THE MAGNETIC CIECUIT THE MAGNETIC CIRCUIT- IN THEOEY AND PEACTICE BY DE H. DU BOIS u " PRIVATDOCENT IN THE UNIVERSITY OF BERLIN TRANSLATED BY DR ATKINSON FORMERLY PROFESSOR OF EXPERIMENTAL SCIENCE IN THE STAFF COLLEGE, SANDHURST WITH 94 FIGURES IN THE TEXT OP THE , UNIVERSITY LONGMANS, GBEEN, AND CO. LONDON, NEW YOEK, AND BOMBAY 1896 All rights reserved ^^K fc UNIVERSITY - >ITY) A^X THE AUTHOR'S PREFACE THE plan of the present work arose out of a lecture on the Magnetic Circuit and its Measurement; which I delivered on the occasion of the International Congress of Electricians at Frank- fort in September 1891. From several sides the desire was ex- pressed for a systematic and critical account, from the physical point of view, of the more important; developments in this direc- tion. Hitherto such an account had been wanting, and this deficiency I have endeavoured to supply, however imperfectly. The important action which the rapid rise of electro- technology in the last decade has exerted on those branches of physics which form its basis, has been abundantly insisted on, and almost universally recognised. Electro-technology seems at present to have entered on a phase of quieter development ; and from the scientific point of view the time appears suitable to survey the position, critically to investigate results of very unequal value, often hastily brought to light amidst the bustle of practical work, and to blend, the older as well as the more recent results into one consistent exposition. I have in general directed my efforts chiefly to describing the actual state of theoretical and experimental inquiry. Any attempt to chronicle completely the manifold phases of its previous development seemed the more unnecessary since a new edition of Wiedemann's comprehensive ' Lehrbuch der Elek- tricitat ' is now appearing, in the third volume of which the sub- ject in question is historically treated in an exhaustive manner. VI PREFACE I have made an exception in the seventh chapter, in which I have endeavoured to give the history of the analogy between the magnetic circuit and various other kinds of circuits, an analogy which, though long known, has been specially dwelt upon in recent times. The book forms two parts. The two introductory chapters are intentionally concise, and yet as elementary as possible, so as to enable anyone not completely familiar with mathematical physics to understand the later parts of the book. I have made no attempt to state completely the results of investigations into ferromagnetic induction, which of late have been so much pur- sued, since a full account of them has recently appeared from the authoritative pen of Professor Ewing. Hence the knowledge of these results is assumed, and the theory of the magnetic circuit based on them, as well as on earlier theoretical investigations. The fundamental type of these circuits is the radially divided toroid introduced at the outset, which is discussed in detail in Chapter V., and continually referred to afterwards. The expla- nation of ferromagnetism by pre-existing elementary magnets capable of being directed, and the comparison of these magnets to rotatory processes, whether vortices (Lord Kelvin), molecular currents (Ampere) rotating electrical particles (Weber), or ionic charges (Richarz), could only be treated very superficially. In Chapters III. and IV. the outlines of the theory of ' rigid ' magnets on the one hand, of absolutely ' soft ' cores on the other, are briefly summarised. The mode of treatment is similar to that of Maxwell, which had been followed among others by Professor Chrystal in his article on Magnetism in the ' Encyclo- paedia Britannica,' and by MM. Mascart and Joubert in their Treatise. From the nature of the case this treatment could not be elementary. But by adopting a geometrical or graphical mode of representation, as well as by avoiding purely analytical refine- ments, I have endeavoured to attain that clearness and distinct- ness to the want of which it may be due that in many cases .PREFACE Vll ignorance or doubt exists as to the long-established theory in question. I have made no use of actual quaternion methods in their original form or in the modified one advocated by Mr. Heaviside, as their knowledge can scarcely be considered sufficiently widespread. Unlike the purely scientific method previously employed, the second part of the book is treated more from the point of view of applied physics. In Chapter VI. the general pro- perties of the magnetic circuit are discussed. Chapter VII., in which historical treatment preponderates, has been already mentioned. Chapters VIII. and IX. treat briefly the application of the principles developed to the principal machines and apparatus used in actual practice or in the laboratory. Such a concise account of the chief applications of the science may be welcome to many physicists on the one hand, and on the other may interest practical electricians, as giving the scien- tific foundation of their special subject. The last two chapters, finally, are devoted to experimental methods of measurement. Wherever the more important re- sults of allied branches of mathematical or experimental physics are assumed to be known, I have referred to the original passages in the text-books mentioned above or in others. Numerous references facilitate, moreover, the study of details in original papers. The chief contents of 81, 94, 95, 109, 124, 139, 154, 158, 179 have hitherto, so far as I know, either not been published at all or only briefly and without proof. In other places I have given new proofs, wherever necessary, but have not always mentioned them. I have throughout endea- voured to adopt a suitable nomenclature, which has not always been an easy matter, considering the confusion which prevails in this respect. The nomenclature and notation of all the more important conceptions are logically retained throughout, and are given at the end in the form of a tabular summary. Vlll PKEFACE To Professor EWING, Lord KELVIN, the late Professor KUNDT, Dr. H. LEHMANN, Dr. LINDECK, Dr. NAGAOKA, Professor PLANCK^ Dr. KAPS and Dr. RUBENS, who have each been good enough to look through portions of the proof sheets, I am indebted for much valuable advice, and in conclusion I wish to express my best thanks to them. H. DU BOIS. BERLIN : February 1894. TEANSLATOE'S PEEFACE THE Preface by the Author explains so completely the scope of the present work, that no addition in this respect is required from me. In offering the translation I have to acknowledge the valuable assistance of Dr. C. V. BURTON, of University College, London, who translated a considerable portion, and revised the rest of the more purely mathematical part of the work : to Dr. E. EL BARTON, of University College, Nottingham, I am also indebted for having looked through and revised the more purely technical portions. I must further express my acknowledgments to Dr. E. TAYLOR JONES, now lecturer in University College, Bangor, who was studying under Dr. du Bois while some of these sheets were passing through the press, and who kindly read them. I am still more indebted to Dr. du Bois himself, who, an ex- cellent English scholar, read the proofs and added to the trans- lation a series of notes embodying the latest results of scientific investigation in magnetism. E. ATKINSON. CAMBERLEY : April 1896. OF THE ^^\ UNIVERSITY) CONTENTS PART L THEORY CHAPTER I INTRODUCTION ARTICLE PAGE 1. The Electromagnetic Field 1 2. The Magnetic Condition as a Directed Quantity ... 2 3. Elementary Conceptions of Quaternions 3 4. Magnetic Intensity 4 5. Magnetic Field of Straight Conductors 5 6. Magnetic Field of a Circular Conductor 6 7. Diamagnetic and Paramagnetic Substances . . . . 8 8. Ferromagnetic Substances and Interferric .... 9 9. Magnetically Indifferent Toroid ...... 10 10. Ferromagnetic Toroid ; Magnetic Induction .... 12 11. Saturation; Magnetisation 13 12. Summary ........... 15 13. Curves of Magnetisation ; Curves of Induction .... 16 14. Susceptibility, Permeability, Reductivity ..... 19 15. Perfect and Imperfect Magnetic Circuits 21 CHAPTER II ELEMENTARY THEORY OF IMPERFECT MAGNETIC CIRCUITS 16. Action of a Narrow Transverse Cut 23 17. Shearing ; Backward Shearing . . . . . . .25 18. Action at a Distance of the Ends 25 19. Action at a Distance of a Single End 26 20. General Remarks about the Law of Action between Points . 28 21. Attraction or Repulsion between the Ends .... 29 22. Action at a Distance of a Pair of Ends ..... 30 23. Mechanical Action of External Fields on Pairs of Ends . . 31 24. Demagnetising Action of a Bar 32 25. Demagnetising Factors of Circular Cylinders .... 34 Xii CONTENTS ARTICLE PAGE 26. Short Cylinder ; Lines of Magnetisation 35 27. The End Elements as Centres of Action at a Distance ; Hy- pothesis of two Fluids ........ 36 28. Uniform Field ; Ellipsoid . . . . . . . 37 29. Ellipsoid of Revolution ; Ovoid ; Spheroid . ... .38 30. Further Special Cases ; Solid Sphere and Solid Cylinder . 39 31. Tabular Summary , . . 40 32. Graphical Representation ......... 41 33. Hyperbolic Curves of Magnetisation . . . . . 42 CHAPTER III OUTLINES OF THE THEORY OF RIGID MAGNETS A. Geometrical Theory of Vector Distribution 34. Vector Distribution . . .44 35. Surface Integrals and their Properties . . . . 45 36. Complex Solenoidal Distribution . -. . '" 48 37. Solenoidal Distribution . . . . ... 48 38. Complex Lamellar Distribution . . . ... 50 39. Lamellar Distribution . . . 51 40. Line-Integrals and their Properties . . ./ . .53 41. Lamellar- Solenoidal Distribution . . .. . . . 54 42. Complex Lamellar- Solenoidal Distribution .... 55 43. Uniform Distributions : General Law ..... 56 B. Conductors conveying Currents and Permanent Magnets 44. Solenoidal Character of the Electromagnetic Field ... 57 45. Magnetic Potential in the Field outside Conductors . . 58 46. Action of a Permanent Magnet at External Points ... 60 47. Distribution of Magnetic Intensity 62 48. Potential of a Permanent Magnet # . . .64 49. Analogy with Gravitation Potential 66 50. Local Variations of Magnetic Strength as Centres of Magnetic Force .67 51. Magnetic Intensity and Magnetic Induction within Ferro- magnetic Media 69 52. Magnetic Induction is Distributed Solenoidally ... 70 CHAPTER IV OUTLINES OF THE THEORY OF MAGNETIC INDUCTION 53. Magnetic Intensity due to Magnetisation and to External Causes \ 73 54. KirchhofPs Assumptions . . . . . . . .74 CONTENTS xiii ARTICLE PAGE 55. Line-Integral of the Demagnetising Intensity .... 76 56. Properties of the Eesultant Magnetic Intensity ... 79 57. Properties of the Magnetisation ...... 80 58. Properties of the Resultant Magnetic Induction ... 82 59. Practical Approximation 83 60. Ferromagnetic Body conveying a Current . . . .84 61. Conservation of the Flow of Induction 85 62. Refraction of the Lines of Induction ..... 86 63. Representation of the Field by means of Unit Tubes . . 89 64. Induced Electromotive Force 91 65. Faraday's Lines of Force 92 66. Statement of the Problem of Magnetisation .... 95 67. Similar Systems ; Lord Kelvin's Rules 96 68. Uniform Magnetisation 98 69. Magnetisation of an Ellipsoid ....... 99 70. Further Special Cases 101 71. Solution by Successive Superposition 103 CHAPTER V MAGNETISATION OF CLOSED AND OF RADIALLY DIVIDED TOROIDS A. Theoretical 72. Peripheral Magnetisation of a Solid of Revolution . . . 105 73. Kirchhoffs Theory 106 74. Rings of Rectangular and of Circular Section .... 108 75. Fundamental Equation of a Radially Divided Toroid . . 109 76. First Approximation ; Limiting Case Ill 77. Divergence of the Lines of Induction ..... 114 78. Leakage Coefficient 116 79. Magnetic End-Elements on the Boundary Surface . . . 117 80. Second Approximation 118 81. Toroid with several Radial Slits .119 82. The Functions v and n are approximately Reciprocal . . 121 B. Experimental 83. The Iron Toroid Examined 123 84. Standardisation of the Ballistic Galvanometer . . . 124 85. Tracing the Normal Curve of Magnetisation . . . 125 86. Arrangement of the Slit 127 87. The Curves of Magnetisation 129 88. Discussion of the Principal Results 130 89. Comparison of Theory and Experiment 134 90. Empirical Formula for the Leakage 137 XIV CONTENTS PART II. APPLICATIONS CHAPTEE VI GENERAL PROPERTIES OF MAGNETIC CIRCUITS A. Non- Uniformly Magnetised Rings ARTICLE 91. General Eemarks 92. Experiments of Oberbeck with Local Coils 93. Further Experiments by Von Ettingshausen and Mues . 94. Theoretical Explanation of the Experiments . . . 95. Self- Compensating Effect of Leakage .... B. Hopkinson's Synthetic Method 96. Principles of the Method . 97. Application to Kadially Divided Toroids . 98. Graphical Representation ; Transformation of Curves 99. Second Approximation ; Correction for Leakage . . 100. Generalisation of the Method . . . . , C. Electromagnetic Stress , 101. Specification of the state of Stress ..... 102. Besultant Tension in the Gap 103. Theoretical Lifting Force of a Diametrically Divided Toroid 104. Eesolution and Interpretation of Maxwell's Equation D. Magnetic Lifting Force 105. Older Investigations . . . . . . . 106. Wassmuth's Experiments 107. Bidwell's Experiments ; Sources of Er-.'or 108. Bosanquet's Experiments . . * . 109. Conclusions from Maxwell's Law 110. Load-Eatio of a Magnet CHAPTEE VII ANALOGY OF THE MAGNETIC CIRCUIT WITH OTHER CIRCUITS A. Historical Survey 111. Older Developments ; First Stage 170 112. Continuation (Faraday, Maxwell) 171 113. Continuation (Lord Kelvin) ^ . 172 114. Summary 174 CONTENTS I'AGK 115. More Recent Developments (Rowland) ..... 175 116. Continuation (Bosanquet) ........ 176 117. Continuation (W. von Siemens) .... . . . 176 118. Continuation (Kapp, Pisati) 178 B. Modern Conception of the Magnetic Circuit 119. Definitions . 179 120. Ohm's Law 181' 121. The Magnetic Reluctance-Function 182 122. Summary 184 123. Leakage. Magnetic Shunts 184 124. Comparative Tables 186 CHAPTER VIII MAGNETIC CIRCUIT OF DYNAMOS OR ELECTROMOTORS 125. Dynamo with a Single Magnetic Circuit 189 126. Predetermination ; Total Characteristic 190 127. Armature yielding a Current. External Characteristic . . . 192 128. Investigations of Dr. Hopkinson 194 129. Graphical Construction . . . . . . .. .195 130. Experimental Determination of the Leakage .... 197 131. Introduction of Magnetic Reluctance 199 132. Calculation of Air Reluctances ^ . 201 133. Other Determinations of Air Reluctances 203 134. Influence of the Position of the Brushes 204 135. Calculation of the Armature Reaction 206 136. Experiments on Reactions of the Armature .... 208 137. Empirical Formulae 209 138. Frolich's Formula . . 210 139. Relation of Frolich's to other Formulae 211 140. General Arrangement of the Magnetic Circuit . . . .213 141. Arrangement of the Field Magnets 214 142. Continuation. Pole-Pieces. Material 215 143. Arrangement of the Armature 217 144. Arrangement of the Interspace . . . . . . . 218 145. Machines with Multiple Magnetic Circuit .... 219 146. Diagrams of Various Magnetic Circuits 220 CHAPTER IX MAGNETIC CIRCUITS OF VARIOUS KINDS OF ELECTROMAGNETS AND TRANSFORMERS A. Physical Principles 147. Magnetic Cycles 224 148. Dissipation of Energy by Hysteresis 226 XVI CONTENTS ARTICLE 149. Influence of Shape ; Eetentivity ; Coercive Intensity 150. Permanent Magnets ..,..*... 151. Magnetic Eeluctance in Joints 152. Influence of Applied Longitudinal Pressure 153. Time Variations of the Magnetic Conditions . . . 154. Discussion of the Function d 23 ld<%> e . ... 155. Simplification with Constant Self- Induction . . . 156. Influence of Variable Self- Inductors . . . . 157. Sinusoidal Electromotive Forces ...... B. Electromagnets for exerting Different Kinds of Pull 158. Principle of Least Eeluctance , 159. Mechanisms depending on Electromagnetism .... 160. Small Iron Sphere in a Magnetic Field . . . _~7~~~ , 161. Attractive Action of Circular Conductors on Sphere . . , 162. Attractive Action of Coils on Spheres .... 163. Attractive Action of Coils on Iron Cores .... 164. Polarised Mechanisms . . . . -. ~ * . 165. Electromagnets with Large Lifting Power * , 166. Description of some Types of Electromagnets *...--. C. Electromagnets for producing Strong Fields 167. Eeview of the Usual Types 168. Principles of Design . . . ~ . 169. Description of the Electromagnet ..... 170. Coils of the Electromagnet 171. Method of Investigation 172. Confirmation of the Theory 173. Influence of Leakage 174. Theory of Conical Pole-Pieces 175. Experiments with Truncated Cones . . ... D. Inductors and Transformers 176. Discussion of Mutually Inducing Coils 272 177. Mutual Induction 273 178. Action of Induction Coils 275 179. Magnetic Circuit of Induction Coils 276 180. Simultaneous Differential Equations of Transformers . . 277 181. Action of an Ideal Transformer ...... 279 182. Influence of Saturation and of Hysteresis .... 281 183. Influence of Leakage 282 184. Transformer Diagrams 283 185. Core and Shell Transformers 285 186. Magnetic Circuit of Transformers ~ . 286 187. Eddy Currents. Screening Action 287 CONTENTS xvil CHAPTEE X EXPERIMENTAL DETERMINATION OF FIELD INTENSITY PAGK ARTICLE ~Qq 188. General Introduction 189. Distribution of Magnetic Fields A. Magnetometric Methods 190. Plan of Gauss's Method . 191. Observations of Deflection B. Electrodynamic Methods 192. Measurement of a Dynamical Force 193. Measurement of a Torque 194. Measurement of a Hydrostatic Pressure . C. Methods of Induction 195. Arrangement of the Exploring Coil . 196. Ballistic Galvanometer 197. Standard Flux of Induction 198. Measurement of a Field by Damping . ^ . D. Magneto-optical Methods 199. Rotation of the Plane of Polarisation 200. Standard Glass Plates .... E. Hall's Phenomenon. Magneto -Electrical Alteration of Resistance 201. Hall's Phenomenon . 202. Measurement of a Field by Bismuth Spirals . F. Magneto-Hydrostatic Method 203. Principle of the Method . 204. Practical Execution . CHAPTEE XI EXPERIMENTAL DETERMINATION OF MAGNETISATION OR OF INDUCTION O-| 17 205. General Eemarks 206. Discussion of the Shape of the Test- Piece 207. Details of the Method .... 31 ^ 208. Determination of Distribution A. Magnetometric Methods OQ-J 209. Plan of the Experiments .... *" 210. Virtual Length of the Magnet . a OF THE TT-KTTHSITY xvill CONTENTS ARTICLE 211. Helmholtz's Method. Compensating Coil 212. Searle's Curve Tracer ...... 213. Eickemeyer's Differential Magnetometer . . ; B. Electrodynamic Methods 214. Swing's Curve Tracer 215. Apparatus of Koepsel and of Kennelly C. Induction Methods 216. The Ballistic Method . .... 217. Isthmus Method . . . . . . . 218. Yoke Method . . . . -...-.. 219. Various Forms of Closed Yoke . .... 220. Case of great Self- Induction . ; , . . 221. Methods of J. and B. Hopkinson and of T. Gray D. Magneto-optical Methods 222. Kerr's Phenomenon . . . < 223. Kundt's Phenomenon \ ' . . . . . E. Hall's Phenomenon. Bismuth Spiral 224. Hall's Phenomenon. Bismuth Spiral ... 225. Bruger's Apparatus . . . . . " - . ; . .. F. Traction Methods 226. Thompson's Permeameter 227. Magnetic Balance 228. Use of the Balance 229. Magnetohydrostatic Methods INDEX OF NAMES INDEX OF SUBJECTS NOMENCLATURE . .... THE MAGNETIC CIRCUIT PART I THEOEY * * ^ *0NI"V CHAPTER I INTRODUCTION 1. The Electromagnetic Field. ' The electric current, or, more generally, electricity in motion, is the only known source of any kind of magnetism, and more particularly also of terres- trial magnetism,' as may, with great probability, be assumed. 4 Magnetic iron ore and other bodies occurring in nature in the magnetic condition manifestly owe their magnetism to that of the earth, or, in some cases, no doubt, to the direct action of electrical discharges.' l We start, therefore, by taking the fact as known, that a conductor along which a current is passing produces in its vicinity a peculiar condition which is called an electromagnetic, or ; more briefly, a magnetic field. The air which in the ordinary conditions of experiment occupies this space plays only a very subordinate part, to which we shall afterwards refer ( 7). In the phenomena to be subsequently described we shall assume that they take place in a vacuum. The condition in question manifests itself among other things by the fact that, on the one hand, forces are exerted in the mag- netic field on other conductors carrying currents ; and, on the other, that momentary currents are induced in conductors when, and only in so far as, the condition in question is- altered, either as regards position or value, and particularly when it either suddenly appears or completely vanishes. Movable conductors conveying a current are therefore put in motion. On the other hand, momentary currents are induced in movable con- 1 Compare W. von Siemens, Wied. Ann. vol. 24, p. 94, 1885. B an ^ 5 i n general, very considerably. If we lay the chief stress on the phenomena of magnetic in- duction, as their practical importance requires, we may obviously proceed as follows. Just as as we have considered the expres- sion Q R j n 2 S as a measure of the magnetic condition in the indifferent coil, we are by analogy entitled to use Q' Rfn 2 S to determine that condition in the ferromagnetic substance. We put, therefore, and call the value, thus defined, the induction. This name was introduced by Maxwell ; 2 it recalls that the vector can be arrived at by considering one of the most important manifesta- tions of varying magnetic condition : the induction of electro- motive impulses in adjacent conductors. Yet there are several such manifestations, and among them processes which occur not merely at the moment of change. And although induction has hitherto undoubtedly played the chief part, especially from the practical point of view, there is no physical reason for assigning to it a preferential position under 1 Compare in this respect Chapter IV. 64. 2 Maxwell, Treatise, 2nd edition, vol. 2, 400. This term, of such general use, has for a strictly definite idea the disadvantage of being applied in a perfectly indeterminate manner to a group of physical phenomena, not to speak of non-physical branches of knowledge. Mistakes are, however, not likely to occur, although two of these groups, as follows from the above, are in direct relation with the idea in question (compare 113). SATURATION. MAGNETISATION 13 all circumstances. It is usual to assert in reference to such a ferromagnetic ring as the above that its magnetic condition can in no way be externally perceived. This is only correct if we are thinking of the apparent actions at a distance which we have been accustomed to observe as arising from bar magnets or bodies of similar shape. For, in the first place, the perimeter of the toroid changes on magnetisation, though only to an insignificant extent. If, secondly, a small portion of the surface is polished, so as to reflect, and a pencil of light is reflected from this, a change of the state of polarisation in the reflected light is observed on magnetisation. 1 In the particular case in which the light is polarised in the plane of incidence (that is, of the plane of the figure, fig. 1, p. 11), and the angle of incidence is about 60, a simple rotation of the plane of polarisation is observed. It might, thirdly, be added that on magnetisation, internal stresses ( 101), peculiar (rotatory) properties of electric and thermal conductivity, as well as changes of thermo-electric behaviour, of specific heat, &c. are met with. From this we see that the existence of the magnetic con- dition in the ferromagnetic substance is manifested in various ways, even when no action at a distance in the ordinary sense is to be observed. We shall even see that it is the absence of this in the case of a toroid which marks this case as typical ; for an apparent action at a distance forms no criterion for the existence of a uniform magnetic condition ; it proceeds, in fact, only from places where there is a local variation, or even a sudden cessation of such conditions, as will be more fully explained further on. 11. Saturation. Magnetisation. The phenomena last men- tioned all show the peculiarity that with an unlimited increase of the magnetising field they do not increase in a corre- sponding manner. They undergo, on the contrary, continually smaller increments, until finally they become practically constant. The magnetic condition appears then, as it were, saturated, to use the ordinary expression. The induction 95, defined and deduced as above from magneto-electrical phenomena, never, on the contrary, attains 1 This phenomenon was first observed by Kerr, Pliil. Mag. [5], vol. 3, p. 321, 1877, and vol. 5, p. 161, 1878. 14 INTRODUCTION such saturation- values. By using more intense magnetic fields, on the contrary, it has been possible to force the induction to higher and higher values. But if we consider the difference of induction and intensity, that is the expression 23 , it is apparent that this shows the characteristic course which the processes described present. It has even been established that one of the phenomena which is most easily and accurately accessible to quantitative deter- mination, the rotation of the plane of polarisation, is in all circumstances proportional to 93 <. It is probable, further, from all our present experimental observations, that the other phenomena also depend either on the difference 23 $ or on its square (93 ) 2 , according as they are even or odd func- tions of 95 < It is natural, therefore, to introduce this expression, or one proportional to it, as a measure of the magnetic condition, as manifested in the physical properties of the ferromagnetic sub- stance. Following the historical development, and taking into consideration the usual absolute electro-magnetic system, we shall choose as factor the number 1 / (4 TT) (compare note p. 59), and put therefore The value 3 defined by this equation we briefly call the magnetisation : 1 it will in the sequel play 'an important part. By transforming the equation (12) we obtain the fundamental equation (13) .... 23 = +477-3 In the typical cases of ferromagnetic rings which have hitherto been considered, those three quantities S3, <, and 3 all have the same direction, so that the interpretation of equa- tion (13) presents no difficulties. We shall subsequently (51) return to its more general character as a vector equation. In order to give some idea of the numerical values in question, the following data may be mentioned. In ordinary 1 This abbreviated expression is already frequently used instead of intensity of magnetisation ; the latter, besides being longer, has the disad- vantage of being liable to be confounded with the intensity .>. SUMMAKY 15 circumstances the magnetic field of a coil cannot be driven beyond a few hundreds of electro-magnetic C.G.S. units. Only by using special arrangements for cooling (ice or currents of water) is it possible in this way to attain at most 1,500 C.G.S. The saturation-value of magnetisation in the case of iron, which as a ferromagnetic substance has by far the most ex- tended application, is in the most favourable case from 1,700 ta 1,750 C.G.S. units; the product 4?r3 is then between 21,500 and 22,000. Hence, according to equation (13), it has but little influence- if a few hundreds are added to the latter number. With small values of magnetisation the second term 4 TT 3> still more outweighs the first. We arrive, thus, at the practically impor- tant result, that in ordinary circumstances we may with close approximation put (14) . . . . 93 = 47r3 In those cases in which > is intentionally raised to higher values than are attainable with simple coils, this simplification of course does not hold. 12. Summary. Once more summarising our results, the consideration of experimental facts leads to the following conception. The physical condition of a ferromagnetic substance is completely defined by the magnitude which we have called magnetisation, and denoted by 3. The electromotive impulses in the surrounding conductors are not, however, defined by the magnetisation 3, but by the induction 93, for this reason, that they depend, not only on the condition of the ferromagnetic substance, but also on that condition which would hold if it were removed, or if it lost its specific properties. To illustrate this by an example, let us assume that the ring in question consisted of nickel, while the coils were of platinum wire insulated by asbestos. If this arrangement is heated above 300, a change will be perceived ; the specific ferromagnetic properties of nickel rapidly diminish and disappear altogether at 350 , 1 the metal becomes nearly 1 This phenomenon also occurs with other ferromagnetic metals, but at a much higher temperature. 16 INTKODUCTION indifferent in the sense explained in 8. The property of acting inductively on adjacent conductors also diminishes considerably, but does not disappear ; in the expressions used above the quan- tity of electricity induced by opening or closing the primary current would on heating fall from Q r to Q, but not to zero. In this state of things there would be no essential change on further rise of temperature, not even if the melting-point of nickel were reached, and the metal flowed away. By closing or opening the primary current, the quantity of electricity Q is always induced ; it measures the condition which magnetisation produces in an indifferent medium or in heated nickel. But if we again cool the nickel, the ferromagnetic properties also return, and at the same time the value Q rapidly increases to the value Q r . We are accordingly compelled to admit that the specific condition, which alone produces those properties, is superposed on that condition which previously existed in the space in question, whether it was void or was occupied with indifferent matter. It is the superposition of the two conditions, the variations of which can under suitable conditions produce electromotive efforts. From the great importance of this inductive action, the induction 33 was long regarded, particularly by some English authors, as the more fundamental magnitude. Herein there has, however, already been a change of opinion, and now, as at first, the magnetisation is considered as the physically more important conception in dealing with purely scientific questions. The idea of induction of course loses none of its great value for the mathematical treatment of cognate problems on the one hand, and for technical applications on the other. We shall in the sequel have frequent opportunity to recognise this, when we enter on the field of applied physics (Chapter VI. et seq.). 13. Curves of Magnetisation. Curves of Induction. In a ring of any given dimensions, consisting of a definite ferro- magnetic material, both 3 and 93 are only functions of <>, apart from any question of hysteresis. The graphical representations of those functions for the case of rings are therefore to be con- sidered as normal curves, since they only are characteristic for the material in question. For such a curve, as we shall see further on, is, in general, considerably, often, indeed, prepon- CUEVES OF MAGNETISATION 17 deratingly influenced by the form of the ferromagnetic sub- stance. As regards the experimental details of these curves, we must refer to works which treat magnetic induction with com- pleteness. 1 We shall here only represent and discuss such a curve of ascending reversals for an iron ring ; the curves for other ferromagnetic substances show always the same general character, even when they present marked deviations in their individual features. Generally speaking three parts can be distinguished in all normal curves of magnetisation. The first corresponds to the smallest value of the magnetising intensity ; the magnetisation then increases about in proportion to that vector, so that this portion deviates but little from a straight line through the origin of co-ordinates. The curve then bends strongly away from the abscissa axis, corresponding to a more rapid increase of magneti- sation ; this second portion corresponds to mean intensities. The curve then reaches a point of inflexion, increasing at last continu- ally more slowly. The last portion, which corresponds to the highest intensities, even to infinite values, so far as at present known, gradually approaches the asymptote parallel to the axis of abscissse ; this represents a maximum value of magnetisation to which this quantity tends without strictly speaking ever attaining it. The three portions of the curve correspond to three different stages of the processes of magnetisation represented by it. And this division is not merely an arbitrary one, but is based on the very nature of the process. In special circumstances it is pos- sible to put the ferromagnetic substance in a molecular condition in which the three stages of magnetisation are quite remarkably developed, and appear distinct from each other. 2 Fig. 2 represents, for instance, the normal curve of magne- tisation, 3 = funct. ( ) for a variety of cast iron ; the } indicates that for infinite values of the abscissa the ordinate 3 would attain the saturation value 3 m = 1100 C.G.S. units ; it does, in fact, approach this asymptotically, if the intensity is more and more increased. 1 Compare, for example, Ewiug, Magnetic Induction in Iron and other Metals, chaps, iv. vi. and vii. 2 Compare Nagaoka, Journ. Coll. Science Imp. Univ. Japan, vol. 2, pp. 263, 304, 1888; iUd. vol. 3, p. 189, 1889. C 18 INTRODUCTION In order to transform a normal curve of magnetisation [3 = funct. (<)] into the corresponding normal curve of induc- tion [23 = funct ($)], we take into account equation (13). SB = 4-77-3 + $ in which the first term on the right usually very much exceeds the second; it will therefore in most cases be sufficient to read off the curve of magnetisation on a scale of 1 / 4?r (right scale of ordinates fig. 2). In cases in which the approximation 93 = 4?r3 is not sufficient, we have to add a portion to that noo 900 700 -eoo 00 400 300 zoo 100 S9 FIG. 2 7500 50OO 2500 2500 ordinate, which is manifestly equal to the numerical value of the corresponding abscissa. This is most conveniently done by a kind of transformation of co-ordinates ; that is, by drawing through the origin a straight line Q, the equation of which is 93 = , and then measuring the ordinates from this new axis of abscissae. If, however, we are to return to the ordinary orthogonal system of co-ordinates, we must distort the whole plane of the figure, which we assume to be completely extensible, in such a way that each point moves upwards parallel to the axis of ordinates, so that the straight line Q ultimately coincides with the original axis of abscissae. SUSCEPTIBILITY, PERMEABILITY 19 Such an operation Ewing calls a shearing of the curve, parallel to the ordinates, from the directrix Q to the axis of abscissae. Such shearing of curves, which, however, is usually done parallel to the abscissae, and whose directrices need not necessarily be straight, we shall frequently have to deal with for other purposes. The operation can be conveniently effected by means of a pair The normal curve of induction deduced from fig. 2 is repre- sented in fig. 3. While, however, in the former, the abscissae are plotted to $ = 1000, in the latter this is carried up to the value 50OOO WOOO 30000 10000 100 JOD00 Wt 00 WOO 2000 FIG. 3 woo of the intensity $=40,000, the highest which has hitherto been reached. For this reason it was necessary to use two scales of abscissae. The first, which is already one-fourth that of fig. 2, extends to $=4000 ; the continuation of the curve can be easily read on the accessory axis by the -^ scale. These curves will clearly show how the vector 3 tends towards a maximum, whereas the vector 95 increases finally without limit. 14. Susceptibility, Permeability, Reluctivity. Besides the chief quantities 3 and 5 S, both which, apart from hysteresis, are single- valued functions of $, some other less important c 2 20 INTRODUCTION quantities have been introduced which are frequently useful. They are thus briefly defined : /e, magnetic Susceptibility, defined as //,, Permeability, f, Keluctivity, These scalar quantities are all pure numbers at least, they are thus to be regarded, in all cases in which the electro-magnetic system is used. In this system the vectors 3, 93, and <$ have the same dimensions [Ir* Ml 2 7 - 1 ] and the permeability of a vacuum is supposed equal to unity, as follows implicitly from the definitions in question ; the same holds for its reciprocal, the magnetic reluctivity of a vacuum. These three numbers, like the vectors 3 and 35, are only functions of the independent variable < ; they may, however, just as well be considered as functions of 3 or 95, and as such be graphically represented. In different authors we find, for instance, representations of /c=f(3), /A=f(95), /t=f($), f = f (<). The form of these curves, their singular points, and other properties may be simply deduced and discussed from those of the normal curve of magnetisation ; they are interesting as guides in forming a judgment on many questions. But to enter here upon all the . developments would lead us too far without any corresponding advantage. We confine ourselves therefore to the curves of fig. 4, which represent the permeability //, (left scale of ordinates) and their reciprocal the coefficient of reluctivity f (right scale of ordinates) in so far as they depend on the intensity &. They refer to the same cast iron as figs. 2 and 3. The permeability, as will be seen, attains a maximum which may be very considerable. For the best wrought iron it may, for instance, attain a value of several thousands ; it then falls off gradually until with an un- limited increase of <% it would presumably have the value 1. Conversely, the reluctivity passes through a sharply defined minimum, and then gradually increases, and would like- wise at last approach the value 1, though far outside the region represented in fig. 4. (Compare 121.) PEEFECT AND IMPEEFECT MAGNETIC CIECUITS 21 If in equation (13) 11 both sides are divided by f? we obtain or if the above values are introduced (15) //,= 1 -f-47T K as the relation between permeability and susceptibility. The latter number would probably vanish if the intensity in- creased indefinitely; but it is to be observed that of course only supposi- tions can be made for this case, based upon what actually happens, at the highest intensities attain- able ( 13). The suscep- tibility then appears as the quotient of a finite quantity, the saturation of the magnetisation, divided by an infinite quantity. We have therefore follows that yit given above. As we shall afterwards make no distinction between mag- netically indifferent bodies and a vacuum, we shall consequently suppose the permeability of the former unity, whereas it actually differs in the fourth or fifth decimal place. The reluctivity is then also unity, but by equation (14) the susceptibility is zero. 15. Perfect and Imperfect Magnetic Circuits. Our pre- vious considerations referred to the case of a uniformly wound circular ring, having also a circular section; a body thus shaped we have called a toroid. But we may at once extend them to rings of any given section and any shape. The fJU 1 \ \ $ 008 \ S* ,' \ / \ S s 005 \ s' s oot \ S s 003 so - \ s s s, \ ^ s '^ .^ nm \s $ fa V & A JO * X) M FIG 10 6L L 4 v) n 70 Si to att a /6 W) =0; from equation (14) it 1, and in like manner 1/^=^=1 as 22 INTRODUCTION centroid ( 6), that is, the curve corresponding to the central circle of the ring ( 9), may be either a plane or a solid curve, provided only that its radius of curvature be always large com- pared with the dimensions of the section. With uniform wind- ing there will be no apparent magnetic action at a distance l that is, the magnetic condition is restricted to the body of the ferromagnetic ring. Such an arrangement is called a perfect magnetic circuit. We assume that the condition for this per- fection is the absence of any action at a distance, and hence an arrangement in which such an action does occur is aptly called an imperfect magnetic circuit. 2 1 In the few places in which action at a distance has been mentioned, it has always been spoken of as apparent ; for in the present state of science it may be considered as almost certain, that direct actions at a distance do not take place, but are transmitted through the intervening medium. But since, for our present object, we need not concern ourselves with this deeper insight into the mechanism concerned, the expression 'action at a distance' will, with this reservation, be used without further specification. 2 Compare 100. 23 CHAPTER II ELEMENTARY THEORY OF IMPERFECT MAGNETIC CIRCUITS 16. Action of a Narrow Transverse Cut, The reason for the comparatively simple behaviour of perfect magnetic circuits in the sense of the previous paragraph, is their geometrical property of having no ends, so that in the literal sense of the words, they are endless. This endlessness entails, as will be seen, the absence of action at a distance, and may there- fore just as well be regarded as that property which directly determines the perfection of the magnetic circuit. The correctness of this conception is seen when we provide the perfect circuit with ends by cutting it. Any cut, however fine, then manifests itself by the occurrence of an action at a distance, which is strongest in the immediate neighbourhood ; in the part surrounding the cut magnetic conditions are pro- duced, the intensity of which increases with the width of the cut. The occurrence of actions at a distance, conversely, points to the presence of transverse cuts, even when these cannot be seen, as for instance in the case of concealed cracks, or such as have been soldered, or joints in the ferromagnetic substance itself (Chapter IX., 151). The air-gap which occupies the cut differs in no respect from the indifferent environment, and therefore the action at a distance cannot proceed from it ; we must accordingly conclude, as a geometrical necessity, that the action at a distance is due to the ends which the cross cut has produced. In order to acquire an insight into the action of such cuts from a totally different point of view, let us consider the typical case of a uniformly wound toroid. We suppose this to be cut transversely in the direction of the radius say at A (fig. 1, p. 11), in such a manner that the cut is very narrow, compared with the dimensions of the section ; let its width be d. Let (A) be the 24 ELEMENTARY THEORY OF IMPERFECT MAGNETIC CIRCUITS normal curve of magnetisation of the closed ring before it has been cut, as represented in fig. 5 for a particular variety of soft iron. If then the curve is determined for the divided ring, it will be found to run more to the right, like the curve (B) for instance. We thus see that with a given magnetising intensity as abscissa, the ring which has been cut has a smaller magnetisation than the closed one, and that conversely, in order to produce the same magnetisation, a more powerful field is required. By how much must this be greater ? To answer this question we have to determine for given values of the ordinates, the differences of ab- scissae A <> of the two curves, and then to try in what way these depend on the ordinates, that is functions of the ordinates. For narrow to examine them as transverse cuts we thus obtain, as a first approximation, a straight line C, which is so inclined that for each given ordinate Absc. (0) = Absc. (5) - Absc. (A). The answer to the question is, therefore, that to obtain a given magnetisation with divided rings, an increase of intensity is necessary, which, as a first approximation, is proportional to the magnetisation to be attained ; that is to say, is a certain constant fraction of it. The theory and experiments by which these results are arrived at will be discussed in detail in a special section (Chapter V.) ; we deal here only with the general qualitative results, in so far as they afford an insight into the action of cuts. SHEAEING OF CUEVES OF MAGNETISATION 25 17. Shearing; Backward Shearing. This is the appro- priate place for discussing the graphical operations by means of which, by transforming co-ordinates, normal curves of magnetisa- tion may be transformed into curves for divided rings, or, in general, for imperfect magnetic circuits, and conversely. With this object we draw a straight line G' through the origin (fig. 5) into the upper left of the quadrant, which is symmetrical with G with respect to the axis of ordinates. It is clear from what has been said above that we obtain the new curve (B) from the normal curve (A), when, in conformity with 13, we effect a shearing from the directrix (77? to the axis of ordinates. By the converse operation, which we may call backward shearing from the axis of ordinates to the line G 0', we transform again the curve (.B) into the normal curve (A). 1 The equation of the line 00' we may write generally in which N is a factor increasing with the width of the cut, to which we shall afterwards refer in more detail. The increase in the intensity of the magnetising field, A, which has to be applied in consequence of the cut, is, in fact, according to equation (1) proportional to the value 3 of the magnetisation to be induced, as was also mentioned in the previous paragraph. 18. Action at a Distance of the Ends. As regards the expla- nation of the facts described, we have already seen that actions at a distance are directly produced by a cut, and the more markedly the wider is the cut. These actions extend over the whole surrounding space, and it is, therefore, natural to suppose that they will also be produced in the space occupied by the ferro- niagnetic substance of the ring, and will, therefore, act induc- tively. Now it may be readily shown that such an inductive action must be opposed to that of the field due to the current in the coil (fig. 6, p. 27), so that the latter must be increased, in order to compensate the former and thus to obtain the same total intensity, and, hence, the same magnetisation. It is' this increase in intensity which is expressed by the difference of the abscissae of the curves, and which, therefore, must be equal and 1 This construction was first given by Lord Kayleigh, Phil Mag., 6th series, vol. 22, p. 175, 1886, and generalised by Swing. 26 ELEMENTARY THEORY OF IMPERFECT MAGNETIC CIRCUITS opposite to the mean intensity of the action at a distance of the cut within the space occupied by the ring itself. As, further, we have already seen that the air-gap itself can exert no action, and that this must proceed from the ends, we con- clude that such a pair of ends acts at great distances proportion- ally to the value of the magnetisation. We have thus arrived at a conception which necessitates the closer investigation ot the action at a distance of pairs of ends. We are, at the same time, led to a point of view which was formerly the general starting-point, and the one almost exclu- sively taken into consideration. A noteworthy reaction has, in some quarters, recently set in, which at once considers as in- appropriate, antiquated, and useless the ideas of magnetic fluids, of poles, attractions, and Coulomb's law, &c., which were pecu- liar to that mode of view. 1 We shall proceed most safely in this respect if we take a middle course, and endeavour, as far as possible, to combine and utilise the advantages of both views. 19. Action at a Distance of a Single End. Let us imagine the ring cut through, and then stretched out so that it has the form of a long straight bar, and thus its ends are at the greatest distance possible. In the divided ring we assumed that the coiling was uniform, notwithstanding the gap. We shall here make the same assumption, and imagine a uniformly wound closed coil, like that in fig. 6, which does not produce of itself an external magnetic field, but within the windings produces a uniform field in the direction of the feathered arrows. Magne- tisation in the same direction is then produced in the ferro- magnetic bar, as can be proved by means of secondary coils closely wound round it. Owing to the action of the ends a magnetic field will now be formed in the surrounding space. We will now investigate this somewhat more closely, by means of an exploring coil as in 2. If now we investigate the space near one end, NfoY example, where its own influence decidedly preponderates, we shall find that the field has everywhere a radial direction, and from the end outwards, as shown by the unfeathered arrows. Its numerical value is inversely proportional to the square of the distance from 1 Compare for instance, Prof. Silv. Thompson, Cantor Lectures on the Electromagnet, London, 1890. ACTION AT A DISTANCE OF A SINGLE END 27 the ends as long as this is small compared with the length of the bar. Near the other end the condition is the same, except that the radial fielcj is directed inwards at the end, and is again represented by plain arrows. The latter (8) is usually called the negative, and the former (N) the positive, end. 1 The positive direction of magnetisation in ferromagnetic substances is always from the negative towards the positive end. The intensity of the field near the ends increases propor- tionally to the value of magnetisation, as we proved with the divided ring, and is also proportional to the section 8 of the bar. It is independent of the length, provided this is considerable in comparison with the cross section, as is always supposed to be FIG. 6 the case. The absolute value of the intensity of the field is therefore given by the equation ( 2 ) # = 7r where r is the distance from the end. Accordingly the power of the ends to exert actions at a distance depends upon the value of the product (3/S), which may be called the magnetic strength of the bar. 2 1 With a freely suspended bar the positive end sets nearly north, the negative south. The positive end is by most writers called the north pole, the negative S, the south pole. (See Maxwell, Treatise, vol. 2, 393.) 2 Sir W. Thomson, Reprint of Papers on Electricity and Magnetism, p. 354, 454. 28 ELEMENTARY THEORY OF IMPERFECT MAGNETIC CIRCUITS 20. General Remarks about the Law of Action between Points. The relation expressed by equation (2) is a somewhat modified conception of what is called * Coulomb's law,' to which we shall afterwards return. An essential condition for the validity of that law is, that the end of the bar is regarded as a point, and that, therefore, its dimensions are infinitely small compared with the distance r ; the latter, again, is supposed to be small compared with the length of the bar. The experimental demonstration of the relations in question can therefore only be made with very long thin bars. We may here observe that Coulomb's law is by no means a specific magnetic law. It is only a special case of the perfectly general, purely geometrical, law which governs all actions that proceed and are propagated to a distance from points, or rather from centres which may be regarded as points. These actions all diminish as the distance increases, in such a way that their intensity is inversely proportional to r 2 , and for this simple geometrical reason that the surface of a sphere is proportional to the square of the radius. As the total action, which remains constant, has to be distributed over continually increasing sur- faces of concentric spherical shells, the above law of the decrease of its intensity, that is of the action per unit surface, follows at once. Among the known examples of this general relation are the law of gravitation, Coulomb's electrostatic law, the photo- metric law for the decrease with the distance, of the intensity of light from luminous points. In all these cases the assumption of points as centres is the essential basis for the application of the (I/?