism , 
 
 Polarisation of light, 498; rotation of plane 
 of, by a magnetic field, 578. 583 ; rotation 
 of plane of, by reflection from a magnet, 
 578 
 
 Pole, magnetic, see Magnet 
 
 Potential, convection, in moving electrical 
 systems, 627 
 
 Potential, electromotive, Volta's .difference 
 of, for substances in contact, 286; elec- 
 tron theory of, 320 
 
 Potential, electrostatic , of the various types 
 of charge distribution, 46-54; the physi- 
 cal significance of, 55-58 ; convergence 
 of integrals for the, 51, 194; of line 
 charge, 67 ; of spherical shell, 68 ; of 
 
 circular disc, 69; of homogeneous ellip- 
 soid, 73; of homoeoidal shell, 74; cha- 
 racteristic properties of, 77, 94; of spher 
 cal harmonic distributions, 83; of plane 
 circular distributions, 84; of concentric 
 spherical distributions, 86; of distribu- 
 tion on a conducting ellipsoid, 87 ; of 
 various two dimensional distributions, 
 103-110; characteristic properties of, 
 on the Faraday-Maxwell theory, 159; 
 of a polarised medium, 192; Poisson's 
 transformation of, for a polarised me- 
 dium, 195 
 
 Potential energy, see Energy 
 Potential, magnetic, 237 ; convergence -of 
 integral for, 238; of uniformly magne- 
 tised body, 238 ; characteristic equations 
 for, 253; for various induced distribu- 
 tions, 253257 ; of a closed qurrent, 345 ; 
 of various specified current distributions, 
 371-381 
 
 Potential, scalar, see Scalar potential 
 Potential, vector, see Vector potential 
 Pressure of radiation, see Radiation 
 Pyro-electricity, 232 
 
 Radiation, mechanism of, 502 : from a group 
 of electrons, 511; the flux of energy in 
 a plane field of, 561 ; the flux of energy 
 in the, from a Hertzian oscillator, 
 563-565 ; the flux of energy in the, from 
 an oscillating spherical charge, 565; 
 pressure of, on any absorbing medium, 
 595, 649 ; pressure of, on a mirror, 595, 
 650 ; reaction of, on its source, 596 ; 
 pressure of, when mirror or source is 
 in motion, 599-600, 648-649; thermo- 
 dynamics of, 600-613; propagation of, 
 rn a moving medium, 645-647, 650 
 
 Radium, 44 ; velocity of emission of electrons 
 from, 44 
 
 Ray of light, energy definition of, 561, 650; 
 absolute and relative paths of a, in 
 moving media, 650 ; difference in lengths 
 of absolute paths for propagation of a, 
 in different directions in a moving me- 
 dium, 655 
 
 Rayleigh's theorem on the distribution of 
 alternating currents in a network of 
 conductors, 453 
 
 Reaction, principle of action and, 62, 413: 
 Kelvin's kinetic, 440 
 
 Reflexion, of electric waves, Hertz's experi- 
 ments, 498; Fresnel's formulae, 517; 
 at a plane conducting surface, "520 ; 
 power of metals, 523 
 
 Refraction of electric waves, Hertz's ex- 
 periments on, 499; Snell's law, 517; at 
 a dielectric interface, 515-520 
 
 Relativity, theory of, 656-661 
 
 Resistance of metals to the flow of electri- 
 city ,"282 ; the mechanism of, in metals, 
 309-323; of a metal cable to the trans 
 mission of an oscillating signal, 561 
 
 Retarded scalar potential, see Scalar potential 
 
 Retarded vector potential, see Vector potential 
 
 Reversibility of path, involved in the defini- 
 tion of potential, 55 
 
General Index 
 
 111 
 
 Roentgen rays, conductivity of gases under 
 the influence of, 334; from discharge 
 tube, 339; constitution of, 333, 510 
 
 Rotation of plane of polarisation of light, by 
 a magnetic field, 579 ; by reflection from 
 a magnet, 575 
 
 Rotation of a conductor in a magnetic field, 
 619 
 
 Scalar, definition of, 1 ; gradient of, 8 ; differ- 
 entiation of, 6 
 
 Scalar potential, of Maxwell's theory, 357 ; 
 retarded, 357 
 
 Self-induction, electrostatic, 128; electro- 
 magnetic, 383, 429 
 
 Sky, Rayleigh's theory of colour of, 533 
 
 Specific heat of electricity, 308 
 
 Stokes's theorem, 13; in spherical and cylin- 
 drical polar coordinates, 15; for moving 
 circuits, 16 
 