* 2 ) law of action between points or of inverse squares. A gravitating, an electrified, or a luminous infinitely long straight line, on the contrary, does not act inversely as the square of the distance, but inversely as the distance itself; and here again, for the simple reason that now the surface of a cylinder of given length is proportional to its radius. A further example of this general (1/r) law of the action between lines is Biot and Savart's law of the electromagnetic action of long straight portions of conductors, which has been already dis- cussed in 5. The action, finally, of a gravitating, electrified, or luminous infinite plane does not depend on the distance; it may, in ATTRACTION OK REPULSION BETWEEN THE ENDS 29 fact, be considered as enclosed by two planes, also infinite, the surface of which is obviously independent of the distance. For the sake of a general expression, we may also, in this case, speak of a (1/r^J laiv of action between planes. * 21. Attraction or Repulsion between the Ends. The end of a magnet not only produces a field in its vicinity, but it is in- fluenced by an existing extraneous field in such a manner that a mechanical force is exerted on it. This force has either the same or the opposite direction, according as that part of the original field where the end is placed is positive or negative. Its numerical value $ is equal in absolute measure to the product of the intensity of the field into the magnetic strength (3$) of the end (19): (3) . . . . ffrr^Stf), If, as a particular case, the field in question radiates from another magnetic end, the resultant action may obviously be explained by saying that according as their signs are the same or opposite there is a repulsion or an attraction, $ 12 , between the ends, which is the direction of the line joining them, directly proportional to the product of their magnetic strengths, and in- versely proportional to the square of the distance. The mathe- matical formula for this law follows directly from equations (2) and (3), and is as follows : This is the original form of Coulomb's law, 1 which was dis- covered by Coulomb by experiments with the torsion balance. An exact confirmation was first obtained by the measurements which Gauss made for this purpose. 2 As the existence of direct mechanical actions at a distance, which a verbal interpretation of this law entails, can scarcely be accepted by modern science, the tendency is to explain these apparent actions at a distance by stresses in the medium which transmits the action. We shall 1 In the notation already mentioned ( 19, note), it is usual to formulate Coulomb's law as follows. Like (unlike) poles, repel (attract) with a force which is directly proportional to the strength of the poles, and inversely pro- portional to the square of their distance. 2 Gauss, Intensitas vis Magn. Terrestr. ad Mensuram Absol. Revocata, 21, Werke, vol. 5, p. 109 ; 2nd reprint, Gottingen, 1877. 30 ELEMENTAKY THEOEY OF IMPERFECT MAGNETIC CIRCUITS discuss these stresses in the gap more in detail in articles 101-110. The introduction of this conception, besides afford- ing a more satisfactory theoretical explanation of the facts, has the advantage of providing a far more suitable basis for most practical problems than Coulomb's law does. The latter, indeed, still gives the simplest representation of the mechanical actions in all cases in which it is a question of the reciprocal influence of a small number of ends of bars, as is still the case in many of the usual experimental methods. 22. Action at a Distance of a Pair of Ends. We will now return to the action at a distance of our bar magnet, and con- sider this at distances which are great compared with the length of the bar ; the numerical value and direction of the magnetic field is obtained for all points by the superposition of the com- ponents due to the two ends as given from Coulomb's law. Two special cases are to be distinguished, the proof of which need not here be gone into. In the first place, the intensity of the field at a point on the prolongation of the geometrical axis of the bar is directed along the axis, and its value is given by the equation (5) .-. - -? in which L is the length of the bar, D the distance of the given point from the centre of the bar. Secondly, at points in the equatorial plane of the bar, traced by the line EE, in fig. 6, the value is //,\ (< -6 S L (6) - * = ~IF- where the direction is still parallel to the axis. We thus see how the length of a bar now enters as a factor into the equations, and accordingly (3 8 L) and not (3 8) is the measure of its power to produce a field at distant points. But 8 L is the volume of the bar, and therefore the product of the magnetisation into the volume is the determining quantity. In accordance with the usual practice we shall call this the magnetic moment of the bar, and denote it by 2ft ; it is analogous to the corresponding quantity for coils, which is also thus designated ( 6). If the action at a distance is used to measure ACTION AT A DISTANCE OF A PAIE OF ENDS 31 the magnetic properties of bodies as in what are called mag- netometric methods the magnetisation is equal to the magnetic moment measured divided by the volume. In case the point in question is neither in the prolongation of the axis nor in the equatorial plane of the bar, the equations for the intensity of the field are less simple than (5) and (6). As regards the shape of the lines of intensity we refer to fig. 7 in which they are represented outside the bar by the dotted lines ; the figure in question holds for a short bar ; with longer bars, as assumed in the above, the action at a distance proceeds almost exclusively from the geometrical ends, and the lines accordingly diverge radially. In fig. 7 these lines proceed not only from the end surfaces +~E and E, but also partially at FIG. 7 least from the neighbouring parts of the sides. The lines either proceed from the positive to the negative end, or they direct their course towards an infinite distance. 23. Mechanical Action of External Fields on Pairs of Ends. We have already seen in 21 how in an extraneous field a force is exerted on a single end. The occurrence of single ends is, however, excluded from the nature of the case ; we can at most assume that in very long bars one of the ends may be considered singly by neglecting the actions proceeding from the other end, or exerted on it, as being too distant. But, as a matter of fact, in bar magnets, we have always to deal with a 1 Compare 26, where the lines inside the bar, * the lines of magnetisation,' are further discussed. 32 ELEMENTARY THEOKY OF IMPERFECT MAGNETIC CIRCUITS pair of ends, whether the magnetisation is induced or residual. The area 8 and the magnetisation 3 being constant, both ends have the strength 3 8, though with opposite signs. In an extraneous field, the intensity of which is constant within the space occupied by the bar, and is in the same direction, according to equation (3) 21, equal and opposite forces are exerted on the ends. $ = $ 3 8) If the positive axial direction of the bar ( 19) makes an angle a, with the positive direction of the field, these two forces act at points which are at a distance of L sin a from each other, L again being the length of the bar. They thus exert a torque, the moment of which is given by the equation (7) . . . & = (3 S) L sin a or introducing again the magnetic moment 2ft, as in the previous paragraph (8) . . . = < 2ft sin a The couple therefore in general acts on the pair of ends in such a manner as to tend to draw the bar into the position of stable equilibrium, which corresponds to the value a = 0. It will make oscillations about this position the period of which r is given by the equation in which K is the moment of inertia of the bar. A position of unstable equilibrium corresponds to the value a = 180. These conclusions are most completely confirmed by experi- ment. A single force is never exerted on a magnet in a field of constant value, and constant direction, but always a torque ; and this holds not only for bar magnets of constant strength, but also for bodies of any form, and magnetised in any way whatever. 24. Demagnetising Action of a Bar. It has already been explained in the case of the divided ring that each of the ends DEMAGNETISING EACTORS OF CIRCULAR CYLINDERS 33 will also act according to Coulomb's law in the space occupied by the bar itself. This ' self action,' as is at once seen from the plain and feathered arrows in fig. 6, p. 27, will always be opposed to the action of the coil. Both ends exert therefore a demagnetising action on the bar, the intensity of which we will call >. This is a minimum at the middle of the bar and in- creases towards each end ; from what has been said, it is pro- portional at each point to the magnetic strength(3 8) of the bar. Hence this proportionality also holds for the mean value of the vector 4? i5 which we shall distinguish by a bar over the letter, thus ^ Let us now consider a circular cylindrical bar with plane ends. The dimensional ratio, by which is meant the ratio of the length to the diameter, we will call m. If we now assume that the cylinder becomes gradually thinner, the length remaining the same, this would correspond to an increase of the ratio m, and to a decrease of the cross-section proportional to that of 1/tn 2 . Hence $< also decreases, and, from what has been said, in the same proportion as the magnetic strength that is, proportionally to(3/m 2 ). If we now consider the quotient &/3I that is, the mean demagnetising intensity per unit of magnetisation and if, as in equation (1) ( 17), we denote it with JV, we shall call the number thus defined the mean demagnetising factor. It follows, then, from the preceding that JV must theoretically be proportional to 1/m 2 that is, I. The demagnetising factor of a circular cylindrical bar is, theoretically , inversely proportional to the square of the ratio of the dimensional ratio. If C is the factor of the proportion, then (9) .... ^m 2 = C must be constant. From an analysis of experiments with cylinders, it is found that this, as a matter of fact, does hold, provided the ratio of the dimensions exceeds 100. C has then the constant value 45. The mean demagnetising factor of a cylinder, whose length is at least 100 times the diameter, may be simply calculated by dividing the square of the dimen- sional ratio into the number 45. D 34 ELEMENTAEY THEOBY OF IMPEEFECT MAGNETIC CIRCUITS 25. Demagnetising Factors of Circular Cylinders. The experiments mentioned were made with cylinders of varying lengths, but with a given diameter and given material. Owing to the unavoidable heterogeneity of the material, it is better to cut the cylinders gradually shorter from one and the same piece. It would be best of all, retaining the length constant, to gradu- ally turn it down to a smaller diameter, as we have supposed in our theoretical investigation. The curves found thus experimentally for various values of m are then plotted side by side, and the corresponding demagnetising factors l are then deduced from the differences of the abscissae in the manner above described (5 1.6)- It appeared that for short cylinders for which m < 100, the value C is not constant but diminishes. In Table I., which, for more convenient comparison with other numbers, is printed on page 41, a general view is given of the mean demagnetising factors found as described. The values C = m 2 N are also given, as being better fitted for interpolation owing to the slight extent to which they vary. It has further been experimentally established that ferro- magnetic prisms or bundles of wire tied together, of any given profile, differ but little from circular cylinders of equal length and equal section. 2 The table furnishes, therefore, a means of reducing the results of experiments with bars and bundles to the proper normal case of endless shapes; in other words, by back shearing, to obtain the normal curve of the material investigated, which is alone characteristic. This is the more necessary as by far the greater number of the existing and, in part, very valu- able experiments have been made with bars ; and from the nature of the case this will, in the future, also often occur. The results and the curves obtained in this way will also be difficult of direct interpretation, and hence lose much of their value. 26. Short Cylinder. Lines of Magnetisation. We have already seen that for short cylinders (for which m < 100) the 1 Such experiments have been made partly by Ewing, Phil. Trans, vol. 176, p. 535, partly by Tanakadate", Phil. Mag. vol. 26, p. 450, 1888. -The theoretical deductions are due to the author (Wied. Ann. vol. 46, p. 497, 1892). 2 Von Waltenhofen, Wien. Ber. vol. 48, part 2, p. 578, 1863. [Extended series of experiments on bundles of wires have lately been undertaken by Ascoli and Lari. R. Acad. dei Lincei, Kome, 1894. H. d. B.] SHORT CYLINDER. LINES OF MAGNETISATION 35 principle laid down in 24 no longer holds, but that their de- magnetisation factor is smaller than that principle prescribes ; one or more of the assumptions on which it was deduced do not, therefore, hold. In fact, if a secondary coil is displaced along the cylinder, and the momentary current excited in it on magnetisation is investigated, we find that this does not suddenly cease when the coil is pushed beyond the ends, but becomes smaller at a certain distance from the ends. This distance is relatively more important in the case of a short cylinder than in that of a long one. We conclude from this that the induction is greater in the middle of the bar than towards the ends. This, therefore, is also the case with the magnetisation. If we examine the action at a distance due to the ends, we shall find it smaller than would correspond to the value of magnetisa- tion in the middle of the bar. At the same time the distribution of the external field is of the same nature as if an action proceeded not only from the geometrical ends, but also from the adjacent parts of the bar : as if in a certain sense there were ends there also. We are, then, thus led to generalise the idea of 'ends.' We no longer consider it as a purely geometrical conception in the sense of meaning by it the terminal faces of the bar ; we rather conceive the magnetisation of the short cylinder as if the stronger magnetic condition which exists in the middle of the bar only gradually ended, and therefore shows a great number of magnetic end-elements, each of which exerts its own elementary action at a distance in accordance with Coulomb's law. Just as a positive and a negative end were previously distin- guished, so we must now distinguish as positive and negative the end-elements, which are distributed on the corresponding halves of the bar. In this the algebraic sum of the strengths of all the end-elements is always zero, even with a body of any given shape, and magnetised in any given way. This follows, among other things, from the fact adduced in 23, that an external field of constant value and invariable direction exerts a torque on such a body and never a single force. Closely connected with this is the course of the lines of magnetisation that is, of those lines whose tangent at each point coincides in direction with the vector 3, as was established D 2 36 ELEMENTARY THEORY OF IMPERFECT MAGNETIC CIRCUITS before for the lines of intensity for the vector >. A group of such lines of magnetisation has already been sketched in fig. 7, p. 31, with their approximate course inside a short cylinder. It will be seen that many lines do not terminate in the base, but do so previously in the cylindrical surface. These terminal points represent in a tangible manner, what at first sight is an un- familiar idea, namely the end-element which we arrived at above. We have hitherto only regarded systems of lines in general, and especially of lines of magnetisation, as a means of representing the directions of the vector considered, but we shall afterwards see that in many cases they also allow conclusions to be drawn concerning the numerical value of the vector. 27. The End-elements as Centres of Action at a Distance. Hypothesis of Two Fluids. We have thus arrived at the con- ception of a magnetic body as possessed of end-elements each of which appears to act at a distance according to Coulomb's law. Such apparent centres of action at a distance preponderate on the surface of bodies, and in some cases are confined to them ; but, as we shall see, they occur in the most general case in the interior also. With the introduction of an infinite number of end-elements the problem in question passes, of course, from the region of elementary treatment to that of the higher mathe- matics. In this respect we must refer to Chapters III. and IV., although, in what follows, we discuss a few special cases, the analysis of which can only be given afterwards ; we are here concerned only with results. The conception at which we have arrived by two distinct methods is connected with a classical hypothesis, still widely diffused, to which we must devote a few words. Two magnetic fluids were assumed, a positive (north) and a negative (south), each elementary particle of which exerts an action at a distance according to Coulomb's law. These are distributed in exactly equal quantities over the magnetised body, to the greatest extent over the surface, but also in the interior. The mathematical treat- ment of these principles has led to results which are capable of exact experimental confirmation. In consequence of this, there has long been a disposition to regard the assumptions, which were the starting points, as also correct, without being restrained by the fact that those conceptions were from the outset devoid UNIFOKM FIELD. ELLIPSOID 37 of any plausible physical basis, having been developed at a time in which comparatively small weight was laid on such a thing. The theory of north and south fluids here sketched has great similarity with the conception according to which end- elements must be regarded as the essential agent for actions at a distance. The older mathematical theory of Poisson may therefore be used just as before, without essential modification, as results from the following chapters ; the adjustment of that theory to actual facts is chiefly the work of Lord Kelvin, F. Neumann, Kirchhoff, and Maxwell. We have no occasion in this book to go minutely into the various hypotheses which from time to time have been made to explain the essential nature of magnetism ; we may , however, men- tion that, before Poisson's theory, Euler put forth the hypothesis that magnetism is matter flowing in closed paths ( 111). With our present views, neither hypothesis is probable. The theory which best suits the facts, and is most capable of development, is Weber's assumption of elementary particles whose directions are controllable, as further developed chiefly by Maxwell, Wiedemann and Ewing ; l in this the pre-existing magnetism of the ele- mentary particles may be either due to vortices or to Amperian molecular currents, or as assumed by Bicharz and Chattock, to rotating ions. 28. Uniform Field. Ellipsoid. A magnetic field through- out a definite space is said to be uniform when it has everywhere the same value and the same direction ; the lines of force are then parallel and straight. It is usual, in like manner, to speak of the uniform distribution of any vectors in space ( 43). In our most recent considerations we have assumed uniform fields, as, for instance, that which a coil like the one in fig. 6, p. 27, pro- duces in the interior. Such coils are scarcely used in practice, but straight solenoids at least three times as long as the body to be magnetised ; in the space occupied by this latter in the middle of the coil the field is then sufficiently uniform for most purposes. The magnetisation induced in a uniform field is not necessarily also uniform, as we have already seen in the case of the short cylinder. It may be mathematically proved ( 69) that such 1 Wiedemann, EleUridtdt, 3rd edition, vol. 3, 1883 ; Ewing, Magnetic Induction, chap. xi. Berlin, 1892. 38 ELEMENTAEY THEOEY OF IMPEEFECT MAGNETIC CIECUITS uniformity only holds for bodies of an ellipsoidal shape l ; if one axis coincides with the direction of the field, the magnetisation is in the same direction as that axis. The demagnetising intensity for unit magnetisation we shall again speak of as the de- magnetising factor for the axial direction in question. The demagnetising intensity in the interior of the body ( 24) is in this case also uniformly distributed, as is also the resultant intensity due to this and to the original field of the coil ( 68). The ellipsoid, and the geometrical shapes directly derivable from it, then become simple types of imperfect magnetic circuits, as the closed toroid is for perfect circuits. 29. Ellipsoid of Revolution : Ovoid ; Spheroid. We shall here confine ourselves to the case of an ellipsoid of revolution, which can be turned in the lathe, and is therefore practically important. Let 2c be the length of the axis of revolution, CC r (figs. 8, 9), 2a that of the equatorial diameter at right angles to this axis. The ratio of the axes is then m = c/a. fft ' V' r / i \ f i \ * I \C C y / ! / V 7 \ ' ' Direction of magnetisation FIG. 8 FIG. 9 It is assumed that the axis of revolution is parallel to the direction of the field, so that, from what has been stated above, the magnetisation is also in this direction. We distinguish two cases according as the axis of rotation is longer or shorter than the equatorial axis A A' . A. Ovoid. Prolate ellipsoid of revolution (elongated 1 Compare also Maxwell, Treatise, 2nd edition, vol. 2, 437, 438. The formulae to be afterwards given are there deduced. FUKTHER SPECIAL CASES 39 ellipsoid of revolution) c > a, consequently m > 1. The ec- centricity of the meridian ellipse is given by the expression JV 2 , the demagnetising factor in the direction of the axis of revolution, is given as a function of the eccentricity by the equation (10, . ._*(-.) & or, if we take the ratio of the axes m as the argument, If m in this expression has greater values, and we have to deal with more elongated ellipsoids, it approaches (as is at once seen if 1 is negligible as compared with m 2 ) the simpler form (12) . . N = i^L(lg 2 m - 1) m which, with ratios of the axes exceeding 50, gives values correct to within a few thousandths, and with still more elongated ovoids, almost absolutely correct values for N z . B. Spheroid (oblate ellipsoid of revolution) c < &, consequently m 40 38-7 0-0238 0-0266 42-5 50 40-5 0-0162 0-0181 45-3 60 42-4 0-0118 0-0132 47-5 70 43-7 0-0089 0-0101 49-5 80 44-4 00069 0-0080 51-2 90 44-8 0-0055 0-0065 52-5 100 45-0 0-0045 0-0054 54-0 t 150 45-0 0-0020 0-0026 58-3 , 200 45-0 0-0011 0-0016 64-0 , 300 45-0 0-00050 000075 67-5 , 400 45-0 0-00028 0-00045 72-0 , 500 450 0-00018 0-00030 75-0 , 1,000 45-0 0-00005 0-00008 80-0 00 Endless 32. Graphical Representation. It follows from Table I. that the demagnetising factor increases from to 4?r (= 12-5664) if, starting from an endless figure (whether an infinitely long 1 Compare W. Weber, Eleldrodyn. Maassbestimmungen, vol. 3, p. 573, 1867 ; Kirchkoff, Ges. AM. p. 221; Oberbeck, Fogg. Ann. vol. 135, p. 84, 1868. [Recent unpublished experiments, however, render it probable that the values of Nva the table are correct for wire bundles, but rather too small for iron bars. H. d. B.I 42 ELEMENTAEY THEOEY OF IMPERFECT MAGNETIC CIRCUITS one or a closed annular one), we gradually shorten it until we have ultimately a thin plate, in which the influence of the ends that is, of the bounding surfaces on the two sides manifestly attains its greatest possible value. We may apply this to the curves of magnetisation, repre- senting the influence of the demagnetising factor by the directrix A = N% [equation (1), 17], and performing the shearing process explained above. It is then manifest how, as the values of N increase, the directrix will be continually more inclined to the left from the axis of ordinates. The curve of magnetisation will therefore appear more displaced towards the right, the greater N is that is, the shorter the ellipsoids, or the greater the gap in the rings. It is advisable to perform this construction for a concrete case, for a wrought-iron ellipsoid, for instance, and not on too small a scale. It will then be seen that, as the ellipsoid is shortened, the curve gradually alters its shape, and soon seems closely embraced by two straight lines. The first passes through the origin, and makes with the axis of ordinates the same angle towards the right as the directrix does towards the left. The second straight line proceeds parallel to the axis of abscissae at the distance 3 m , where 3m is the maximum magnetisation. We see now how the first straight line is determined by the value of N alone that is, by the shape and the second by the nature of the material alone, but the point of intersection by both these factors. This holds also almost exactly for those parts of the curve which lie close to each straight line ; by this property the curve will be at once held for an hyperbola. 33. Hyperbolic Curves of Magnetisation. For shorter ellipsoids, or for rings with wider gaps, in short for greater values of the factor N, the curve of magnetisation indeed differs but little from a hyperbola, the asymptotes of which are the straight line in question, and the equation of which may be most simply written /i Q\ AT P Ny + (P - Nf) (18) . x Ny + , or x = y -/. l-y ly Here P denotes a second constant ; the maximum magnetisa- 1 Compare fig. 21, p. 131, which represents the influence of cuts of in- creasing width on the course of the curves of magnetisation. HYPEKBOLIC CUKVE OF MAGNETISATION 43 tion is the unit of length ; the equation may be simply deduced by the methods of analytical geometry. It must be kept in mind that this is only an approximation to the true form of the curve. For the inclined part can never be the branch of an hyperbola, because it has a point of inflexion, though not very marked, and it must pass through the origin. As regards the horizontal branch, according to certain optical ex- periments, the law of its approach to the asymptote is, in fact, hyperbolic, at any rate in very intense fields. 1 This curve is not to be confounded with 0. Frolich's purely empirical curve of c effective magnetism/ to which we shall return in the eighth chapter. We see thus how, as the demagnetising factor increases, the curves of magnetisation for ellipsoids or divided rings gradually recede from the axis of ordinates ; at the same time the second asymptote parallel to the axis of abscissas always remains the same. The limiting curves are, on the one hand, the normal curve (N = 0), and that for a thin plate, magnetised transversely, on the other (N = 4 TT). The former characterises the material and is of cardinal importance ; the latter demands a somewhat separate position, since by its means various physical properties ( 10) have become accessible to quantitative reasoning. The whole of the curves obtained with bodies of different shapes of a given ferromagnetic substance must lie between these two limiting curves, and this holds not only for the two shapes, ellipsoids of revolution, and divided rings, but can be extended to any shape whatever in which the material is formed. The magnetisation is thus indeed no longer uniform, but what has been said applies to its mean component as before. The mean demagnetising factor for each shape may be empirically con- structed from the difference of abscissas of the corresponding curve found empirically, and the normal curve (compare Chap- ter V.) 1 This appears to follow from optical measurements by the author (Phil. Mag. vol. 29, p. 302, 1890), even if it cannot in general be considered established with certainty. 44 CHAPTER III OUTLINES OF THE THEORY OF RIGID MAGNETS A. Geometrical theory of Vector-distribution 34. Vector-distributions. In the beginning of the first chapter it seemed appropriate to premise some elementary con- siderations respecting quaternions ; here also we must deal, as far as necessary, with geometrical considerations of a general character. Their introduction will produce greater clearness, in treating what is to follow, than would be attainable by the use of purely analytical methods. 1 We will first agree upon the conventions as to the notation used. 2 The quaternion expressions scalar and vector we have already defined ( 3) ; we shall denote a vector magnitude in general without reference to its special nature by ^. In order to prevent the excessive accumulation of symbols to be used, suffixes will be added, such as a?, y, z, v, r, in order to indicate the components of a vector in the directions of the .X, Y, Z axes, and its com- ponents, normal and tangential, to given surfaces. The former is geometrically speaking the projection of the vector on the normal to the surface, the latter that on the tangent plane to the same at the point in question. The normal to a surface we shall generally denote by 01 ; it has to be determined in each case which of its two directions shall be positive. The angle between two vectors is expressed by enclosing them in brackets and separat- ing them by a comma. S will stand for areas of surfaces, and 1 In the following Chapters III. IV. and V. we must abandon the use of elementary methods. 2 We may here refer to two chapters of a general character in the first volume of Maxwell's Treatise ; the introduction and chapter iv. SURFACE-INTEGRALS AND THEIR PROPERTIES 45 L for lengths of curves, d 8 and d L being the corresponding elements. The meaning of this notation is sufficiently shown by the following expressions : ff,*9oaB(9,ft) and 8f y =8fsin(8?,0l) which at the same time indicate that the numerical value of a vector-component is obtained by multiplying that of the vector itself into the cosine of the angle between the two directions (3). If we now consider a limited, or an infinitely extended region, within which a vector-quantity has a finite value, this may be called the field of the vector. Its direction, and its numerical value, will in general vary continuously from point to point ; the existence of surfaces of discontinuity is not, however, excluded. The region may be either simply- or multiply-connected ; the former is always assumed unless the contrary is expressly mentioned. The manner in which the direction and value of the vector at a point are related to the position of the point is called the geometrical distribution of the vector in the region to be investigated. There are several different modes of dis- tribution characteristic of the vector quantities occurring in nature. Each of these is conditioned by definite relations the analytical expression of which involves the derived functions (differential coefficients) of the vector-components with respect to the co-ordinates. It will be our immediate object to investigate these more thoroughly ; but we have first to prove an important general principle. 35. Surface-Integrals and their Properties. The double integral S = ftcos (ff, W)d8 = $ v d 8 taken over 8, is called the surface-integral of the vector ft over this portion 8. It is obtained by multiplying each surface element into the component of the vector perpendicular to it, and integrating this product over the entire surface. Let us specially consider the surface-integral over a closed surface 8'. 46 OUTLINES OF THE THEORY OF RIGID MAGNETS Within the space enclosed, let ft,,, ft,,, ft 2 , be continuous and finite, except at a surface of discontinuity, F (x, ?/, z) where these vector components experience an abrupt change of value ; let their values on one side of the surface be simply denoted by ft z , ft y and ft 2 , and those on the other side by ft*, ft^, and ;. Let the positive direction of the normal $l s to the closed surface be always that which is drawn inwards ; we shall for shortness denote its direction cosines by I. = cos (01,, JX), m. = cos (0l t , Y), n. = cos (01., Z) In like manner we write the direction cosines of the normal to F l f = cos (0t /5 X), m f = cos (0t/, Y), n/ = cos (0*,, Z) If now we draw a straight line parallel to the X axis, this in general must cut the surface S' in an even number of points ; first assume that there are only two, and that between these the straight line cuts the surface of discontinuity F. Following this line in the positive direction, it will first enter the surface S' at any point x=x^ at which ft* = ft^, and l.dS' = dydz it will then cut the surface F, where ft* changes abruptly to the value ft* ; further, l f dF = dydz the straight line finally will emerge from S' at somepoint=a? 2 , where ft* = ft* 2 and l t d S' = dy dz We have then, as is known, (1) . . &, - ft^ = J |^ dx + (K - ft,) We are now in a position to calculate our surface integral ; for from the well-known equation cos (ft, 01) = cos ( ft, X) cos (01, X) + cos (ft, Y) cos (01, Y) + cos (ft, Z) cos (01, Z) SUKFACE-INTEGKALS AND THEIK PEOPEKTIES 47 it follows that (2) . JJff cos (,.)(' = in which all the double integrals are to be taken over $'. Let us now consider one of the members on the right, the first, for example. We observe that, in accordance with the above, dy If we introduce this, together with the similar expressions for the two other members, we obtain finally from (2) (3) - &) l f + (8-, - $;) m,+ (8-,- K n dF the expression _(!& + lj*? +3* \ 9 a? By 3z we may call, with Maxwell, the convergence of the vector at the point considered. If now we state equation (3) in words we arrive at the following fundamental theorem. I. The surface-integral of a vector over a given closed sur- face is, apart from discontinuities, equal to the volume-integral of the convergence of the vector over the whole region enclosed. If there is a surface of discontinuity, a term must be added which on closer consideration may be described as follows : it is the surface-integral of the difference of the normal components of the vector on the two sides of the surface of discontinuity integrated over that portion of the latter which is enclosed by S'. Our straight line parallel to the axis of X may moreover cut the surface S' in more than two points, and any number of sur- faces of discontinuity may occur without the proof being essentially different. We have reproduced the general proof simplified as 48 OUTLINES OF THE THEORY OF BIGID MAGNETS much as possible, for the development here given can scarcely be considered to be as well known as the principle itself. 1 36. Complex Solenoidal Distribution. A method suitable for exhibiting the distribution of a vector in space consists in supposing vector lines to be introduced that is, curves at each point of which the tangent line is coincident in direction with the vector. We have already used this method more than once, having introduced lines of intensity ( 4) and lines of magne- tisation ( 26). Change of direction along such curves will generally be continuous, but at surfaces of discontinuity they may be abruptly bent, or may terminate. We may further consider the field as divided into wider or narrower tubes fitting closely against each other, the surfaces of which have the curves in question for their generating lines. These figures, which we shall call vector-tubes, will then extend over the whole of the region in question, their direction and their sectional area being in general variable from point to point. These variations may in general be quite unrestricted ; such a vector-distribution of the most general kind may be called a complex solenoidal one 2 (from crco\r}v, tube). In dealing with the physical vector quantities which occur in nature we meet with modes of distribution in which the variations of the cross-section, and of the direction of the vector-tubes, are restricted by special relations, to the consideration of which we will now turn our attention. 37. Solenoidal Distribution. There is in the first place an important kind of distribution in which the surface-integral of the vector over any given closed surface is zero. A glance at equation (3) shows that this is equivalent to two other necessary 1 For further mathematical details it will be sufficient to refer to Maxwell (Treatise, vol. 1, 21, Theorem III), which states the theorem, together with the proof in the form given, and names, as the discoverer in 1828, the Kussian mathematician Ostrogradsky (Mem. de VAcad. de St.-Petersbourg, vol. 1, p. 39, 1831). This important theorem is connected, moreover, with the equation of continuity, as we shall presently see, and may be regarded as a special case of Green's more general theorem which was made known in the same year (Green, Essay on the Application of Mathematics to Electricity and Magnetism, Nottingham, 1828). 2 Sir W. Thomson, Reprint of Papers on Electricity and Magnetism, 509 ; from which source most of what immediately follows is taken. SOLENOIDAL DISTRIBUTION 49 and sufficient conditions. In the first place, throughout the space enclosed by the surface we must have (4 ) ... 3fc + 3. + 8. = o ox a y 9 z that is, the convergence of the vector at each point must vanish. In the second place, at all surfaces of discontinuity or, what is the same thing, (5) . . . . &, = The former equation (4) may be called the volume-equation of continuity, since in hydrodynamics, and in the theories of diffusion and of thermal and electrical conduction, it expresses the condition for the continuity of the corresponding components of flow (Chapter VII.) In like manner equation (5) is to be regarded as the boundary-equation of continuity. We shall now suppose our integration to extend over a sur- face 8 f of special form, choosing for this purpose the surface of a finite vector-tube, whose ends are closed by surfaces 8 l and S 2 (fig. 10). The in- termediate portion of the surface enclosed by the vec- tor lines we shall call S ; its share in the value of the sur- face-integral S is obviously FlG , 10 zero, since that surface is every where tangential to the vector, and the latter has consequently no component perpendicular to the surface. The whole surface- integral is then seen to be the sum of portions furnished by the end-surfaces, which we designate by IL andjj^. But now, from the assumption at the beginning of this paragraph, but since J f = 0, as explained above, 50 OUTLINES OF THE THEOKY OF KIG-ID MAONETS As we previously assumed that the direction of the normal drawn inwards was positive, the sign of the term on the right side is to be reversed if now, on the contrary, we consider the direc- tion of the vector as positive at both surfaces 8^ and $ 2 (fig. 10). In the mode of distribution here considered, the whole region may be divided up into vector-tubes, which have the following property : II. The surface-integral of the vector is the same over every cross-section of the tube. Such a vector-tube is called a simple solenoid, and the cor- responding distribution a solenoidal one. Let us now consider an infinitely thin vector-tube, so that the value of the vector does not vary appreciably over the cross- section, and draw the surfaces 6' 1 and $ 2 at right angles to the direction of the tube. If then the values of the vector at the two terminal surfaces are $ } and 3f a , we may write the two portions of the surface-integrals as follows : The product (^ 8) may be called the strength of the infinitely thin vector-tube. The contents of this paragraph may then be expressed in the following proposition : III. When a vector is distributed solenoidally, the field may be divided up into infinitely thin solenoids of constant strength. Equations (4) and (5) constitute the necessary and sufficient conditions for this mode of distribution. In a thin vector-tube of constant strength (8r 8), the numeri- cal value of the vector is obviously inversely proportional to the normal cross-section. A solenoid, the strength of which is everywhere unity, may be called a unit-tube \ its section is every- where numerically equal to the reciprocal of the vector ; hence the greater the value of the latter, the more unit-tubes will be cut by a given surface. The ' density ' with which the unit- tubes are crowded together in space furnishes a direct measure of the value of the vector, since the number which falls on a normal surface of unit area is numerically equal to the mean value of the vector over that surface. 38. Complex Lamellar Distribution. As regards the varia- tion of the direction of the vector from point to point, we premise that in general it is not possible to construct a system LAMELLAR DISTRIBUTION 51 of surfaces such that at each point the vector is perpendicular to the surface through that point ; that is to say, that the pencil of vector lines is throughout orthogonal to the system of surfaces. The necessary and sufficient condition for the existence of such an orthogonal system of surfaces is expressed by the well-known equation (6) . (^ - :) + %, (^ - 8*-) +R ( 9 ^- 3 />-) = \oz dyJ \ox ozj \dy dx/ which is also the condition that the equation & dx + 8f, dy + & dz = may be integrable. 1 The terms on the left may be transformed by a scalar in- tegrating divisor / (x, y, z) into an exact differential, the integral of which (B) (x, y, z), equated to a number of constant parameters j, 2) 3> represents the system of surfaces in question that, namely, which is orthogonal to the pencil of vector lines. The surfaces corresponding to successive values of the para- meter enclose shell-like spaces, into which the region in question may be divided ; such figures are said to be complex shells. They have only the geometrical property that at every point the vector meets them at right angles ; its value has generally no connection with their thickness. Distributions of the kind here considered are said to be complex lamellar. 39. Lamellar Distribution. If the above-mentioned in- tegrating divisor / (a?, y, z) is equal to unity -in other words, if the expression ~ , ~ , ~ , 8f dx + $ v dy + $ z dz is already an exact differential the distribution belongs to an important special group. The necessary and sufficient couditions for the immediate integrability of the above expression are given by the equations (7) 9^ = 9ft a_& = cMk 38^ as, dz dy' dx " 9 z 3 dy dx which in hydrodynamics are the equations characteristic of irrotational motion. The integral of the exact differential raken with reversed sign is called the scalar potential of the >ctor; we shall denote it by <. 1 Schlomilch's Handbuch der Mathematik, vol. 2,p.B71 t et seqq. :reslau, 1881. [Forsyth, Differential Equations, 151, 152.] 52 OUTLINES OF THE THEOEY OF EIGID MAGNETS We have accordingly (8) . . d(-3>) =8* and + d , and if the portion of the normal intercepted between them that is, the thickness of the shell is called d 91, then throughout the entire shell $ d 01 = d = constant The thickness at any point is inversely proportional to the value of the vector, since the product of the two quantities, which is called the strength of the shell, is everywhere constant. Such a shell is called a simple shell, and the corresponding vector-distribution a lamellar one. We may express these con- siderations by the following theorem : IY. When a vector is distributed lamellarly, its field is divided into infinitely thin shells of constant strength. This lamellar character is bound up with the equations (7), which at the same time condition the existence of a scalar potential. In a thin vector-shell of constant strength (g- d\ the numeri- cal value of the vector is inversely proportional to the variable thickness d. A shell whose strength is constant and equal to unity may be called a unit-shell ; its thickness is everywhere numerically equal to the reciprocal of the vector; hence the greater the value of the latter, the more unit-shells will be in- LINE-INTEGRALS AND THEIR PROPERTIES 53 tercepted on a given length at right angles to the equipotential surfaces. The c density ' with which unit-shells succeed each other is a direct measure of the value of the vector, as the number intercepted on a line of unit length, drawn perpendicu- larly to the equipotential surfaces, is numerically equal to the mean value of the vector over that length. 40. Line-Integrals and their Properties. The definite integral B L = ( B ft cos (& L) dL = taken on the curve L between two points A and B is called the line-integral of the vector 8f, along the length A B. It is obtained by multiplying each element of the curve into the tan- gential component of the vector, and integrating this product over the whole length. If we consider A as the starting-point, and define B (whose co-ordinates we may call a?, y, z) by its distance L from the point A, as measured along the path of integration, then, con- sidering the equation cos ($,) = cos ( ff, X) cos ( L, X) + cos ( ff, Y) cos (L, Y) + cos ( ft, Z) cos (L, Z) we may write (10) B L -j: The value of this expression differs, in general, with the path of integration by which we pass from A to B. If, how- ever, the distribution of the vector is everywhere lamellar, so that &da> + 9 9 dy + 9,de = d (- 3>) it follows from equation (10) that under all circumstances dL That is, expressed in words : V. When the distribution of a vector is lamellar, its line- integral is equal to the difference of the potentials at the two 54 OUTLINES OF THE THEOEY OF KIGID MAGNETS terminal points, independently of the path of integration between those points. It follows directly from this that in the case of a lamellar distribution the line-integral taken round any closed curve must vanish. For imagine any two given points on the curve, the portions of the integral corresponding to the two paths between them are equal and opposite, and accordingly their sum is zero. In these investigations it is tacitly assumed that the region considered is a simply -connected one. If the distribution in a multiply-connected region is lamellar, while at external points the vector does not possess this property, the potential becomes, in general, a multiple-valued function of the co-ordinates, and the theorems mentioned only hold under certain restrictions. This is not the place to pursue any further general investiga- tions, whose interest would be chiefly theoretical. 1 We will, however, mention one special case, which we shall often meet with in the sequel. It is that of an annular space in which a vector has a lamellar distribution. The line-integral of the vector is not zero around a closed curve, which makes a complete circuit of the annular space, but with the completion of each circuit the integral increases by a constant, which is independent of the position of the curve within the ring so long as no part lies in the external space in which the distribution is supposed to be no longer lamellar. 41. Lamellar-solenoidal Distribution. A vector can be dis- tributed both lamellarly and solenoidally, provided it satisfies the two necessary conditions. The first property implies the exist- ence of a potential ; that is, in accordance with the foregoing, If we insert these values in the equation of continuity [37, equation (4)] 3 which is the condition that the vector may possess the second property, we obtain 1 See Helmholtz, Crelle's Journal, vol. 55, p. 25, 1858 ; Wiss. Abhandl. vol. 1, p. 101 ; Maxwell, Treatise, introduction ; Lejeune-Dirichlet, Vorlesungen ub&r COMPLEX LAMELLAK-SOLENOIDAL DISTRIBUTION 55 Accordingly, when the vector $ is distributed in this lamellar-solenoidal manner, the potential must satisfy the last- written equation, which expresses the necessary and sufficient condition for such a distribution. This equation, which is of great importance in all branches of physics, is often written for shortness y 2 = 0, and is called Laplace's equation. Maxwell also applies the name of * Laplacian distribution ' to such a distribution of the vector $ as satisfies equation (12). At surfaces where discontinuity arises, the normal com- ponent of the vector must preserve its continuity as we pass from one side of the surface to the other, just as in the case of a simple solenoidal distribution. Since the distribution is solenoidal, the cross-section of the vector-tube is at each point inversely proportional to the value of the vector ; and since the distribution is also lamellar, the same holds true for the distance between consecutive equi- potential surfaces. The system of equipotential surfaces, together with the orthogonal system of vector-tubes, divide the field into ' cells,' whose volume is inversely proportional to the square of the vector. 42. Complex Lamellar-solenoidal Distribution. We have already seen that, in the case of a complex lamellar distribution, the expression ftxdoo 4- $ y dy + $ z dz can be rendered integrable by a scalar integrating divisor; that is, it can thus be made an exact differential d . We must therefore have ^_8_e %, = 8 & = 'dx dy "dz and this is the condition for a solenoidal distribution of the vector [ 37, equation (4)]. It can also be shown that < satisfies the superficial equation of continuity at the surface separating the conductor from the surrounding medium. 45. Magnetic Potential in the Field outside Conductors. In close connection with the equations of the last paragraph we have the following : (16) dz 8 x dy which also hold good for all points throughout the field. If the point considered lies within the conductor, where & has a finite value, these equations show that the distribution of cannot there be lamellar, since a potential does not exist. But outside the conductor, where there is no current, and where, consequently, ( is zero, the terms on the left will vanish, and the terms on the right then express the condition that the vector $ may be distributed lamellarly (the condition, that is, MAGNETIC POTENTIAL IN AN OUTSIDE FIELD 59 that its components may be the space-derivatives of a potential). We have, in fact, _ 82 9 ?/ ' 9 a? 9 z ' 9 1/ 9 a? [Compare 39, equation (7).] We accordingly arrive at the following theorem : VIII. In the region surrounding a conductor (not within the substance of the conductor itself) the magnetic intensity is dis- tributed lamellarly^ and has consequently a scalar potential. This potential of the magnetic intensity we shall denote by T, and we shall call it the magnetic potential. But the region where 4? is lamellar is now no longer simply connected, since it is interrupted by a non-lamellar region at least doubly con- nected, which includes every separate closed conducting circuit round which a current is flowing. Thus the present case differs from that previously discussed ( 40). In fact, the line-integral of 4? taken along any closed curve increases by a ' cyclic constant of integration,' 0, for each time that the closed curve embraces the conductor, and it is only along such closed curves as do not embrace the conductor that the line-integral vanishes. In using the term ' cyclic constants ' we must, of course, understand that these can only depend on the electric currents embraced by the path along which the line-integral is taken, and not on any purely geometrical relations. We can now determine d priori the value of G by observing that its relation to the current 1 must be one of proportionality, since the same is known from experiment to be true of 4?. The completion of the value of G can involve nothing further than the introduction of a numerical factor, and to this again, as in a former case ( 11), we assign the value 4 vr, 1 following the historical development of the sub- ject, and in conformity with the system of electromagnetic units in general use. Hence, finally, (17) . . . C = 47rl. 1 The introduction of the factor 4ir is condemned by several authors, especially by Heaviside. Its elimination, however, could only be effected by remodelling the system of units already adopted, so that this comparatively unimportant simplification would be rather dearly bought; moreover, the factor would probably reappear in another place. 60 OUTLINES OF THE THEOKY OF KIGJD MAGNETS We can easily verify this equation from one of the elementary examples already considered ; we shall choose the case of a long straight linear conductor, in the neighbourhood of which the electromagnetic action follows the law of Biot and Savart. The magnetic intensity 4? at a distance r from the conductor is [ 5, equation (3)1 The line of force through the point considered is a circle of cir- cumference 2 TT r ; and consequently the value of the line-integral of $ along a circuit embracing the conductor once (that is, the value of the cyclic constant C which we have to determine) is 27T7- f 9 T' 9 c = ( - 7T ~ a conclusion identical with (17). We can combine these results in the following fundamental theorem : IX. Every time the circuit along which we integrate embraces a conductor conveying a current J, the line-integral of magnetic intensity increases by the cyclic constant 4 TT J, independently of any other variables whatever. This important general law is abundantly confirmed by ex- periment, as in the present case is almost self-evident. 46. Action of a Kigid Magnet at External Points. We have already ( 26) introduced the conception of magnetic end-elements, referring to the present chapter for a fuller mathematical development of the method. We have also given [in 19, equation (2)] a statement of the elementary law of Coulomb, that the force apparently exerted by a magnetic pole, at any point in its neighbourhood, is given by * = -^ where <$ is the magnetic intensity at the distance r from the end, and is directed away from the end when the end is positive, towards the end in the contrary case. 8 is the cross-sectional area of the bar-magnet, and 3 the vector which we have called the magnetisation ( 11). ACTION OF A RIGID MAGNET AT EXTERNAL POINTS 61 Let us consider now, instead of a simple bar-magnet, a ferro- magnetic body of any shape, magnetised in any arbitrary manner without reference to the cause of such magnetisation. When we wish to express this total independence of the magnetis- ation on external causes, we shall speak of such a body as a rigid magnet ; the phenomenon of magnetic retentiveness shows us the possibility of realising such a rigidly magnetised body with some degree of approximation. Within the region occupied by the body, therefore, we are to consider the distribution of the vector 5 to be entirely arbitrary (and in general, consequently, com- plex-solenoidal, 36) ; at the bounding surface the component along the normal drawn inwards is 3 M while along the normal drawn outwards into the magnetically indifferent surrounding medium, the component of magnetisa- tion is zero. To calcu- late the magnetic intensity at external points, we may divide the body up into elementary paral- lelopipeds dx dy dz, write down the expression for the effect due to the three pairs of opposite faces of an infinitely small paral- lelepiped, and integrate this expression throughout the whole volume of the body. Let us consider the parallelepiped cfe dy dz as a short bar parallel to the axis of z ; then the component of magnetisation in a direction parallel to this axis becomes + 3 Z . Since the magnetisation is a vector, we make what use we please of the principle of resolution into components. Let us 'now fix our attention on the upper face of the parallelepiped (shaded in fig. 11), its co-ordinates being , y, z, and the area of the face in question doody; its magnetic strength, which ^determines its influence at external points, is therefore -f- 3 2 cfe<%, by the definition already given ( 19). At a point P (f , 77, ?), whose distance from (x, y, z) is r, this FIG. 11 62 OUTLINES OF THE THEOEY OF RIGID MAGNETS end-element will exert the magnetic intensity < given by the equation and the components of will have the values x) * ._3,& * - 8 * ~ so a s< " and "aV ay DISTKIBUTION OF MAGNETIC INTENSITY 63 The equations (7) of 39 are consequently fulfilled, so that everywhere without exception throughout the whole field the distribution of <% is lamellar ; even at the place where the end-element itself is situated. In the next place, let us form the derivatives of the com- ponents of magnetic force with respect to the co-ordinates of the point P, by differentiating the expressions given in 46 with respect to f , 77, f ; thus we find 9fe 3, (to dr* - 3 - g d% y _3 z dxdy [r 2 -3 (77 - y)] 8*7 r 5 while for the sum of the right-hand terms of these equations we have the expression ,,, i. 3 r* - 8 (g - *) - 3 (, - y) - 3 (f - ,) Eeferring to (18), this expression is seen to be zero for finite values of r, while for infinitely small values of r it assumes the form 0/0, and moreover it can be shown that in the latter case the expression does not vanish. Thus we have that is, the distribution of is solenoidal throughout the field, except in the place where the end-element itself is situated. In accordance with the law of superposition VII ( 43), we can now extend by summation to any number of elements the property just proved for a single end-element. Thus we arrive at the following theorem : X. The magnetic intensity due to a rigid magnet arbitrarily magnetised lias everywhere a lamellar-solenoidal distribution, except in those places where end-elements are situated; there the dis- tribution is only lamellar. 64 OUTLINES OF THE THEOKY OF RIGID MAGNETS 48. Potential of a Rigid Magnet. In accordance with what we have just seen, the magnetic intensity due to a rigid magnet has everywhere a scalar potential, which satisfies Laplace's equation at every point, except in those places mentioned in Theorem X. This function is called the magnetic potential of the rigid magnet, and, as before ( 45), we shall denote it byT. Let us now turn our attention once more to the magnetic field due to the elementary parallelepiped. Let the face which has hitherto been considered (that which is shaded in fig. 11, p. 61) be distinguished by the number 1, the face immediately opposed to it by 4 ; and in the same way let the two remaining pairs of opposite faces be numbered 2 and 5, 3 and 6, respec- tively. We shall denote by 8 T\ the magnetic potential at the point P, due to the face 1, and from the general theory of potential it immediately follows that y dr= + Consider next the elementary face 4 ; let its distance from P be r + drj its strength is % z dx dy. It produces at P a magnetic potential S T 4 , which is found exactly as in the case of 1 ; we have ~ 3 Z dx dy " r + dr If we add together the two terms S T\ and 8 T 4 of the poten- tial, we obtain the magnetic potential due to the pair of faces (1 4), which we denote by 8 T r4 , and which has accordingly the following value : T,. 4 = 8T I + ST 4 .3,d( or, as it may also be written, s ^ cy 7 j j 1 , 3L dx dy dr oT 1<4 = 3 z dx dy d- = + - ^~ On comparing this expression with that for 8 T\ or S T 4 , it will be seen that the magnetic potential of a pair of opposite end-elements is an infinitesimal of the third order, while that POTENTIAL OF A KICKED MAGNET 65 of a single face belongs only to the second order of small quantities. When we pass from the face 1 to the face 4 (compare fig. 11, p. 61), the ^-co-ordinate increases by , 3, = K (ft) ti< z The fundamental relation connecting the three vectors SB, ti, and 3 [ 51, equation (27)] becomes in the present case (4) . . . 33* = tit + 4 TT 3 From this it follows that the resultant induction 93< must be in the same direction as its two terms ti t and 4 TT 3, which are themselves coincident in direction. To express this formally, we must introduce the permeability p (ft), that is, the ratio of the induction to the magnetic intensity ( 14). Thus we obtain And since 23 t and tit are in the same direction, this is equiva- lent to (5) SB, = /* (ft) ft,, 33, = fj, (tit) ft*, 33. = /* (tit) tit, 55. Line-integral of the Demagnetising Intensity. That part of the magnetic intensity which arises from the magneti- sation of the body itself we shall denote by ft in the space out- side the body ; in accordance with 47, its distribution is therer lamellar and solenoidal. When we are dealing with the interior of the body, we shall further distinguish this quantity by adding an accent, til being called the demagnetising intensity, since in general its tendency is to oppose the external field ft, and so to diminish the magnetisation of the body. Now, within the sub- LINE-INTEGBAL OF DEMAGNETISING INTENSITY 77 stance of the body the distribution of <*, though still lamellar, is no longer solenoidal, since the convergence of the magnetisa- tion has, in general, a finite value, which we have shown ( 50) to be equal to 4 TT times the convergence of the vector J. Now, since & (or &, as the case may be) has everywhere a lamellar distribution, the magnetic intensity due to the magnet- isation of the body must always be derivable from a magnetic potential T t , and in accordance with what we have already proved ( 40), the line-integral of & or K taken round any closed curve must vanish. Let us consider more particularly such a closed path of inte- gration, of which part lies within the ferromagnetic substance, and part in the region outside, the curve being traversed in the di- rection indicated by the arrow (fig. 12). For the portion E A of the path which lies within the ferromagnetic sub- stance, the line-integral of the demag- netising intensity is - IL a J-> ; E where $'& denotes, as usual, the component of the vector in question in the direction of the tangent to the curve _L, so that #; L = & cos ($, L) On the other hand, the portion AE of the path which lies without the ferromagnetic substance furnishes the line-integral A which is equal to the increment experienced by the magnetic potential due to magnetisation as we pass from the point where the path of integration leaves the body to that where it enters again. Now, we have already seen that l A A {% iL dL = 1 Since, in integrating along a closed curve, we have the initial and final extremities coincident, we shall in the sequel denote such an integral by tl symbol A \ 78 OUTLINES OF THE THEORY OF MAGNETIC INDUCTION so that (6) . This equation shows that I. Between any two points, E and A, of the bounding- surf ace of a ferromagnetic body, the difference of value of the magnetic potential due to its own magnetisation is 7iumerically equal to the line-integral of demagnetising intensity, reckoned along a path from E to A, which lies within the body. 1 The applicability of this equation will become apparent later on, when we have to effect the solution of special problems. It will be seen that, in virtue of Theorem I, the difference of magnetic potential JT i5 which can be calculated or measured in the accessible external medium, may be made to furnish a basis for calculating the demagnetising influence within the ferromagnetic body. It is easy to see that the proposition may also be extended to the case where two or more separate portions of the path of integration lie in the surrounding medium, so that the curve in question cuts the bounding surface in more than two points, though, of course, always in an even number. If the points where the curve passes out of the ferromagnetic body, taken in order, be denoted by A l , A v .... A n ,fhe points where it enters the body being called _E/ n E 2 , .... E n , we shall have, instead of equation (6), the following, which may be proved analytically in -the same manner : _ EI T - E T - - E "T Ai i A z L i ..... A n x t //2 "\ d L + ____ + E, E 2 E n By sketching a figure to illustrate this more general case, the reader may immediately satisfy himself of the truth of equa- tion (6a). Thus, we see that : I. A. If any circuit be drawn, of which some portions lie within the ferromagnetic body and some portions in the surrounding medium, the sum of the increments of the body's own magnetic potential corresponding to the latter portions is numerically equal 1 This theorem was given by the author in Wied. Ann., vol. 46, p. 489, 1892. Throughout the demonstration, however, ^ was wrongly printed in the place of 4?i- PEOPEETIES OF EESULTANT MAGNETIC INTENSITY 79 to the siim of the line-integrals of demagnetising intensity reckoned along the former portions. 56. Properties of Resultant Magnetic Intensity. In accor- dance with the law of superposition (VII, 43), &, being the sum of two lamellarly distributed vectors & and <, has itself a distribution which is everywhere lamellar, though not neces- sarily solenoidal. Again, the line-integral of the resultant magnetic intensity, reckoned along a closed curve (indicated by i/) 3 becomes A ,dL + = - in the case where the curve embraces q times a conductor con- veying a current I. The expression for the magnetic potential T< due to the body itself has already been compared with that for the gravi- tation-potential, and from this it follows that at surfaces of dis- continuity the magnetic potential must remain continuous, since this property is known to appertain to the gravitation-potential. Accordingly, at any point of the surface which separates the ferromagnetic from the external medium, the space-derivative of T 4 in any direction lying in the tangent-plane must have the same value within and without the surface ; that is, On the other hand, we have already shown ( 52) that, as in the case of gravitation-potential, so also in our case, the space- derivative along the normal at a point of the bounding-surface, experiences an abrupt change of value, equal to 4 TT 3,,. Thus we have $ = $ + 4 IT 3, And, further, since there is nothing which could cause the vector % e to exhibit any special peculiarities at the bounding- surface of the body, the vector-sum & = <> + %i must there show the same discontinuity as its second term alone. Thus, (7) . . . & T = &r While, on the other hand, (8) ... &> = ; + 4 ir 3, 80 OUTLINES OF THE THEORY OF MAGNETIC INDUCTION From equations (7) and (8) we see that : II. The tangential component of the resultant magnetic intensity is continuous at the bounding-surface of the body; while the normal component is discontinuous at the surface wherever its value (and consequently also that of the normal component of magnetisation) is finite. 1 57. Properties of Magnetisation, In accordance with our first proposition, the vector 3 is at each point coincident in direction with ft ; so that the lines of magnetisation everywhere coincide with the lines of resultant magnetic intensity. Since the last-mentioned vector is lamellarly distributed, and is derivable therefore from a potential T M the corresponding lines of magnetic intensity cut the system of equipotential surfaces (T f = const.) orthogonally ; and the numerical value of ft is inversely propor- tional to the distance between*two consecutive equipotential sur- faces ( 39). Evidently, therefore, the lines of magnetisation must cut the same system of equipotential surfaces orthogonally, though in general the same relation does not hold good between the numerical value of this vector and the distance between neighbouring surfaces of the system. The existence of a system of surfaces, which are everywhere orthogonal to the direction of the magnetisation, shows that the distribution of this vector must at least be complex-lamellar ( 38). This can also be proved analytically by taking as our starting-point the relation pre- viously established 3 = * ft, ft and substituting this value in the equation which conditions the complex-lamellar distribution of 3 [ 38, equation (6)]. Thus we obtain $* 3* 3ft. 8 "" ~ , * . ^ 8^ ' fc ffS ~ TiT " " ^ cv /a 3, 93A ^ ^9^ + fi 8^ 8^ O Z I -rS - - "TV I - K 'Vtz [ K ~r\ - T Vte 5 - - K -r\~ - e and < become indefinitely great ; it is evident, that this case cannot be completely realised in practice, though we may approximate to it more or less closely. In accordance with the theorem of Kirchhoff already given (III, 57), the distribution of magne- tisation is, in the limiting case, neither lamellar nor solenoidal, for it tends everywhere towards the same saturation-value, without at the same time having at all points the same constant direction, which is a necessary condition for a uniform distribu- tion ( 43) ; but the magnetisation must still have a complex- lamellar distribution, as we have already given (57) a perfectly general proof of this property. 60. Ferromagnetic Body conveying a Current. The case in which the ferromagnetic substance itself is conveying an electric current has so far been specifically excluded from con- sideration ; but in connection with some problems it has both a theoretical and a practical interest, 1 and accordingly we shall devote this paragraph to the question, without, however, entering upon a detailed investigation. We have already seen that the magnetic intensity arising from the electromagnetic action of a current (and expressed, there- fore, by $ e in our present notation) is distributed solenoidally within the conductor, but not lamellarly, as is the case outside. It further satisfies the equation [ 45, equation (16)] dy 9 z ^ ' I 9 z 8 x \ dx dy in which ( denotes the electric flow (as defined in 44) at the point considered. To determine whether the distribution of <% e is perchance complex-lamellar, we must insert the above values in the 1 In this connection, we need only mention iron telegraph wires, and those dynamos in which the iron armature serves to convey a current, as, for example, those constructed by Fritsche. (Compare 144.) CONSERVATION OF THE FLUX OF INDUCTION 85 equation which conditions this property [ 38, equation (6)1, which then, after simplification by division, assumes the form This is easily seen to be equivalent to which would imply that at every point the two vectors & and <% e must be in directions perpendicular to one another. This, however, is not, in general the case; and it is possible to imagine cases where the directions of the two vectors are inclined to one another at any angle. Thus <% e has not even a complex-lamellar distribution. Again, the distribution of < is always lamellar ( 47), but the vector-sum & = 4 &, being thus the resultant of two vectors, one distributed solenoidally and the other lamellarly, has a distribution of no especial character; we may therefore say that its distribution is complex-solenoidal. Hence, in accordance with the principles already established, the same must be true of the magnetisation 3, which at every point is coincident in direction with ,. Since S3 is identical with &,, and is therefore also dis- tributed solenoidally, and the same was shown above ( 52) to hold good for 3^, it follows that even in a ferromagnetic body through which a current is flowing the distribution of the vector S3* is still solenoidal. The solenoidal property, or, as it is also called, the continuity of the resultant induction, is thus seen to constitute a perfectly general fundamental principle, 1 which we shall consider some- what more fully in the following paragraphs. 61. Conservation of the Flux of Induction. In explaining the general properties of any solenoidal distribution of a vector ( 37) we have already shown that the whole of the region under consideration is divisible into vector-tubes, which are of such a form that the surface -integral of the vector over any cross-section whatever of the tube has a constant value. To vector-tubes possessing this property we gave the name of solenoids. 1 Compare Janet, Journal de Physique [2], vol. 9, p. 500, 1890. 86 OUTLINES OF THE THEOEY OF MAGNETIC INDUCTION If we apply this result to a ferromagnetic body subject to inductive influence, the whole of infinite space must be included in the scope of our enquiry, since the . distance to which the magnetic action extends is unlimited. In the special case with which we are now concerned the tubes of induction appear in place of the vector-tubes of the more general case, their generating lines being the lines of induction. The surface-integral of the resultant magnetic induction over any cross-section of a tube of induction we shall call the flux of induction through the tube, denoting its value by <. This nomenclature is founded on a hydrodynamical analogy (compare Chap. VII.). Since, in the case of an incompressible fluid, the velocity (that is, the quantity of fluid flowing across a normal surface-element per unit area per unit time) is known to satisfy that form of the equation of continuity which conditions a solenoidal distribution, it follows that the surface-integral of the velocity over any area depends only on the boundary of the latter, and measures the flux of fluid, or quantity which flows per unit time, across the area in question. The tubes of magnetic induction into which the whole of space may be mapped out must be either in the form of closed solenoids, re- entering into themselves after traversing a finite distance, or else in the form of endless solenoids which proceed to infinity, spreading out wider and wider without limit. But in both cases the flux of induction corresponding to any given tube alike remains constant, so that, in the case of infinitely long tubes, as the cross-sectional area becomes indefinitely great, the induction becomes indefinitely small, the surface-integral of the induction over the cross-section thus assuming the form oo x 0, corresponding to the constant finite value of the flux of induction for the tube in question. Thus, the fundamental principle, established in all its generality in the last section, expressing that the distribution of the resultant induction is everywhere solenoidal, and named accordingly the principle of the continuity of the resultant induction, might equally well be called the principle of the conservation of the flux of induction. 62. Refraction of the Lines of Induction. The interpreta- tion of this principle is sufficiently obvious in any continuous EEFEACTION OF THE LINES OF INDUCTION 87 region, whether we are concerned with closed solenoids or with those which extend to infinity ; but we must now further enquire what relations hold at a surface of discontinuity that is, in the present investigation, at the surface separating a ferromagnetic body from the surrounding medium. Owing to the discontinuity of the tangential component 98^, the direction of the lines of resultant induction will, in general, change abruptly at the bounding surface ; we shall have what is called a refraction of the lines of induction. Since $? t possesses tangential continuity ( 56), the com- ponents ^ and %tri i n the immediate neighbourhood of the bounding surface on its two sides respectively, will be coincident in direction ; and, in accordance with what we have already proved, this must also hold good for the tangential components SSi,. and 95^ at the ferromagnetic and outer sides of the bounding- surface respectively. The tangential components of the resultant induction may be represented geometrically as the projections of. this vector upon the tangent plane to the bounding- surface at the point considered ( 34). If in addition to these we know the value of the normal component, which is at right angles to them, the vector will be com- pletely determined ; its direction must evidently FlG 13 lie in that plane which contains the directions of its two components. Accordingly the lines of resultant induction within and without the ferromagnetic body will lie in the same plane, which contains also the normals 91' and 91 ; or at least this will be true when we are dealing with such short portions of the line in the immediate neighbourhood of the bounding-surface as may be considered sensibly straight. On this plane, which may by analogy be called the plane of incidence, OG (fig. 13) is the trace of the bounding surface, while 3K and 93< are the directions of the lines of resultant induction on either side of the surface, the angles made by these lines with the normals 91' 88 OUTLINES OF THE THEOKY OF MAGNETIC INDUCTION and 9t being denoted by a' and a respectively. The lines of resultant magnetic force ($ and <><) are further known to coincide with the lines of resultant magnetic induction, and, moreover, within the ferromagnetic substance the lines of magnetisation (3) have the same direction in common with these. Starting from the known relations I T = < T and %$' tv = $, ( 58), we have at the point : A. In the ferromagnetic substance SB; T = /*$;, = /&,&,. and %' iv = Therefore, .() ' ;">.'- * B. In the surrounding medium : Thus, finally, from (a) and (b) (12) . . . tana' = where, as usual, JJL denotes the permeability of the ferromagnetic substance at the point 0. The relation expressed by equation (12) may be called the tangent-law of refraction of the resultant induction. In the cases which most frequently arise, p is a large number, its value being sometimes as high as several thousand, so that the value of a is always very small, even when that of a' is considerable. In almost all cases, therefore, the lines of induction which pass from the ferromagnetic body into the surrounding medium leave the surface of separation in a direc- tion nearly coincident with the normal. On the other hand, as the magnetisation rises to a very high value, the value of the permeability approaches more and more nearly to unity ( 14), and the refraction becomes continually less marked. In the limiting case, when //, = 1, there will no REPRESENTATION OF THE MAGNETIC FIELD BY UNIT-TUBES 89 longer be any refraction, for then, by equation (12), we have a' = a. From these rules governing the refraction at the bound- ing-surface of a body we may often obtain an approximate estimate of the distribution of the lines of induction in the neighbourhood of the surface. Let us imagine two very narrow tubes of induction, whose normal sectional areas are very small and equal to S 8 f and S 8 respectively, the directions of the tubes being coincident with those of the lines of induction (33/ or 2B<, as the case may be) ; and, further, let these two tubes meet one another at the bound- ing-surface of the body, each intercepting there the same surface element & 8 T . Then we shall have (fig. 13, p. 87) SS' = S/S T cosa' and SS = SS T cosa Also 9B' ~~ and 9S/ ~~ cos a 1 cos a The product 2SS/S" or 93* $8 is evidently equal to the flux ot induction through the narrow solenoids. Denoting it by 8'< or 8 <, we obtain for its value from the above equations, on multiplying them together in pairs, cos a' ' and cos a Since %' tv = 9B i2/ , > fo is not coincident in direction with that of 3 Z , 3 tf , 3*. In analytical language, fa is a function of 3* alone and independent of 3 tf and 3,, similar relations holding good for $ iy and <$&. We may therefore write : % ix =-N x 3,, % iy =-N y 3,, % iz = - N z 3 2 So that, in accordance with the generalised definition of 24, N x , N y , N z are the factors of demagnetisation for the directions corresponding to X, Y, Z. Hence it follows that, for the solution of the problem of the uniform magnetisation of a body (sup- posing such a condition to be possible), it is necessary and sufficient to know the demagnetising factors for the three principal directions in the body. After this investigation we shall not need to adduce any separate proof, that 3 is not in general coincident in direction with <$ et the external field. 69. Magnetisation of an Ellipsoid. It may be shown that if F is the gravitation potential due to a body of any shape and of uniform density!), dTjdx is the proper expression for the magnetic potential T t due to the same body when magnetised uniformly, its magnetisation having the value 3* = D. 1 In accordance with the theorem of the last section, in order that it be possible to induce such magnetisation (uniform through- out the body), te must necessarily be constant. But we have *" = ~ ' = + Compare Maxwell, Treatise, 2nd edition, vol. 2, 437. H 2 100 OUTLINES OF THE THEOEY OF MAGNETIC INDUCTION Similarly, 8 2 r But if the second differential coefficients of the gravitation- potential with respect to the co-ordinates are to be constant, this function itself must be expressible as a quadratic function of the co-ordinates. And further, from the theory of attractions it follows that this can only be the case when the attracting mass is bounded by a closed surface of the second order. The only case, again, where the mass thus bounded is of finite extent, is that in which the boundary is ellipsoidal. The problem of uniform magnetisation is thus seen to become an extremely limited one. Let the equation of the ellipsoid considered be 9 " 7 9 ' 9 CT 0* C* the principal axes accordingly having the same directions as the axes of reference ; and let us denote by the definite (elliptic) integral 1 = if now we introduce the notation (17) , then the gravitation-potential F, due to matter of uniform density D filling the ellipsoid, has at points within the latter the value r = -~ (A^> 2 + N y y* + N z z>) + const. FUKTHEE SPECIAL ."CASES , 101 Applying this result to the magnetic problem in the manner already explained, we obtain at'r 43- 4- N vw ~>\ o ~~ y The quantities JV^, J^, N z , which are given by equation (17), are therefore the required factors of demagnetisation corre- sponding to the directions of the axes in the ellipsoid in question. 70. Further Special Cases. The formulse (17) give the factors of demagnetisation as differential coefficients of an ellip- tic definite integral. These reduce to elementary functions, however, when we pass to the special case of an ellipsoid of revolution, magnetised in the direction of its axis of revolution. The formulae for the factor of demagnetisation have already been given ( 29) for the two distinct cases which arise, those namely of the prolate and oblate ellipsoids. It is therefore unnecessary to repeat them here. The other shapes which may be considered as special cases of the ellipsoid of revolution were the sphere, a circular cylinder of infinite length magnetised transversely, and a plate of infinite extent magnetised per- pendicularly to its plane. This last case may also be deduced from that of a thin hollow sphere, magnetised by a field which may be regarded as having approximately a uniform radial distribution. There will then also be a uniform radial distribution of magnetisation, and as the radius is made to increase indefinitely we pass from the case of the hollow sphere to that of the plane plate. 1 The magnetisation of a circular cylinder of finite length by a field parallel to its axis was investigated by Green. 2 Starting from several assumptions which were not, however, free from objection, he arrived at an empirical equation which 1 du Bois, Wied. Ann. vol. 31, p. 947, 1887. 2 Green, Essay on the Application of Mathematics to Electricity and Magnetism, 17, Nottingham, 1828. t !02 OU^INES OF THE THEOEY OF MAGNETIC INDUCTION is tolerably accurate for any cylinder of a length considerable in comparison with its radius. The equation gives the ' linear density of free magnetism/ along the cylinder, or, as we should express it, the normal component of "magnetisation at the bounding-surface ; a quantity which is zero at the middle and increases continually in numerical value as we approach either end. But this linear density is of no special interest ; in experimental investigations we are chiefly concerned with the average value of the factor of demagnetisation, which we have already (Table I, p. 41) given as an empirical function of the ratio of the length to the diameter. In addition to the problems in magnetisation already mentioned, wherein we assume the application of a uniform external field, several cases have been investigated in which the magnetising field may be distributed in any manner. Poisson treated in this way the case of a hollow sphere by means of spherical harmonics, 1 which had already been experimentally examined by Barlow. 2 The most interesting result of these investigations is that a hollow sphere of ferro- magnetic material (provided its thickness is not too small) when magnetised by any field whatever, exerts the same effect at all external points as if it were solid throughout. In the inner hollow space, on the other hand, the original external field suddenly generated will be greatly reduced by the action of the ferromagnetic shell. 3 Again F. Neumann has investigated the magnetisation ot an ellipsoid of revolution under the influence of an arbitrarily distributed external magnetic field, especially for the particular case where the ellipsoid degenerates into a sphere. 4 The in- vestigation has further been extended by Kirchhoff to the case of an infinitely long cylinder. 5 For the details of these 1 See Maxwell, Treatise, 2nd edition, vol. 2, 431-434. 2 Barlow, Essay on Magnetic Attractions, London, 1820; Gilb, Ann. vol. 73, p. 1, 1828. 3 This general property of enclosures surrounded by thick shells of ferro- magnetic material is often applied to diminish the effects of external magnetic influences and disturbances ; for example, in marine galvanometers and other apparatus. 4 F. Neumann, CrelW* Journal, vol. 37, 1848 ; Vorlewngen uber Magnetis- mus, 43, p. 112, Leipzig, 1885. 5 Kirchhoff, Gesammelte A Wiandlungen, p. 193. SOLUTION BY SUCCESSIVE SUPEEPOSITION 103 mathematical researches, as well as of many others to which reference cannot here be made, we must refer to treatises which give a complete historical survey of the present subject. 1 71. Solution by successive Superposition. The inherent difficulty of the problem of magnetisation arises from the fact that the demagnetising intensity must be taken into account, and this in turn depends on the distribution of magnetisation which we have to determine. Hence some investigators have applied to this problem a method of successive approximation, which in principle is analogous to that which Murphy employed in electro- static problems for calculating the distribution of electricity. 2 We will conclude this chapter with a brief discussion of the method. Let the distribution of the external field ,, or its mag- netic potential T e , be given. Neglecting for a moment the demagnetising intensity, we calculate the magnetisation which would be induced in the body considered under the influence of the potential T e alone. Let this be 3' ; it may be called the magnetisation of the first order. This will give rise to a potential of the first order T/, which in its turn will induce a magnetisation of the second order 3". This again will produce a potential of the second order T/', which in turn will cause a magnetisation of the third order, and so on. The successive (vectorial) superposition of these magnetisa- tions of different orders gives with a continually increasing degree of approximation the actual distribution of magnetisation 3, so that (18) . . 3 = 3' + 3" + 3'" + ..'.. If this method is to be suited for solving problems, we must seek to express the magnetisations of successive orders by means of the most rapidly converging series possible. This question was first considered by Beer, to whom the method is due, and subsequently by C. Neumann, L. Weber and Eiecke. 3 1 Especially Wiedemann, Lehre von der EleUricitat, vol. 3, pp. 354-390 ; < Nachtrag,' p. 1320. 2 Mascart and Joubert, Electricity and Magnetism, vol. 1, 86; Wiedemann, Lehre von der EleUricitat, vol. 1, 85 ; vol. 3, 387. 3 Beer, Einleitung in die LeJire von der EleUricitat u. d. Magnetismus, pp. 155-165 ; C. Neumann, Das Logarithm. Potential, p. 248 ; L. Weber, Zur 104 OUTLINES OF THE THEORY OF MAGNETIC INDUCTION Quite recently Wassmuth has shown how all these expansions can be deduced from a common form, whose physical interpre- tation is easy. 1 The greater number of the above researches are chiefly of mathematical interest ; at least their physical results bear no sort of proportion to the mathematical skill and ingenuity that have been lavished on them. But amongst them are several results which furnish us with valuable guidance for the prose- cution of experimental enquiries. For the greater number of the important modern applications of electromagnetism they are, on the other hand, entirely unfruitful. We shall now turn our attention to problems whose solution appears to promise more in this respect. Theorie der magnetisclien Induktion, Kiel, 1877 ; Riecke, Wied. Ann. vol. 13, p. 465, 1881. 1 Wassmuth, ' Losung des Magnetisirungsproblems durch Reihen,' Wiener JSerickte, vol. 102, part 2, p. 65, 1893 ; Wied. Ann. vol. 51, p. 367, 1894. 105 CHAPTER V MAGNETISATION OF CLOSED AND OF RADIALLY DIVIDED TOROIDS A. Theory 72. Peripheral Magnetisation of a Solid of Revolution. In the first two chapters we have already considered, in an elementary manner and from different points of view, the magnetic properties of closed rings, as well as of those which are divided by a radial slit ( 9, 10, 16), and we have seen how these may in a measure be considered as typical forms. The problem of their magnetisation must now be more closely examined in the light of the results obtained in the last two chapters, since this problem furnishes a basis for the further elaboration of the theory of the magnetic circuits. Kirchhoff l was the first to investigate mathematically the magnetisation of a ring, or, to speak more definitely, of a solid of revolution which is not intersected by its axis of symmetry. It is assumed that each single winding of the magnetising coil lies in a ' meridian plane ' which passes through the axis ; all the windings, taken collectively, constitute a hollow ring, which encloses the ferromagnetic solid of revolution, and is disposed symmetrically with respect to the same axis. Let n denote the number of windings, and I the current flowing in the circuit. Any arbitrary closed curve which runs completely round within the hollow annular space is then evidently embraced n times by the current-conductor ; so that when we pass once round such a path of integration, the line-integral of the magnetic intensity 4?, produced by the current within the region in question, increases by the value 4>7rnl ( 45). In particular, let us choose as paths of integration, circles whose centres lie on the axis ZZ of the 1 Kirchhoff, Gesammelte Abliandlungen, p. 223. 106 MAGNETISATION OF CLOSED AND DIVIDED TOKOIDS body of revolution (fig. 14) and whose radius may be denoted by r. It will then be evident, from considerations of symmetry, that % e must be at each point peripheral in direction that is, tangential to the circle of integration and at right angles to the meridian plane (plane of the paper in fig. 14) ; while along the circumference of any one and the same circle of integration the value of % e must be constant. The line-integral in ques- tion may therefore be expressed as the product of the value of the vector into the circumference 2 TT r of the circle of integration. Thus we have FIG. 14 2nl The magnetic intensity is there- fore inversely proportional to the distance of the point considered from the axis, so that its value diminishes as the point is taken further from the axis. 73. Kirchhoff's Theory. - Whatever be the form of the meridional section of the solid of revolution, we may conceive it to be divided up into rectangular elements dr dz (fig. 14), each of these, on rotation about the axis ZZ generating an ele- mentary ring, for which <% e has then a sensibly l uniform peripheral' distribution ( 43). We next assume that this would give rise to a uniform peripheral distribution of mag- netisation, so that there will be no demagnetising influence, magnetically effective end-elements being absent. 1 The re- sultant magnetic intensity is therefore identical with that (j? e ) which is directly due to the current in the coil, and must consequently, like the latter vector, have a uniform peripheral 1 Since a uniform peripheral distribution, as explained in the place already quoted, is also a solenoidal one, the convergence of the magnetisation will be everywhere zero, and there will consequently be no internal centres of action which can produce any effect at distant points ( 37, 50). KIKCHHOFFS THEOKY 107 distribution. Thus it follows from the assumption that we have made, in accordance with a theorem already given ( 68), that there is a uniform peripheral distribution of magnetisation within the thin elementary ring. Just as there is an absence of demagnetising effect within the elementary ring, so also this latter remains without magnetic influence at external points, or upon neighbouring elementary rings, which are thus seen to be wholly independent of one another as regards their magnetic properties. Throughout the entire ring, therefore, the mag- netisation will have a constant value for each given value of the radius, but will decrease with c as the radius increases ; it is given by an equation which follows directly from (1) : /o\ p may be called a hoop (Reifring), while a flat ring (Flachring) is one for which p > f. (B) Circular section [fig. 14 (J5), p. 106]. Our solid oi revolution now comes under the definition of a toroid ( 9) . Let r z be the radius of the cross-section, r 15 as before, the radius of the circular axis that is, of the circle which is the locus of the centres of all the (circular) cross-sections. Then (7) . . ,= 2 Here again, if r 2 is small in comparison with r n the expres- sion approximates to the following simple form : (Qa) 4 = 2 "A Iwr 8 which is identical with that given by equation (6) above. KINGS OF RECTANGULAR AND OF CIRCULAR SECTION 109 Kirchhoff, in the paper already quoted, expressed the opinion that this special case of magnetisation, first theoretically inves- tigated by himself, might form the basis for a convenient method of practical measurement. We shall return to the more particular consideration of this method in 83, where it will be illustrated by an example. Kirchhoff's proposal was first realised by Stoletow in 1872 ; and soon afterwards Kowland conducted a series of fruitful researches by means of the method, which has since been made the basis for a large number of investigations connected with this subject. 1 75. Fundamental Equation for a Radially Divided Toroid. We shall here confine our attention to the case of a ring of circular section that is, of a toroid and we shall assume that the dimensions of the cross-section are small in comparison with the diameter of the toroid, so that r 2 jr l is a small quantity, as we have already supposed it to be in equation (Qa). Let us now make a radial cut through the toroid, the resulting slit having throughout a constant width, which we denote by d (fig. 15). The slit is supposed to have no influence on the regularity of the winding, which we assume to be entirely uniform as before. If n is the number of wind- FIG ings, the intensity of the magnetic field exerted by the current at all points of the circular axis (dotted in fig. 15) will be, in accordance with equation (1), 1 Stoletow, Pogg. Ann. vol. 146, p. 442, 1872; Kowland, Phil. Mag.\\, vol. 46, p. 140, 1878. Up to the present time these investigations have given us no reason to doubt the correctness of Kirchhoff's theory. Researches of G. vom Hofe (Diss. Greifswald, 1889 ; Wied. Ann. vol. 37, p. 482, 1889) on three rings of rectangular section, for which the ratio p : was different, appeared to show some divergence from the theory, but these can be fully explained by the differences which are known often to exist between materials which are supposed to be identical ; such differences arising from the manner in which the substances have been heated and annealed, as well as from a variety of causes less easily recognised. Compare also Hues, Uber den Magnetismus von Eisenringen, &c., p. 2 (Diss. Greifswald, 1893). 110 MAGNETISATION OF CLOSED AND DIVIDED TOROIDS and will not differ greatly from this value at any point of the cross-section. Let us now choose the centroid as the. path of integration, and apply Theorem I, already enunciated [55, equation (6) ] which states that the increase of the magnetic potential due to magnetisation (8) . iT. In the present case the points E and A are those in which the circular axis intersects the two faces of the slit. From considerations of symmetry we see that the circular axis must be a line of magnetisation that is, at each point of the circular axis the direction of the tangent will coincide with that of the vector 3. Moreover, the value of 3 will not vary more percep- tibly from point to point along the circular axis, any more than it varies from point to point over the cross-section of the toroid. For purposes of approximate calculation therefore and more than this cannot be effected in the present case we may introduce a mean value of the magnetisation, which we distin- guish by the symbol 3. The same holds good for the demag- netising intensity $, whose mean value is denoted by & Since the length of the path of integration from E to A through the ferromagnetic body is (2 TTT I cZ), we have in accor- dance with the ordinary rules for the evaluation of definite integrals We may consider this equation as the definition of the mean value >< If we further introduce a mean factor of demagneti- sation, defined by the equation (10) . . . &= -N5 we obtain finally from the three equations (8), (9) and (10) above (i) - jr^-ST This elementary formula may be in a certain sense regarded FUNDAMENTAL EQUATION FOE A DIVIDED TOROID 111 as the fundamental equation for a radially divided toroid. 1 The left-hand member denotes the magnetic difference of potential due to magnetisation measured within the ferromagnetic body between the two faces of the slit; it is approximately equal to the mean value <> which the magnetic intensity due to mag- netisation has within the slit, multiplied by its width. 76. First Approximation; Limiting Case. Another ex- pression must now be found for the left-hand member of the fundamental equation (I). To this end, in our first approxima- tion we make the assumption that the magnetisation 3 is constant over the entire cross-section of the toroid, and at right angles to the plane of the section, so that it has, in accordance with the definition of 43, a uniform peripheral distribution. 2 According to the ' law of saturation ' III ( 57) the actual state of things must approximate more or less closely to this limiting case of our assumption, when we suppose the intensity <% e of the inducing magnetic field to increase without limit, so that finally its value becomes very great in comparison with that of the demagnetising intensity ',. In accordance with this assumption there will be no magnetically effective end- elements on the convex bounding-surface of the toroid, such elements being confined to the two plane surfaces which bound the slit on either side. These will produce an external field determined by their magnetic strength. But this latter quantity has per unit area of the surface in question the value 3 V ( 49), and since, in the present case the magneti- sation is at right angles to the bounding-surfaces of the cleft, 3, = 3. Let us now consider an element of one of the faces of the slit, taken in the form of a plane circular ring of infinitesimal breadth dy and of mean radius y (fig. 16); its area is Zvrydy, and its magnetic strength is therefore 2 TT 3 y dy. Thus at a point P which is on the normal, drawn from the centre of the bounding-face of the slit, at a distance a? from that face, our elementary plane ring exerts an infinitesimal magnetic intensity 1 du Bois, Wied. Ann. vol. 46, p. 494, equation (7), 1892. 2 It is unnecessary in this case to introduce a mean value 3, since the value of 3 is everywhere the same. 112 MAGNETISATION OF CLOSED AND DIVIDED TOROIDS d <;, which in accordance with Coulomb's law is given by the equation /i i \ j K. <-> TT ->$y dy (* 1; . d & = 2 cos a . Here z 2 = x 2 + ?/ 2 is the distance of each separate point (such as Q) of the elementary plane ring from P. The angle 7 2, T V ^ ~^~~^r~~^ <*$ # ^ f >J? FIG. 16 Q P is denoted by a, so that cos a = x/z ; and consequently the above equation can be written in the following form: (12) . . . d$ t =- From the relation z* = x 2 + f it follows that, for a given position of the point P, that is, for a constant value of x, zdz = y dy And if we substitute this in (12), we obtain This expression has to be integrated over the entire surface which forms one face of the slit, in order to find the resultant magnetic intensity & at P, arising from the action of that face ; we shall have, then, The two limits of this definite integral correspond to the centre and the circumference of the surface in question ; per- forming the integration we obtain / FIRST APPROXIMATION. LIMITING CASE 113 or (13) . . & = 2 TT 3 |l - We are now in a position to calculate the change of magnetic potential in passing from the centre of the surface 1 to the point P: denoting this quantity by T n , we have X x f-\A\ T ( K J^ 9 CV f f x & x (14J . 1ft = \ 3 ; &t &' t /t = v \lvt/'t = 1-6 69 2197 _ 2-4 117 3720 2045 1-82 0-550 3-8 206 6520 3660 1-78 0-561 5-1 279 8830 4880 1-81 0-553 6-8 383 12120 6720 1-81 0-553 8-9 492 15580 8610 . 1-81 0'553 11-9 628 19910 11230 1-78 0563 14-9 748 23680 13400 1-77 0-565 18-0 852 26970 15250 177 0-565 20-9 923 29260 16650 1-76 0-568 24-3 988 31300 18000 1-74 0-575 27-2 1036 32850 18900 1-74 0-575 36-9 1141 36190 21400 1-69 0-591 49-0 1200 38120 -23200 1-65 0-608 64-5 1250 39800 24750 1-61 0-621 78-5 1285 40870 25800 1-59 0-631 99-5 1325 42230 27030 1-56 0-641 181-0 1428 45600 30550 1-50 0-670 267-0 1500 48120 32800 1-47 0-681 (300) 1525 48900 32500 1-50 0-667 It must be noticed that for the abscissae $? e three different scales had to be used, so as to avoid unduly extending the height of the diagram (fig. 21). The first scale for abscissas corresponds to ordinates ranging from 3 = to 3 = 1000 ; the second scale, one -fifth of the former, from 3 = 1000 to 3 = 1300 ; and the third scale, one-twentieth of the first, from 3 = 1300 upwards. Thus, the inclinations of the curves to the axes of reference are throughout kept within convenient limits. The diagram shows at a glance how the influence of the gap gradually increases as its width is made greater. 88. Discussion of the Principal Results. The experimental data may be considered from three different points of view : I. Lines of Demagnetisation If we suppose each of the curves 1, 2, 3, 4, 5 (fig. 21) to suffer separately a shear parallel to the axis of abscissae until it is brought into coincidence with the normal curve (0), the DISCUSSION OF THE PRINCIPAL RESULTS 131 K 2 132 MAGNETISATION OF CLOSED AND DIVIDED TOROIDS directrix corresponding to each width of the gap employed will thus be constructed. These directrices, or, as we shall call them for distinction, lines of demagnetisation, are drawn to the left of the axis of ordinates through a number of observed points, and are distinguished by the same numbers as the corresponding curves of magnetisation . Within the limits of experimental error, as will be easily seen, the points lie on straight lines about as far as the ordinate 3 = 875 C.G.S. (which corresponds to half the saturation value for the specimen of iron examined). The lines of demagnetisation express the relation between the mean mag- netisation 3 and the mean demagnetising intensity ^. Their approximately rectilinear form in the neighbourhood of the origin thus furnishes an experimental proof that the ratio of the ordinate to the abscissa, that is, the mean factor of demag- netisation N, remains constant until the magnetisation has reached about half its saturation value. The following are the values of the factor in question for those parts of the different curves over which it remains approximately constant : No. 1 2 3 4 5 N: 0-0079 0-0102 0-0140 0*0203 0-0246 We shall return to the consideration of these numbers in II. Leakage-Coefficients So far we have confined our attention to the numbers con- tained in the first two columns of Table III. From the numbers in the last column, it follows that the leakage- coefficient, as determined by experiment, likewise remains con- stant until the magnetisation reaches about half its saturation value, but that for higher values of the magnetisation it slowly diminishes. The following table gives the mean values of the coefficient over its range of constancy, corresponding to the various widths of air-gap : No. l 2 3 4-5 v: 1-00 1-52 1-79 2-48 3-81 Thus, the leakage-coefficient attains a value differing con- siderably from unity, even wbile the air-gap is still compara- DISTRIBUTION OF LEAKAGE 133 lively narrow. In fig. 22 the leakage-coefficient v is plotted as a function of the mean magnetisation 3 (lower scale for abscissas) for the gaps (2), (3), (4), (5) ; in the case of the narrowest gap, (1), no experiments on leakage were made. The initial con- stancy of v here corresponds to the circumstance that the curves continue parallel to the axis of abscissas until the magnetisation has reached about half its saturation value. As we pass to the higher values of the abscissae, the curves begin to bend down- wards. They are drawn as continuous lines as far as the experimental numbers extend, but if they are prolonged, as in the diagram, by the dotted parts, it appears that, as we approach the saturation point for the specimen of iron in question (about V H / / r i h / 7 tr ~~- ^x S ft 1 y \ \ / -*. K^ \ (3 2 X x t \ W /i -- \> -a, ?