 Stress, in elastic solid, specification of, 183; 
 Maxwell's dielectric, 183-184; insuffi- 
 ciency of Maxwell's dielectric, 183-185; 
 in a polarised medium, 186, 214, 590; 
 magnetic stress, 270, 592 
 
 Temperature, in condition for existence of a 
 potential function, 57; critical, for ferro- 
 magnetic effects, 263, 276; neutral, for 
 Peltier effect, 307; the Wiedemann- 
 Franz law of variability of the ratio of 
 electrical and thermal conductivities 
 with, 318; variability of a voltaic cell 
 with, 329; Stefan's law for the depen- 
 dence of radiant energy density on the, 
 604 
 
 Thomson thermoelectric effect. 307 ; electron 
 theory of, 323 
 
 Tube of force, 92 
 
 Uniqueness theorem, 127 
 
 Units, for electromagnetic quantities, 352 
 
 Vector, definition of, 1 ; addition and sub- 
 traction of, 2 ; multiplication of, 3 ; unit, 
 3; scalar product of two, 4; vector pro- 
 duct of two, 4 ; product of three, 6 ; 
 differentiation of, 6 
 Vector operator, the Hamiltonian, 8 
 Vector potential, defined, 240; of a particle, 
 240 ; of a finite magnet, 241 ; of a uni- 
 form shell, 243 ; derivation of magnetic 
 
 induction from, 241 ; expression of elec- 
 . trie force in terms of, and a scalar poten- 
 tial, 355 ; value of, in the general case, 
 357; the retarded, 357; of various speci- 
 fied current distributions, 371-384; the 
 retarded, of the field of an electron, 505 ; 
 the expression for the energy in the field 
 in terms of the, 555 
 
 Velocity of electromagnetic waves, 462, 473 ; 
 of light, 474; of propagation of electric 
 waves along wires, 456, 539; dependence 
 of mass of a moving system on its, 643 ; 
 of electromagnetic waves in moving 
 media, 647 
 
 Voltaic cell, 289 ; electromotive force of, 292 ; 
 explanation of action of, 327 ; thermal 
 relations of, 329 
 
 Waves, electric, generation of, 434, 476; 
 the equation for propagation of, along 
 cables, 455, and its solution, 457 ; the 
 general equations of propagation of, 
 461-462 ; solution of equation of propaga- 
 tion, 465-471; solution of equation with 
 dissipation, 472; physical significance 
 of the equation, 464-466 ; generated by 
 Hertz's oscillator, 478-481 ; Hertz's me- 
 thod of detecting, 482 ; on the mechanism 
 of the establishment of fields of, 485-493 ; 
 conditions at the wave front in the pro- 
 pagation of, 487 ; plane waves, 494 ; 
 transverse character of, 495 ; stationary 
 waves, 500; Sarasin and de la Rive's 
 experiments with. 500; the mechanism 
 of the emission of, in general, 502-514; 
 reflection and refraction of, 499, 515- 
 523 ; diffraction of, 499 ; diffraction by 
 a cylinder, 526; diffraction by a sphere, 
 529; propagation of, along parallel con- 
 ductors and cables, 534-548; plane, in 
 a parallel slab of dielectric between two 
 infinite conductors, 542 ; the flux of 
 energy in, 561-567 
 
 X-Rays, diffraction of, 511; constitution of, 
 510; wave length of, 511 
 
 Young-Poisson principle of the mutual com- 
 pensation of molecular forcives, 196 
 
 Zeeman effect, 578; electron theory of, 579 
 
 CAMBSIDGE- PRINTED BY J. B. PEACE, M.A., AT THE UNIVERSITY PRESS. 
 
( 
 
RETURN PHYSICS LIBRARY 
 TO ^ 351 LeConteHall 
 
 M - 
 
 ^\\ ^ 
 
 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 
 Overdue books are subject to replacement bills 
 
 DUE AS STAMPED BELOW 
 
 193 
 
 FORM NO. DD 25 
 
 UNIVERSITY OF CALIFORNIA, BERKELEY 
 BERKELEY, CA 94720 
 
 /86ft 
 
 !fo6 
 
 LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF GAL 
 
--. ___ 
 
 iRY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA 
 
 C5\\ 
 
 LIBRARY 
 
 / 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY 
 
 (HE HiiEimr OF OUIFHIII UIIIIT OF m ,, OF tll|F , 1IU " lj||m > 
 
m