/l X' , M' A ff to < OO'J to 00 S 00 6 to i (to f 10V 9 -" TO & ti fj~s E 1 FIG. 22 1750 C.G.S.), they will all converge to the same point, whose ordinate corresponds to a leakage-coefficient equal to unity. III. Distribution of Leakage In the case of the three widest gaps employed, and for three values of the magnetisation (about 500, 1000 and 1500 O.G.S.), observations were made on the distribution of leakage along the circumference, using the movable auxiliary secondary coil mentioned in 86. The results obtained are given in Table IV, p. 134, where the positions of the auxiliary coil on the circumference are entered as points of the compass, together with the corresponding values obtained for the flux of induction through this coil. From this table it follows that even for 134 MAGNETISATION OF CLOSED AND DIVIDED TOKOIDS TABLE IV g 5 &8X8W fff ( S)E = w @A T E=A T IF N Gap (3). d = 0-103 cm. 492 988 1525 17100 33560 48000 16650 32850 48000 15850 32100 48000 14400 29400 48000 8610 18000 32500 Gap (4). d = 0-202 cm. 487 997 1520 17630 34750 46250 17030 33550 46000 15950 32100 46300 13800 28700 46100 6230 13150 24400 Gap (5). d = 0-357 cm. 503 1015 1455 18800 36200 46350 17900 34500 46100 16200 | 13500 32700 28300 45800 4160 8800 16800 2TW. E feeble magnetisations, for which the leakage is most considerable,. the greater part of the leakage takes place within the short length which lies between ^TF and NE, and which contains the air-gap. In the case of the strongest magnetisation employed (the leakage-coefficient being considerably smaller), the property in question may be expressed by saying that up to the point NE or NW no considerable change occurs in the flux of induction through the cross-section of the toroid. The distribution of the induction is therefore sensibly uniform- peripheral over more than three-fourths of the circumference, and the uni- formity of distribution will be the greater the higher the value which is reached by the induction. 89. Comparison of Theory and Experiment. We are now in a position to compare the results of the experiments described above with the conclusions of the theory previously developed. In equation (III) ( 80) FIG. 23 27T COMPARISON OF THEORY WITH EXPERIMENT 135 we have a relation between the mean factor of demagnetisation N and the function it, which is the reciprocal of the leakage- coefficient v. This last quantity, moreover, is represented graphically in fig. 22, p. 133, as a function of the magnetisation for the four widths of the gap (2, 3, 4, 5) which were employed. Again from (III) we easily obtain for the lines of demagnetisa- tion the following equation : 2eZ 5 (28) . . . -- ---- 7T which now enables us to construct the lines in question from the curves v = funct. (3) of fig. 22. On the 'left-hand portion of fig. 21, p. 131, the lines ot demagnetisation constructed in this manner are shown. The lines (2) and (3) are continued as far as the ordinate 3 = 1500, because for higher values of 3 the assumed reciprocity of n and v ceases to hold good with sufficient approximation (compare 82). On the other hand (4) and (5) are only drawn for the range covered by the directly observed points. 1 As will be seen, these points lie approximately on the lines of demag- netisation. Thus, the theory leads to a satisfactory coincidence of the lines of demagnetisation plotted from measurements of the leakage with the curves of magnetisation which were deter- mined by an entirely independent method. Fig. 21 also gives for the three narrowest air-gaps, (1), (2), (3), between the values 3 = 1000 and 5 = 1750 C.G.S., the straight lines of demagnetisation whose equation is (29) . . . "; = N* 3 where JV , denotes that factor of demagnetisation which is to be found from equation II ( 76), and which, in accordance with the assumption there made, is, strictly speaking, only applicable for infinitely high values of ^> c . From fig. 21, p. 131, it will now be observed how the values of ^V, calculated from the measurements of leakage by means of 1 Fig. 22 does not give the function v = funct, (3) for the gap (1), but, as we shall see in the next section, this can be found by interpolation. The line of demagnetisation (1) in fig. 21 was obtained in this manner. 186 MAGNETISATION OF CLOSED AND DIVIDED TOROIDS equation III, and. as we have seen, according well with obser- vation, tend also to the limit N <# as the saturation value 3 OT = 1750 C.Gr.S. is approached. 1 The lines of magnetisation may, in fact, be produced as in the dotted curves, so as to pass through the points A 19 A v A y In Table V the somewhat complicated relations under dis- cussion are collected in a form convenient for reference, so far as they correspond to the range of magnetisation 3 = to 3 = 875, in which both the leakage-coefficient v and the factor of demagnetisation N may be considered as constant. The meaning of each column will be sufficiently clear from its heading. TABLE V Calculated No. d d r. V n N (observed) Percentage difference ^00 N 1 0-040 0-045 1-31 2 0-765 0-0098 0-0077 0-0079 -1- 2-5 2 0-063 0-070 ]-52 0-660 0-0151 0-0105 0-0102 - 3 3 0-103 0-115 1-79 0-558 0-0242 00145 0-0140 - 3 4 0-202 0-226 2-48 0-403 0-0451 0-0205 0-0203 - 1 5 0-357 0-400 3-81 0262 ! 0-0726 0-0236 0-0246 + 4 The agreement between the calculated and the observed values of N is as good as could be expected, when we remember that, on the one hand, the theory deals only with mean values and approximations, and that, on the other hand, the sources of experimental error, especially in relation to the exact form of the gap, may easily introduce an uncertainty of several per cents. Finally, then, we may consider the theory here developed as confirmed by experiment with sufficient accuracy for most purposes ; while the experimental data furnish us with the means of determining the function n or its reciprocal v, the 1 To express the condition that magnetisation of the body is to be near the point of saturation, so that KirchhofF s law of saturation becomes approxi- mately applicable, it was supposed in 57 that ^ was small in comparison with e (compare Culmann, Wied.Ann.vol. 48, p. 380, 1893). From fig. 21 it will be seen that in reality for 3 = 1500 C.G.S., Q e was of an order of magni- tude about tenfold as great as Q. 2 Va^ue found by interpolation ; compare the following section. EMPIRICAL FORMULA FOR THE LEAKAGE 187 leakage-coefficient. These two quantities had to be provisionally introduced into our theory as unknown ( 78, 80). 90. Empirical Formula for the Leakage. On introducing the function n, we denoted it ( 80) symbolically by which expresses the fact that it depends only on the ratio r 2 fd determined by the shape of the gap, and not on the radius of the entire toroid. In the following discussion we shall con- fine our attention to the range of magnetisation 3 = to 3 = 875, which for practical applications is the most important ; we can then put n = 1 / j/, and consider both these quantities to be independent of the magnetisation. The question then arises how the leakage-coefficient v (or the function n) depends on the shape of the air-gap, as deter- mined by the ratio djr z (or r 2 /d). In order to obtain the experimental answer to this question, the ordinates v in fig. 22, p. 133, are also plotted as a function of djr z ; the second (upper) scale for abscissas is introduced so as to allow this relation to be read off. We thus arrive at the empirical rule that, within the limits of experimental error, the four observed points lie on a straight line. This line cuts the axis of ordinates for the value v = 1 , the corresponding law being that, when the width of the gap is reduced to zero, the leakage vanishes. The equation to the straight line is empirically found to be (30) . v =l+7- which may also be transformed as follows : 2 (31) . n = d 7 + r f d This formula for n = funct. (r 2 jd) is represented graphically by an hyperbola. In the square brackets following equations (30) and (31) is given the range of the independent variable within which the corresponding formulas hold good. Too much weight must not be attached to such purely empirical relations 138 MAGNETISATION OF CLOSED AND DIVIDED TOROIDS as these. For our physical insight into the phenomena they are quite useless ; but, on the other hand, in practical applications it is useful to have at least some method for roughly estimating leakage-coefficients. We shall return to the consideration of this empirical formula for the leakage in 173. In the present case, the formula can be applied to find by interpolation the value of the leakage-coefficient v = 1*31 for the gap (1), for which it was not directly measured. This has already been done in Table V. From the researches here described, it is clearly established that in a radially divided toroid the leakage for moderate values of the magnetisation (0 <^ < 875) remains nearly constant, while beyond this range the leakage decreases with increasing magnetisation. It also follows from our theory that this must be the case. To show this, let us consider once more figs. 13, p. 87, and 17, p. 115. The acute angle (90 a'), which the lines of induction within the toroid make with its bounding-surface, will become still more acute as the magnetisation increases beyond a certain value, since the peripherally directed distribu- tion of the magnetisation ( 76), and therefore also of the induction, tends to become more completely established in accordance with Kirchhoff's law of saturation. Let us further consider the tangent law of refraction for lines of induction, in accordance with which tan a = tan a' f* , We have just seen that, as the magnetisation increases, a r becomes greater, while, on the other hand, the permeability becomes smaller ( 14); that is, I///, becomes greater. These two causes conspire to increase the value of a, so that the lines of induction in the external medium will deviate further and further from the normal to the bounding-surface, the leakage thus becoming smaller, as was found by experiment to be actually the case. PART II APPLICATIONS 141 CHAPTER VI GENERAL PROPERTIES OF MAGNETIC CIRCUITS A. Non-uniformly Magnetised Ring 91. General Remarks. In the preceding chapter we have treated as rigorously as possible the typical case of a uniformly wound radially divided toroid, though only with a certain degree of approximation, and we have found our theoretical conclusions sufficiently confirmed by experiment. We shall now turn to cases in which the coils, or the ferromagnetic substance, are of a form less simple than that which has hitherto been assumed for the sake of theory, but which does not in general suffice for the applications of electromagnetism. In this, however, we may and must be content with rough approximations, for a rigorous mathematical treatment of such problems is as impracticable as it is useless. If we have hitherto attempted to treat the questions which arise from the purely scientific point of view, we must now realise that it would be illusory to pursue that method any further, as it seems almost wholly out of the question to obtain in this way results of practical utility. The new standpoint to which we refer is that of applied physics. The magnetisation 3 at each point of the ferro- magnetic substance, which from the physical point of view is the vector of most fundamental importance, is now no longer to be regarded as of the first consequence (compare 12). We shall now rather concern ourselves with the induction 35 or the flux of induction ; the latter vector is of the greatest practical importance, since its variations determine the electromotive forces induced in a conductor embracing a bundle of lines of induction. 142 GENERAL PROPERTIES OF MAGNETIC CIRCUITS 92. Experiments of Oberbeck with Local Coils. We first attack the case of a closed toroid, which, however, is not now under the influence of a peripherally uniform magnetic field. Instead of the uniform winding hitherto assumed, there will only be one coil extending over a portion of the circumference, at 8 for instance. We shall first give the experimental investi- gation of this special case which has been made by Oberbeck. The toroid of soft iron which he used had a mean radius ^ = 9-5 cm., the radius r 2 of the cross-section being 1 cm. 1 (fig. 15, p. 109). A primary coil of 145 turns was wound on it, and occupied one-fifteenth part of the circumference, so that the angle a (fig. 24) was 24. A secondary coil of a few turns FIG. 24 could be moved over the whole circumference, and clamped in any given position. By reversing the magnetising current, a momentary current was produced in this secondary coil, which could be determined by means of a ballistic galvanometer, and this measured the flux of induction through the secondary coil, in the position which it then occupied. We again designate the position of the secondary coil on the circum- ference in fig. 24 by points of the compass as we did in 1 Oberbeck, Fortpflanzung der magrnetischen Induction im Elsen, Habilit. Schrift, Halle, 1878. The statement of dimensions on p. 5 of that paper does not seem quite clear. EXPEEDIENTS WITH LOCAL COILS 143 fig. 23, p. 134, and distinguish the corresponding values of the flux of induction by suffixes as in Table IV, p. 134. We start from the highest value of the flux of induction, which is obviously at 8, and therefore put s = 100. Oberbeck found in one case, for example, E = w = 93 and Having regard to the very unequal values of the magnetising field of the coil at different parts of the circumference of the toroid, the flux of induction may be said to be almost constant, especially over that half of the toroid WNE which is not over- wound. From the fall in the value of , towards that part which is directly opposite the coil, it follows that some induction- tubes ( 61) must emerge from the ferromagnetic substance, and spread out into the surrounding indifferent medium ; in other words, there is a leakage, which manifests itself by magnetic action at a distance (compare 15, 78). On using one coil at 8 and one at JV, each of which occupied an angle of about 18 that is, the twentieth part of the circum- ference and each of which tended to produce magnetisation in the same direction, the result obtained was : s = y = 100 E = w = 98 The variation was therefore extremely small. The case in which the magnetising coils acted in opposition to each other, whether they were of the same or of different strengths, was also exam- ined by Oberbeck, but has less interest for our present purpose. 93. Further Experiments by von Ettingshausen and Mues, Soon after the publication of Oberbeck's investigations, von Ettingshausen l published measurements which were quite 1 Von Ettingshausen, Wied. Ann. vol. 8, p. 554, 1879. These experiments were made with the view of testing an equation which Boltzmann ( Wiener Anzeiger, No. 22, p. 203, 1878; Wiedemanris BeiUatter, vol. 3, p. 372, 1879) had deduced for the present case. This formula has a purely mathematical interest, as it is based on the assumption of constant susceptibility, which, under no circumstances, is accurate, and especially in the present case, as von Ettingshausen himself observes, does not appear capable of giving even approximately correct results. ^r&^ (UNIVERSITY 144 GENERAL PROPERTIES OF MAGNETIC CIRCUITS similar. He first used a welded toroid, of ordinary bar iron (TJ = 12*26 cm., r z = 0*7.7 cm.), in which the coil at S covered one forty-fifth of the circumference (a = 8). Greater differences of the flux of induction between 8 and N were now observed than had been found by Oberbeck. These differences were, however, less the stronger was the primary current. After this a second toroid was turned from a plate of Styrian soft iron so as to have no weld (r l = 10-95 cm., r 2 = 0-75 cm.) ; it was wound in just the same way as Oberbeck had dene that is to say, with 145 turns on a fifteenth part of the circumference (a = 24). The results agreed better with those of Oberbeck, and the more closely the stronger was the magnetising current. In the experiments of von Ettingshausen the absolute system of measurements is adopted, which was not the case with those of Oberbeck . The latter probably employed stronger magnetis- ing fields than the former (taking into consideration the mean value of the magnetic intensity which varies along the circum- ference). This is probably the origin of the difference in the results of the two investigators, as will be more completely discussed in the following paragraphs. More recently, Mues, 1 under the guidance of Oberbeck, has investigated the case of a ring magnetised by two local coils in a somewhat different manner by measuring the action at a distance due to leakage. The annealed iron rings were rect- angular in section, and were wound at two places diametrically opposite each other with coils which tended to magnetise the ring in the same sense and to the same extent. In deter- mining the action at a distance only points in the plane of the ring (for instance, P, fig. 24, p. 142) were considered, and at these points the radial components of the field <,. were especially determined. From considerations of symmetry it is evident that these radial components must vanish at all points which lie in the straight lines NS and WE? and this was also verified by 1 Louis Mues, Mag netismus von Eisenr'mgen. Dissertation, Greifswald, 1893 ; also Wiedemann BeiUdttfir, vol. 18, p. 592, 1894. 2 It scarcely needs mention that the terms derived from the compass, especially ..Yand 8 t have nothing to do with north and south magnetism, but only serve for orientation. THEOEETICAL EXPLANATION OF THE EXPERIMENTS 145 experiment. If the position of the point is denned firstly by its azimuth < measured from N8 (fig. 24, p. 142), and secondly by its distance 9ft from the centre C of the ring, then from the nature of the case the value of the radial component must be a periodic function of the azimuth <, the period of which is TT, and which can be expanded in a Fourier's series. Now experiment showed that the first term of such a series, which of course is proportional to sin 2^>, represented the measurements with sufficient accuracy. The value of <$ r was a maximum therefore for c/> = 45, 135, 225, 315, while it was zero for = 0, 90, 180, 270. If, for example, a radial com- ponent of magnetic force directed outwards from C resulted from a given direction of current in the NE quadrant, this was also the case in the SW quadrant, while on the contrary the- radial components in the SE and NW quadrants would be directed towards the centre of the ring. It was finally established that within a certain range < r was inversely pro- portional to 3ft 4 . 94. Theoretical Explanation of the Experiments, These ex- periments of Oberbeck and of von Ettingshausen show that the total flux of induction along the circumference of the toroid is approximately constant ; the section being constant, this holds also for the resultant induction $$ t . This finally proves that the resultant magnetic intensity <% t which produces that induction 93, is likewise free from perceptible variations. Since l <$ t = $c -f & (in the algebraic sense, in accordance with (1) 53), it follows finally from experiment, that the enormous variation of the member <&,, which is due to the magnetising coil, is almost com- pensated by that of the demagnetising force &, which also varies along the circumference ; this vector is opposite in direc- tion to the field of the coil, and, owing to the comparatively small values of the latter, it will also be of the same order of magnitude (compare 18, 53, 54). On the other hand, as the flux of induction is only approximately constant, it follows that a few tubes of induction will always emerge, and that, in consequence, magnetically effective end-elements will exist at the curved boundary of 1 Concerning the meaning of the suffixes e, i, t, reference should be made to p. 80, where they are fully discussed. L 146 GENERAL PROPERTIES OF MAGNETIC CIRCUITS the toroid, and the strength of these will determine the necessary compensating variations of the demagnetising force <. We finally come to what, at first sight, is a surprising result, namely, that just because the toroid is only magnetised by a local coil, and therefore a leakage of induction-tubes occurs, the magnetic end-elements thereby produced tend to compen- sate for the inequalities of the field of the coil. Consequently the distribution of the total intensity, and with it that of the magnetisation, as well as of the induction, will be little different from a uniform-peripheral one, which as a matter of fact is what takes place. It is further intelligible why, in the experiments described, a uniform-peripheral distribution of the resultant induction is the more nearly reached the stronger is the magnetising current. For since, so far as we can judge, the maximum susceptibility was not attained with the current employed, this quantity must increase with the current. It is, however, obvious that, other things being equal, the greater the susceptibility, the more completely must the field arising from the magnetisation of the toroid compensate for the inequalities of the field which is due to the magnetising coil. This is in accordance with von Ettingshausen's observations, who found, with a welded toroid of ordinary bar iron, less complete uniformity than with one turned out of a Styrian soft iron plate, of undoubtedly far higher susceptibility. 95. Self-Compensating Effect of Leakage. This self- opposing action of leakage in magnetic circuits which we have observed in the comparatively simple case of the toroid, is a perfectly general phenomenon. Differences in the flax of induction at different parts of the circuit cause some tubes of induction to emerge that is, leakage occurs, so that magnetic end-elements make their appearance, their strength being greater the more unit- tubes emerge. The direction of the magnetic intensity due to these elements will evidently be opposed to the external field in those places where the flux of induction is greatest, and in agreement with the external field in places where the flux is least. In this way the variations of the flux of induction at different places along the circumference are, in a certain measure, automatically diminished. SELF-COMPENSATING EFFECT OF LEAKAGE 147 The self-compensating effect of leakage in magnetic cir- cuits affords a certain analogy with the demagnetising tendency of the faces which bound the interspace. Both actions may, as follows, be brought under one point of view. We have seen ( 50) that any apparent action at a distance is due to local variations in the ' strength ' of the magnetisation. Now it may easily be shown that the magnetic action at a distance arising from these variations, is opposite in direction to the magnetisation at places where the strength is greater, while at places of less strength it is in the same direction; the local variations will therefore indirectly tend to partially neutralise themselves. In the experiments which have been described the field was so weak as to satisfy the conditions of 11 (equation 14), and therefore the magnetisation was proportional to the induction, and, like this vector, it had a solenoidal distribution. It is ac- cordingly sufficient to assume magnetic end-elements on the lateral surface as the direct result of leakage. The theoretical formulation last mentioned includes also, such variations in the strength, as take place in the interior of the ferromagnetic substance when the distribution of the magnetisation is no longer solenoidal, so that its convergence is finite, as may readily occur with higher intensities of field (compare 11, 59). The leakage in the latter case cannot, in general, be deter- mined, nor can we fix the limiting conditions corresponding to a magnetising field whose intensity increases without limit. In any case, however, Kirchhoff's law of saturation will then hold. The question, then, is how the lines of magnetic in- tensity of the external field would run in reference to the geo- metrical configuration of the magnetic circuit ; for in the limiting case, in accordance with the principle in question, these alone will completely determine and direct the other vectors concerned. The answer to this question depends on the nature of the special case we consider, and will usually present considerable difficulties. Even in the simple case of a toroid magnetised by a local coil covering one special part, it depends on the relation of the dimensions of the coil to the diameter of the toroid ; l so that any general enun- 1 Compare in this connection the graphical representation of the lines of intensity of a single circular conductor, Maxwell, Treatise, vol. 2, Plate XVIII. L 2 148 GENERAL PROPERTIES OF MAGNETIC CIRCUITS ciation, not to speak of a solution, of the problem, applicable to all cases, seems out of the question. Finally, since it has been experimentally established and theoretically explained how the value of the vector , is sen- sibly constant throughout a ring magnetised by a local coil, the question arises how the mean value of that vector can be calculated. To this end, we observe that the line-integral of <$ e along a path of integration lying within the toroid is, as before, 4>7rnl, in which n is the number of turns of the local coil, /the current in deca-arnperes (i.e. in absolute C.G.S. measure). The line-integral of <>* is, however, zero in this as in all other cases. The mean value & is found by dividing the mean cir- cumference of the toroid (2 TT r t ) into the sum of the two line- integrals just mentioned, which sum, in this case, evidently also amounts to 4 TT n L Accordingly, ; 1N the same expression which we previously found for the case of a uniform winding [ 72, equation (1)], that is for a uniform peripheral distribution of the magnetising field. We shall con- sequently drop, in the sequel, the tacit or express assumption that the winding is uniform. B. Hopkinsoris Synthetic Method 96. Principles of the Method. We now turn to a mode of treating the magnetic circuit, as ingenious as it is fruitful, which was published in 1886 by Drs. J. and E. Hopkinson. 1 It depends on two fundamental ideas, each of which, again, is based on a mathematical theorem capable of rigorous proof. The starting-point is the consideration of the total flux of induction, the ' conservation ' of which is the first of the principles referred to. This we have already fully discussed or, what amounts to the same thing, we have established the general solenoidal property of the resultant induction, as 1 J. and E. Hopkinson, Phil. Trans, vol. 177, 1. p. 331, 1886. Reprinted in J. Hopkirson, Original Papers on Dynamo Machinery and Allied Subjects, p. 79, New York 1893. HOPKINSON'S SYNTHETIC METHOD 149 expressed by the equation of continuity, and have shown how this principle governs a series of phenomena ( 60-65). In the second place, the fundamental principle is applied that the line-integral of the resultant magnetic intensity & along any closed curve is 4*7rnl, where n is the number of turns of the conductor which embrace the curve, I the current which passes through all of them ( 56). The magnetic circuit is separated into its natural parts, through which the curve of integration successively passes. To each such part of the circuit corresponds a portion of the line- integral in question, the integral being calculated by multiply- ing the mean value & for each part by the corresponding length of the path of integration. An essential assumption here tacitly made is the tendency of the resultant induction SB,, already dis- cussed, to become distributed as uniformly as possible, so that its variations along any one such portion of the curve are but small. From 95 t we find & by equation (3o) p. 147. The several portions of the integral are finally added, and their sum must amount to 4nrnl. We can thus ascertain synthetically the value of the line-integral corresponding to any given or prescribed value of the total flux of induction ,, which we will call M. The relation between these two quantities M and we can represent graphically; and, according to the practice of the Drs. Hopkinson, it is usual to choose the values of the former as abscissge and those of the latter as ordinates. The curve thus obtained, which represents the ' Hopkinson's function/ or t = <& may be called the magnetic characteristic of the corresponding magnetic circuit. In order that the method, here represented in its general features, may be available, the relation between the vectors % t and ,, which, as we know, are coincident in direction ( 54) at each point, must be given for the ferromagnetic substance in question. We can represent it by the equation (3) . . . $< = or, conversely, by 150 GENERAL PROPEETIES OF MAGNETIC CIRCUITS The functions f and <, as well as F H and 3> H , are * inverse functions.' l The former are assumed to be given empirically by the normal curve of induction for the. material in question. 97. Application to Radially-divided Toroids. As it may at first prove somewhat difficult to understand Hopkinson's syn- thetical method, we will once more explain the mode of applying it in the typical case of a toroid with a radial slit. We shall then ultimately see that it leads to the same results as the method which we have developed in Chapter V., and which, at first sight, appears to be totally different. Using the same notation as before ( 75, fig. 15), we assume as a first approximation that the width of the slit d is small, so that the leakage may be neglected. Or, as the Drs. Hopkinson say, we suppose that, by some miracle, the tubes of induction are prevented from emerging from the surface of the toroid, so that they can only pass from one face to the other through the gap. Let 8 [= 7rr 2 2 ] be the cross-section of the toroid, as well as of the slit ; then, on the above assumption, 2 Let us now evaluate the separate portions of the line-integral mentioned above. For this we first consider the slit, where 23, = jQ t ; consequently (see fig. 15, p. 109), E (4) ( Next, in the remaining ferromagnetic part of the toroid, by introducing the mean value of the resultant intensity 4?l, A (5) &dL = & (2 irr, - d) = (2 TTT, - d) f ' in which / is the function defined by equation (3ft). The sum of the two portions (4) and (5) of the integral must, from the 1 In Hopkinson's paper is put =/~ 1 in conformity with the usual nota- tion for inverse functions. 2 It should be remembered that when a symbol is accented, it denotes the value of the corresponding quantity within the ferromagnetic substance. GRAPHICAL EEPEESENTATION 151 preceding paragraph, amount to M = 4?r In. Hence we finally obtain (I) M = 4 irnl = | ( d + (2 7TT, - d) f ( ') = Fx( t ) This equation represents Hopkinson's solution of the problem of the radially-divided toroid. 98. Graphical Representation. Transformation of Curves. In fig. 25 this solution is graphically represented for a concrete case. The toroid is assumed to be of the specimen of iron whose normal curve of magnetisation is represented in fig. 5, FIG. 25 p. 24, by the curve A. Let its dimensions in round numbers be as follows : r t = 10 cm. r 2 = 1 cm. d = O05 cm. Circumference L = 2 7rr T = 62-83 cm Cross-section 8 = 2 7rr 2 2 = 3-14 cm 2 . Katio dr 2 = 0-05. The two terms of equation (I) corresponding to the two por- tions of the integral are now represented by curves (A) and (#), the first of which is obviously a straight line through the origin. If then we add the abscissae of these curves (A) and (#), we obtain a third one (0), which represents the relation sought for between , and M, and which is therefore the magnetic characteristic of the divided toroid. 152 GENERAL PROPERTIES OF MAGNETIC CIRCUITS It is at once evident that this addition of abscissae amounts to the same thing as if we had sheared the curve (B) parallel to the axis of abscissas, by starting from a directrix to the left of the axis of ordinates, symmetrically related to the straight line (A), to the right, and moving the former into coincidence with the axis of ordinates. This points directly to an analogy with the method followed in the last chapter (compare fig. 21, p. 131). In order to explain more clearly these analogies, we shall endeavour to transform the curve of magnetisation pre- viously l given [3 = funct. (&)] into the magnetic characteristic [ t = F H (If)] represented by curve (0), fig. 25. We have already seen, in 13, how curves of magnetisation may be converted into curves of induction by altering the measure of the scale of ordinates. and then by shearing parallel to the ordinates starting from a straight line under the axis of abscissas up to that axis. But the curve of induction may further be transformed into the magnetic characteristic by merely altering the scales for ordinates and abscissae, if we remember that in the latter the ordinate : t = 93, 8 the abscissa : M = <$ e L where 8 is the cross-section and L the circumference of the toroid. It may be left to the reader to carry out graphically the complete transformation of the curve (I?), fig. 5, into curve (0), fig. 25, in the manner described. It will then be seen that the two entirely different modes of representation come to exactly the same thing, as the curves in question finally coincide. 99. Second Approximation. Correction for Leakage. In the further course of their investigation the Drs. Hopkinson drop the simplifying assumption that leakage is prevented, and in- troduce the coefficient of leakage v, which we have already had occasion to employ. We have given as the equation defining it [ 78, equation (17)] (6) ..- " = | [-1] 1 That is in fig. 5, p. 24, the curve (J5) which not only as observed in the text refers to the same material, but is also drawn for the same dimensions as in the example chosen. SECOND APPKOXIMATION. COKKECTION FOE LEAKAGE 153 where ' t is the mean total flux of induction in the ferromag- netic substance, f that in the gap. The resultant induction here is (7) ... *, = J = On the other hand, the mean value of this vector in the ferromagnetic substance is (K\ 9V '>-"< (8) . . . SB,= -g- -5- If we take into account this modification due to leakage in the investigation of 97, equation (I), as is readily seen, becomes to a second approximation (II) M=4 i 7rnl= ^ d + (2 7 rr 1 -^)/ - ^ } = F> S \ 8 ' This is the fundamental equation of ^the Hopkinson method, which may be immediately extended to complicated cases, as we shall see in the following paragraphs. But if, for the sake of better comparison, we take as independent variable 1 in the ferromagnetic substance instead of < in the gap, we have (Ha) M = 4 irnl = d + (2 TTT, - d) f V O \ O If we also plot this latter function F H graphically, and for the same concrete case as the above, we see how, instead of the straight line (J.), which represented the first term in equation (I), we have now the dotted line (A 1 ) (fig. 25, p. 151). The value of Vj in so far as it is constant, is taken from the empirical equation [ 90, equation (30)] : -, . H d v = 1 + 7 - r 2 And hence, since d/r 2 = 0'05, we shall have v = 1*35. We know, further, that it more and more approaches the value unity the more nearly the ferromagnetic substance approaches the point of saturation. 1 The Hopkinson function F H is then 1 The ultimate decrease of v manifests itself in fig. 25 only by the fact that the original straight line (J.') for values above = 40000 is somewhat inclined to the straight line ( A~). 154 G-ENEKAL PEOPEKTIES OF MAGNETIC CIRCUITS obtained by adding the abscissae ; it is now represented by the dotted curve (C r ). The construction formerly given ( 17), the analogy of which with the Hopkinson method was thoroughly discussed, under- goes quite similar modifications by taking leakage into account (fig. 21, p. 131), so that the agreement of the two methods is the same as before. We can. therefore regard the confirmation of the author's theory (Chapter V.) by the experiments of Lehmann, made expressly with this view, also as a confirmation of the Hopkinson method at any rate in so far as it deals with the simple case of a toroid with one radial slit. Of the two methods, which at first appear totally different, sometimes the one has the advantage, and sometimes the other, from the point of view of practical application. 100. Generalisation of the Method. The Hopkinson method thus has the advantage that it can be directly applied to imperfect magnetic circuits of a more general kind than the typical example previously considered. We have already treated theoretically the case of a toroid with several radial slits, taking leakage into account ( 81). Introducing the generalisa- tion of 15, according to which the curved axis of the ring may be any arbitrary plane or tortuous curve, provided only its radius of curvature is always great compared with the greatest diameter of the normal cross-section, this latter may have any arbitrary though invariable form. If, finally, we remove these latter restrictions also, so that the curve may now be sharply bent, and the shape, as well as the area of the cross-section, may be variable, and if we further assume that the ferromagnetic parts consist of different materials, w'e have obviously an im- perfect magnetic circuit of the most general kind possible. We now start from the mean flux of induction in one of the gaps, the length and sectional area of which are L Q and $ respectively. In the other parts, 1, 2, 3, &c., into which we can separate the magnetic circuit, let the values of the flux of induction be v l , v 2 , v 3 , and so on. In like manner, let the particular functions /, which are characteristic of the various ferromagnetic substances forming those parts, be/j,/ 2 , / 3 , &c. [96, equation (3a)]. If we call the corresponding lengths of path J/ n jC 2 , L 3 , &c., and the sectional areas $ 15 8%, $ 3 , &c., we ELECTEOMAGNETIC STEESS 155 ultimately obtain from equation (II), p. 153, the more general equation (Ill) H = 4 TT nl = L + L, f } o + .... = ;p a ( ) In so far as, besides the parts of the magnetic circuit represented by the first term, there are others of indifferent material for instance, the nth part to each of these parts there will correspond a linear term /Y\\ T V V2'r> ( 9 ) L n~T- n since in this case the function f n is simply equal to its inde- pendent variable. This most general equation has, so far, been scarcely tested by purely magnetic experiments. Any attempt at an accurate test would, moreover, be useless, as the method is intended for practical requirements, and, of course, only allows of mean values and rough approximations. We shall revert to this in Chapter VIII., in discussing the chief applications in connection with dynamos, and show how the Drs. Hopkinson succeeded in applying their determination of the electromotive force of the machine as a function of the current in the field-magnets to give a test of the theory. By their measurements the approximate correct- ness of the synthetical method could, on the whole, be confirmed (compare especially 128, 129). C. Electromagnetic Stress 101. Specification of the state of Stress. We will now consider more closely the state of stress which occurs in the ferromagnetic parts of magnetic circuits, or which prevails in the gaps between them. The latter explains the well-known apparent action at a distance, which appears, in general, as an attraction or repulsion between the ferromagnetic parts. To these considerations we will preface a few elementary definitions. 156 G-ENEKAL PEOPEKTIES OF MAGNETIC CIRCUITS By stress is understood, in general, a system of forces which tend, not to move a body, but to strain it. ' The stress produces, in general, a strain. The closer investigation of the latter, or of its relations to the stress, constitutes a problem of geometry, or of the theory of elasticity respectively. Every stress is to be expressed as a force per unit area, and has therefore, in absolute measure, the dimensions [L~^MT~' r \. There are various elementary forms of stress ; the most impor- tant are shearing stress, pull or tension, and thrust or pressure. That being premised, we may mention that in the theoretical part ( 65) we have already expressed in a few equations the most general state of stress in a magnetic body, as deduced mathematically by Maxwell. Making the simplification that the induction S3 and the magnetic intensity * have the same direction, which, on the assumption always made of isotropy and absence of hysteresis ( 54), does actually hold, those equations assume an elementary form ; and, in accordance there- with, the stress may be completely specified in terms of the two following elementary forms : I. A (hydrostatic) pressure, the same in all directions, and equal in absolute measure to ' 2 /S7r. II. A simple tension in the direction of the lines of induc- tion, whose value in absolute measure is 3B''/47r. The investigation of the strains 2 due to the electromagnetic stress concerns the theory of elasticity. The test whether such strain is admissible, from the point of view of construction, is a question of strength of materials. From neither of these points of view need we follow the question in this book; but its importance is apparent from what has been said, for, as we shall see, the stress may in some circumstances be very great. 102. Resultant Tension in the Gap. In the preceding paragraph the stress in a ferromagnetic body is fully specified. In the sequel we shall consider only that one of its manifestations which is of greatest experimental and practical interest ; we 1 Since, in the sequel we shall only deal with the resultant intensity, and the resultant induction, we shall drop the prefix ' resultant ' as well as the corresponding index t from the corresponding symbols ( 63). 2 We have already alluded to stresses which occur in magnetisation, and to the small charges of size and shape of ferromagnetic bodies due (at any rate partly) to them. .RESULTANT TENSION IN THE GAP 157 shall confine ourselves to an investigation of the resultant tension in the direction of the lines of induction. This vector (in the ferromagnetic substance) we shall call 3'- We obtain its value if, from that of the simple tension in the direction of the lines of induction (2), we subtract the pressure mentioned in (1), which of course acts in the direction in question, as well as in all others. We accordingly obtain the equation ' 3 ' = Let us now consider an infinitely narrow slit in the ferro- magnetic substance at right angles to the direction of the lines of induction and intensity, which does not produce either demagnetising actions or leakage. As this will in practice be always more or less the case, we shall speak of such a break in the continuity of the ferromagnetic substance as an ideal slit. and 95 are there identical, as in an indifferent substance, and are equal to 23' in the ferromagnetic substance, as follows from the principle of the normal continuity of induction ( 58) ; hence the resultant tension 3 in the ideal slit itself is (IV) 3 = j 1 - & - ^- & = jp- & = o ' 2 4-7T 07T 07T O7T This value for 3 i n the narrow gap is considerably greater than that for 3' in the ferromagnetic substance. We now take the difference of the two quantities and introduce the magnetisa- tion, remembering the fundamental equation $8' -' = 4 IT 3 [ 11, eq. (13)] By subtracting the above equation (10) from (IV) we obtain or (11) 3 -3' = (*' -*') 2 = This difference 2 TT 3* between the values of the resultant tension in the ferromagnetic substance and in the ideal slit, 158 GENERAL PROPERTIES OF MAGNETIC CIRCUITS can only be due to the two terminal faces of the latter. On these, magnetic end-elements are present,. the strength of which per unit surface is + 3 or 3. As we have already proved ( 21), an apparent attraction is exerted by the magnetic end-elements of one surface on those of the other, the latter being of opposite sign. In the language of the old theory, the two faces, being charged with imaginary fluids of opposite signs and uniform density it 3, attract one another, just like the two plates of a plane air-condenser charged to the electric surface-density 3. This attraction may be simply calculated, 1 and has the value, 2 TT 3 2 per unit area, which we found in equation (11). 103. Theoretical Lifting Force of a Diametrically-divided Toroid. Let us first consider for the sake of simplicity a toroid uniformly and closely wound, and having a uniform peripheral dis- tribution of magnetisation, while at the same time it is cut through radially at two diametrically op- posite places (fig. 26). Let us assume the faces quite smooth, and polished, so that the two halves of the toroid fit closely to each other. The width of the cut is then as small as possible, so that it differs as little as pos- sible from the ideal cut postu- lated in the last paragraph. Its demagnetising action 2 as well as the leakage we may assume to be infinitely small ; how far this is actually the case will be sub- sequently discussed. Let us now investigate the attraction which the two halves exert on each other that is, supposing we determine it by hanging weights to the lower half, 1 See, for instance, Mascarfc and Joubert, Electricity and Magnetism, vol. 1, 81. 2 Compare the theory of toroids with more than one radial slit, 81. FIG. 26 LIFTING FORCE OF A DIVIDED TOROID 159 the greatest lilting force of the diametrically-divided toroid. The double sectional area being $, the force of attraction 3r of both slits is (12) - O 7T The value of the force 3f is given by (12) in absolute measure, i.e. in dynes, provided SB' and S are expressed inC.G.S. units. If, on the contrary, it is to be expressed in kilogrammes- weight, and denoted by ^ l for distinction, we have (12.) . . 8f I = 3 l S where g is the acceleration of gravity ; if we take the integral number 1 981 cm. per second per second for this we have or with an approximation 2 sufficient for most purposes sooo Considering that under ordinary circumstances the value of induction practically attainable in a soft iron toroid is scarcely 20,000 C.G-.S. units, it follows from the approximate formula (126), that the corresponding tension, that is the lifting force per unit area, is about 16 kilogrammes- weight per cm 2 . This 1 Strictly speaking, the lifting force of a magnet depends on the latitude, owing to the variability of g. 2 As the dimensions of a tension are the same as those of a hydrostatic pressure, or more generally as those of any stress ( 101), it can also be ex- pressed in atmospheres; one atmosphere is equal to 1*0136 megadynes per cm. 2 ; introducing this into the equation, and distinguishing the vectors by the suffix 2, we have ~S 25400000 or again, with tolerable approximation, Thus from equation (12&) we obtain the tension in kilogrammes-weight per cm. 2 about 1-5 per cent, too small (compare Table VI, p. 166), and from the equation(12c-) in atmospheres too great by about the same percentage. 160 GENERAL PROPERTIES OF MAGNETIC CIRCUITS value is therefore practically the upper limit for the tension. The highest attainable value 33 = 60,000 C.G.-S. would theoretically correspond to a pull of almost three hundredweight (144 kilo- grammes-weight) per sq. cm. (compare 175). 104. Resolution and Interpretation of Maxwell's Equation. Maxwell's equation (IV) p. 157 for the resultant tension in the gap 3=6- S' 2 O 7T may be resolved into three terms by once more taking into account the fundamental relation We have then 3 = O 7T or (13) 7T Now it may be shown that, assuming for a moment a direct action at a distance, each of the three terms of equation (13) has its own significance, which can be physically interpreted, and which in conclusion we will briefly discuss. 1. 2 TT 3 2 corresponds to the true magnetic pull between the two halves of the toroid, which can be 'represented by the attraction of the fictitious fluid on the faces bounding the slits ( 102). This term, for instance, would alone come into play if the magnetisation were solely due to hysteresis of the material; that is if the value of 45 1 , being zero, 3 had a finite value depending on magnetic retentiveness. Under ordinary cir- cumstances this first term of equation (13) far exceeds the two others. 1 2. 3<' corresponds to the electromagnetic pull of the lower half of the magnetising coil on the upper half of the ferro- magnetic toroid and vice versa. 1 Stefan (Wiener BericJite, vol. 81, p. 89, 1880) has given a formula for the lifting force, which contains this term only. Its approximation to the complete equation (13) is accordingly of the same order as that obtained when we write S3' = 4*3 (11, 59). MAGNETIC LIFTING FORCE 161 3. <' 2 / 8 TT represents the purely electrodynamic action of the two halves of the magnetising coil upon each other. If the windings are not closely coiled on the toroid, as has been assumed above, so as to form as it were a single rigid body, it may become necessary to distinguish carefully the three terms. A detailed discussion l of all possible conditions of experiment would lead too far. It need merely be observed that in practice the last (electrodynamic) term 1 / 8 TT ' 2 is very small in com- parison with the second (electromagnetic) one, which in turn, as we have said, is considerably exceeded by the first (magnetic) term ; the right-hand side of the equation is thus arranged according to decreasing values of its three terms. D. Magnetic Lifting Force 105. Older Investigations. The theory of the lifting force of magnets discussed in the foregoing paragraphs is as com- prehensive as it is simple; it is fully expressed in Maxwell's fundamental equation (12) or (I2a). But, however simple the theory, it is difficult to test it by experiments not open to objection, so that it can scarcely be considered as having up to now been rigorously established by experiment throughout the entire available range of induction. It seem indeed probable that the divergencies which up to the present exist are not to be ascribed to the insufficiency of the theory, but to the peculiar difficulties met with in the method of detachment which at first sight appears so easy. A great number of investigations of all kinds have been made on the lifting force of magnets of the most varied form, electromagnets as well as permanent magnets. Only those of Dub, Lament, Nickles, du Moncel, dal Negro, Joule, von Walten- hofen, W. v. Siemens, &c., need be mentioned, 2 some of them having led to a series of empirical formulae for the lifting force. We shall here restrict ourselves to a discussion of some of the more recent investigations; in the first place, because these 1 Compare S. P. Thompson, Phil. Mag. [5], vol. 26, p. 70, 1888 ; du Bois, iUd. vol. 29, p. 294, 1890 ; Shelford Bidwell, ibid. vol. 29, p. 440, 1890. 2 See the summaries in the following works : Dub, Elektromagnetismus, Leipzig, 1861; G. Wiedemann, LeJire v. der MeJttriritat, 3rd edition, vol. 3, 666-682 and 717-745, Brunswick, 1883 ; Silv. P. Thompson, The Electro- magnet, Cantor Lectures, London, 1890. M 162 GENERAL PEOPERTIES OF MAGNETIC CIRCUITS have been made either from the point of view of the theoretical principles laid down in the preceding paragraphs or with a view of testing them ; and, in the second place, because they are based on the absolute system of measurements, and this gives them far greater value than the older researches. For in- superable difficulties in interpreting or critically discussing those earlier results arise from the circumstance that in these, for some reason or other, either only relative measurements or none at all are given by their authors. 106. Wassrnuth's Experiments. Wassmuth was the first to make accurate observations based on absolute measurements, and his experiments were arranged l with the express object of satisfying the conditions of theory as far as possible. He used 1. An iron toroid (r l = 5'84 cm. ; r 2 = 0'30 cm. ; 2 8 = 0*565 sq. cm. ; see fig. 15, p. 109). 2. A welded hoop of rolled iron (r { = 5'69 cm. ; p = 0*55 cm. ; ?= 1-97 cm.; 28= 2-17 cm 2 , fig. 14, p. 106). Both were cut through diametrically, and the surfaces of contact were carefully ground and polished ; it was then possible to detach both ends of the armature (that is, the lower half of the ring) ' almost simultaneously ' by means of a spring balance. Both the upper and the lower half were closely wound with primary windings, the magnetisation being determined by measuring the electromotive force induced in secondary coils wound on the armature. The following maximum values were obtained, which are comparatively small, especially when compared with those given in the following paragraphs : With 1: = 93 C.G.S.; 3 = 1308 C.G.S. ; 3=8'4 kilo- grammes-weight per square centimetre. With 2: =138 C.G.S. ; 3 = 1157 C.G.S. ; 3=6'6 kilogrammes-weight per square centimeter. Wassmuth found with reference to the relation between 3 and 3 that the former quantity does not increase so rapidly as 3 2 , as it should do from the first term of equation (13), but that it increases more rapidly than 3 ; at the same time a peculiar 1 Wassmuth, Wten. Ber. vol. 85, p. 327, 1882. BID WELL'S EXPEKIMENTS 103 behaviour was observed in the region of maximum susceptibility. Similar results had been obtained by W. von Siemens in the investigation mentioned in 105 on what may be regarded as very broad hoops that is to say, pieces of tubing cut parallel to their axis. This investigation need not be further discussed here. 107. Bidwell's Experiments. Sources of Error. Experi- ments have further been made by Shelford Bidwell with a welded toroid of soft charcoal iron (r l = 3'76 cm. ; r 2 = 0'24 cm.) cut through diametrically. 1 The contacts were finely polished, but had retained nevertheless a slight convexity. Each half was closely wound with nearly 1,000 turns of wire ; in conse- quence of this the relatively strong field intensity of 585 C.G.S. units was obtained, the pull amounting to 15'9 kg.-weight per cm. 2 , and being therefore almost equal to the value given above ( 103) as the maximum which could in practice be obtained ; such high values as these had, at that time, not been reached by any observer. Bidwell made no measurements of magnetisation or of induction by means of secondary coils, so that his experiments are not suited for testing equation (13) experimentally. He rather assumed this as correct (restricting himself to the first two terms), and by its aid he calculated from the observed values of the pull the corresponding magnetisation as a function of the magnetising field. In 226 we shall return to this method, which forms the foundation of certain modern methods of measurement and apparatus. It will now be proper to consider the sources of error which always affect the apparently simple determination of the lift- ing force by detachment. We can no more realise in practice the theoretical assumption of an infinitely narrow slit ( 102) than we can neglect the demagnetising and leakage-effects due to the slits ; in short, their treatment as ideal cuts is in- admissible. For, however carefully the ends in contact are planed and polished, experiment shows that each slit gives rise to irregularities of that kind which can only be got rid of by using strong external pressure, and which perhaps the 1 Shelford Bidwell, Proc. Roy. Soc. vol. 40, p. 486, 1886. M 2 164 GENERAL PROPERTIES OF MAGNETIC CIRCUITS existence of a surface of separation necessarily entails. (Com- pare 151.) On the other hand, the grinding and polishing of the surfaces have the disadvantage that the natural adhesion may be con- siderable, so that the lifting force appears too great, especially with feeble magnetisation. Yet the chief source of error may perhaps be sought in the indefiniteness of the ' detachment ' of such surfaces in contact, especially where two places of con- tact are concerned. Wassmuth, as stated above, speaks of an 1 almost simultaneous ' detachment ; but this seems undoubtedly to mean that, in fact, one place of contact is broken first, by which an air-gap is formed which must at once demagnetise the entire toroid ; owing to the decrease of induction thereby produced the other contact must also soon yield. It is questionable whether, with the otherwise favourable conditions of the toroid, less reliable results are not attained than with a bisected bar in which there is only one place of contact ; an elongated ovoid divided equatorially would probably be most suitable [see 108]. Yet even with a single contact there is uncertainty, as has always been shown since the earliest of such determinations. The more recent experiments on the action of such surfaces to be men- tioned further on ( 15, &c.), will partly explain this uncertainty, but we are not in a position to remove it. It must be assumed that a loosening of contact gradually occurs, and that in general this will first give way in one place ; in this case we determine, as in all experiments on strength of materials, the resistance of the weakest part, and not the mean resistance, which is the really important matter. Just as magnetisation gives rise to a condition of stress in the ferromagnetic substance, so also a stress produced by external forces exerts an influence on a magnetisation already existing. 1 This action is however comparatively small, so that errors arising in this manner from the feeble external forces which come into play in experiments on detachment may be neglected in comparison with the sources of error already mentioned. 108. Bosanquet's Experiments. Bidwell has likewise (loo. cit.) described some measurements on divided bars, but in this 1 For the details of this phenomenon we refer to Ewing, Magnetic Induc- tion in, Iron (chapter ix.). BOSANQUET'S EXPERIMENTS 165 case also he did not determine the induction. This case has been the subject of measurements by Bosanquet. 1 The arrange- ment he adopted is represented in fig. 27, the cylindrical iron core consisting of two pieces each 20 cm. long by 0'526 cm. in diameter, and each wound with 1096 turns of wire. The upper electromagnet was rigidly fixed to a table, while the lower one, together with its magnetising coil, had freedom to move up and down between two brass guides (not shown in the figure) with as little friction as possible. The ends of the two pieces were ground against each other so as to ensure perfect contact. To the lower electromagnet was attached a scale- pan ; the weight was counterpoised, as shown in the figure, so that the weights placed in the pan gave a direct measure of the lifting force. Near the division was a small secondary coil, so that the induction could be measured by reversing the primary current. Weights were added until the lower electromagnet was detached, and struck against a stop, at a few millimetres distance. The highest value obtained for the pull was 14-6 kg.-weight per square cm. with an induction of 18,500 C.G.S. units. On the whole, the equation 1 8000 irg may be regarded as roughly confirmed by Bosanquet's experi- ments. The deviations may be attributed to the various recog- nised sources of error. 2 It follows from all this that Maxwell's fundamental law of magnetic tension as expressed by equation (12) 103 is 1 Bosanquet, Phil. Mag. [5], vol. 22, p. 535, 1886. 2 Besides the irregular deviations, which are to be referred to the un- certainty of the detachment already discussed, Bosanquet constantly found too high lifting forces, especially for small values of the induction. This might be due to friction against the brass collars besides the ordinary adhesion of the surfaces in contact. The mutual electrodynamic action of the two coils also was greater than $' 2 $/8 tr (corresponding to the last term of equation 166 GENEKAL PEOPEETIES OF MAGNETIC CIRCUITS to be considered as experimentally confirmed with tolerably close approximation; experiments perfectly free from objection would, however, be very desirable. 1 We give finally in Table VI a series of corresponding values of the induction -35' and of the tension 3 ; in the fourth column the values 2 are given according to the rigorous equation (12a), in the third according to the convenient approximate equation (126) ; the two sets of values agree closely, as will be seen. (Compare the remark p. 159.) TABLE VI S3' C. G. S. unit 25' 5000 3 Kg.-weight per sq. cm. Eq. (12&) Eq. (12) 1000 0-2 0-04 0-041 2000 0-4 0-16 0-162 3000 0-6 0-36 0-365 4000 0-8 0-64 O649 5000 1-0 100 1-014 6000 1-2 1-44 1-460 7000 1-4 1-96 1-987 8000 1-6 2-56 2596 9000 1-8 324 3-286 10000 2-0 4-00 4-056 11000 2-2 4-84 4-907 12000 2-4 5-76 5-841 13000 2-6 6-76 6-855 14000 2-8 7-84 7-550 15000 3-0 9-00 9-124 16000 3-2 10-24 10-39 17000 3-4 11-56 11-72 18000 3-6 12-96 13-14 19000 3-8 14-44 14-63 20000 4-0 16-00 16-23 (13)), for, owing to their considerable diameter, their sectional area may be far greater than that of the bar. 1 In connection with this it may be mentioned that the corresponding Maxwellian equations for paramagnetic and diamagnetic substances of con- stant susceptibility, which, for our present purpose, we regard as magnetically indifferent, have been completely confirmed by experiment ; so far, at any rate, as the experiments of Quincke and others with the ' CT-tube ' methods extend. See du Bois, Wied. Ann. vol. 35, p. 137, 1888, where the literature on the subject is collated. (Compare also 203.) 2 Taken from the Cantor Lectures on the Electromagnet by Professor Silv. Thompson. [Since the above was originally published, experiments by Threlf all appeared in the Phil. Mag. for July 1894, made by a method essentially like that of CONCLUSIONS FKOM MAXWELL'S LAW 167 109. Conclusions from Maxwell's Law. Maxwell's law constitutes the common foundation on which are based all considerations as to magnetic tension, attraction, lifting force, &C. 1 Maxwell, who only mentions the law casually in a few sentences, 2 has thus laid the foundation for a rational under- standing of this question, while the great number of experi- mental investigations previously made were unable even remotely to suggest an explanation. We shall discuss in conclusion some aspects of the question, which are to be kept in view, in applying the law to the various kinds of magnetic circuits occurring in practice. At each point of the surface of an ideal slit [equation (IV) 102] the magnetic tension that is, the pull per unit surface of the gap is 3 = _L^ I - ,| If now the induction 93' has the same value over the whole area 8 of the surface, as was assumed in the theoretical treatment of the toroid ( 103), the flux of induction through it is' ' = 95' S hence the whole force $ due to this tension will be 1 1 '2 This equation may be expressed in words as follows : When the induction is given, the tension is directly proportional to the area; when the flux of induction is given it is inversely pro- portional to the area. The latter principle, which at first appears somewhat strange, Bosanquet. Mr. Taylor Jones, acting on the suggestions contained in the text, made a complete and careful investigation on the validity of Maxwell's law by the divided-ovoid method. He found it confirmed for inductions up to 19,000 C.G.S. units, so that his experiments appear to settle the question so far, it being left to the investigation of a higher range of induction to pro- duce further evidence (PHI. Mag., March 1895). H. du Bois.] 1 In many special cases, Coulomb's law is as before conveniently made use of, 21. This, however, is closely related to that of Maxwell. 2 Maxwell, Treatise, 2nd edit. vol. 2, 642. 168 GENEEAL PEOPEETIES OF MAGNETIC CIECUITS is a simple deduction from the quadratic law of tension. To under- stand this more clearly, let us imagine a- magnetic circuit, and first of all without any leakage, so that the flux of induction has the same value in every part of it. The total tractive force on a section will then be greater the smaller the section, and will in fact be inversely proportional to it. The unavoid- able leakage entails, however, an important limitation, so that actually, as the cross-section is diminished, the tractive force will attain a maximum value, and will then again decrease. It is here expressly assumed that the flux of induction is in some way or other kept constant, so that when the section is diminished at one or more parts of the magnetic circuit, a greater mean magnetic intensity is required. For with given values of the magnetising current, and of the number of windings, every such { throttling ' of the magnetic circuit will directly produce a diminution of the mean flux of induction ; that follows directly from the Hopkinson theory, or from the considerations in the following chapter. 110. Load-Ratio of a Magnet. There are numerous experi- mental confirmations, or simple examples of the conclusions in the foregoing paragraphs, for which we may refer to Chapter IX. as well as to the work of Professor Silvanus Thompson already quoted. A further deduction from the fundamental law 7T is this, that the question so frequently occurring in the older literature as to the ratio of the lifting force to the weight of a magnet or electromagnet is quite unimportant. For with similar electromagnetic systems, with currents proportional to the linear dimensions ( 67), the values of the magnetic intensity, and consequently of the induction, will be the same in corre- sponding places. Hence the lifting force of corresponding surfaces of contact in the two systems, will be proportional to their section that is, to the square of their linear dimensions. On the other hand, the weight of the magnet itself is pro- portional to the third power of its linear dimensions ; and it follows from this that the load-ratio (lifting- force /weight) in similar magnets will be inversely proportional to the linear LOAD-KATIO OF A MAGNET 169 dimensions. Thus large magnets have in this respect less favourable dimensions, while the relative load-ratio may theoreti- cally be infinitely increased by making the magnet small enough. This is in accordance with experiment. Silv. Thompson mentions l a small electromagnet weighing 0*1 gramme which could lift 250 grammes that is, 2500 times its own weight. If the induction over the area of contact is not constant but variable, the resultant force $ is manifestly given by the surface integral taken over the section 8. This value, by a well-known theorem, will be greater than 1 1 ' 2 $' = __ ro/2 .0 __ _ STT* b ' Sir 8 which would be obtained by introducing the arithmetical mean value of the induction An unequal distribution of the induction is therefore in a certain sense advantageous, provided care is taken to ensure the constancy of the value of its surface-integral over the cross- section of the magnetic circuit the constancy, that is, of the flux of induction. 1 Silv. Thompson, loc. cit. p. 34. It is there shown how the relation deduced above may be so expressed that the lifting force is proportional to the two-third power or the f root of the magnet's weight which agrees with the older Bernouilli-Hacker empirical formula (see also Phil. Mag. [5], vol. 26, p. 70, 1888). 170 CHAPTER VII ANALOGY OF THE MAGNETIC CIRCUIT WITH OTHER CIRCUITS A. Historical Survey 111. Older Development. First Stage. The idea of an analogy of magnetic systems with systems of circuits of various kinds (hydrokinetic, thermal, galvanic) is already, in 1761, met with in Euler. 1 In contradistinction to Poisson's hypothesis of two fluids ( 27), which at that time had not yet been elaborated, he assumed a single subtile kind of matter, which was supposed to flow with great velocity through the magnets, as well as through the surrounding air-space. The ferromagnetic sub- stance offering a far less resistance than the indifferent sur- rounding medium, it tends preferably to pass through that substance. The paths of that subtile matter are assumed by Euler to be identical with the lines mapped out by iron filings, and he therefore lays considerable weight on the determination of these figures. From the tendency of the magnetic matter to take its course, as far as possible, through the ferromagnetic substance, which is supposed, for this purpose, to possess innumerable fine channels, he explains a number of phenomena in a manner which surprisingly suggests our present mode of expression. Those very elementary letters and figures contain a remarkable forecast of views which were only to be completely developed more than a century later. In the later literature we repeatedly find indications very obscurely conceived, it is true of magnetism . being some- thing flowing in closed paths, combined with the assumption 1 Euler, JBriefe an eine deutscJie Prinzessin, vol. 3, pp. 95-150, Leipzig, 1780. These considerations apply exclusively to permanent magnets, since the connection between electricity and magnetism was at that time unknown. FAKADAY, MAXWELL 171 that this flow takes place more easily in iron than in indifferent media. Gumming, 1 so long ago as 1821, made researches on what, in this sense, was called magnetic conductivity. In the various writings, moreover, of Ritchie, Sturgeon, Dove, Dub, and de la Rive the theory of such closed magnetic circuits is more or less clearly discussed. Joule, who, in the first years of his scientific activity, occupied himself greatly with the investigation of electromagnetic machines, enunciates in one place 2 the following principle : The maximum power of an electromagnet is directly propor- tional to its least transverse sectional area. On the other hand (loc. cit. p. 36), he maintains that the { resistance ' to induction varies in the direct ratio of the length of the (closed) electromagnet. If we compare these and some other statements in Joule's writings, we obtain an approximately correct representation of the modern views on this subject, which only wants the mathematical clothing. 112. Continuation (Faraday, Maxwell). The question entered into a more advanced phase by the conceptions intro- duced by Faraday, to illustrate which, he was fond of using lines offeree ( 65) . 3 His theoretical views were, indeed, neither under- stood nor appreciated by most of his contemporaries, although his experimental inquiries evoked the highest admiration. It can- not, indeed, be denied that much want of clearness attached to Faraday's views, which was only afterwards gradually removed, mainly by the exertions of Maxwell. Faraday showed, in the first place, that his lines of force must always form closed curves, the course of which is influenced by the magnetic < conductivity ' of the medium traversed. He, also, was the first to compare an electromagnet with a voltaic pile, and, in order to complete the analogy, he supposed it immersed in an electrolyte, the finite conductivity of which was the analogue of the finite permeability, equal to unity, of the air surrounding the ferromagnetic substance ; for in the ordinary arrangement of electrical circuits the conductivity of the sur- rounding medium is obviously zero, or, at any rate, excessively 1 Gumming, Cambridge Phil. Soc. Trans. 1821. 2 Joule, Reprint of Scientific Papers, vol. 1, p. 34, London, 1884. 3 Faraday, Exp. Res. vol. 3, pp. 328-443. 172 ANALOGY OF THE MAGNETIC WITH OTHEE CIRCUITS small. The analogy in question was frequently referred to by later authors ( 123, 133). To Maxwell is due the merit of having elucidated Faraday's views, and of clothing them in a mathematical form. Instead of lines of force, he introduces the conception of tubes of induc- tion ( 63). In one part of his work he expresses himself as follows : 1 ' The problem of induced magnetism, when considered with respect to the relation between magnetic induction and mag- netic force, corresponds exactly with the problem of the conduc- tion of electric currents through heterogeneous media. 1 The magnetic force is derived from the magnetic potential, precisely as the electric force is derived from the electric potential. The magnetic induction is a quantity of the nature of a flux, and satisfies the same conditions of continuity as the electric current does. ' In isotropic media the magnetic induction depends on the magnetic force in a manner which exactly corresponds with that in which the electric current depends on the electromotive force. The magnetic inductive capacity in the one problem corresponds to the specific conductivity in the other.' 2 These principles form the kernel of the later development of the subject. 113. Continuation (Lord Kelvin). In his theory of mag- netism Lord Kelvin has repeatedly 3 dwelt on the complete analogy which exists between the mathematical theories of magnetic induction, of dielectric polarisation, and of Fourier's theory of thermal conduction, on the one hand, and the theory of certain hydrokinetic processes on the other. He has further shown, referring to the above-mentioned speculations of Euler, that the vector which we have called the magnetic intensity $ cannot be likened to the velocity of an incompressible liquid, for this leads logically to the assumption of a creation or destruction of that liquid in the places where the 1 Maxwell, Treatise, 2nd edition, vol. 2, p. 51. 2 Ibid. ; the last passage obviously refers to the assumption of a constant permeability ; its significance, as we shall presently see, is essentially limited by the fact that this assumption is not in accordance with the facts. 3 Sir W. Thomson, JRepi'int of Papers on Electrostatics and Magnetism, Arts. 27, 31, 32, 41, 42. LOKD KELVIN 173 magnetic end-elements would exist in the corresponding electro- magnetic system. Lord Kelvin showed at that time (1872), on the contrary, that it is the induction 33 l which must be taken in this case as the analogue of the velocity of an incompressible liquid, and he enunciates the following principle, among others (loc. cit. 576) : The resultant force defined electromagnetically for the space occupied by the magnet, and the resultant magnetic force according to the unambiguous definition for space not occupied by the magnet, agree everywhere in magnitude and direction with the velocity in a possible case of motion of an incompress- ible liquid filling all space. In order to make the analogy more complete Lord Kelvin assumed a special kind of medium (loc. cit. art. 42) ; that is to say, a porous solid of infinitely fine-grained texture, through which filters an incompressible frictionless liquid ; its motion must be irrotational, or in other words it must possess a velocity potential. 2 The more permeable is such a porous medium, the greater will be the flow 3 of the liquid in comparison with the kinetic energy of unit volume of the space occupied by the body in question and the liquid. The ratio of these two quantities affords therefore a measure of the permeability of the porous body ; it may be called the hydrokmetic permeability. Having regard to the existing analogies, Lord Kelvin pro- posed to extend the conception of permeability and to denote 1 The vector Jg> was called by Lord Kelvin (loo. cit. 479) the 'resultant magnetic force,' provided we are dealing with some point in the magnetically indifferent region. At points within the ferromagnetic substance, he further distinguishes between the ' polar definition ' (<>) and the ' electromagnetic definition' (S3) of magnetic force; according as the cylindrical cavity (Chap. III. 51), with its axis in the direction of magnetisation, has a diameter very small compared with its length, or a length very small compared with its diameter. He expressly opposes (JMatli. and Phys. Papers, vol. 3, p. 478) the name 'induction' given to 33 by Maxwell. (Compare Chap. I. p. 13, footnote.) 2 If the components of velocity satisfy the hydrodynamical equations of irrotationality (eq. (7), 39), the distribution of the velocity, from what has been said, is a lamellar one, and this vector has therefore a 'velocity potential.' 3 By flow is understood the quantity of liquid which per unit time passes through unit section of the space occupied by the porous body and the liquid. 174 ANALOGY OF THE MAGNETIC WITH OTHER CIRCUITS by the same expression the analogous property in the four analogous theories adduced by him ; in addition to the hydro- kinetic permeability already defined, he therefore distinguished (loc. cit. art. 31). 1. Magnetic Permeability, ratio of magnetic induction to magnetic intensity. 2. Dielectric Permeability; identical with dielectric constant, or [specific inductive capacity]. 3. Thermal Permeability ; identical with thermal con- ductivity. These analogies are then illustrated by special examples, with calculations and graphic representations. The discussions in question comprise (loc. cit.) more than fifty articles, of which we can here only give a short sketch. Having regard to the modern developments and the wide diffusion of the ideas which form its basis, those researches are at the present time of the greatest interest. 1 114. Summary. It may from all this be reasonably maintained, that the mathematical analogy of the theory of electromagnetic systems with other well-known physical theories, has not only been suggested by the publications of Euler, Faraday, Maxwell and Lord Kelvin, but even worked out in all particulars, especially by the last mentioned, and in an un- assailable manner. Correct interpretation only was needed to apply these mathematical results to practical problems. The introduction of these views into the domain of applied science, to which we shall now turn our attention, has only been effected within the last decade. If we consider that many errors have thus crept in which might lead to obvious incon- sistency, we can understand why this modern development has hitherto been little considered by some physicists, being re- garded as not even new, and at the same time partly incorrect. We shall proceed most safely if we maintain from the outset that that modern development plays, and will always 1 It may finally be observed that besides the purely mathematical hydro- kinetic analogy a more profound hydrodynamical one is discussed (loc. cit. art. 41) ; in connection therewith the experiments of Schellbach and of Guthrie on repulsion and attraction due to vibrations in a liquid are discussed ; this subject, as is well known, has recently been subjected by Bjerknes to detailed mathematical and experimental treatment. RECENT DEVELOPMENT (ROWLAND) 175 play, only a subordinate part from the purely scientific point of view. Its undoubtedly great success in practical applications compels us, however, to deal with it more in detail ; we shall, therefore, endeavour at the outset to anticipate such errors as are most likely to arise, and thereby prevent our attaining incorrect results. We will, however, first extend the historical notices over the last decade, giving them in the chronological order of their publication. 115. Recent Development (Rowland). In September 1884 Rowland proposed a formula for the number of lines of force in the field magnets of a dynamo. 1 He formed a frac- tion the numerator of which was the product of the current into the number of turns (what is now called the ampere- turns) ; the denominator was a complicated expression which was to represent the ' magnetic resistance ' of the iron and of the air ; the leakage being taken into account. We have spoken of this proposal of Rowland, in the first place, because mention of it is to be found in an extended experi- mental investigation, published as long ago as 1873. 1 He then wrote [loc. cit., p. 145, equations (3) and (4)] the fraction in question as follows : (i) . . . . Q' = f which in our notation, to be afterwards described, would be expressed by the formula (2) . . . . = Rowland's R, our ~, is defined by him as the magnetic resist- Jj ance per unit length. Hence in the above formulge numerator and denominator have only to be multiplied by L, the length of the portion of magnetic circuit considered to bring them to the form which is now most usual. [ 119, equation (I).] The principle, so important in practice, that the magnetic 1 H. A Rowland, Electrician, vol. 13, p. 536, 1884. 2 Ibid., Phil. Mag. [4], vol 46, p. 140, 1873. 176 ANALOGY OF THE MAGNETIC WITH OTHER CIRCUITS resistance is to be made as small as possible was applied in 1879 in the Elphinstone-Vincent six-polar dynamo machine, 1 and was expressly discussed by the inventors ; but that this machine, owing to practical difficulties in the construction, has found no general reception, by no means diminishes its historical interest. 116. Continuation (Bosanquet). In March 1883 Bosan- quet developed the mathematical analogy of magnetism with elec- trical conduction, 2 and made experiments in this direction with closed rings. He first used the expression 'magnetomotive force' as an analogue of electromotive force. He denned the former quantity simply as a difference of magnetic potential, just as in many cases electromotive force is only another expression for difference of electrical potential. Instead of taking the mag- netic intensity <> as the starting-point, he took its line-integral that is, the magnetomotive force M just defined. Against the adaptation of this conception, which had long before been intro- duced, nothing was to be said. The name alone was new, and we cannot consider it as a happy choice. The same objection may of course be as well raised against its prototype, ' electro- motive force,' which is, however, so generally adopted that there can be no question of changing it. Bosanquet further retained the induction 95, and called Mj 95 the magnetic resistance, although it seems from his paper as if he saw that the cross-section should also have been taken into account. In fact, in the analogy carried out by him, the flux of induction , and not the induction 95, corresponds to the electrical current. If, then, a magnetic resistance is at all to be introduced, we must choose thv3 ratio M/@, as has since then been done by all authors, and not Mj 95. 117. Continuation (W. von Siemens). In 1884 Werner von Siemens, in his ' Contributions to the Theory of Magnetism,' 3 occupied himself with this subject, experimenting with closed 1 See Silv. Thompson, Dynamo- Electric Machinery, 3rd edition, p. 211 ; 4th edition, p. 172. A comparison of the different editions of this book gives a true picture of the historic development of this subject. 2 Bosanquet, Phil. Mag. [5], vol. 15, p. 205, 1883; and vol. 19, p. 73, 1885. 3 W. von Siemens, Berl. Sitzungsberickte* Oct. 1884 ; Wied. Ann. vol. 24, p. 93, 1885. W. VON SIEMENS 177 and open electromagnetic circuits. He summarises his results as follows in the appendix to his ' Recollections ' : 'We may thus, after Faraday, also conceive the magnetic action at a distance as proceeding from molecule to molecule, or from volume-element to volume-element, and are then justi- fied in applying to magnetism the laws of molecular transfer which hold for heat, current-electricity, and electrostatic dis- tributions. ' This theory requires, again, the assumption that magnetism, like the electric current and electric distributions, can only exist in closed circuits in which the magnetic moment is in- versely proportional to the resistance of the circuit. This view leads, consequently, to the introduction of the ideas of " magnetic distributive resistance " and " magnetic conductivity " of space and of magnetic bodies. Hence, only as much magnetism can be developed in an iron bar by an electrical current circulating round it as can pass or be conducted from one end of the bar to the other through the space round the magnet. My experiments have confirmed this view, and they have shown that the mag- netic conductivity of soft iron is, approximately, five hundred times as great as that of non-magnetic substances or of vacuum. ' Hence, Ohm's law may be applied in the construction of electromagnetic machines to ascertain the most suitable dimen- sions which may be of advantage to the electrician. The idea of magnetic conductivity, which, so far as I know, I first introduced, has meanwhile been frequently used and further developed in technical works, without, however, any mention of my name.' As regards the latter passage, it must be left to the reader to form a judgment as to the relative merits of the investigators concerned. The scientific epoch in question is too near for an historical and critical treatment, so that we must here restrict ourselves to giving as impartial a review as we can of the literature in question. In order further to elucidate the above quotation, which is expressed in popular language, we add the following explana- tion, and refer to the original paper for the rest. The following is enunciated as a general law (loc. cit. p. 95) :- N 178 ANALOGY OF THE MAGNETIC WITH OTHEK CIRCUITS The ' Strength of Magnetism ' [6] is equal to the ' Sum of the magnetising forces ' [e], divided by the 6 Sum of the opposing Kesistances' [fj. (Compare 119 and 124.) In our notation we must read instead of [6] : flux of induc- tion : instead of [_e] line-integral of magnetic intensity ; p] is put proportional (loc. cit. p. 98) to the length, and inversely pro- portional to the section and the magnetic conductivity of the iron. The variability of the latter quantity is discussed, and its highest value is found by experiment to be equal to about 500, the conductivity of air being assumed as the unit. 118. Continuation (Kapp, Pisati). Starting from the above considerations, Gisbert Kapp in 1885 gave empirical rules for the construction of dynamo machines, and these have been much used in practice with great success. 1 Kapp's rules have rather less claim to scientific value, partly because the inch and minute are used as fundamental units instead of the usual C.G.S. units. In this mixed system of measurement the reluctivity of air is expressed by the number 1440 ; that of iron was first considered constant ; for convenience the number 2 was taken for wrought iron, and 3 for cast iron. This rule used to give a value for the number of lines of force which, under certain circumstances, was as much as 40 per cent, in excess of that actually observed; this, no doubt, arose partly from leakage. Kapp then introduced a tan' 1 formula for magnetisation, like those which had been given in 1850 by J. Miiller and in 1865 by von Waltenhofen. 2 More recently Pisati 3 has again brought forward the analogy between the theory of the magnetic circuit and Fourier's theory of the conduction of heat, and has experimentally investigated it. From what has been said above, this is only a special case of the general analogy. Induction must then logically, as we shall afterwards see, correspond to the rate of flow of heat (that is, to the quantity of heat passing per unit cross-section per unit time), magnetic intensity, on the other hand, to the temperature- 1 Gisbert Kapp, Electrician, vols. 14, 15, 16, 1885. 2 J. Miiller, Pogg. Ann. vol. 79, p. 337, 1850, and vol. 82, p. 181, 1851 ; von Waltenhofen, Wiener Berichte, vol. 52, p. 87, 1865. 3 Pisati, Send. R. Ace. Lincei, vol. 6, pp. 82, 168, 487, 1890. DEFINITIONS 179 gradient. The ratio of the two quantities is, in one case, the magnetic permeability ; in the other, Fourier's coefficient of thermal conductivity. The latter is, of course, not constant, as Fourier originally supposed, but depends on the temperature. This analogy then is less faulty than that with Ohm's law, in which the electrical conductivity is absolutely constant ( 120). In order, however, that the analogy may be mathematically correct, thermal conductivity ought not to be a function of the temperature, but of the temperature-gradient, which, so far as is known, is not the case. In Fourier's theory a second coefficient is introduced, which allows for the loss of heat from the surface by convection, radiation, and conduction. This phenomenon, according to Pisati, is the analogue of magnetic leakage. The case of bars and rings with local magnetising coils he compares to Fourier's well-known problem of a conducting body in which there is a given distribution of heat, and which is embedded in a cooling medium. The experiments of Pisati show, in part, a good agreement with his theory. Still more recently papers by Steinmetz, Kennelly, Corsepius, R. Lang and 0. Frolich have been added to the literature of the subject. 1 After this historical review we turn to the exposition of the present state of the views on this subject. B. Modern Conception of the Magnetic Circuit 119. Definitions. We shall first give the definitions of the new conceptions, and illustrate these as usual by the typical example of a closed toroid uniformly magnetised. Let its mean perimeter be L as before, and the area of its cross-section be S. The whole of the quantities to be defined may be de- rived in the simplest manner from the vectors 33 and $, and the geometrical quantities L and S. 1. Flux of Induction has been already defined ( 61) ; it is (3) . . . . = 35 S 1 Steinmetz, EleMrotecJin. Zeitsclirift, vol. 12, pp. 1, 13, 573, 1891 ; vol. 13, pp. 203, 365, 1892; Kennelly, ibid. vol. 13, p. 205, 1892; Corsepius, ibid. vol. 13, pp. 243, 414, 1892 ; R. Lang, ibid. vol. 13, pp. 734, 485, 495, 510, 522, 1892 ; 0. Frolich, ibid. vol. 14, pp. 365, 387, 403, 1893. N 2 180 ANALOGY OF THE MAGNETIC WITH OTHEE CIRCUITS 2. The line-integral of magnetic intensity along the circum- ference of the toroid is (< L) ; this quantity is after Bosanquet pretty generally called magnetomotive force ; we shall denote it by M. In the present case it cannot directly be considered as a difference of magnetic potential, because the magnetic potential within the space occupied by the toroid is a many- valued function. We have thus 3. Permeability //, has already been defined ( 14) ; it is (5) .:.. . . /* = f Hr 4. Reluctivity g has also already been introduced ; it is (6) . - ^ f-J-g- :"' V '- We shall now have to consider two other quantities which are derived from the last two by purely geometrical pro- cesses. These latter numbers only characterise the magnetic properties of the substances in question, while in the scalar quantities now to be introduced, the length and transverse dimensions of the body to be magnetised have also to be taken into account. 5. Permeance Fis defined by the equation (7) .... 7=l and its reciprocal. 6. Reluctance X by the equation (8) ' Z = = F Starting now from equation (5), which we write in the following form, and multiplying both sides by S/L, we obtain after a simple transformation <>- L OHM'S LAW 181 If we now introduce the quantities denned above we have (I) ... = MV=^ wA. This equation means : I. The flux of induction is equal to the magnetomotive force multiplied by the permeance, or, what is the same thing, divided by the reluctance. 120. Ohm's Law. This proposition, in the form given, resembles Ohm's law, but is in fact different in the cardinal point, as will afterwards be demonstrated. Hence to apply Ohm's law to electromagnetic circuits without further enquiry, as has frequently been done, must be stigmatised as an un- scientific proceeding. Such an application no doubt offers cer- tain practical advantages, by allowing us to retain some familiar ideas which are associated with Ohm's law ; this, however, is of course no sufficient reason for their unconditional acceptance, which cannot be admitted, more especially for the following reasons. The quintessence of Ohm's law is the constancy of electrical resistance that is, its perfect independence of the strength of the current. That quantity depends, as is well known, only on the nature of the conductor, its temperature, 1 and its geo- metrical dimensions. So far as our present experimental knowledge goes, it is also independent of the surrounding medium. 2 Maxwell, after stating Ohm's law, expresses himself as follows : ' Here a new term is introduced, the Resistance of a conductor, which is defined to be the ratio of the electromotive force to the strength of the current which it produces. The introduction of 1 That the temperature may in certain circumstances be raised by the heat developed by the current, and thereby the resistance may be indirectly dependent on the current, is quite unimportant, and can be prevented by the suitable application of ice-baths, &c. The temperature is not com- pletely determined by the current. It is not, in fact, a function of the current at all, but is to be considered as an independent variable. This obvious remark is inserted here, since in some quarters opposite conclusions have been based on the heat developed by the current. 2 Sanford's recent experiments (Phil. Mag. [5], vol. 35, p. 65, 1893) are here disregarded, as they are open to a series of critical objections, and in any case appear to require confirmation. 182 ANALOGY OF THE MAGNETIC WITH OTHER CIRCUITS this term would have been of no scientific value, unless Ohm had shown, as he did experimentally, that it corresponds to a real physical quantity that is, that it has a definite value which is altered only when the nature of the conductor is altered.' l Maxwell's warning that the introduction of new ideas, or of modern expressions for old ideas, had better be omitted so long as their scientific value is not demonstrated, seems not in recent times to have been sufficiently laid to heart. After the experiments mentioned by Maxwell as having been made by Ohm himself, countless others have been tried to test this law. We need mention only those of Fechner, Pouillet, Beetz, K. Kohlrausch, and Chrystal. 2 The latter states, as follows, the results of his experiments, in which a method proposed by Maxwell was used : ' Within the very extended range of current-strength employed, the electrical resistance of a conductor does not vary by more than the billionth part of its value/ Ohm's law is therefore confirmed with an accuracy which has scarcely its parallel in physics. 121. The Magnetic Reluctance-Function. Let us now state Ohm's law in its usual simple form : (9) - , - , I=| and compare with it the above magnetic equation (I) M Here, apart from hysteresis, is a function of the magneto- motive force M alone. The two quantities are, however, not proportional, since their ratio X is by no means constant, and cannot, therefore, be regarded as a resistance in Ohm's sense. For if the resistance were no longer to be regarded as constant, we might logically divide any function, however complicated, into its independent variable, and consider the quotient as a mathe- matical resistance to the increase of the function. We should then be in a position to represent almost every process occurring in nature by an equation corresponding to Ohm's law. 1 Maxwell, Treatise, 2nd edition, vol. 1, 241. 2 Wiedemann, LeTire von der EleUricitat. 3rd edition, vol. 1, 329-353. THE MAGNETIC EELUCTANCE-FUNCTION 183 As regards the actual form of the magnetic reluctance- function X for a given ferromagnetic body, we may recall the definition given above [equation (8)] Since L and S are geometrical constants, it is sufficient to consider the law of the reluctivity f (the reciprocal of the permeability yu,), as was already done in the first chapter. We refer, therefore, to the curve f = funct. (<>) (fig. 4, p. 21), and to the discussion of its shape. Any given curve may, as is well known, be represented approximately, within a short range, by a straight line; and the reluctivity-curve, after it has passed its minimum-point, is very nearly straight. Hence, for cast iron, for the range of abscissae between 25 and 500 C.G.S., within which limits the field used in practice will mostly lie, the curve can be represented by the equation (10) . . . f=a + 6 [25<<500] where a and b are two constants. In the present case, a will only have a small value, as the straight line in question almost passes through the origin. The latter would be exactly the case if the induction 95 were constant within the region in question; for, according to the definition, f = $/98, and there- fore, when the denominator is constant, is proportional to the numerator. Its graph is in that case a straight line through the origin. The induction within a certain interval varies in fact but little, yet it is nowhere perfectly constant, and does not tend towards any constant value. Especial prominence has recently been given by several writers l to the empirical linear equation (10). It is in reality of subordinate importance, from a scientific point of view. If it is introduced into the equation 58-* "F we obtain (11} 3$ = ^ = See Kennelly, EleJttrotech. ZeitscJirift, vol. 13, p. 205, 1892. 184 ANALOGY OF THE MAGNETIC WITH OTHER CIRCUITS which equation, except for a constant factor 1/&, agrees with the older formula of Frolich, to be presently discussed ( 138). The above relation, which is purely empirical, and is to be restricted to the same range as equation (10), obviously holds only for closed rings, for the curve f = funct. (<>) (fig. 4, p. 21), is deduced from the normal curve of magnetisation (fig. 2, p. 18), in which self-demagnetisation is not allowed for. 122. Summary. We have not mentioned, or but briefly, the empirical methods which still hold their sway in magnetism ; this is the more justifiable as we are now in possession of rational bases for attacking the solution of most practical questions. The theory of Drs. J. and E. Hopkinson, treated in Chapter VI., 96-100, has in this respect great importance. It unites precision with brevity and elegance; no essential objection can be raised against it. In the memoir of Drs. Hopkinson men- tioned above the expressions ' magnetomotive force ' and ' resis- tance ' will be sought for in vain, and still more so any reference to a law corresponding to that of Ohm . The idea denoted by the former expression, the line-integral of intensity, is, however, introduced as an independent variable, along with the flux of induction. In consequence of this, the Hopkinsons' equation may at once be transformed into a series of terms, each of the form of equation (I), 119, as we shall afterwards show ( 131). Moreover, it must again be urged that it is quite permissible, though it has no scientific value, to divide the magnetomotive force by the flux of induction, and to call the quotient the magnetic reluctance or, its reciprocal the magnetic permeance. The perspicuity which is attained by introducing these two quantities cannot be denied, and in the following chapters we shall repeatedly have occasion to be convinced of this. The error first comes in when the magnetic reluctance thus defined, which depends, in the most arbitrary manner, on the magnetising field (or on the induction) is now compared with electrical resistance, the essential characteristic of which is its absolute constancy. 123. Leakage, Magnetic Shunts. The above remarks are not vitiated by the fact that in some cases approximately cor- rect results may be obtained by applying an imitation of Ohm's LEAKAGE. MAGNETIC SHUNTS 185 law to magnetic circuits. This may be the case especially when the permeability of the ferromagnetic substance is very great in comparison with that of the surrounding indifferent space (which is taken as unity). It will in many cases be indifferent whether the permeability has the value 200 or 2000 ; both numbers are so large that the practical results will not be appreciably affected. On the other hand, the validity of Ohm's law is obviously restricted by such a state of things. The deviations will be greater the more the permeability of the ferromagnetic substance decreases that is, the more the magnetising field increases. Assume for a moment, for the sake of argument, the complete analogy of a magnetic circuit with a voltaic pile immersed in an electrolyte ( 112). The decrease of permeability, and its gradual approximation to unity ( 14), would then have to find its analogue in the fact that the conductivity of the pile is in some way lessened, and con- tinually approximates to the smaller conductivity of the sur- rounding liquid. The electrical current would then spread more through the electrolyte ; that is, the stream-lines would leak more from the pile into the electrolyte. In accordance with this analogy, it has hitherto been generally concluded that the leakage of lines of induction must, under all circumstances, Increase with the magnetising force, so that the coefficients of leakage attain higher values. This is, however, opposed to fact. We have already ( 88) mentioned that, according to Lehmann's experiments on a uniformly -magnetised toroid divided by a radial gap, the leakage finally diminishes. This experiment must be regarded, in a certain sense, as a crucial one, in opposition to the view just explained. We have explained the decrease in question by Kirchhoff's law of saturation, in combination with the tangent- law of refraction of lines of induction. Electromagnetic systems might, however, be easily devised in which leakage finally in- creases with the magnetising field. This, indeed, is probably the case with most of the actual arrangements in practice, although no complete experiments on the subject are yet recorded. Every- thing depends on the distribution of the magnetising field of the coils with reference to the geometrical form of the ferromagnetic substance, as was previously thoroughly discussed. In the elec- 186 ANALOGY OF THE MAGNETIC WITH OTHEK CIRCUITS trokinetic analogue there is obviously no single feature which corresponds to these phenomena of saturation. Our conclusion, therefore, from all this must be that it is inadmissible to apply Ohm's law to magnetic circuits. In a still higher degree is this the case with Kirchhoff's laws for branched circuits, which necessarily presuppose Ohm's law. The application of those rules to branched magnetic circuits and magnetic shunts will nearly always lead to results which are quantitatively false, and which are only correct qualitatively or as a rough approximation within a narrow range, 1 but can in no case claim a more general applicability. 124. Comparative Tables. Although, from the above con- siderations, the physical theory of the magnetic circuit is not completely identical with the theory of circuits of the most general kind, the purely formal analogy goes very far. This mathematical analogy, which extends to a number of branches of physics, is of great interest. We give, therefore, in this paragraph a comparative table from which the essential features of the analogy will be at once apparent. In six different columns the conceptions belonging to the six theories con- sidered are stated, and in the following order : I. Filtration of incompressible frictionless liquids through porous bodies (Lord Kelvin). II. Diffusion of dissolved substances in solutions (Fick). III. Conduction of heat (Fourier). IV. Dielectric polarisation (Maxwell). V. Electrical conduction (Ohm). VI. Ferromagnetic induction (Faraday, Maxwell). In any one line of Table VII. the mutually corresponding conceptions are stated ; in line [a] The real or fictitious substance which flows. [6] The quantity of this substance which per unit time flows through any section 8 ; that is, the current or flux through this section. 1 Compare Ayrton and Perry, on magnetic shunts, Jouvn. Soc. Tel. Engi- neers, vol. 15, p. 539, 1881. According to a private communication from the Eoman physicist Pisati, who unfortunately has since then died, the irregular results obtained by him with magnetic shunts confirm what is stated in the text. So far as the author could learn, these results have not been published. COMPAEATIVE TABLES TABLE VII 187 i Filtration ( H3) II Diffusion in Solutions Ill Thermal Conduction IV Dielectric Polarisation V Electrical Conduction VI Ferro- magnetic Induction w Quantity of Liquid Mass Quantity of Heat Quantity of Elec- tricity Q - [*] Current Mass- Current Thermal Current Flux of Dielectric Induction Electric ' Current / Flux of Induc- tion w Flow Mass- Flow Thermal Flow Dielectric Induction Electric Flow (. Induc- tion S3 M Kinetic Energy per Unit Volume Concen- tration- Gradient 2 Tempera- ture- Gradient Electric Intensity Electro- motive In- tensity e Magnetic Inten- sity [] Concen- tration 2 Tempera- ture Electric Potential Electro- motive Force E Magneto- motive Force M [/] Hydro- kinetic Permea- bility Coefficient of Diffusion Thermal Conduc- tivity Dielectric Constant Electric Con- ductivity Magnetic Permea- bility p. 0] Electric Resistivity, Magnetic Reluc- tivity w Electric Con- ductance Magnetic permeance KJ Electric Resis- tance R Magnetic Reluc- tance X [c] The current per unit of section ; that is, the flow. If this is uniform throughout the section in question, then (12) M = H# 1 Mascart and Joubert, Electricity and Magnetism, vol. I, 115 ; London, 1883. 2 According to the modern theory of diffusion, the osmotic gradient the osmotic pressure would be introduced, which is proportional to the con- centration-gradientor to the concentration respectively. In the diffusion of gases, on the other hand, the partial pressure would have to be introduced. 188 ANALOGY OF THE MAGNETIC WITH OTHER CIRCUITS [d] The agent which causes the current. [e] The line-integral of this latter along a length L. Hence, in case of uniformity over this distance, (18) H = M [/] The property which for a given intensity of the agent [cT] determines the value of the flow [c]. In all cases (14) . - - This ratio is quite constant in (V), and, so far as at present known, also in (IV). It is approximately constant in (II) and (III), and perhaps also in (I). It is entirely variable in (VI), [] p] By definition equal to [ in an imperfect magnetic circuit. We have, how- ever, shown on a former occasion how two such apparently different functions may without difficulty be transformed into each other ( 98). In the original form, Frolich's formula (16) had also a term in $ 2 in the denominator; this was subsequently omitted, as it was found to be superfluous, since the simpler form could equally well represent the case. This latter formed for many years the only guide for interpreting many obscure points which the action of dynamos then presented. The value of so simple a formula with which the complicated action of a machine could be represented, even empirically and ap- proximately, was not to be under-estimated for practical re- quirements. The connection of the empirical linear equation (10) 121 for the magnetic reluctivity , as function of the in- tensity , with Frolich's formula we have already discussed; it strikes one at first sight, if we compare equation (11) p. 207 with equation (16) p. 211. Of course, the latter then only GENERAL ARRANGEMENT OF THE MAGNETIC CIRCUIT 213 applies to closed magnetic circuits without air-gaps, and holds exclusively for the range within which we may write f = a + b $ for the ferromagnetic substance in question. In accordance with this Frolich thus expresses himself in a recent paper 1 : 'My older formula (16), as appears from our considerations, is valid for electromagnets with small air-spaces, and therefore precisely for the newer dynamos in which the air- spaces are reduced to a minimum.' In connection with this Frolich then develops a newer formula with a member added, which holds also with a certain approximation for electromagnets with interspaces of any given extent. But since equation (17), as expressly observed, only holds for imperfect magnetic circuits with a considerable demagnetisa- tion-factor, its want of agreement with Frolich's older formula need not cause surprise, its justification being found rather in the very different range of applicability of the two expressions. 140. General Arrangement of the Magnetic Circuit. Hitherto we have always elucidated our discussion (mainly theoretical) of the magnetic circuit of dynamo machines and electromotors, by reference to an example which has become classical, the Edison-Hopkinson machine (fig. 28, p. 189). We now proceed to discuss the very varied forms which such mag- netic circuits present in the machines occurring in practice. Notwithstanding the extraordinary variety in the types of construction which have been developed in the designing of dynamos and electromotors during the last twenty years, and which have in part established themselves, and the correspond- ing difference in the arrangement of the magnetic circuit, three main parts may always be discriminated in the latter. 1. The Field-magnet, the object of which may be stated in the most general terms to be, the permanent conduction of the flux of magnetic induction of the machine for the greater part of its course ; at the same time, in the great majority of cases it is the seat of the magnetomotive forces which produce that flux of induction. 2. The Armature, the chief part of the dynamo which supports 1 O. Frolich, EleUrotechn. ZeitscJirvft. vol. 14, p. 403, 1893. 214 MAGNETIC CIRCUIT OF DYNAMO MACHINES the current conductors, 1 in which the electromotive forces are induced which cause the production of the current. 3. Tlie Air-gap, which allows the passage of the flux of induction between the two chief ferromagnetic parts of the machine when moving relatively to each other. These three parts may be recognised in all dynamos, whether used for producing direct current, ordinary two- phase or polyphase alternating current ; 2 in discriminating these parts, their function is to be exclusively considered, and not their geometrical or mechanical arrangements. In almost all direct-current machines, the field-magnet is fixed, while the armature rotates. In many alternating current machines this is also the case, while in many others it is just the reverse, experience showing that a fixed armature has many advantages in practice. As to the question which part is to be considered the field- magnet and which the armature, the rotation of the parts does not at all come into consideration. In every case the field- magnet is that part with reference to which the flux of induction remains unchanged, independently of any rotation. The mag- netisation of the armature, on the contrary, undergoes periodic changes of its value, direction, or distribution, which cause the induction of electromotive forces. We will now consider more especially the arrangement of the chief parts of the magnetic circuit. 141. Arrangement of the Field-magnets (Framework), The most important consideration in designing the field-magnet arises from the necessity of making the reluctance of its mag- netic circuit as small as possible. To this corresponds a minimum value of the magnetomotive force necessary for producing a given flux of induction that is, as small a number of ampere-turns as possible. The unavoidable waste of energy by the heat developed in these latter is thereby reduced to a minimum. Accordingly, the field-magnets should be as short and of as large a section as possible, and constructed of iron of 1 These usually consist of copper ; compare, however, 144. 2 Direct current armatures do not materially differ, as regards magnetisa- tion, from those for alternating currents. The difference is almost exclusively in the mode of winding and commuting. ARRANGEMENT OF THE FIELD-MAGNETS 215 the highest permeability. The constructive requirements affect, therefore, the form of the frame of the magnet ; and, secondly, the material of which it is constructed. As regards the shape of the framework, the section of the limbs should be circular, for then its perimeter is a minimum for a given section. In that case, the length of wire needed for magnetising a given section with a prescribed number of ampere- turns is as small as possible, as is also the loss of energy by the heat produced by the current. Limbs of circular section also are most easily turned and wound. Nevertheless, oval or rect- angular sections often occur ; in the latter, the edges, for several reasons, are best rounded. In accordance with the principle, universally recognised at present, not to lengthen the magnetic circuit unnecessarily, the frames of the magnets of all well-constructed modern machines show compact forms. Besides the lessening of the magnetic reluctance, there is the advantage that the leakage is diminished as much as possible. The allowance for this latter circumstance influences the design of a framework in many other details, for which it is difficult to give general rules. It need only be mentioned that the general lines of the frame should follow as closely as possible the course of the lines of induction. In accordance with this, sharp bends in the magnetic circuit are to be avoided, as these cannot be followed by the lines of in- duction. Further, two points of the circuit whose magnetic potential is markedly different, and between which, accordingly, there exists a considerable magnetomotive force, must not come too close anywhere, lest induction lines should pass between them, through the intervening air, and so be lost. In those parts, too, where there are considerable magnetomotive forces, no ferromagnetic bodies must be placed which serve for other purposes such as base-plate, axes, bolts, &c. for these may form injurious mag- netic short circuits. 1 Finally, all projecting points, corners, edges, must be rounded off as far as possible, for experience shows that they always produce a certain leakage. 142. Continuation. Pole-pieces, Material. The shape of the pole-pieces which adapt themselves to the armature requires 1 Compare the statement about short circuit through a base-plate ( 130). 216 MAGNETIC CIRCUIT OF DYNAMO MACHINES particular care. They must allow the induction lines to pass as uniformly as possible over the whole extent of the air-gap. They thereby directly influence the distribution of potential around the commutator, which must not deviate too much from the ideal sine curve. 1 A direct passage of induction lines between the neighbouring horns of the pole-pieces, instead of through the armature, must also be avoided. The economical reasons for desiring to lessen the size of the field-magnet or, rather, of the ( field ampere-turns ' find their natural counterpoise in the requirement that they must always dominate the armature ampere-turns, so that reaction of the armature and sparking do not increase too much ( 134). This holds for the current-generator as well as for the electromotor. For the latter, lighter field-magnets are sometimes constructed, but only in cases in which it is important that they should weigh as little as possible. In order to satisfy the requirement of as high permeability as possible, the best material for the field-magnets is the softest annealed wrought iron. As, however, the conditions previously discussed often lead to very complicated forms for the frame, which cannot well be forged, it is frequently preferable to cast it, by which, at the same time, joints are avoided ( 144). Owing to the smaller permeability of cast iron, all sections must then be larger. Several kinds of steel, too, what is called ' mitis metal,' 2 or ' malleable cast,' 3 are used. In many cases wrought-iron cores for the limbs are used, with cast-iron casings and pole- pieces. Owing to the almost complete constancy of the flux of induction through the field-magnets, the hysteresis of the material has not much influence. This has even a certain advantage in 'starting' the machine. Eddy currents are just 1 Compare 135 ; the self-induction of the armature coils is there disre- garded. See also note, p. 278. 2 Wrought iron made fusible by a slight addition of aluminium ; this alloy is recommended by Silv. Thompson, Dynamoelectrical Machinery, 4th edition, p. 149, London, 1892. 3 The author found (EleJitrotecJin. Zeitsclirift, vol. 13, p. 580, 1892) that this material has considerably higher permeability, and at the same time less hysteresis than ordinary cast iron. (Compare fig. 94, p. 349.) [Quite recently several kinds of cast steel have been brought out, which almost equal wrought iron, though not tne best Swedish specimens. H. d. B.] ARRANGEMENT OF THE ARMATURE 217 as little to be feared, so that it is not necessary to split up the material, but it may be made solid. 1 143. Arrangement of the Armature. The contrary of what has been stated in the last article holds for armatures, since in this case the magnetisation is, in every respect, an essentially variable quantity ( 140). This part is therefore exclusively built up of thin stamped sheet-iron, or iron ribbon (in some cases also of round or, better, square iron wire), the thickness of which is a smaller or larger fraction of a millimetre, according to the quicker or slower variations of magnetisation (187). In accordance with the object of this division, that of breaking the paths of parasitic eddy currents (< Foucault cur- rents ') in the body of the armature, it must be in planes at right angles to those paths that is, parallel to the lines of induction, as well as to the direction of the motion. As the section of the iron is lessened by the division, 2 the magnetic reluctance undergoes an unavoidable increase. On the other hand, demagnetising actions scarcely occur, as would be the case if the induction lines were at right angles to the planes of division ( 30 _E/), instead of parallel to them. Taking that circumstance into account, the section of the armature should be such that its magnetisation is never near saturation, but rather remains on the steep ascending part of the curve of magnetisation. Its highest allowable value is then, for good wrought iron, about 3 = 1200 C.G.S. units, corresponding to an induction 25 = 15,000 C.G.S., as stated before ( 126). As to material, the purest possible soft wrought iron is always used, the best being Swedish, which combines high per- meability with small hysteresis. The plate, after being stamped, is carefully annealed. As it is never possible entirely to sup- press the heating due to hysteresis and eddy currents, to which must be added the heat due to the current itself, care must be 1 For further details as to the construction of field-magnets, reference must be made to Kittler, Handbuch der EleTitrotecTimli, 2nd edition, vol. 1, chapter ix. 2 The volume of the iron in an armature thus divided is, in percentages of the whole, with : Iron plate or strip . . . about 80-90 per cent. Square iron-wire . . . 70-80 Round , 60-70 218 MAGNETIC CIRCUIT OF DYNAMO MACHINES taken to allow the heat to escape. This is not difficult with rapid rotation, since every armature naturally acts as a centri- fugal ventilator. Good insulation of the separate iron discs is not of much importance ; thin paper, or a coating of shellac or varnish, is used. 1 The armature core thus constructed is pressed together to form a compact mass by means of strong bolts, &c., and is then turned and wound. Hoop iron is almost exclusively used for flat ring armatures. We cannot here enter further upon the details of the construction and winding of the other chief forms of armatures (drum, ring, disc, toothed-ring, and pierced- disc armatures) made from sheet iron, and of the most varied patterns. 2 144. Arrangement of the Interspace. The interspace always gives rise, as we have seen ( 131), to the chief part of the magnetic resistance. As it is desirable to lessen this as much as possible, for economical reasons, the measures to be taken with this object must first of all be applied to the inter- space. Useless air-spaces, especially joints between the separate parts, are, in the first case, to be avoided as much as possible, for they always produce magnetic reluctance and, moreover, leakage. In case joints cannot be avoided in the construction, the two faces, if possible, should be ground against each other, and the corresponding parts fastened tightly together by strong bolts ( 151, 152). As regards the useful air-space, its width should be lessened as far as possible. With armatures which are completely wound, the width is prescribed by the section of the copper and the insulation, as well as by the space necessary for the free play of the armatures rapidly revolving between the pole-pieces. 3 In consequence of this, a lower limit is defined for the width of clearance. 1 The iron scale produced in the annealing is sufficient for this purpose ; it is, however, better to remove it, as it has been observed that such armatures often show remarkably high hysteresis, probably owing to the formation of Fe 3 O 4 , which is identical with the highly hysterestic magnetic iron ore. 2 Silv. Thompson, loc. cit. chapter xiii ; Kittler, loo. cit. chapter vii. 3 Disturbances in working often occur owing to the irregular motion of the armature coils against the pole-pieces ; an armature entirely covered by the winding can, of course, not be so accurately centred as one whose perimeter is entirely or partially formed by the iron core itself, which can, in being turned, be accurately centred, as will be seen in the types to be afterwards mentioned. MACHINES WITH MULTIPLE MAGNETIC CIRCUIT 219 In imitation of the original Pacinotti ring, what are called toothed-ring armatures are often used. Their section is like that of a toothed wheel ; the coiling is in the notches, while the magnetic reluctance is considerably decreased by the projecting teeth. More recently, perforated armatures have been intro- duced in which the conductor lies in a series of holes parallel to the axis of rotation, and near the perimeter of the armature. In this arrangement the air-gap is practically restricted to those cavities. The obvious idea of using iron for the conductors instead of copper, has been realised, by which the air-gap, and therewith the resistance, is reduced to the smallest degree. The consider- ably higher electrical resistance of iron is, it is true, an objection. Although such iron coiling has hitherto been but little used, it is interesting from the purely magnetic point of view. We have mentioned it in the theoretical treatment of a ferro- magnetic substance through which a current passes ( 60). The construction of Forbes may be mentioned, which, indeed, represents the simplest plan of a dynamo. A solid cylinder or disc of wrought iron rotates with as little clearance as pos- sible within a thick iron jacket, which completely encloses it. The field is produced by a few peripheral windings, which are firmly bedded in the iron jacket. Radial electromotive forces are induced in the rotating iron core, producing currents in the external circuit attached by sliding contacts to the axis and periphery. 1 We may, in conclusion, mention Fritsche's recently- constructed 2 dynamo, in which the conductors of the armature are also of soft iron. 145. Machines with Multiple Magnetic Circuit. In our theoretical developments ( 125135) we have, for the sake of a readier survey, tacitly confined ourselves to those machines in which the useful flow of induction enters one part of the armature, and emerges at the opposite side, without closing further branch circuits through the frame of the magnet. Such machines are said to have a single magnetic circuit. The 1 Forbes calls his machine a ' non-polar one ' ; it belongs to the type of machines without commutator, whose armatures continually cut the lines of induction, and the very earliest representative of which is Faraday's disc. They are still frequently called by the unsuitable name unipolar machines. * W. Fritsche, Die Gleichstrom-Dynamomaschine, Berlin, 1892. 220 MAGNETIC CIRCUIT OF DYNAMO MACHINES consideration is thus simplified as much as possible, as seen in the case of the Edison-Hopkinson dynamo. It has now, indeed, been shown by Rowland that theoretically it is better if the magnetic circuit of a dynamo is simple than if it is a mul- tiple that is, a branched circuit. Nevertheless, theoretical reasons are not always the only ones to be considered in such cases. As regards ordinary direct-current machines, with a divided magnetic circuit, the magnetic field is symmetrical. It is, further, easier to cope with the constructive difficulties which arise from the very considerable magnetic tractive forces. A continuous-current machine with simple magnetic circuit, if deprived of its commutator, constitutes a two-phase alternating current, the frequency l of which is equal to the number of turns. But as the latter cannot be in- creased so that the former is sufficiently high, it is necessary, in alternating machines, to increase the number of periods n times by arranging n magnetic circuits, and it is immaterial whether the armature or the field-magnets are movable. Fig. 34 gives the plan of a machine with a quadruple magnetic circuit. The direction of the lines of induction is indicated by the arrows. In planning this machine, each of the four magnetic divided circuits must be designed on the principles which have been developed in the preceding articles. The product X of the prescribed flux of induction into the calculated magnetic reluctance JC, which is to be kept as low as possible, gives the requisite magnetomotive force M which is to be assigned to the partial circuit in question. If that product is divided by 0-4 TT (or multiplied by 0*8), the necessary number of ampere- turns is obtained. 146. Diagrams of various Magnetic Circuits. In fig. 35, p. 221, the magnetic circuits of fifteen types of machines, 1 That is the number of complete periods in a second (equal to half the number of alternations), which, with the usual alternating currents now in use, is between 40 and 120; at the same time there is a decided tendency to diminish the number to the lower value. DIAGRAMS OF VARIOUS MAGNETIC CIRCUITS 221 D 222 MAGNETIC CIRCUIT OF DYNAMO MACHINES actually constructed and in more or less extensive use, are represented diagrammatically, according to Silv. Thompson ; 1 and they are so chosen that those diagrams are not given against which serious objections may be raised from a magnetic point of view. The order of the figures A-0 is progressive from the simplest to the most complicated magnetic arrangement. The ferromagnetic parts which produce the field are cross-shaded, and the current-producing armatures are simply shaded. The names of the constructors are not given, since many of the arrangements are used in various machines with slight modi- fications, and this is a matter of less moment for the objects of the present book. Single magnetic circuits are met with in the machines repre- sented in figures A to E, several of which are extensively used. The arrangements are sufficiently clear from the figures, so that any further description is needless. Double magnetic circuits are represented in figures F to J. The latter shows an arrangement in which the magnetising coils are wound round the armature instead of round the limbs of the field-magnets. This interesting mode of winding has been recommended by different electricians on the ground that the total flow of induction is thereby utilised, and there is no useless leakage. Armatures thus wound do not, however, allow of sufficient ventilation. Multiple magnetic circuits are, finally, shown in the diagrams K to 0. In K the last-mentioned principle of winding is, in a certain sense, met with in hollow cores of magnets 2 and spherical armatures. The magnetic circuit is closed by a number of iron rods, the top and bottom ones of which are represented in the figure. L and M represent fourfold magnetic circuits ; in the latter machine the armature is outside the field-magnets, as has 1 Silv. Thompson, Dynamoelectric Machine 1 )*}/, 4th edition, chapter viii., from which the substance and the figures of these last paragraphs are taken with kind permission of the author. In Kittler, loc. tit. figs. 462-466, over 60 such diagrams of magnetic circuits are depicted. 2 Compare Grotrian, Wied. Ann. vol. 50, p. 737, 1893, and du Bois, ibid. vol. 51, p. 536, 1894, where the magnetic circuit of such machines is subjected to discussion. (See also note 2, p. 263.) DIAGRAMS OF VARIOUS MAGNETIC CIRCUITS 223 recently been often arranged ( 140). N, finally, represents a sixfold, and an eightfold, magnetic circuit. 1 1 The classification of machines according to the number of magnetic circuits is perhaps the most rational from the magnetic point of view; in practice, indeed, we still frequently speak of non-polar, unipolar (compare note, p. 219), bipolar, multipolar, as well as external polar, internal polar, con- secutive polar machines, and the like. 224 CHAPTER IX MAGNETIC CIRCUIT OF VARIOUS KINDS OF ELECTROMAGNETS AND TRANSFORMERS A. Physical Principles 147. Magnetic Cycles. In the present chapter we shall continue to treat practical examples of the applications of the principles which hold for magnetic circuits ; but from the great variety of arrangements we shall only select those which are typical, and have special theoretical interest. We have, how- ever, previously to discuss the results of experimental investiga- tions and theoretical considerations which have come into account, and which refer more particularly to the alteration of magnetic states. We shall first consider the phenomenon of magnetic hysteresis. In 8 we have defined its general character, but in our subsequent developments we have always disregarded hysteresic processes. We shall limit ourselves, at present, to stating the main features of this important phenomenon, referring for ex- perimental details to works in which ferromagnetic induction is more completely dealt with than is consistent with the object of the present book. 1 We will, after Warburg, 2 subject a ferromagnetic substance of endless shape to a magnetic cycle, whereby we cause the mag- netic intensity to pass through all values from <$ G to -f <$ G , and back again to < + 100), while the corresponding range of magnetisation amounts to ( 1000 < 3 < + 1000). The arrows denote the so-called increasing or decreasing branches of the curves corresponding to the ascending or descending values of magnetisation respectively. The curve 6r, that with no arrow, is what is called the ' curve of ascending reversals,' which has always formed the basis of the preceding considerations ( 85). On sufficiently frequent repetition, a definite loop of greater or less extent, according to the range of magnetisation, corresponds to each such cyclical change of intensity between given limits. 148. Dissipation of Energy by Hysteresis. The chief pro- perty of hysteresis loops lies in the fact, that their area furnishes a measure of the energy it transformed into heat in the cycle in question, per unit volume of the ferromagnetic substance. For, as was shown by Warburg (loc. cit.), and soon afterwards independently by Ewing, (1) . . u= f 3d% = A [ J 4 7T J taken between the limiting values of the cycle. 1 The proof of this fundamental principle, which can be given in various ways, would here lead too far. The heat disengaged produces, in certain circumstances, a rise of temperature of the ferromagnetic sub- stance, which has, however, but a small value for a single cycle being of the order of about the thousandth of a degree but is very appreciable with frequent repetitions of the process. A complete cycle is, strictly speaking, one which ranges between the limits of intensity oo and + oo , in which the maximum magnetisation obviously represents the range of or- dinates. For most purposes, however, it is sufficient to keep in view high finite values of intensity. Neither the limiting value 1 The second expression for u not only holds with the same approximation with which we can often write 95 = 4 ir3, but is perfectly exact, since the integral / d$ must necessarily vanish for all closed loops. Strictly speaking, equa- tion (I) holds only under definite assumptions as to interchange of heat ; for instance, for isothermal or for isentropic fadiabatic) cycles (see Warburg and Honig, loc. cit., p. 817 ; Ewing, Proc. Hoy. Soc., vol. 23, p. 22, 1881, and vol. 24, p. 39, 1882). For most purposes, however, this is of small importance. DISSIPATION OF ENEKGY BY HYSTEKESIS 227 of magnetisation, nor the quantity of energy dissipated, can then greatly increase, and it is therefore sufficient to give the value of the latter for such approximately complete cycles. A great number of such determinations have been given by Ewing in his comprehensive investigation of this subject. 1 The value of it varies from about 10,000 ergs per cubic centimetre for the softest annealed iron, up to over 200,000 ergs per cubic centi- metre for glass-hard tungsten steel. With incomplete cycles of restricted range of magnetisation, the dissipated energy exhibits a correspondingly smaller value. In his fundamental experiments Warburg had endeavoured to make clear its relation to the limiting value of magnetisation, but he found no definite law, and particularly no proportionality, between u and 3 2 G . Taking Swing's data as basis, Steinmetz has recently found that the process may be represented ap- proximately by the following empirical relation : u = C (3 t - V 6 or u = B ($! - 33 2 ) 1>6 in which C and B are constants, depending on the nature of the material ; 3j and 3 2 (or 93 1 and 35 2 ) represent the, upper and lower limiting values, which need not be numerically equal. The difference (3 t 3 2 ) to be understood in the algebraical sense represents the entire range of magne- tisation, or the corresponding section of ordinates, which, moreover, need not be symmetrical to the axis of abscissas ; 3 } and 3 2 may even have the same sign. 2 In special cases Ewing was able to represent the dissipation of energy as a linear function of the range of intensity. 3 W. Kunz 4 has recently thoroughly investigated the influence of temperature on hys- teresis. The chief result he finds is that, on the whole, the dissipation of energy corresponding to a given range, decreases as the temperature increases, until at the temperature at which magnetisation almost vanishes, it would obviously altogether cease (compare note, p. 15). Shocks and vibrations, also, as 1 Ewing, Phil Trans., vol. 176, II., p. 523, 1885. 2 Steinmetz, Elektrotecli. Zeitschrift, vol. 12, p. 62, 3891; vol. 13, p. 519 et seq., 1892. 3 Ewing, The Electrician, vol. 28, p. 635, 1892. 4 W. Kunz, AVhangiglteit der magn. Hyst. von der Temperatur, Programm- Beilage Gymn. Darmstadt, Easter, 1893, and Dissertation, Tubingen, 1893. Q 2 228 MAGNETIC CIRCUIT OF ELECTROMAGNETS well as rise of temperature, tend to counteract hysteresis, as already mentioned ( 8). The above discussions refer to the case in which the cycle is slowly performed, as is the case in the ordinary { statical ' method, by means of which the hysteresis loop may be deter- mined ( 207). The question whether, with rapid cycles, their duration would have an essential influence on the value of the dissipation of energy must still be regarded as an open one. It is connected with the phenomenon of magnetic time-lag, which has been but little investigated (note, p. 205). From the calorimetrical experiments of Warburg and Honig, and of Tanakadate, as well as from the investigations, by various methods, of J. and E. Hopkinson ( 221), Evershed and Vignoles, Ayrton and Sumpner, 1 it must be considered probable that that influence is by no means an important one, at any rate, provided the duration of the cycle is not less than the one-hundredth of a second, which is about the lower limit for the period of alter- nating currents hitherto usual in practice (note, p. 220). The quantity of energy dissipated per unit time in a mag- netic cycle in consequence of hysteresis that is, the power which must be practically regarded as lost 2 is obviously obtained by multiplying the values of the hysteresis integral, or the area of the corresponding loop, into the volume of the ferromagnetic substance and into the number of cycles which are performed per unit time. 149. Influence of Shape. Retentivity, Coercive Inten- sity. The shape of the ferromagnetic substance exerts a considerable influence on the form of the hysteresis loop, as well as on the curve of magnetisation, which may be allowed for in the same manner by corresponding ' shearing ' of the curve ( 17). In fig. 36, A, p. 225, the full-line loop CGCG is supposed to be obtained with endless shapes, and therefore, in a certain sense, represents the normal loop for the material. Now 1 Warburg and Honig, loc cit., Tanakadate, Phil. Mag. [5], vol. 28, p. 207, 1889 ; J. and B. Hopkinson, Electrician, vol. 29, p. 510, 1892 ; Evershed and Vignoles, ibid., vol 27, p. 664, 1891 ; vol. 29, pp. 583, 605, 1892 ; Ayrton and Sumpner, ibid., vol. 29, p. 615, 1892. 2 The C.G.S. unit of power or activity is the erg per second ; the practical unit is the watt or the kilowatt. Between these units we have the following rela- tions : 1-34 British horse-power = 1 kilowatt = 1,000 watts = 10 10 ergs per second- INFLUENCE OF SHAPE. KETENTIVITY 229 let the straight line EOE correspond, for instance, to a demagne- tising factor N O05 1 ; if the normal loop is sheared from it to the axis of ordinates, the dotted loop C G' G' is obtained, which holds for the given value of N. If during a cycle, after magnetisation in a definite say positive direction, the inten- sity be allowed to diminish to the value 0, it is known that the magnetisation will not quite disappear, but will retain a certain positive value. This, the residual magnetisation 3 B , is equal to the ordinate OR of the point of intersection R of the descending branch of the loop with the axis of ordinates. In the present example, % E = 700 C.G.S. units. The ratio t of the residual magnetisation to the previous limiting value is called the retentivity ; this amounts, hence, to f = 3.R/3> 90 per cent., and > e > 1 C.G.S. unit. With steel t is always less ; on the contrary, 4? c is far greater than the values given above, while, e.g., with hard tungsten steel the coercive intensity in certain circumstances amounts to more than 50 C.G.S. units. 2 Older statements as to the production of permanent magnets are met with in Lament, Handbuch des Magnetismus, 1867. See further Jamin, Compt. Mend, vol. 76, p. 1153, 1872, and vol. 77, p. 305, 1873; Strouhal and Barus, Wied. Ann., vol. 11, p. 930, 1880, vol. 20, pp. 525, 621, 662, 1883 ; Holborn, Zeitsclirift fur Instrument en- Kunde, vol. 11, p. 113, 1891, as well as many MAGNETIC EELUCTANCE IN JOINTS 233 the finest joints and cracks exert in magnetic circuits, and which shows itself by actions not confined to the joints only that is, by demagnetisation and leakage. We shall here go more minutely into the recent literature of this subject. J. J. Thomson and H. F. Newall first showed l that transverse joints in iron bars exert a considerable demagnetising action and leakage. They first determined the magnetisation in a given field : after cutting through the bars and putting them together again a surprising decrease of magnetisation was observed, which was only partially removed by grinding the ends ; by interposing an increasing number of thin indifferent layers, the decrease of course always became greater. They also determined the leak- age by means of iron filings ( 4, 189), and then depicted the magnetic figures obtained (loc. cit.*) Ewing and Low 2 then worked at this subject ; they first investigated an iron bar (12- 7 cm. long by O49 sq. cm. cross- section) by means of the ballistic bar-and-yoke method ( 218). After the curve of magnetisation of the undivided bar had been determined, the bar was cut through transversely, and the ends fitted as usual by careful scraping and testing on ~~s, plane surface. Both halves were then put together again in as close contact as possible, and the curve of magnetisation again determined. It was found sheared in respect of the first (fig. 37). It follows from this that a joint acts like an air-gap, although there must undoubtedly be an actual contact of the ends of the iron rod at many points of the surface of separation. . The de- magnetising action of the joints is best expressed by stating the width d of the equivalent air-gap, between two geometrically plane faces, which would produce the same demagnetisation that is, would offer the same magnetic reluctance, whereby the other data. The most recent collation of the literature, which for the most part is widely scattered, as well as of tables of constants and so forth, is given by Silv. Thompson, the Electromagnet, 2nd edition (3rd edition of his Cantor Lectures), chap, xvi., London, 1892. 1 J. J. Thomson and H. F. Newall, Proc. Phil. Soc., Cambridge, vol. 6, II., p. 84, 1887. 2 Ewing and Low, Phil. Mag, [5], vol. 26, p. 274, 1888 ; Ewing, ibid., vol. 34, p. 820, 1892. [In a recent paper by Houston and Kennelly in the Electrician, vol. 35, p. 160, 1895, the subject is dealt with on much the same lines as in the text, and a 'factor of safety' (against demagnetisation) introduced. H. du B.] 234 MAGNETIC CIECUIT OF ELECTROMAGNETS former may be deduced in the usual way; from the difference of abscissas A*> of the two curves of magnetisation. For the purpose of this calculation we will consider the bar in its closed yoke (fig. 38, p. 235) ; in first approximation as a toroid, slit radially, the perimeter of which is 2 TT i\ = L (the length of the bar). In so narrow a slit the simple equation (VII) 82 may be used putting this value into the equation A< = J$"3, we find (5) d = -A,A 47r3 By means of the last equation Ewing and Low calculated the width of the equivalent air-gap, which for two iron bars was found equal to about 0-03 mm. This value was found to be 500 fairly independent of the magnetisation, as is shown by the almost straight line in fig. 37, which represents a value N = 0-003. It is difficult to decide whether that equivalent air-gap did, in fact, represent the mean distance of the faces, although such a considerable value of the latter is not probable. It can, with certainty, be alleged that a surface produced in the manner described differs materially from a highly polished plane mirror, and still more from a geometrical plane. Ewing and Low have further confirmed that the interposition of a single gold foil had not any further appreciable influence on the magnetic reluctance. The thickness of such a foil amounts, as we know, to only a fraction of the wave-length of sodium light that is, it is of the same order of magnitude as that by which a good metal mirror differs from an absolute geometrical plane. INFLUENCE OF LONGITUDINAL PKESSUEE 235 152. Influence of Applied Longitudinal Pressure. The same experimenters then investigated the influence of longitu- dinal pressure on the magnetic behaviour of a joint between two halves of a bar by the apparatus represented in fig. 38. At the same time they examined how far the magnetic properties of the material itself were affected by pressure ( 107). They found that the demagnetising action or, in other words, the magnetic reluctance of a joint arranged as above decreases with increase of pressure, and that for a pressure of over 200 kg. -weight per sq. cm. no difference could be detected between a divided and an undivided bar. In this case, as was to be expected, the interposition of a gold foil had a small, FIG. 38 but appreciable, influence. It is here to be observed that the magnetic pull ( 102) was less than 1 kg.-weight per sq. cm., so that it need scarcely be considered of any influence in com- parison with the above value of the external pressure. Ewing and Low further made experiments with rough joints that is to say, such as were bounded by surfaces simply turned and not further fitted. It was found that the width of the equivalent air-gap for such rough joints was about 0*05 mm., and was only reduced to - 04 mm. by a pressure which would have entirely nullified the magnetic reluctance of a joint between properly prepared faces. Experiments were made with bars which had not one transverse joint only, but three to seven joints. 236 MAGNETIC CIKCUIT OF ELECTROMAGNETS The results given offer considerable practical interest. They lead to the rule that the less a magnetic circuit is to differ from a closed one, the more carefully must the detrimental magnetic reluctance of superfluous joints be avoided ; that, moreover, where, for constructive reasons, joints cannot be avoided, the surfaces must be carefully fitted and pressed against each other with great force. In magnetic circuits with a wide air-gap an additional slit of -^ mm. equivalent width is not of much moment, so that in such cases the joints have scarcely any influence. 1 153. Time-variations of the Magnetic Conditions. We have hitherto limited ourselves to the consideration of invariable lasting magnetic conditions that is, to the case of stationary magnetisation. We have, however, incidentally ( 64) dis- cussed the induction of electromotive forces E in consequence of varying magnetisation ; it was there mentioned that this may be expressed in absolute measure, as the time-rate of variation of %-times the flux of induction where it is encircled ?i-times by the conductor ; that is, We have then, further, exclusively considered the time- integral of Z or, in other words, the total current impulse corresponding to a variation S quite apart from its time- variation, and which affords the most suitable method of measuring it. We shall now investigate more minutely variations of this kind, and we shall again elucidate the somewhat complicated phenomena which occur by the example of a toroid, either closed or divided radially, wound uniformly with n turns (resis- tance .R, radius of the toroid r n perimeter 2irr l = L, cross- section 8). At the beginning of the time (T= 0) let a constant external electromotive force E e act suddenly on such an 1 induction coil ' ; this would correspond to the steady current I e = E e /R, to the immediate establishment of which there is, however, an impediment in the form of a self-induced counter- 1 Compare an investigation by Czermak and Hausmaninger, Wiener JBerickte, vol. 98, 2 Abth., p. 1142, 1889. TIME-VAKIATIONS OF MAGNETIC CONDITIONS 237 electromotive force E.. 1 The corresponding differential equa- tion is /*7\ 77? v T? 7? dn -- dn dl (7) IE = E e - Z. = E e - T being the time, and J the actual current. Besides the constants E and R, and the single variable 7, the current flowing in each instant, the derived function or differential coefficient d (n)/d /occurs, which plays an impor- tant part in what follows. It is quite generally called the coefficient of self-induction, or the self-inductance, A of the coil, independently of the simple arrangement assumed in the present example. From the well-known equations 4 TT n I -. ^ <& e = jr and = S 93 Jj we obtain, in accordance with the above definition, fff> A = dl L The variable coefficient A depends then, in the first place, on the geometrical configuration of the coil, and has the dimen- sion 8/L that is, a length. 2 It is, further, proportional to the differential coefficient of the induction 93 in the ferro- magnetic core, with respect to the intensity % e of the field of the coil. We shall, therefore, at once specially consider this differential coefficient. 154. Discussion of the Function d ^/d^ e . In the first place, we get from the fundamental equation [11, equation (13)] 95 = 4 TT 3 + & 1 We shall here use suffixes similar to those in 53. Parasitic (Foucault's) eddy currents in the ferromagnetic substance itself will be expressly dis- regarded in the sequel ; for this purpose we may consider it as having in- finitely small electrical conductivity, or as being divided so that it has an infinitely fine-grained texture. Further, the electrostatic capacity of the coil will be disregarded, and its resistance R will be considered constant that is, the current will be assumed constant for the whole length of the conductor, and uniformly distributed throughout its cross section. In the extremely rapid alternations of current, to which the attention of experimenters is now being pre-eminently directed, these assumptions are inadmissible ; but in the present case we are considering variations which are comparatively slow. 2 The C.GLS. unit for self -inductance, therefore, is of course the centimetre; the official delegates to the International Congress of Electricians at Chicago in 1893 fixed the henry (10 9 cm.) as practical unit, a length which hitherto has also been termed the Secohm or Quadrant. 238 MAGNETIC CIECUIT OF ELECTKOMAGNETS the farther equation (9) d = 47T^ +'1 d$ e d*. In so far as we may neglect the second term on the right of the former equation, in comparison with the first, we can do so with the corresponding unit in the second equation. From the value of [cZSBj/d^J or of [efcSJ^&f], for a given value of the induction ^B l or the magnetisation 3j with a closed toroid, 1 we shall deduce that which, ceteris paribus, corresponds to a finite demagnetising factor N. Apart from the algebraical sign [ 53, equation (1)], we have & = & + & = & + By differentiation with respect to 3 we obtain which also results from the fact that by ' shearing,' the tangent of the angle of inclination of each element of the curve to the axis of ordinates is increased by an amount proportional to N. If now the latter equation is put in (9), we have The general graph of the function [dSB/d$J,or of which, when there is hysteresis, is no longer a single-valued func- tion, is represented in fig. 36, B, p. 225. These curves exactly correspond to the hysterestic cycle for annealed steel wire repre- sented in A ( 147), and, therefore, scarcely require any further explanation. The full-line curves for a closed toroid show, in the first place, a characteristic prominence, which is still more strongly marked with soft iron, and is therefore difficult to represent (c/. fig. 41, p. 243) ; and, in the second place, a discontinuity corre- sponding to 6r, the top of the hysteresis loop. The full-line curve 1 The expressions referring to closed toroids, N = 0, are in the sequel put in square brackets. Knott has proposed the names differential susceptibility,' or respectively ' permeability,' for the differential coefficients [< 3 / Z tne ' impedance.' 1 In acoustical and optical discussions, differences of phase or of path are usually expressed as fractions of the period or of the ware-length respec- tively, as has been done in the text. In the literature of alternating currents, phases are frequently expressed as parts of the circumference, given in de- grees ; the above fractions would in that case have to be multiplied by 360. INFLUENCE OF VARIABLE SELF-INDUCTION 241 By the latter the throttling action opposing the sinusoidal electromotive forces is expressed analogous to that which Ohm's resistance offers to steady differences of potential. 1 The relation between jB, Y, and J may be represented in an instructive manner by a right-angled triangle (fig. 39), the acute angle of which, expressed as fraction of the circumference, is equal to the difference of phase. It is not, however, our object in this paragraph to discuss the simple processes with constant self- induction. 2 We turn, rather, to the influence which the intro- duction of ferromagnetic cores with variable or multiple values of d3$ld$ e has on the character of those phenomena, and which we can most clearly represent graphically. We shall in this, again, consider in the above order the first problem of the growth and decay of electric currents, as well as, further on, in 157, the properties of alternating currents under the influence of sinu- soidal electromotive forces. 156. Influence of Variable Self-induction. Let the curve OP in fig. 40, p. 242, be a given curve of rise of current /= funct. (T). Its asymptote M f M then corresponds to the steady current I e . It follows from the general differential equation (13) of 154 which, as stated, holds also for variable self-induction that (19) . . . = ^(1. -I) The time-ratio for a point P is therefore equal to the tangent of the inclination of the curve to the axis of ordinates, multiplied into the portion QP of the ordinates, which repre- sents the deficiency by which the actual current falls short of the final steady current. The integration of the latter curve up to the time T l (which corresponds to the point P of the curve) at 1 The contrivance frequently used for this purpose in dealing with alter- nating currents is called a ' choking coil.' 2 Compare, for instance, the elementary graphical representations in Fleming, Alternate Current Transformer, vol. 1, pp. 95-116, London, 1890. R 242 MAGNETIC CIRCUIT OF ELECTROMAGNETS once shows that the area of the shaded surface enclosed by the line PQM'OP, that is (see the fundamental equations of 153), TI . A (20) (PQM' O) =\(I e - I) dT = 1 _\AdI = n - gSj J -fl J fi is proportional to the induction 93 1? which corresponds to the current J t passing at the time T lt or to the intensity &, = 47rwZ 1 /J>. From this follows a theoretically interesting method proposed by T. Gray for deducing curves of induction from curves of growth of current, to which we shall revert in 221. The dotted (J, T)-curve ON (fig. 40) was obtained by the experimenter in question l with a large electromagnet. To this 40" S" 0' FIG. 40 corresponds in the left quadrant the dotted curve N', which represents the variable time-ratio 9 = A/R (absc.) as function of the current (ord.) (see fig. 36, B, p. 225). For the sake of better comparison the full-line curve OP was calculated from Helmholtz's equation (14), taking as basis a constant time- ratio 6 = 3", which is obviously represented by the line 0' P r parallel to the axis of ordinates. A glance at the figure shows 1 T. Gray, Phil. Trans, vol. 184, A, p. 531, 1893 ; the principal constants of the electromagnet were : L = 265 cm., n = 3,840, R = ITS ohms ; in the experiments here given the magnetic circuit was closed, the measurements on open circuit are not free from objection. It scarcely needs mention that the investigations of 153, et seq., though elucidated by the instance of the toroid, are not confined to this simple form. For shortness sake, an (A' Y) -curve denotes in the sequel one which represents X as a function of Y. INFLUENCE OF VAKIABLE SELF-INDUCTION 243 the characteristic deviations which are due to the variable self-induction, especially the slower or quicker increase of the current, according as the variable time-ratio is greater or less than 3" (compare 170). It is further to be observed that in either case the tangent PM in a point P of the (I, 2 T )-curve runs parallel to the straight line, which connects the corresponding point P' of the (#, J)-curve with the fixed point M f . Sumpner has based on this a graphical method by means of which the (J, T)-curve can be constructed for an electromagnet, the induction-curve and dimensions of which are known ; and this, too, not only for ~SOOO #3000 ' 1SOOO >T '(seconds) o r^ z" a" V & FIG. 41 the case of the simple rise and fall of the current, but also when the external electromotive force is one given by any arbitrary function of the time. 1 We shall, in conclusion, reproduce in fig. 41, as example, one of the many diagrams given by Gray (loc. dt.). It embraces, as is seen, an entire cycle. In this the (7, T) -curves are full ; - (0, J)- or (A, /)-curves are dashed ; (3B, J)-curves are dot and dash lines. The scales for the various curves are drawn like them, only 1 Sumpner, Phil. Mag. [5], vol. 25, p. 470, Plate III., 1888. See also Fleming, loc cit., vol. 1, p. 263. R 2 244 MAGNETIC CIRCUIT OF ELECTROMAGNETS thinner to avoid confusion. On all the curves the points of the same height of the ordinates correspond. The figure needs, then, no further explanation. For further details we refer to the paper of T. Gray which has been quoted. 157. Sinusoidal Electromotive Forces. If the impressed electromotive force acting on the induction coil is a sinusoidal one, the form of the curve of the alternating current discussed in 155 is very considerably modified by the introduction of a ferromagnetic core. Let us first consider the simpler case that the self-induction is extremely small, which, according to equa- tion (12), would be effected by making the number of turns and the section as small as possible. Its influence, accordingly, will at first be neglected, so that the alternating current can be represented by a simple sine-function. The same holds for the field of the coil, which is represented as a function of time by the dotted sine-curve of arbitrary period in fig. 36, C, p. 225. If now the values of the magnetisation taken from the hysteresis diagram A above are plotted, the (3, T)-curve of the same period is obtained, this shows, as is seen, a retardation of its zero- points in respect of the (<>, jT)-curve, while the maxima coincide. 1 The (3, T)-curve becomes thereby unsym metrical and flattened owing to incipient saturation. The relatively small displace- ment of zero-points in the special case represented is, obviously, the more considerable the smaller is the range of the intensity in the cycle as compared with the value of the coercive intensity. Taking now a considerable value of self-induction, it may be theoretically proved, and partly also confirmed by experi- ment, that the (/, T)-curve is chiefly influenced by it in the following points : As soon as the magnetisation begins to approach satura- tion, characteristic projections, standing out on the sine-curve, are shown on the (/, jP)-curve, which, as the saturation increases, pass into sharp peaks. The choking action of the coil con- siderably diminishes after this. Hysteresis occasions an asymmetry of the ascending and 1 We can, strictly, only speak of a retardation of phase in the case of two sine- functions of equal period. It may also be remarked that retardation of the null point is in consequence of purely statical hysteresis, and is not necessarily connected with the question of the magnetic retardation. PRINCIPLE OF LEAST MAGNETIC RELUCTANCE 245 descending branches of the (/, 2 7 )-curve, as well as in the retardation of zero-points of the (3, T)-curve in respect of them. Their altered form reacts in turn on the latter, which thereby differ from the shape previously discussed, and again approach the form of a sine-function. The periodical processes, moreover, which come into play in consequence of self-induction, show a certain tendency to revert to the simpler original form repre- sented by the sine-curve, as the higher harmonic components are more choked than the fundamental periodic. An exhaustive discussion of this peculiar feature would lead too far. There is, finally, in the ferromagnetic substance a continual dissipation of energy into heat, which may be calculated by the data in 148. We shall revert to the phenomena previously discussed in dealing with the magnetic circuit of induction coils and transformers. B. Electromagnets for exerting different kinds of Pull 158. Principle of Least Reluctance. Electromotors, in the narrower sense of the word, as discussed in the previous chapter, which on continuous rotation transform a large amount of energy, and may be considered as inversions of dynamo machines, have almost entirely driven from the field the arrangements previously devised for this purpose. There are, however, in the various applications of electromagnetism many forms of apparatus for instance, measuring instruments, arc lamps, regulators, relays, bells, telephones, and many others in which what is usually a very small transformation of energy is scarcely of any im- portance ; the action of which is, however, to produce a relative change of position of their parts. We shall now consider such ' electromagnetic mechanism ' from some general points of view. There can, of course, in this book be no attempt at even an approximately complete review of the special contrivances used in an infinite variety of particular cases. Their action is ruled by the following principle, which has been more or less distinctly expressed by several authors, 1 and which in many practical cases serves as a useful guide, although it can scarcely claim any special theoretical importance : 1 Fleming, loo. cit., vol. 1, pp. 28, 71. Silv. Thompson, The Electromagnet, 2nd edition, p. 277, 1892. Part of the figures and the contents of the previous 246 MAGNETIC CIRCUIT OF ELECTROMAGNETS I. The configuration of an electromagnetic system exhibits a tendency to vary in such a way that its reluctance reaches a minimum value. This condition is obviously tantamount to saying that, with a given magnetomotive force assumed, the flux of induction tends to a maximum. The latter principle, again, might be deduced, with certain restrictions, from a consideration of the general equa- tions of electromagnetic energy. Considering the subordinate scientific importance, and the want of definiteness of the idea of magnetic reluctance ( 119), we may dispense with any rigorous mathematical proof of the above principle. Its practical appli- cation will, in the sequel, be explained in reference to a number of examples. 159. Mechanisms depending on Electromagnetism, in which, in conformity with the above principle, an open magnetic circuit tends to become closed, have been constructed in great num- bers. Let us consider, for in- stance, a ring divided dia- metrically (fig. 42, A 0), * each half of which attracts the other ( 103) that is, tends to FIG 42 diminish the magnetic reluctance of the whole. The variation of the reluctance for a small virtual alteration of the width of the whole gap forms, by well-known mechanical analogies, a measure of the attraction between the two halves of the ring. 2 This attraction, therefore, with a vanishing or a very narrow gap is considerable, but rapidly diminishes as the distance between the halves of the ring increases. The range within which the attraction exhibits or exceeds a prescribed value is, therefore, a small one. Various plans have been devised to get over these defects of the rapid decrease in the attraction, and the small range of the motion, some of which we will mention. section are taken, with the kind assent of the author, from the latter book. This affords the latest and most exhaustive account of the appliances met with in this special branch of electrotechnics. 1 In the following figures the ferromagnetic parts in the section are repre- sented by cross shading. 2 [See J. Hopkinson, the Electrician, vol. 33, p. 100, 1894.] MECHANISMS DEPENDING ON ELECTROMAGNETISM 247 For instance, the approach can be prevented from taking place freely, as would be represented by the plain arrows of fig. 42, but by suitable guides can be made to take a slanting direction in the direction of the feathered arrows, for instance. Or the free motion may be increased or be equalised by any of the well-known kinematic arrangements for transmission different kinds of levers, toothed wheels, &c. ; the variations in attraction may also be partially compensated by suitable springs. Another method consists in closing the circuit (fig. 42, (7), more or less, by means of an iron wedge, which is moved in a direction at right angles to the centroid. By suitably choosing the section of the wedge, an approximately unifornj increase of the reluctance may be obtained when the wedge is drawn out in the direction of the arrow. Where no great attraction is required, a better equalisation may be produced by never com- pletely opening the circuit, so that the continuity of the ferro- magnetic substance is never entirely broken. The plan of such an arrangement is seen in fig. 42, B, where the halves of the ring turn about a joint S, and therefore always touch in this point. The magnetic attraction will exert a pretty uniform torque on the upper half of the ring, in its rotation, as repre- sented by the arrows. Figs. 43, 44, and 45 represent various types of electro- magnets used for the most varied purposes, the action of which FIG. 43 FIG. 44 FIG. 45 is at once seen. The range here is tolerably extensive, and can partially be regulated ; the attraction is, however, not uniform. These forms are the transition to appliances in which a soft iron core is drawn into a coil, which may be either iron clad or uncovered, to the discussion of which we will now turn. 248 MAGNETIC CIKCUIT OF ELECTKOMAGNETS 160. Small Iron Sphere in a Magnetic Field. Let us first investigate generally the mechanical forces exerted by an electro- magnetic field on small ferromagnetic bodies in it. For sim- plicity's sake we will take the case of a small iron sphere, as it offers no preferential direction. In an external field of arbitrary distribution, its magnetisation will, by symmetry, be in the direction of the intensity of the field &,. From 33, the curve of magnetisation of an iron sphere scarcely differs from a straight line through the origin of co-ordinates, the equation of which, since the demagnetising factor of a solid sphere is 4 Tr/3 ( 30), is the following : (21) . . . 3 = A^ e 4-7T and this equation will hold with sufficient approximation up to values of something like 3 = 1500 C.G.S., that is, & = 6000 C.G.S. If V is the volume of the sphere, its magnetic moment may be written (22) . . m = 3 V = ~ F& 4 7T Now it may be shown that the mechanical force exerted by the field on a very small sphere in a particular direction for instance, that of the .X-axis has the following component $ x : (23) 9. = = dx 4?r dx 8-7T ox If we now imagine a surface through all points of the field in which the intensity, quite apart from its direction, has a prescribed numerical value, this will form a magnetic isodynamic surface on which <$ e , and therefore also , is constant. If we now consider, in the usual way, a group of such surfaces in space, which correspond to an arithmetical series of values of a constant surface parameter ( 38), the resultant force on the sphere at each point is directed along the perpendicular -ft to the corresponding surface passing through the point, and amounts to (24) . . fr.i.r ATTEACTIVE ACTION OF CIECULAE CONDUCTOES 249 and the force for a ferromagnetic sphere is in the direction of increasing values of . We may sum up those considera- tions in the following : II. In any given field a small ferromagnetic sphere tends alivays to pass from places of weaker to places of stronger intensity ; and this quite independently of the direction of this vector. The mechanical force ft exerted on the sphere further has evidently the scalar potential ( 39) o (25} - F< 2 ( } 8 TT The above principle was propounded by Faraday on the basis of his experimental investigations. Its mathematical enunciation is due to Lord Kelvin. 1 161. Attractive Action of Circular Conductors on Sphere. Let us apply this fundamental principle to the simple case of a rri FIG. 46 plane circular conductor carrying the current 1. Let r be the radius of the circle, x the distance along the axis (fig. 46), z = *J x 2 + r 2 , the distance of a point on the axis from the 1 Faraday, Exp. Jfes., vol. 3, series 21, especially 2418 ; Sir W. Thomson, Eeprint Electr. and Magnet. 643-646. This potential * of the mechanical force $ must be carefully distinguished from the magnetic potential T ( 45, 48) ; neither are the magnetic isodynamic surfaces directly connected with the ordinary equipotential surfaces. 250 MAGNETIC CIRCUIT OP ELECTKOMAGNETS circumference. The numerical value of the field intensity in a point of the axis (x) [ 6 C, equation (4)] is (26) ^=- hence = + Let the small iron sphere be restricted to motion along the X-axis, for instance, by being compelled to move without friction along a tube. The component of force then acting upon it amounts, according to equations (23) and (26), to (27) . *. F 8?r In the top half of fig. 46 the function z~ 3 is graphically represented. According to (26) this is to be multiplied by the constant 2 TT Jr 2 , in order to have the intensity in a given place. This, as will be seen, attains a maximum in the plane of the circular conductor [_<$ e = 27r//r, according to equation (5), 6] ; at a distance x = 2 r that is, equal to the diameter of the circle it amounts to only eight per cent, of that maximum. In the lower half of fig. 46 the fraction x/z s is represented. This is in the plane of the circle itself, and then rapidly increases. As a repeated differentiation shows, it attains a maximum for x = r/\/7 = O38r while the steepest part of the curve of intensity is at a;=0'5r and then gradually diminishes to very small values. Multiplication of that function by 9 TT V P r 4 gives the component of force in absolute measure. This is always directed towards the conductor, that is in the sense of increasing values of the intensity of the field ; and this inde- pendent of its direction. The left half of fig. 46 is omitted, as it is symmetrical with the right half represented. 162. Attractive Action of Coils on Spheres. The field in the axis of a long, uniformly- wound coil ( 6 D) may be re- garded as the superposition of the fields due to individual turns. The transition from the portion of the field in the middle of the coil, which is known to be uniform, towards the outside is represented by a curve, which is like that of fig. 46. The field at the opening attains half the value of its value in the middle, ATTRACTIVE ACTION OF COILS ON SPHERES 251 and then rapidly diminishes. A small iron sphere, with its motion again restricted to the axis of the coil, 1 will, as with the circular conductor, be drawn into the coil. The attractive action is at its maximum near the opening, and then decreases as we approach the region of appreciably uniform intensity, within which a mechanical action is of course no longer exerted on an iron sphere. An accurate representation of the forces by equations would be as difficult as it would be without object, as they always depend on the particular dimensions of the coil. If we dispense with uniform winding, we may influence as we like the distribution of the field, and therewith that of the attractive force along the axis. This can be attained by altering the pitch of the winding, so that the number of turns per unit length is a variable quantity. If, for example, we desire for any purpose to produce a steady attractive action within a pre- scribed range on the axis of the coil, then we must have dx hence & = ^Cdx = Cx + B and in which C and B are constants. No general directions can be given as to how the coil must be wound so as to produce a given field. In any given case this must be determined by actual trial, in which the principles laid down may serve as some guide. For a given position of the sphere the attractive force, according to equation (27), is, ceteris paribus, proportional to the square of the current in the coil. Saturation can never be obtained with a sphere under the influence of the field of a coil. The discussions in this paragraph apply not only to small spheres, but, approximately, also to other pieces of iron, the dimensions of which are nearly equal in all directions, and are 1 Such a restriction is here necessary as well as with the circular conductor, because the iron sphere, in accordance with Faraday's principle, would other- wise move towards places of higher intensity on the surface of the conductor ; that is, over the inner surface of the coil. 252 MAGNETIC CIRCUIT OF ELECTKOMAGNETS small in comparison with those of the coil. But when one dimension preponderates, and is comparable with the length of the coil, the theory is rather complicated, and we are at present compelled to return to empirical investigation. 163. Attractive Action of Coils on Iron Cores. We possess numerous measurements on the attraction of ' short ' or ' long ' cores that is, of such as are shorter or longer than the coil. We shall pass over the older researches of Hankel, Dub, von Feilitzsch, von Waltenhofen, &C., 1 and confine ourselves to the discussion of the more recent systematic ones of Bruger. 2 His experimental arrangement will be sufficiently clear from fig. 47 ; the weight of the core in each case was compensated by one sliding weight, and the attraction was measured by another weight for various relative positions of the core and of the coil. Equilibrium is obtained, that is to say, the attraction ceases, when the middle of the core M, supposed to be symmetrical, coincides with the middle of the coil m. That follows from symmetry, as well as from the principle of least magnetic reluctance ( 158). We determine, therefore, the relative position by the height Y of the middle of the core over the middle of the coil. If the short or long core is moved downwards, the force of suction only attains an appre- ciable value just before the lower end U of the core enters the upper opening o of the coil. It then rises to a maximum, and falls 1 G. Wiedemann, Lehre von der EleJctricitat, vol. 3, 651-665, 1883. 2 Bruger, Action of Solenoids on Iron Cores of various Shapes. Inaugural Dissertation. Erlangen, 1886. [See also Mr. Mahon, the Electrician, vol. 35, pp. 293, 604 ; 1895. It would appear possible at present to give a theory de- scribing these actions with more or less approximation. H. du B.] FIG. 47. scale POLAEISED MECHANISMS 253 off again until the above position of equilibrium is attained. Brug*er restricted himself to investigating long coils, and found, in agreement with older statements, that the maximum suction corresponds to about the position in which the lower end of the core U is just on the point of emerging from the lower opening u of the coil (as represented in fig. 47) ; and this holds for cores the length of which is twice that of the coil. Bruger used, among others, a coil with the following con- stants : length L s = 13 cm.; number of turns n = 266 ; field % m in the middle of the coil, per ampere, = about 30 C.G.S. He obtained, for example, with three cores of about 39 cm. (= 3L S ) length the curves given in fig. 47. Y defines the position of the core, X gives the corresponding suction in gramme-weight. In the diagrams of the curve : JT3, full curve : cylindrical core ; $ m = about 180 C.G.S. 1/5, dot and dash curve : double cone ; <% m = about 180 C.G.S. CO, dotted curve : moniliform core ; <% m = about 250 C.G.S. For the details we must refer to the paper. Conical cores are used in some arc lamps. The moniliform core gave, as intended, a constant pull over a great range. As long as saturation is excluded, the attraction is roughly approximately proportional to about the square of the current in the coil. Bruger gives curves like those above for various currents. The phenomena for iron-clad coils, as represented in fig. 45, p. 247, are similar ; the field then between the inner edges of the shell is more intense and more uniform than in a coil without shell, but outside the coil it diminishes so much the more rapidly, by which the character of the traction curve is somewhat but not materially altered. The case of coils without shells but with a lining of iron is quite different. These do not attract iron cores, but, on the contrary, repel them. Having regard to the principle discussed in 158, this appears at once intelligible. 164. Polarised Mechanisms. The phenomena are markedly simpler if the core of the coil is not previously magnetised by the current, but has from the outset permanent magnetisation 254 MAGNETIC CIRCUIT OF ELECTROMAGNETS ( 46). This case we will discuss. We have seen in 21 that a mechanical force is exerted on the single end of a magnet in a field, which is in the direction of the latter, and is equal to the product of the intensity of the field into the strength of the end. This may be experimentally realised by bringing one end of a long permanent magnet into the field of the coil, so that the action of the latter on the other end of the coil may be neglected. In fig. 47, p. 252, let UO be a powerful steel magnet, the lower end U of which would move in the direction of the field of the coil or against it, according as its sign is positive or negative. In the middle part of the coil, where the field is uniform, the mechanical force exerted on the end will also be invariable, while it decreases through the openings outwards in proportion to the intensity of the field. It is assumed that the field of the coil is so weak that it exerts no appreciable inductive action on the permanent magnetisation of the steel core. The arrangement described belongs to the group of what are called polarised mechanisms, in which permanent magnets may in any way be used. Two properties present themselves, which in certain circumstances it is desirable to develop as far as possible. In the first place, a ' bilateral ' action, in consequence of which, with currents of opposite directions, actions about proportional to these currents and opposite to one another may be obtained, which is excluded in purely electromagnetic mechanisms. Secondly, the possibility of attaining greater sensitiveness of the attraction with weak currents or rather with small changes of current d I as results from the following considera- tion. According to Maxwell's law [ 103, equation (12)] the attraction is From this it follows, by differentiation, that the sensitiveness = 24,500, and an induction ?8 = 45,350 C.G.S., the highest which had at that time been obtained in soft iron. As regards the field intensities observed in the air, it follows from the published statements that until recently values above 28,000 to 30,000 C.G.S. units had not been reached. 168. Principles of Design. There seemed no a priori reason why the production of still stronger fields should be impossible. The author accordingly attempted the construction of an electromagnet for this purpose, and in this attempt he was guided by the following considerations. The first point is to begin with the production of as high a value as possible of the flux of induction, which then, by i throttling ' the magnetic circuit by means of suitable pole-pieces, may, as it were, be concentrated as described below ( 175). Accordingly the magnetic reluctance which, especially owing to the unavoidable air-space between such pole-pieces, cannot be indefinitely diminished, must be overcome by as great a number of ampere-turns as possible ( 173). In all the electromagnetic apparatus and machines we have hitherto discussed, and indeed in the great majority of such, it was sufficient from the nature of the case to consider only the first two stages of the process of magnetisation. But in the present problem the third, or stage of saturation, alone need be considered. In consequence of this, and of the circumstance that considerations of economy, certainty of working, facility of repair, and the like, are of less account in the present case, the conditions of construction are, to some extent, different. The discussion of 95 showed that the field of the coil finally tends to completely direct and dominate the distribution of the vectors in the magnetic circuit ; hence the coiling must be such that there is everywhere, and especially between the pole-pieces, a field in the desired direction that is, tangential to the centroid of the magnetic circuit. In such an arrangement leakage will ultimately decrease as the saturation increases, and the induction - tubes so gained will be utilised ; this result was confirmed by experiment ( 173). As regards the shape of the ferromagnetic substance, the theoretical conditions already mentioned are best satisfied by a toroid divided radially. In other respects, the points discussed in 141 for the construction of the frames of 262 MAGNETIC CIRCUIT OF ELECTROMAGNETS electromagnets may be referred to ; they only hold, however, in the present case mutatis mutandis. Starting from the principles developed, the author has constructed an electromagnet which we shall proceed to describe. 1 The experiments made with this, to be discussed afterwards, must be considered as a confirmation of the validity of the principles followed in its design. 169. Description of the Electromagnet. As' it was found that, with proper tooling appliances, the construction of a heavy iron toroid was neither more difficult nor more costly than building up a frame of several parts, the former was decided on. Fig. 56 represents the electromagnet in T ^- the natural size, partly in outline, and partly in section. In the notation of fig. 15, p. 109, ^ = 25 cm., r 2 = 5 cm. ; hence L = 157 cm., and 8 = 78-5 sq. cm. At $ the toroid is divided tangentially to the inner circle. A hori- zontal sliding motion is here introduced, so that the right side of the toroid can be dis- placed in reference to the left by means of a handle or wheel G, and thus the upper air- space Z be conven iently adjusted. The slide is so constructed that the break of continuity of the ferromagnetic substance is as small as possible. However, with the preponderating magnetic resistance of the air-gap Z a few joints are not of much account ( 152). In order to prevent any bending in consequence of the con- 1 169, du Bois, WiedAnn., vol. 51, p. 507, 1894 ; Elektrotechn. Zeitschrift, vol. 15, p. 203, 1894. COILS OF THE ELECTROMAGNET 263 siderable tractive force, a brass holder M l D M 2 is fitted, which, by means of a screw, can be adjusted to the width of the gap at any time. By using flat pole-pieces, separated by a narrow slit, the tractive force is so great that only discs of metal placed between can resist it. 1 The perforation L } , L 2 in the direction of the field allows of magneto-optical observations if desired, but iron plugs K } and K. 2 are usually inserted, since an unnecessary increase of reluctance in this place is not desirable. 2 The toroid rests on bronze bearings, which, in turn, are supported by a massive wooden tripod F 19 F 2 , F 3 , provided with rollers E^R V B 3 and levelling screws J^, 2J7 2 , and E y The table TT serves for placing on it accessory apparatus. The axis of the field may, by tilting the whole apparatus, be set vertical, which is desirable for certain experiments. 170. Coils of the Electromagnet. We have hitherto omitted to discuss general rules for winding and plans for connecting because in every special case this is simply determined by pre-existing conditions in a comparatively Dimple manner. 3 It may perhaps be mentioned that the traditional rules for the use of batteries that is, sources of current, whose electromotive force and internal resistance are assumed constant (which, how T - ever, seldom occurs) have less interest at the present time. In most cases we are now concerned with self-regulating dynamo machines, street mains, or accumulators that is, sources of current which furnish a more or less constant difference of potential, and in using which a definite limit of current may generally not be exceeded. 1 Assuming a tension of 16 kg.-weight per sq. cm. ( 103), the total pull is g = 3 S = 16 x 78-5 = 1250 kg.-weight, whereas the entire weight of the whole electromagnet is 270 kg. Suitable discs are provided, 1, 1, 1, 2, 5, 10, 10 mm. thick like a set of weights. 2 When the air-space Zis not too small, it has little effect, as observation shows, whether the poles are filled with iron cores or not, as the reluctance of the air then preponderates ( 175). Similar statements are made by Leduc (Jowrn. de Physique [2], vol. 6, p. 239, 1887). As to the properties of hollow iron cores in general, reference must be made among others to von Feilitzsch, Pogg. Ann., vol. 80, p. 321, 1850 ; Silv. Thompson, loc. tit., pp. 86, 184 ; Leduc, La Lumiere eleotrique, vol. 28, p. 520, 1888 ; Grotrian, Wied. Ann., vol. 50, p. 705, 1893 ; du Bois, ibid., vol. 51, p. 529, 1894 (see also note 1, p. 235). 3 Compare Silv. Thompson, loc. tit., chapter vi., where this question is thoroughly discussed for steady currents ; in chapter vii. follows a discussion 264 MAGNETIC CIRCUIT OF ELECTROMAGNETS In the present case the former amounted to 108 volts, and the latter to about 50 amperes. Each individual coil comprises a sector of the circumference of 20 ; its 200 turns have about 0*2 ohm resistance when warm. If the 12 coils are arranged in series, they have, accordingly, 2*4 ohms resistance, and they cover 240 that is, two-thirds of the circumference. With that total resistance the difference of potential, 108 volts, produces a current of 45 amperes; this corresponds to a magnetomotive force of 108,000 ampere-turns, or 136,000 C.G.S. units. Dividing the latter number by the perimeter L 157 cm., we get 860 C.G.S. for the mean intensity of the field of the coils. Of these only about 380 C.G.S. is to be considered as a direct inductive agent. With the iron actually used 1 the magnetisation attained is 1600 C.G.S. The excess of intensity (480 C.G.S.) serves exclusively for counteracting the demag- netising action. If the value of magnetisation given is to be maintained, a demagnetising factor up to is admissible. This, in fact, is its value with the widest air- spaces which occur in use that is, with pointed pole-pieces. The power necessary for exciting the maximum effect of the electro-magnet is 108 x 45 volt-amperes = 4860 kilo-watts = 6-5 H P Its greatest self-inductance, with closed magnetic circuit, if we disregard the demagnetising action of the sliding guides, as well as of other joints, which, however, may scarcely be neglected, may by 153 (eq. 8) be calculated as follows : r l d% e 25 of the winding and connecting of coils for variable currents as required in electromagnetic mechanism, when as rapid an action as possible is an essential requirement. 1 This was the same brand as that from which the toroid described in 83 was turned, and the normal curve of which is represented in fig. 21, p. 131. 2 From the curve of ascending reversals (0), fig. 21, p. 131, we find the maximum value of the differential quotient d 3 / d a' > 57, No appreciable advantage was found in having the pole-pieces concave. The observed value of the field was always several 272 MAGNETIC CIKCUIT OF ELECTROMAGNETS thousand C.G.S. less than the theoretical. The following measurements were obtained : for r 3 = 2-5 mm. 36,800 C.G.S. for r 3 = 1-5 mm. 38,000 C.G.S. 1 It follows, from the formula of the previous paragraph, and is confirmed by experiment, that high intensity of the field is only attained at the cost of its extent. For many experiments, however, an extent of several sq. mm. is sufficient, or else the methods of investigation must be adapted to satisfy this condition. The bore-holes, which are indispensable in magneto-optical experiments, produce relatively more weakening of the field the wider they are as compared with the distance of the faces. This weakening is, however, less than would follow from the equations in 174, since there is a kind of internal leakage from the edge of the openings towards the axis. The external leakage and action at a distance when truncated cones are used are similar to that described for plane poles. What has been stated in the previous section may be summed up by saying that a ring electromagnet of manageable size, with truncated conical pole-pieces of 120 aperture, enables us to have fields up to, say, 40,000 C.G.S. over an extent of some square millimetres. To exceed this to any material extent could at present be only accomplished by an undue expenditure of means out of all proportion with the end in view. That follows already from the formula given in which log (r) comes in ; while the weight and cost of an electromagnet are determined rather by the third power of its linear dimesions. D. Inductors and Transformers 176. Discussion of Mutually Inducing Coils. We have already explained the principal manifestations of self-induction by the example of a uniformly coiled toroid, either closed or divided 1 This field would produce in a piece of thin soft iron between the trun- cated cones the approximate induction equal to S3 = 38,000 + 4?r x 1750 = 60,000 C.G.S. to which would correspond a total longitudinal stress equal to ( 103) /60,000> \ 5000, \ 2 ) = 144 kg.-weight per square cm. MUTUAL INDUCTION 273 radially (mean radius r lt perimeter 2?rr 1 = Zf, section S). Besides that primary coil 1 (resistance R v number of turns 7i p self-inductance A^, let there be a secondary one 2 (R v n 2 , A 2 ), also wound uniformly on the toroid, as would be the case with the experimental ring described in 83. Taking the case of such a pair of mutually inducing coils, we will explain as briefly as possible the most important facts in mutual induction, in so far as'they are essential for understanding what follows. To a variation of the flux of induction l produced by the primary current corresponds an electromotive force J7 42 in the secondary, which, from equation (6), 153, will have the follow- ing value : , _ dT dl, dT The partial differential quotient of (n 2 j) in respect of the primary current Jj, which here occurs, is called, in analogy with the definition of 153, the mutual inductance H 12 between the coil 1 and the coil 2. We find, in the present case, /QO\ w _ 4 ^ n \ n 2 8 d SB _ w \OJ I . > *12 - - T ~ T7^~ - "1 L d% e n } If now the parts played by each coil in the phenomenon are interchanged, so that 2 now acts inductively on 1, we have, in an analogous manner, dT IdT ~ ~ 21 dT in which, again, the mutual inductance is _ n - - <* 177. Mutual Induction. Now it is obvious from the above that H 12 = H 21 . Hence this quantity may be called, without further definition, the * mutual inductance ' of the two coils. 1 From (39) and (41) we see, directly, that for ?^ 1 r n 2 we have A l = H = A 2 1 Mutual as well as self inductance has the dimensions of a length, and like this is to be expressed in henries (compare note, p. 237). T 274 MAGNETIC CIRCUIT OF ELECTROMAGNETS Further, for any given value of S3 we have (42) g = These conclusions are based on the assumption that in (39) and (41) 8 as well as d 99 / d<$ e are identical ; that is, in other words, the common flux of induction threading through the two coils is the same. When a ferromagnetic core is used, this is always the case with sufficient approximation, since the induc- tion tubes outside it produce no appreciable difference (compare, however, 183). 1 As the mutual inductance H differs from the differential quotient d 95 / d <& e by a constant factor only, it will be sufficient to refer to 154 and fig. 36, B, p. 225, where the character of that differential quotient is completely discussed. For closed magnetic circuits there can be no question of a constancy for H, any more than there can for A ; for unclosed magnetic circuits, on the contrary, that assumption is here also the more admissible, the greater the value of the demagnetising factor. By means of an arrangement quite similar to that here assumed, in which, however, each of the two coils was severally placed on one half of a ring, Faraday, as is well known, dis- covered, in 1 83 1 , the existence of induction phenomena. 2 Almost simultaneously, and quite independently of him, J. Henry 3 also carried out his fundamental researches in this department, more especially on ' extra current,' as it was called ; he already introduced the expression * self-induction.' The kind of ap- paratus used by these experimenters had undergone a long course of development, 4 until, after the lapse of half a century, a division into two classes was made, inductoriums, or induct ion coils, and transformers, of which the first has mainly a scientific interest ; in the decade which has elapsed since that separation, the latter class has from its technical importance undergone an 1 The circumstances are quite different with two coils without cores, in any given relative position ; such cases, however, may be disregarded for our present purposes. 2 Faraday, Exp. Res. vol. 1, Series I. 27 (Plate I, fig. 1), 3 J. Henry, Collect. Sclent. Writings, vol. 1, p. 73 et seq. ; Silliman, Amerie. Journ. vol. 22, p. 403, 1832. 4 G. Wiedemann, loc. cit. vol. 4, 409-430 ; Fleming, loc. cit. vol. 2, chap- ter i. ACTION OF INDUCTION COILS 275 extraordinary development, both in the theoretical relations and in the practical details of construction. 1 78. Action of Induction Coils. In an induction apparatus, in the narrower sense of the word, extremely high electromotive forces are induced in a secondary coil, which has a great many turns, by making or breaking the primary current. These are mostly used to produce electric discharges of various kinds between the ends of this coil. The discharge on making corresponding to the gradual rise of a given primary current, as described in 155, starts a quantity of electricity in a closed secondary coil, equal to the break discharge which takes place in the opposite di- rection ; because in both cases the same total variation 8 CM of the flux of induction and the same resistance are concerned [ 64, equation (14)]. Although, then, the time-integral of the secondary electromotive force is in both cases the same, its maximum value is far greater on breaking the primary current ; for this process, though never instantaneous, is, however, con- siderably more rapid than the gradual increase of the current up to a large fraction of its steady value, which takes place on making (15). In the case therefore of spark discharges, only the ' break-spark ' passes across wide spaces of air ; with the same distance of the electrodes the 'make-spark' cannot traverse the air-path. It follows from this, that it is of paramount importance to break the primary as rapidly as possible; for this purpose the following means are more especially applied. In the first place, contact-breakers are used, which reduce as much as possible the primary spark due to the self-induction of the primary coil, which prevents an absolutely instantaneous break of the current. Experience shows that in this respect mercury contact-breakers are best, working under insulating liquids, such as alcohol or petroleum. (Note 2, p. 265.) In the second place, it is usual to adopt Fizeau's plan of interposing a condenser near the break. This diminishes the injurious spark by partially storing up the primary discharge. For any given primary coil the most suitable capacity, that which most effectually extinguishes the primary spark, while it prolongs the secondary one, is to be found by experiment. This probably depends on one of the phenomena of resonance which ' T 2 UNIVERSITY) 276 MAGNETIC CIRCUIT OF ELECTROMAGNETS by the brilliant discoveries of Hertz have recently claimed promi- nent attention. The oscillating primary discharge is then, as it were, thrown back by the condenser, and produces a primary current in the opposite direction ; in this way it is obvious that longer secondary break-sparks are formed than when the primary current merely vanishes without altering its direction. 179. Magnetic Circuit of Induction Coils, It is not sufficient that the break or the reversal be as rapid as possible ; it is also necessary that the corresponding variation of the flux of induction in the core shall directly follow it. Let us assume at first that the primary current only falls to zero ; for the induction in the secondary coil the corresponding decrease of the flux of induction that is to say, the evanescent magnetisation denoted in 149 by % E is of paramount im- portance. Let us imagine, -for example, a closed core of good soft wrought iron ; with an intensity of & = 20 C.G.S., let the magnetisation 3j/ = 1000 be attained ; as the retentivity in this case might easily amount to 90 per cent., 3* would be = 900, hence 3 E would only equal 100 (see 149 and fig. 36,4, p. 225). The case is different with an unclosed core, the demag- netising factor of which we suppose to be, say, N = 0*02, corre- sponding to (m = 45, Table I, p. 41); taking the coercive in- tensity c = 2 C.Gr.S., we have 3 ft = c / JV = 100, and hence, 3 = 900. At the same time a demagnetising intensity & = N% M = 20 will have to be compensated, so that the field of the primary current must be doubled, that is &, must be increased to 40 C.G.S. From the purely magnetical point of view, there would scarcely be any object in making the dimen- sion-ratio of the core less than 45, and thereby needlessly increasing the demagnetising action. But if the primary current does change its direction owing to the action of the condenser, a closed core appears again more advantageous, provided that the opposing primary field attains at least the coercive intensity ; an inspection of fig. 36, A, p. 225, shows the correctness of this contention. On the other hand, the smaller the demagnetising factor the greater is the value of the differential quotient d$$ / d$ e ( 154); the self- induction of the primary coil is hereby increased in a manner MAGNETIC CIRCUIT OF INDUCTION COILS 277 difficult to calculate, but which is scarcely desirable. We can- not at present be said at all to possess a comprehensive rational theory of the induction coil, probably because these apparatus have not hitherto had any extensive practical applications. One great difficulty is that we have no sufficient knowledge of what takes place when the current is broken and produces a spark with or without condenser ; such a theory must necessarily be mainly based on that knowledge. The most practical, efficient and best known forms of induction coils are produced by a few workshops ; their empiri- cal principles of design are doubtless based on experience, and are partly kept secret. 1 We must, therefore, be content with referring to the fact that the cores generally consist of bundles of wire, the dimension- ratio of which varies between 15 and 20 (that is, 0-12 > F> 0-08, Table I, p. 41) ; the thickness of the wire is usually about 1 mm. With reference to the most recent investigations on eddy currents as occurring in transformers (see 187), it might be desirable to have thirner iron wires for the cores, with their much more rapid magnetic variations, and without any corresponding disadvantages inherent to their use. 180. Simultaneous Differential Equations of Transformers. The first transformers of induction coils, introduced about a decade ago, had a general resemblance to the induction coils then in use, and accordingly had an unclosed core of thin iron wire. Following the plan of Zipernowsky (1885), forms with closed magnetic circuit were soon reverted to ; in these Faraday's ring, or the uniformly double-wound toroid discussed in 176, became the prototype for by far the greater number of trans- formers now made. Accordingly the magnetic characteristic ( 96) of the transformer circuit becomes very simple, since neither an air-gap nor leakage has to be considered ( 183). On the other hand, the expression for the magnetomotive force is somewhat more complicated on account of the action of two coils ; for obviously the following equation must be always satisfied : 1 See du Moncel, Notice on RuJimkorff's Apparatus, 5th edition, Paris, 1867. Most induction coils are based on this apparatus with but slight modifications. 278 MAGNETIC CIRCUIT OF ELECTROMAGNETS Moreover, the Hopkinson function ( 97) now becomes (44) . jf=i where L is the perimeter of the centroid, and the indices 1 and 2 refer to the primary and secondary coil respectively. Hence, the equation of the magnetic circuit becomes (45) . 4^( rei 7 1 + 2 I 2 ) Using the notation of 176, and in connection with the discussion there given, the differential equation of the primary coil then becomes 'dI dl it asserts that the sum of the three electromotive forces corre- sponding to self-induction, mutual induction and ohmic resis- tance are at each moment equal to the impressed electromotive force Eei ; the latter is a periodic function of the time, and in a certain number of cases, it may with sufficient approximation be regarded as a sine function. 1 The circuit external to the secondary (resistance 72 2 ') being supposed devoid of self-induction and capacity, and therefore offering no source of E.M.F., we have (47) We cannot expect to arrive at a general solution of these two simultaneous differential equations with the two variables Jj and / 2 , since the inductances A l A 2 and 3 in a closed magnetic circuit are, as our former reasonings show, neither constant nor unambiguous ( 154, 177). The only plan is to consider, in the first place, an c ideal transformer ' in which 1 The time-curve of an electromotive force furnished by an alternator depends on the form of its pole-pieces and coils ; moreover, the self-induction of i he armature, &c. shows a certain tendency to partially annul the deviations from the sine form. A discussion of the curve of periodic electromotive forces given by some of the usual alternators is given by Fleming, loo. cit. vol. 2, pp. 446-475. ACTION OF AN IDEAL TRANSFORMER 279 these functions are supposed to be constant ; by assuming that the primary electromotive force is sinusoidal, it is then found that the primary as well as the secondary current, and there- fore also the secondary electromotive force E 2 , are sine functions showing differences of phase among each other which can be calculated ; it is, in fact, obvious that the above differential equations can be satisfied by such functions. From the behaviour of the ideal transformer, conclusions, more or less pertinent, as to that of a transformer with an iron core may be drawn. This method depends on Maxwell's general method of mathematically treating any given inducing pairs of coils. 1 J. Hopkinson 2 has pursued a somewhat different plan, which depends on the identity, apart from leakage ( 183), of the flux of induction embraced by the two coils (see note 2, p. 240) ; in consequence of this, the differentials may be eliminated in the first mode of writing equations (46) and (47), by multi- plying the former by w 2 and the latter by 7t n and subtracting ; we then obtain the following equation : (48) . n, E el = n, R, I, - n, (R 2 + JB 2 ') J 2 If this is combined with equation (45) of the magnetic circuit, and if some terms are disregarded, we also get data for approxi- mately judging the action of a transformer. 181. Action of an Ideal Transformer. We shall first discuss the case in which the resistance R' 2 of the external secondary circuit is infinite that is, it is broken and the trans- former therefore runs ' empty.' There being no current in the secondary coil, it cannot react on the first, and the latter behaves like a simple induction coil, the properties of which are discussed in 153 et seqq. It follows from the equations there given that, with a sufficiently high self-inductance that is, time- ratio the primary current is considerably choked, and more- over a retardation of phase % as against the primary electro- motive force will be shown, which approaches the value % = (1/2-Tr) tan -1 = 1/4; in the last case we have to deal with what is called a c wattless ' primary current, which represents 1 Maxwell, Phil. Trans, vol. 155, I, p. 459, 1865. See also Mascart and Joubert, Electricity and Magnetism, vol. 1, p. 593 ; vol. 2, p. 834. 2 J. Hopkinson, Proc. Roy. Soc. vol. 42, p. 85, 1887 ; Reprint of Papers, p. 182. New York, 1893. 280 MAGNETIC CIRCUIT OF] ELECTROMAGNETS no electrical energy. 1 With an ideal transformer running empty, the flux of induction has the same* phase as the primary current which produces it alone, while the secondary electro- motive force induced by the variations of the former is in all circumstances retarded by a quarter of a phase in reference to it. If now the secondary current is made, and the transformer is thus loaded, the phenomena are more complicated ; we shall confine ourselves to mentioning the chief points. The self- induction of the primary coil is apparently diminished by that action, so that the primary current is so much the stronger and its phase gains the more on that of the primary electro- motive force, the greater the load. The electrical power sup- plied to the primary coil, which must exceed the power corre- sponding to the sum of the load, and of the loss of efficiency in the transformer, at the same time increases sufficiently to meet the demand. In case the self-induction of the secondary coil is small, and its external circuit free from induction, the phase of the current in it is but little behind the electromotive force ; in respect of the primary current it usually shows a retardation of phase of almost J that is, the two currents with a full load have usually almost opposite phases, and therefore magnetise the core in opposite directions. The ratio of the mean primary, to the mean secondary electromotive force, is called the transfor- mation-ratio or the coefficient of transformation $ ; for the ideal transformer hitherto considered this is (49) . . ^==^ = 22 , and this relation holds very nearly for practical transformers. As, therefore, the electromotive forces are proportional to the corresponding number of turns, it is clear that the products T&! Jj and 7i 2 J 2 that is, the primary or secondary ampere-turns respectively will be but little different ; in consequence of this, their algebraical sum, occurring in equation (45), or, as they have 1 The mean electrical power of a sinusoidal alternating current is known to be equal to half the product of the maximum values of the electromotive force, and the current, multiplied by cos 2irx, and would therefore vanish for X = 1/4- For running empty such a condition is of course to be aimed at ; that it is, however, only approximately realised is a weak point in the use of transformers permanently inserted in at any rate the primary circuits, the secondary coils of which run empty for days together, or are but little loaded. INFLUENCE OF SATURATION AND OF HYSTEEESIS 281 opposite signs, their numerical difference will only be small in comparison with each of them singly. The curve which represents the secondary electromotive force (or the potential at the terminals l ) as a function of the secondary useful current, may be called the < total (or external) characteristic ' of the transformer, just as it was with the dynamo ( 126, 127). This, however, is not the place to discuss it, as it has by no means so simple a connection with the magnetic characteristic (equation 44) as in the case of dynamos. The processes described are made plainer by plotting a trans- former diagram, which represents the four quantities J^, / _E7 2 , 1 2 as periodical functions of the time ; many suitable methods have in recent times been devised for this purpose (see fig. 59, p. 284). After these brief remarks we proceed to discuss the modifications of the ideal processes caused by allowing for magnetic saturation and hysteresis as well as for leakage. 182. Influence of Saturation and of Hysteresis. As regards saturation, the inductances A l , H, and A 2 diminish considerably even before it is attained. Phenomena occur in consequence of this which are similar to those produced in a simple induction coil ( 157) by saturation. The behaviour of the transformer, briefly speaking, would approximate to that of a coreless one ; as this, however, must absolutely be avoided, the induction SB in good transformers at the present time is scarcely allowed to exceed the value 6000 to 7000 C.G.S., at which the inductances have exceeded their maximum, and just begin to decrease (fig. 41 , p. 243). This value, which is usual, in practice amounts to less than half that often reached in dynamos or electromotors ( 126). Accordingly, since even an incipient saturation is nowadays never allowed to occur, there is scarcely any interest in further discussing its influence, which, however, with the older transformers is in many ways characteristically observable. In the second place, hysteresis produces considerable dif- ferences of the coefficients of induction for ascending or for descending magnetisation in the magnetic circuit of transformers. It will be seen from the transformer diagram obtained by 1 In the ordinary winding of transformers, the potential at the terminals moreover usually differs by less than 1 per cent, from the electromotive force. 282 MAGNETIC CIRCUIT OF ELECTROMAGNETS experiment (fig. 59, p. 284) that the curve of the primary current is chiefly affected by it, which, as we have already seen in the case of the simple induction coil, loses the symmetry of the ascending and descending branches, and in some circum- stances acquires a perfectly irregular form, which, however, recurs periodically. As scarcely more than two to three times the coercive intensity are required to produce the induction SB = 7000 in soft iron, the retardation of the zero-point of the (3. T)- or of the (33, 2 7 )-curve, in respect of the (, T)-curve, will be considerable (compare 157 and fig. 36, C, p. 225) ; while, according to the foregoing paragraphs, the resultant field < is already retarded in respect of I I by the action of the secondary current. A very considerable displacement of the (SB, T)-curve in respect of the (J 1? T)-curve results from all this. The zero-points of the former, as appears from many transformer diagrams, are about in the middle, between those of the curves of the primary and the secondary current. According to Ferraris, the action of hysteresis may be compared with that of a ' dead ' secondary coil supposed to be added. 1 Such a one would also retard the (SB, 7 7 )-curve, and, more particularly, like hysteresis, would cause a deleterious dissipation of energy into heat. The latter may, however, be diminished as much as possible by keeping the induction within the narrow limits mentioned. This is an additional reason in itself for working with a low degree of saturation. For a given range of induction the loss of power by hysteresis is, as stated in 148, proportional to the volume of the iron and to the frequency, which is to be remembered in predetermining these quantities. 183. Influence of Leakage. Leakage does not occur at all in the typically shaped and coiled transformer which has formed the basis of the above considerations. The fields due to the currents in each coil, owing to their almost exactly opposite phase ( 181), act in opposition to each other; but this always produces a resultant periodic field, which, if both coils are uniformly wound, will also be peripherically uniform. 1 Ferraris, Mem. R. Ace. di Scienze, Torino [2] vol. 37, p. 15, 1885, and vol. 38, 1887. The action of eddy currents may obviously be considered from the same point of view ; the aggregation of their paths may evidently be regarded as a separate secondary coil. TEANSFOEMEE DIAGEAMS 283 The resultant induction has, therefore, also uniform peri- pheral distribution. The flux of induction will thus, without any leakage, be confined to the core, and be surrounded by both coils in the same manner. In that case only, from 177, will the equation H = V AA be satisfied. This condition is to be aimed at in all transformers. It is not, however, practicable to wind the primary and secon- dary coils uniformly together, for in that case the electrostatic capacity becomes excessive, and an adequate insulation is impossible. If, in the other extreme case, the opposing coils are wound separately on the two halves of the toroid, as in Faraday's ring ( 177) and in some experiments of Oberbeck ( 92), there is obviously considerable leakage. For transformers an inter- diate course is adopted, by arranging the primary and secondary parts of the coils alternately as regularly as possible. There is, nevertheless, always a certain amount of leakage. This can then be measured by a coefficient of leakage v' > 1, which is defined by the equation (50) V . . i/'s This evidently has an entirely different meaning from the coefficient v previously introduced ( 78, 128), and is a periodic function of time. It may be proved that chiefly the symmetry of the (E. 2 , T)-curve is destroyed by leakage, as appears from transformer diagrams, plotted with an unfavour- able arrangement of the two coils that is, when the leakage is considerable. 184. Transformer Diagrams. In order further to explain the processes described, a typical transformer diagram by Eyan and Merritt 1 is represented in fig. 59, p. 284. Of the great number of diagrams plotted by these experimenters we only reproduce that for ' no load ' (A) and for ' full load ' (D). In 1 H. J. Eyan, Trans. Amer. Inst. Electr. Engin., vol. 7, p. 25, 1890; The Electrician, vol. 24, p. 263 and vol. 25, p. 313, 1890. Fleming gives a large number of such diagrams in his book ; loc. cit. pp. 446-478. They are also sometimes called, though less briefly, the indicator-diagrams of a transformer. 284 MAGNETIC CIRCUIT OF ELECTROMAGNETS the transformer examined, ^ = 675, n^ = 35 ; hence, the theoretical transformation-ratio [equation (49), 181] ^ = n l jn 2 A. FIG. 59 = 19 -3 ; the mean length L of the closed magnetic circuit was 3O8 cm., the mean section 8 = 63-3 sq. cm., the volume V 2050 cub. cm. The remainder of the chief data are given COEE AND SHELL TRANSFOKMEKS 285 in Table VIII, columns B and C of which represent inter- mediate loads. 1 TABLE VIII A B C D Values measured No Intermediate Full Unit load loads load Period T 7-3 7-3 7-6 76 Frequency .W= I/T . . . 138 138 132 132 1000 sec - Per second [ Primary E.M.F. E l . . . 1025 1053 1050 1040 Volts 1 Primary current l l . . . 0-14 0-20 0-39 0-63 Amperes ( Power supplied A l ... 96 159 389 608 Watts (Secondary E.M.F. E 2 . . 54-5 52-3 51-0 49-3 Volts Secondary current l z . . 1-26 5-83 1065 Amperes Useful output A^ . . . . 64 301 525 Watts Efficiency g = A^\ A } . . . 41 77 87 Per cent. Ill these diagrams 1. Dotted curves represent primary electromotive force 2. Full lines 8. Dot-and-dash currents J, secondary electromotive force E 2 . Owing to the small number of secondary turns, and to the fact that in the experiments the external resistance consisted of 1 to 10 incandescent lamps in parallel, the entire circuit may be regarded as a non-inductive one. Hence the curve of its current is identical with the (E^ jF)-curve but for the ohmic resistance as a factor, and was therefore omitted from the diagrams. The summits of the curves are marked by lines of ordinates, which enable a judgment to be formed as to the relations of symmetry. Along those lines the scales to the corresponding curves are placed. After the foregoing discussion the trans- former diagrams hardly require further explanation. 185. Core and Shell Transformers. The typical ring transformer which we have hitherto discussed, besides its theo- retical simplicity, also offers advantages for practical use, but would be difficult to wind and to repair. Accordingly, a series of modifications of this fundamental type have been developed, which, in conformity with Kapp's nomenclature, are usually called core transformers. 1 The values given for the periodic quantities are, as usual, the square roots of mean squares. 286 MAGNETIC CIRCUIT OF ELECTKOMAGNETS The reverse of this arrangement is a ring formed of the primary and secondary copper conductors, which, in turn, are wound with a sufficient quantity of iron wire. From this type a number of practical designs are derived, which are usually known as shell transformers. These are the two chief classes of transformers, to which most forms may be assigned. In the course of the last few years a great variety of designs have been proposed, and have more or less come into practical use. A sharp distinction can scarcely be made between core and shell transformers, seeing that many forms of them are, as it were, a transition between the two classes. Given the two coils, which, from what has been stated, are packed together as closely as possible in order to avoid leakage, and which have approximately equal weights of copper, the mutual and self-inductance will be a maximum, when the mag- netic circuit is closed, of as large a section as possible, and of the softest wrought-iron. In this way the efficiency which can be transformed for the given pair of coils is a maximum ; against this advantage, the disadvantage that, on account of the great volume of iron, the loss of power by hysteresis is, ceteris paribus, a maximum, does not come into consideration from the purely magnetic point of view. 186. Magnetic Circuit of Transformers. The form of the magnetic circuit is far simpler with transformers than with dynamos, as it is essentially governed by the following rule : ' The immediate vicinity of the coil, in all places where an appreciable field is produced, ought to be filled up with soft iron sufficiently divided/ 1 By far the greater number of modern transformers have, as stated in 180, a closed magnetic circuit ; there are only a few which have an unclosed one. Besides the points of view discussed in the text, in what is still an open question as to the advantages of both arrangements other con- siderations, more of an economical nature, come into play, upon which we cannot here enter. For further details as to transformers, the following works and papers must be referred to : Fleming, loc. cit. ; Kittler, loc. cit. 1st edition, vol. 2, 177-235 ; Uppenborn, OeschicJite der Transformatoren, Miinchen, 1889 ; Blakesley, Alternating Currents ; Silv. Thompson, Dynamo- Electric Machinery, 4th edition, chap, xxv., London, 1892 ; J. Hopkinson, Reprint, pp. 148-216 ; Ferraris, La Lumiere clectrique,vo\. 10, p. 99, 1885, and vol. 27, p. 518, 1888. EDDY CURRENTS. SCREENING ACTION 287 It will be sufficient to explain by a single example how this rule may be satisfied. It may be premised that the magnetic circuit is filled either with thin iron wire, or, as is most usual, with very thin sheet iron. This is stamped to patterns corre- sponding to the cross-section of the air-space about the coils which is to be filled. The pair of coils of a number of the best known transformers is, for instance, coiled in an shape ; the lines of intensity therefore are in planes at right angles to the plane of the figure. E-shaped iron laminae are now packed up parallel to those planes, so that the tongue of the E is stuck through the 0, and the flaps are again put together on the other side ; such a process is obviously susceptible of a great number of variations. It will be sufficient in this respect to refer to the works cited above. A magnetic circuit like that described can obviously be con- sidered either as core or as shell ; the core may further be regarded as a magnetic circuit divided twice through the shell. Multifold magnetic circuits in the sense of 145 occur in poly- phase (rotary) transformers ; for the sake of simplicity we have tacitly restricted ourselves to two-phase currents in the above ; the considerations apply also in given cases to three-phase or polyphase currents. 187. Eddy Currents. Screening Action. In our previous discussions we have expressly disregarded parasitic eddy currents, since by sufficient division they may be theoretically brought below any assignable limit, in contradistinction to hysteresis, which is not affected thereby. In good transformers, the dissi- pation of energy due to the former cause is therefore but small, compared with that due to the latter. The present question has been examined in detail by J. J. Thomson, 1 and by Ewing. Ac- cording to these researches, the heat developed by eddy currents for a given volume of iron is, ceteris paribus, nearly proportional to the square of the thickness of the iron laminae--as long as this is less than 1 mm. and to the square of the range of intensity ; it also increases with the frequency as well as with the electrical 1 J. J. Thomson, The Electrician, vol. 28, p. 509, 1892 ; Ewiog, ibid., p. 631 ; Fleming, loc. tit., vol. 2, pp. 485-490 and 535-538. [Also J. Hopkinson and E. Wilson, Phil. Trans, vol. 186, A, p. 93, 1895.] 288 MAGNETIC CIRCUIT OF ELECTROMAGNETS conductivity of the iron, according to a law which is represented by complicated hyperbolic functions. The magnetic screening action also which eddy currents exert on the interior of the bodies in which they occur, is also thoroughly investigated in the place quoted ; we shall confine ourselves to giving the practical result : that with 100 periods per second (note, p. 220 and 148) the iron must not be thicker than J mm. With transformers used for that frequency this theoretical result agrees with good practice ; for with smaller frequencies this upper limit is raised. We refer, in addition, to what was said in 148 about build- ing up armatures. This is the place to point out the analogy between the action of the transformer and of the dynamo. In the secondary coil of the former, and in the armature of the latter, electromotive forces are induced by varying magnetic conditions ; in the dynamo the motion relative to the field- magnets produces these variations ; in the transformer, on the contrary, the primary current, which itself is variable, produces them. With a fixed transformer a sufficient loss of heat is not so easily attained as with a rapidly rotating armature, so that in the former case four or five times as great a cooling surface is required as in the latter. A moderate increase of temperature of the core of transformers say to 90 is not without advantages. For hysteresis is then less ( 148), and the loss by eddy currents, owing to diminished electrical conductivity, is so too ; while the magnetisation of the iron under these circumstances shows no appreciable decrease up to that temperature. 289 CHAPTER X EXPERIMENTAL DETERMINATION OF FIELD-INTENSITY 188. General Introduction. The complete determination of a magnetic field embraces on the one hand a topographical survey of its distribution (Chapter III.) within the region in question, and on the other hand the numerical evaluation of the intensity at a given place, which then also gives that in other parts of the field ; the latter problem is by far the more im- portant, as, in general, we are dealing with fields which are either quite uniformly distributed throughout a certain region, or may be approximately considered so. In the present chapter we shall review the various methods of measurement ; in so doing we shall give only the chief features of the older, more or less classical methods, which may be assumed to be known, 1 while several newer methods, less generally understood, will be discussed more in detail. In so far as < electromagnetic ' fields are concerned that is, such as are exclusively produced by electrical currents it will of course, if possible, be simpler to calculate the distribution, and the absolute value of the field from the dimensions of the conductors, and the easily measurable current : the corresponding formula? for the cases of most frequent occurrence are given in 5, 6. In many cases, however, the calculation meets with insuperable analytical difficulties ; any such field is, however, subject to the laws of distribution in 44 and 45. The same holds also for fields due to rigid magnets, the distribution of which was dis- cussed in 47-49. When the calculation is seen to be impracticable, as i See among others, F. Kohlrausch, Leitfaden der prakt. PJiysik, 7th edition, Leipzig, 1892 ; Heydweiller, Hulfsbuch filr elektr Messung., 62-77, Leipzig, 1892 ; ; Mascart and Joubert, Electricity and Magnetism, vol. 2, 1139-1188, London, 1888. 290 DETERMINATION OF THE FIELD-INTENSITY almost always the case in considering ferromagnetic bodies, recourse must be had to experimental methods ; which of these is to be preferred depends on the special circumstances of the case ; that is on the accessibility, the extent, and the order of magnitude of the field ; and also whether its direction is hori- zontal or not ; whether an absolute or a relative, an approximate, or an exact measurement is intended. The elements necessary for forming a judgment will appear in what follows. 189. Distribution of Magnetic Fields. The distribution of a magnetic field in two dimensions may, as already stated ( 4), be represented by means of iron filings. The magnetic diagram, as it is called, obtained in this way not only gives a clear representation of the course of the lines in the plane chosen, but it also gives an approximate idea as to the relative value of the intensity in various places ; as the ' lines are closer, the higher is that value. Lindeck 1 has represented a number of such figures, one of which is reproduced in fig. 60. This is the case of a field in the meridional plane of a circular conductor conveying a current, the position of which appears from the blank places at the top and bottom. The lines of intensity close to the conductors form circles around them, while the field in the middle of the circular conductor is pretty uniform. This figure agrees with the distribution theoretically calculated for this case ( 6, 0). A method, which is more accurate, though, at the same time, more troublesome, than the use of iron filings, and which gives the lines of intensity in three instead of in two dimensions, consists in following them with a small movable magnetic needle, proceeding from point to point always in its own direc- tion. 2 A small cross-piece WW fixed at right angles to the needle gives, according to Searle, the direction of the corre- sponding equipotential surface. The value of the intensity may often be deduced from the frequency of the vibrations, since the square of the frequency is proportional to that value [see equation (I) in the next article]. When using permanent mag- netic needles the superposed induced magnetisation constitutes 1 Lindeck, Zeitsclirift fur Instrument en Kunde, vol. 9, p. 352, 1889. 2 C. Bering, Electrical Engineer, vol. 6, p. 292, 1887. DISTRIBUTION OF MAGNETIC FIELDS 291 a considerable source of error, especially in intense fields. Hence, in many cases permanent magnetism is dispensed with, and a soft iron needle is used. In so far as its magnetisation, in the first stage, is approximately proportional to the intensity of the field, which, from 33, is nearly the case for not too long u 2 292 DETERMINATION OF THE FIELD-INTENSITY needles, the frequency of the vibrations in this case offers a pro- portional direct measure of the numerical value of the intensity. The method of investigating the distribution of a field least open to objection, but which, at the same time, is very tedious, is that by means of an exploring coil ( 2, 4). From the position of maximum induction, the direction of the field may be deduced ; and from the quantity of electricity set in motion, the numerical value of the intensity. For the details of this method, which is very seldom used, we may refer to the sections on ballistic methods ( 195, 196 ; see also Chapter XI., 208). To complete this account we may mention a few instruments in this connection, viz. the declinometer, inclinometer, and local variometer. 1 As these, however, are almost exclusively used for special work on terrestrial magnetic measurements, a detailed description need not be given here, A. Magnetometric Methods 190. Plan of Gauss's Method. The accurate measurement of the absolute value of the horizontal component of a uniform field was first made possible by the classical method which Gauss devised for determining the terrestrial field. 2 The value to be measured is deduced from the deflection which the direc- tion of the horizontal component experiences when it is combined with an auxiliary component of known value, also horizontal, but at right angles to it. To determine the direction of the resultant a magnetometer is used. The essential part of this instrument is a well-damped small system of magnets suspended by a vertical fibre, as free from torsion as possible (quartz fibres are best), and which can be turned, the azimuth being read off" by a mirror. 3 The above-mentioned auxiliary 1 F. Kohlrausch, Leitfaden der pralit. Physik. 7th edition, p. 255 et sqg. Leipzig, 1892. 2 Gauss, Tntensitas vis magnet, terrestris ad mensuram absolutam rerocata. Werke, vol. 5, p. 89 ; 2 Reprint. Gottingen, 1877. See also P. Kohlrausch, loc. cit. pp. 230-236. 3 The following are the chief points in reference to the construction of a magnetometer, of which there are many types, simple and complicated. The system of magnets must be small, so that the auxiliary component in the space occupied by it is sufficiently uniform, and its moment of inertia is small ; on the other hand, its magnetic moment must be as great as possible. Perhaps the best is a thin aluminium disc on both sides of which small mag- PLAN OF GAUSS'S METHOD 293 component is produced by means of the action at a distance of an auxiliary magnet, as will be described in the following paragraphs. The magnetic moment of this magnet may either be known at the outset, or its product into the horizontal intensity, which is the quantity to be measured, may be deter- mined by one of the following methods : A. Observation of Oscillations. The auxiliary magnet, whose (unknown) permanent magnetic moment may be 2ft, is suspended horizontally in the place where the horizontal component is to be determined. In this position its period r, or its frequency _$-__ l/ Tj is observed, from which is obtained [by equation (8), 23] (1) . . 2ft = i^=47r 2 ^ 2 # The moment of inertia K of the auxiliary magnet may be either calculated, or else determined experimentally, by some dynamical method. B. Method of Weighing. 1 The auxiliary magnet is fastened vertically in the middle of the scale-beam. Let the balance swing in the magnetic meridian, and let the difference in weight corresponding to half a turn about the vertical be B M. If D is the length of the scale-beam, g the acceleration of gravity, we obtain (2) ... Wti^lgBMD C. Bifilar Suspension; Torsion. In the arrangements pre- viously described, the auxiliary magnet oscillates about a position of equilibrium that is, the horizontal (A) or the vertical direc- tion (B) in the magnetic meridian. In the former case the horizontal component, and in the latter the vertical component, obviously induces in it a certain magnetisation, which is super- nets are fixed. As electromagnet copper dampers are of little use in such feeble systems, air-damping, which can be regulated, is to be preferred. Although torsion may nearly always be neglected when quartz fibres are used, a torsion head should in all cases be fitted. It is, lastly, very convenient, if the mirror can be turned in reference to the system, and if the case, which must not contain the least iron, is arranged for being set upon a horizontal plane, or else suspended against a wall. 1 Toepler, Wied. Ann. vol. 21, p. 158, 1884. 294 DETERMINATION OF THE FIELD-INTENSITY posed on the existing permanent magnetisation, and therefore increases the moment 2ft. The correction due to this is, however, only small in observations on terrestrial magnetism ; neverthe- less, it is always better to place the auxiliary magnet in a nearly east and west position. The transverse magnetisation then produced has scarcely any effect. For instance, the auxiliary magnet may be suspended in this position to a bifilar or torsion arrangement, the directive torque of which is known (note 3, p. 297). The latter is multiplied with the tangent of half the deflection produced by reversing the magnet to the west and east direction. The product gives then directly the value 5ft <$. t T m N FIG. 61 191. Observations of Deflection. After the auxiliary magnet has been taken away, the magnetometer M is put in its place. Its magnetic system then sets itself in the direction of the field, which we have called the magnetic meridian, and which is marked N8 in fig. 61. In order now to produce a deflection of the magnetic system, the auxiliary magnet is placed in one of two different positions, its own direction in both cases being either west-easterly or east-westerly. 1. First Principal Position. Auxiliary magnet at distance D { in W or at right angles to the meridian. The component <>! induced by it at M is also at right angles to the meridian, and amounts to [equation (5), 22] OBSERVATIONS OF DEFLECTION 295 2 9 fi 1 2 3 L 4 . in which JC is the geometrical (or ' virtual,' 210) length of the auxiliary magnet. 2. Second Principal Position. Auxiliary magnet at N or b at distance D 2 , at right angles to the meridian, as well as the component 2 due to it, which in this case has the following value [equation (6), 2 fc Z] l : g n 3 I? , _16 i 4 _ I * i L ''8 2?. 128 ^ . The deflections a, or a 2 due to the second component are obviously given by the following equations : (4) . . tana^ tana 2 = - From equations (3,) or (3 2 ) and (4), neglecting the factors in the brackets, we get, as a first approximation, (5) . = -LD'tan^ or = I? tan a, After having expressed 3tf and 2tf/ as functions of deter- minate quantities, not only the intensity , but also inciden- tally the moment W of the auxiliary magnet, may be calculated. If the latter is already known, $ is obviously obtained from equation (5) without further determinations. These simplified equations, however, only hold for distances which are very great in comparison with the length of the deflecting magnet ( 22) ; but as the deflections obtained at such distances are usually too small, the magnet must be 'brought nearer the magnetometer, so that several members of the series in equation (3,) or (3 2 ) have to be brought into account. The length of the magnet is, however, usually eliminated by observing at two successive distances (Kohlrausch, loc. Git., p. 231). Gauss's method is used mostly for determining the absolute value of the earth's horizontal intensity. It may, however, in principle be applied to measuring the horizontal components * An elementary deduction of equations (3,) and (3 2 ) is found, for instance, in F. Kohlrausch, loc. tit. pp. 389, 390. 296 DETERMINATION OF THE FIELD-INTENSITY uniform fields due to any agent, provided their intensity does not exceed 1 C.G-.S.-' If the field is stronger, it is scarcely possible to produce sufficient deflection by means of magnets of the usual size. B. Electrodynamic Methods 192. Measurement of a Dynamical Force. It was men- tioned in 1 that the magnetic field can be completely defined by the two chief forms in which it manifests itself the electro- dynamic and the inductive; and for the practical methods of measurement based on this, refer- ence was made to the present chapter. As regards electrodynamic 1 methods, we will mention the following simple arrangement, due to Lord Kelvin. 2 In the field F a metal wire hangs be- tween two pole-pieces, supposed horizontal and at right angles to the plane of fig. 62. By means of two mercury cups C } C a cur- rent of known strength 1 (in dec- amperes) is passed. If then the intensity of the field is , its effective height^, and its direction is such that a force, %, expressed in dynes, is exerted, say, towards the left on the wire, this, from electro-dynamical principles, is given by the equation (6) ... % = I%L This force is held in equilibrium by the tension of the threads t l and 2 , which are fastened to the string pendulums p l P l or p 2 jP 2 . It needs no explanation how that force can be 1 By electrodynamic action all forces are understood which are exerted on conductors conveying currents in the magnetic field, no matter whether the field is due to other conductors, or depends on other sources, such as rigid magnets. 2 A. Gray, Absolute Measurements in Electricity and Magnetism, vol. 2, p. 701, London, 1893. FlG 62 UT MEASUREMENT OF A TOEQUE " 297 determined in absolute measure from the readings on the scales S l and $ 2 , as well as from the weights P l and P 2 . The wire conveying the current may also be stretched. By the action of the electrodynamic forces it will then sag laterally, like a tight string. The elongation, which may be measured by a microscope, or by mirror reading, is, as a first approximation, proportional to the force, and therefore, with a constant current, proportional also to the intensity of the field. This principle has been applied in Ewing's magnetic curve-tracer ( 214). The method described is suitable for fields of the order of 100 C.G.S. units and upwards. 193. Measurement of a Torque. The simple method which has been described is obviously not very accurate. More trust- worthy results are obtained when the conductor is bent into a loop of one or more turns ; and the couple exerted in general on such a loop that is to say, on a coil in the field is determined by one of the known methods, such as weighing, torsion, or bifilar suspension. Steiiger has described a special apparatus based on this last principle, 1 which may be regarded as an inversion of the well- known 'syphon recorder' of Lord Kelvin, and of the bifilar galvanometer of F. Kohlrausch. 2 A small coil is suspended by two wires, which at the same time convey the current I, which is separately measured. Let the plane of the coil be parallel to the field ; let the total area of the windings be S, the directive torque 3 of the bifilar D, and the deflection observed be a. The field-intensity to be measured is then given by the equation (7) ... *-?* Stenger thus succeeded in determining in absolute measure fields of the order of 100 C.G.S. to within O'l per cent, with certainty and convenience. The sensitiveness of such methods may be lowered at will by diminishing the current through the 1 Stenger, Wied. Ann. vol. 33, p. 312, 1888. Compare also Himstedt, Wied. Ann. vol. 11, p. 829, 1880. 2 F. Kohlrausch, Wied. Ann. vol. 17, p. 752, 1882. 3 In a suspended system directed by any appliance we suppose the directive torque referred to unit deflection expressed in circular measure (57 '29 6). 298 DETERMINATION OF THE FIELD-INTENSITY coil, and they may tlius also be used for measuring the most intense fields. Torsion, applied as directive torque, has been used in measure- ments of fields, which A. du Bois-Reymond l has published. The arrangement in this case may be regarded as an inversion of the Deprez-d' Arson val galvanometer. 2 On a similar prin- ciple Edser and Stansfield 3 have recently constructed a con- venient portable instrument for measuring a field. A plate of mica supports a coil of thin copper wire, which is stretched by two German silver wires conveying the current. One of the wires is fastened to a torsion head, which at the same time is ingeniously made to serve as commutator. A Hellesen dry cell furnishes a nearly constant current of, at most, two centiamperes, since the coil itself has a resistance of 50 ohms. By inserting independent resistances the sensitiveness can be diminished, if necessary. The instrument, it is stated, measures fields of any direction from 1 C.G.S. upwards, with an accuracy of about two per cent. It is especially suitable for measuring leakage in and about dynamo machines, as Edser and Stans- field show (loc. cit.) by some examples. 194. Measurement of a Hydrostatic Pressure. The method of measuring force just mentioned may, lastly, be so modified that, instead of a solid wire conductor, a liquid one mercury is used. This is con- tained in a flat insulating cavity perpen- dicular to the direction of the field, which is again assumed to be horizontal, and at right angles to the plane of the figure. The width of the cavity d does not exceed a fraction of a millimetre. Let the current traverse the mercury in a vertical direction, for which pur- pose two platinum electrodes, E l and E 2 (fig. 63), are in contact with it. By electrodynamic action a lateral thrust is exerted on the mercury, driving it out, so that a difference of level results between the tubes E^ and J? 2 , which represents a pressure P 1 A. du Bois-Reymond, EleUrotecJi. Zeitschrift, vol. 12, p. 305, 1891. 2 Deprez and d' Arson val, Compt. Rend. vol. 94, p. 1347, 1882. 3 Edser and Stansfield, Phil. Mag. [5], vol. 34, p. 186, 1892. MEASUREMENT OF A HYDROSTATIC PRESSURE 299 equivalent to that thrust. It may be readily shown l that the condition of equilibrium is represented by the following equa- tion : (8) ... %I=Pd It will thus be seen that, ceteris paribus, the pressure, assumed to be expressed in the equation in dynes per sq. cm., is inversely proportional to the clear width of the cavity, and accordingly the latter is taken as small as possible. Lippmann 2 has devised a galvanometer, depending 011 this principle, in which I in the equation is the quantity to be measured. By reversing this an arrangement is obtained by which - may be measured, as in the previous cases. An appa- ratus of this kind was first described by Leduc. 3 The mercury is contained between two glass plates, which are kept apart by pieces of microscope glass, and cemented by Canada balsam. The tubes R l and R 2 were widened in one place, in which were the surfaces separating the mercury and the supernatant water or alcohol. In this way the sensitiveness could be considerably increased, as also by inclining one of the tubes at the cost, however, of the accuracy of the measurement. Leduc's instrument was afterwards somewhat modified by the author, 4 in order to adapt it for use in the narrow spaces between the pole-pieces of powerful electromagnets. Neither the galvanometer nor the portable instrument for measuring the field is suitable for absolute measurements, from the difficulty of accurately determining the thickness of the layer of mercury ; but its use for the relative determination of horizontal fields of small extent, of the order of 1000 C.G.S., with an accuracy of 0*5 per cent., is so much the more convenient. By using water or alcohol for measuring the pressure, the apparatus is adapted also for a lower order of magnitude of, say, 100 O.G.S. 1 Mascart and Joubert, Electricity and Magnetism, English edition, vol. 2, p. 243. 2 Lippmann, Compt. Rend. vol. 98, p. 1256, 1884. Journal de Physique [2], vol. 3, p. 384, 1884. 3 Leduc, Journal de Physique [2], vol. 6, p. 184, 1887. 4 du Bois, Wied. Ann. vol. 35, p. 142, 1888. See also Field and Walker, The Electrician, vol. 32, p. 186, 1893. 300 DETEKMINATION OF THE FIELD-INTENSITY C. Methods of Induction 195. Arrangement of the Exploring Coil, After describing in the previous section the methods of measurement based on electrodynamic actions, we now pass to the second chief manifes- tation of the magnetic condition, the inductive action, wherein we shall provisionally confine ourselves to indifferent media. In the introduction ( 2, 4) the most important properties of the magnetic field were already elucidated by their aid, experiments with an exploring coil having then been described, only the outlines of which could be explained. After, in 189, having pointed out the use of exploring coils for determining the distribution of a field, we must now describe the mode of carrying out such experiments, in so far as measuring the absolute value of intensity is concerned. Let S be the area of the exploring coil, R the total resistance of the secondary circuit, Q the quantity of electricity conveyed on the sudden pro- duction or cessation of the field , which is assumed to be at right angles to the exploring coil. We have shown [equation (1), 4, and 64] that then (9) ... In making such experiments it must be remembered that by means of the exploring coil we can only detect or measure variations in the flux of induction which in the indifferent air- space is identical with (<> S) and cannot determine its value at a given instant. Accordingly, one or other of the following methods is used : 1. The exploring coil is placed in a fixed position, which is that of the direction of maximum induction. The field is then suddenly produced, stopped, or its direction reversed. If it is exclusively due to a current, this is attained by making, breaking, or reversing. 2. The exploring coil is rapidly removed to a position in which the intensity of the field may be neglected. For this purpose it is usually fixed to a long lever, which springs back at the desired moment, by means of a spring, for example. 3. The exploring coil is rapidly turned, so that the flux of BALLISTIC OALVANOMETEK 301 induction traverses it in the opposite direction. This method is to be recommended, provided the space is sufficient, for it is easily effected, and, moreover, twice the quantity of electricity is set in motion. The exploring coil is fastened to a handle, which can be turned about an axis in the plane of the coil. The area 8 of the windings can be determined by various methods : l 1. By directly measuring the diameter of each layer of windings with the kathetometer or the dividing machine ; or measuring the perimeter by a tape, allowing for the thickness of the wire. 2. By measuring the length of wire used in coiling. The formulae required for this are given by F. Kohlrausch (loc. cit.). 3. By the magnetic action at a distance of the coil con- veying the current. Its magnetic moment is 3R = IS ( 6), and can be measured magnetometrically ( 209). If, further, the same current is passed through a tangent galvanometer, its value may be completely eliminated. The ballistic method may be used for measuring fields of any intensity. The sensitiveness of the galvanometer, the area and the number of turns of the exploring coil, and the resistance of the secondary circuit are factors by which, when suitably fixed, the sensitiveness of the method may be raised or lowered to almost any extent. This general applicability is a chief advan- tage of the ballistic method. 196. Ballistic Galvanometer. The quantity of induced electricity Q set in motion is almost always determined by the ballistic galvanometer, whence the usual name is derived. In selecting a galvanometer the chief point to be attended to is that its period can be made as long as possible in comparison with the time required by the variation of the flux of induction. A galvanometer which swings too rapidly may often be utilised for ballistic purposes by fixing small light bodies to its magnetic (for instance, a bit of match fixed horizontally, or an aluminium stirrup often provided with the galvanometer), so that without any considerable increase of load the moment of inertia, and 1 F. Kohlrausch, Leitfaden, 7th ed. p. 343 ; Heydweiller, loo. cit. 152-155 ; Himstedt, Wied. Ann. vol. 26, p. 555, 1885. 302 DETERMINATION OF THE FIELD-INTENSITY therewith the period, are greater. The efficiency of a galva- nometer is best characterised by the following quantities l : The current sensitiveness S s is the deflection in scale-divisions per microampere, if the distance of the scale is 2000 parts, and the period is 10". The ballistic sensitiveness S b is the deflection in scale-divi- sions of the scale per microcoulomb, if the distance of the scale is 2000 parts, and the period 10". In order to eliminate the resistance R g of the galvanometer, the normal sensitiveness @ then is introduced, which is defined by equations . = Taking the normal period of 10", and assuming that damp- ing may be neglected, the two conditions of sensitiveness are connected by the equation Now in ballistic experiments as those it is unnecessary that the damping be excessively small ; it is, rather, far more convenient to work with as strong damping as possible, the extent of which is limited by one condition only ; the deflection must always remain proportional to the quantity of electricity passing through the galvanometer, a condition which can easily be checked by suitable calibration. Owing to damping the ballistic sensitiveness decreases somewhat ; if in these conditions we denote it by > 6 ', and if in is the ' damping-ratio,' then, ac- cording to Ayrton, Mather and Sumpner, 2 (11) <& b = <&' [1 + 0-500 (m - 1) - 0-277 (m - 1)* + 0-130 (m - I) 3 ] 1 Compare du Bois and Rubens, modified astatic galvanometer, Wied. Ann. vol. 48, p. 248, 1893. 2 Ayrton, Mather, and Sumpner, PHI. Mag. vol. 30, p. 69, 1890. The damping-ratio is the ratio between the amplitudes of two immediate successive half -swings. STANDARD FLUX OF INDUCTION 303 Besides the calculation we have given for deducing / from the sensitiveness to steady currents, this quantity may also be determined experimentally. A condenser of known capacity is charged to a given potential, and the quantity of electricity defined thereby is directly discharged through the galvanometer. This method is, however, not advisable in practice ; it is not usual to measure the absolute sensitiveness of the galvanometer, but to graduate this in the given secon- dary circuit, directly as it were, with regard to flux of induction. That is to say, a factor is determined which, multiplied into the deflection, gives at once the product Q E ; in other words, the corresponding flux of induction. If, further, this is divided by the known area of the exploring coil, the value of the intensity is obtained. For this purpose a convenient flux of induction is required, the absolute value of which is known, and which can at any time be reproduced ; that is a 197. Standard Flux of Induction. Various arrangements may be used for this purpose ; all of them have a secondary coil, FIG. 64 which is inserted in the secondary circuit containing the ballistic galvanometer to be standardised. In the first place, we must mention W. Weber's earth- inductor (fig. 64). By means of a handle the circular coil is moved rapidly from the horizontal (or vertical) position through an angle of 180, against a suitable stop. The total variation 304 DETERMINATION OF THE FIELD -INTENSITY of the flux of induction to be taken into account is equal to twice the product of the area of the coil into the horizontal (or vertical) component of the earth's field ; the latter must therefore be first determined, and as it. varies considerably with time, especially within a laboratory, this is a circumstance which speaks against the use of the earth-inductor. And the general tendency at present is not to introduce factors involving ter- restrial magnetism into measurements wherever this is possible. According to Lord Kelvin, it is more convenient to use a primary standardising coil without a core, the number of turns of which n, and the geometrical dimensions (length L, area S) are accurately determined, and which conveys a standard current I A determined in absolute measure. The reversal of this current then corresponds to the following variation of the flux of induc- tion [ 6, equation (7)]. In 84 the use of this method for graduating a ballistic galvanometer was explained by reference to an example. FIG. 65 FIG. 6G The permanent flux of induction of as constant a magnet as possible N S (fig. 65) may further be used as standard, by letting an exploring coil, rest on a fixed ring RR in the STANDARD FLUX OF INDUCTION 305 middle ; by means of a handle Gr, or by means of a suitable spring, this may be rapidly drawn off. A modification of this simple arrangement has been designed by Hibbert, 1 so as to form a permanent standard field. Fig. 66 represents a vertical section through this apparatus. The magnetic circuit of a good steel magnet jV S is closed by an iron disc A A, and iron shell BB, with the exception of a narrow annular air-gap. On account of the small demagnetising action and by previous artificial ' ageing' of the steel magnet ( 150) a perma- nent flux of induction as constant as possible is obtained. The exploring coil can be moved closely through the air- gap ; for this purpose it is fixed by three radial cross arms, QII fe QS> t an ebonite disc, and by means of a knob If, it can be raised or lowered along the rod C. The vertical displace- ment is accurately limited by means of stops, and therefore corresponds to a definite variation of the flux of induction. The constancy of the standard may be judged from Table IX, in which are given Hibbert's experiments with three of his appa- ratus, extending over a long period. The greatest variation in the instruments I and II amounts, as will be seen, to -J per cent., and with III to 1 per cent. The last measurement denoted by * has been kindly communicated to the author ; it shows that within two years there has been no alteration exceeding the errors of observation. Accord- ing to Hibbert, this constancy depends on the condition that the induction 25 in the steel magnet shall not exceed the value 5,000 C.G.S. He proposes to adjust the flux of induc- tion of the standard to a round number such as 20,000 C.G.S., which would be obtained by suitable stops for the exploring coil ; apart from leakage a steel magnet 4 sq. cm. in section would then be necessary to obtain this value with 95 = 5,000 C.G.S. ; owing to the greater section the mean-field intensity in the air- gap is only about the tenth part of the value of the induction in the steel magnet. 1 Hibbert, Phil. Mag. [5], vol. 33, p. 307, 1892. X 306 DETERMINATION OF THE FIELD-INTENSITY TABLE IX Date Tempera- Flux of induction ture I II III April 16, 1891 20 21790 22 12-5 730 23 13-5 710 32360 May 8 16-5 710 420 > ^" 13 680 330 30 16 720 410 29290 June 6 v 18 720 380 270 12 22 780 470 260 29 21-5 720 345 290 July 10 19 790 510 500 27 20 700 470 550 31 17-5 780 460 530 Sept. 22 16 690 400 470 Nov. 10 13 700 400 480 4, 1893* 720* Maximum variation .... 110 180 290 Mean intensity % in the air-gap . 515 770 700 198. Measurement of a Field by Damping. A method presenting theoretical interest is that of determining the in- tensity by the electromagnetic damping of a coil oscillating in a field. The damping, as is known, is due to the reaction of the current induced in a closed coil on the field : the method must, therefore, be classed among the induction methods ; it has a certain similarity with one of the methods of determining the ohm. The coil has either a bifilar or unifilar suspension in the field, the plane of its winding being parallel to the direction of the field; the period of oscillation r, or the frequency N, is determined under the influence of the directive force of the suspension alone and with an open circuit. The latter is then closed through an adjustable anti-inductive resistance (a rheo- stat for example), and this is gradually diminished until the swinging is just dead-beat, a condition which may be ascer- tained with considerable accuracy. If the corresponding total .resistance is R in C.G.S. units (that is in millimicrohms), the moment of inertia of the coil is K, the total area of its windings is S ; it may be shown that the intensity of the field is given l by (12) . . . 8 ; = V^irNRK 1 A. Gray, Absolute Measurements, part ii. vol. 2, p. 708, 1893. MEASUEEMENT OF A FIELD BY DAMPING 307 The air-damping is here disregarded, and several factors have been neglected in the course of calculation. This method is, as stated, of theoretical interest rather than practical value ; yet in certain circumstances it may do good service in approximately determining or comparing fields of the order of 1,000 C.G.S. In experimental physics, besides the methods of precision which have been worked out in all their details, there is room for simple ones by means of which approximate quantitative de- terminations may be rapidly made such as the above. D. Magneto-optical Methods 199. Rotation of the Plane of Polarisation. A very con- venient method of measuring a field depends on a determination of the magneto-optical rotation of the plane of polarisation of FIG. 67 light in transparent substances, which, as is well known, was discovered by Faraday. Let a plane parallel plate P, of such a substance of thickness d, be placed at right angles to the direction of the field (fig. 67). If then a linearly polarised ray of light 1 1 passes parallel to the latter, its plane of polarisa- tion undergoes a rotation s in the plate, which is proportional to the variation A T of the magnetic potential ( 48) between the point at which it enters and that at which it emerges; if we assume the field to be uniform, then evidently A T = & d, and therefore (13) . . . e = co%d The factor of proportionality co is equal to the rotation per unit variation of magnetic potential ; it depends in addition only on the wave-length \ of the light, as well as on the nature x 2 308 DETEKMINATION OF THE FIELD-INTENSITY and to a slight extent only on the temperature of the sub- stance traversed; it is called Verdefs constant. Its sign defines the direction of the rotation with reference to the direction of the field 1 (plain arrows in fig. 67); the direction of the propagation of light (feathered arrows) is immaterial. If, therefore, the plate is partially coated with silver Ag, and a pencil of rays 2 2 is allowed to fall on it under a very small angle of incidence, and is reflected as shown, the rotation is doubled ; that is, we have (14) . . . . e = 2a>$d The numerical value of Verdet's constant has been accurately determined in absolute measure for the two liquids most used, water and bisulphide of carbon; for sodium light (X = 58*9 microcentimeters) ; its value is in minutes per unit variation of potential [A T] at a temperature of 18 is 2 Water (H 2 0) : + 0-0130' per [AT] Bisulphide of carbon (CS a ) : + 0420' [AT] For several other inorganic and organic liquids (acids, alcohols, ethers, &c.) o> is more or less accurately known, as well as for some tolerably well defined kinds of glass. 3 Hence, in order to have absolute measurements, it is only necessary to determine the thickness of the transparent plane parallel plate used. Liquids are investigated in small glass troughs; the rotation in their glass covers is previously determined while they are empty, and afterwards subtracted. 200. Standard Glass Plates. The magneto-optical method has, among others, the advantage that by the formula = s / cod the intensity is simply proportional to s, the measurement of which is as convenient as it is accurate. It is, moreover, not at all necessary to use a substance for which Verdet's constant is ' It is usual to define the direction of rotations in respect of the direction perpendicular to their plane by saying, for instance, that the direction of the motion of a clock band is positive in respect of the direction from the dial to the clock works. This corresponds to the ' right-hand system,' in which the ordinary screw, for instance, may be classed. See Maxwell, Treatise, 2nd ed., vol. 1, p. 24. 2 Lord Kayleigh, Proc. Roy. Soc. vol. 37, p. 146, 1884 ; Arons, Wied. Ann. vol. 24, p. 182, 1885 ; Kopsel, Wied. Ann. vol. 26, p. 474, 1885. 3 Du Bois, Wied. Ann. vol. 51, p. 545, 1894. STANDAKD GLASS PLATES 309 known. Instead of a layer of liquid, a glass plate will even be generally preferred, the material of which is indeed less well defined, but as to which we are more certain that it is unalter- able. Such a plate is graduated once for all in a field of known intensity, and then represents a convenient standard field which exceeds all others in simplicity, portability, and invariability. In using glass plates which are exactly, or nearly exactly, plane and parallel with transmitted light, the rays twice re- flected internally (fig. 67, 1 1) often exert a disturbing influence, as also, with a silvered glass, do the rays directly reflected from the front surface (fig. 67, 2-2) ; because, in the first case, they experience a threefold, and in the latter no rotation at all. The author has, therefore, had constructed slightly wedge- shaped standards of the densest, strongly rotating Jena flint glass, 1 in which these disturbing images may be screened near jj T\ the principal image. They are coated with black paint (cross-shaded in fig. 68), with the exception of a square window of about 5 x 1 ' cm. ; one half of which, jP, is free and transmits the light, while the other FIG. 68. scale R is silvered at the back, and can be used for reflection. 2 The latter method is twice* as sensitive, and may in many cases be more conveniently used, provided polariser and analyser are arranged for almost normally reflected light. The rotation may be determined by one of the accurate polarimetric methods, which Lippich has . recently greatly improved. Owing to the great dispersion, the source of light must be monochromatic ; the best is as bright a sodium burner as possible, several of which have been described. Such a standard plate of about 1 mm. thickness is adapted for measuring fields of the order of 1,000 C.G.S. ; the latter is of course inversely proportional to the thickness of plate admissible ; that is, to .the free long dimension of the field. 3 1 Du Bois, Wied. Ann. vol. 51, p. 545, 1894. 2 Within the window the small changes of thickness due to the wedge shape have no detrimental influence. 3 For farther details and examples of the magneto-optical methods of measuring a field, reference may be made to a paper recently published by the author {Wied. Ann. vol. 51, 1894). It may be remarked that it is difii- 310 DETERMINATION OF THE FIELD-INTENSITY E. Hall's Phenomenon. Magneto-Electrical Alteration of Resistance 201. Hall's Phenomenon. We now pass to the discussion of two phenomena produced in metals by the influence of a field, which are suitable for measuring it, although they can scarcely be considered as having been hitherto examined in all directions and explained. Briefly stated, the phenomenon discovered by Hall consists in the occurrence of a ' rotatory ' kind of electrical conduction in metals, which has hitherto been observed only under the influ- ence of a magnetic field. It manifests itself chiefly by the fact that the lines of electric flow no longer run at right angles to the equipotential surfaces, as is always the case in ordinary condi- tions. Let us consider, for the sake of simplicity, conduction in two dimensions ; for instance, a thin strip of metal at right angles to the direction of the field (fig. 69) ; the lines of flow of the- primary current entering or leaving through the electrodes E l and E 2 will run, as FlG 69 before, parallel to the free edges ; the equi- potential lines, however, are deflected from their original ortho- gonal direction (represented by the plain lines) when the field is excited, into the slanting position (represented by the dotted lines). In places not too near the primary electrodes the rotation is ceteris paribus the same in all points, and with metals not ferromagnetic ( 224) is in general proportional to the field. cult to prepare thicker glass plates or parallelepipeds so that double refraction shall not exert a disturbing influence. It is necessary, in that case, to have recourse to liquids, notwithstanding their high temperature-coefficient, the formation of striae, and other disadvantages. Attempts have recently been made to utilise the magneto-optical method also for determining in absolute measure currents passing through coils the dimensions of which have been measured geometrically. HALL'S PHENOMENON 311 A difference of potential now exists owing to this rotation between points on the edge which before were equipotential ; by means of what are called Hall Electrodes e l and e 2 , and of a suitable galvanometer 6?, this can easily be measured. Now the deflections of the latter are proportional to the intensity of the field, and may theoretically, therefore, be used for measur- ing it ; l since, further, they are also proportional to the primary current, by gradating this, the sensitiveness of the method may be regulated ; it is, however, specially suited for determining powerful fields ; it is possible that samples suitably prepared may serve as standards. This method has not hitherto been practically developed. 202. Measurement of a Field by Bismuth Spirals. The following phenomenon is connected with that described in the last section. The electric resistance of a metallic con- ductor is in general altered when it is brought into a magnetic field. Without entering upon the details of this phenomenon,, it may merely be mentioned that, as Bighi first found, the be- haviour of bismuth in this respect is very striking ; in an FIG. 70 intense field its resistance may in certain circumstances be more than trebled. Hence, Leduc proposed to use bismuth wire for measuring magnetic fields. To the endeavours of Lenard and Howard we owe a practical method of measurement based on this phenomenon. 2 They succeeded in preparing by pressure a chemically pure bismuth wire of less than 0*5 mm. diameter ; this is insulated, wound into a bifilar flat spiral of 5 to 20 mm. diameter having no appreciable self-induction, and then cemented between two plates of mica. The thickness of the 1 Kundt, Wied. Ann. vol. 49, p. 257, 1893. The proportionality there- mentioned has been confirmed in the case of gold and silver for fields up to 22,000 C.G.S. 2 Leduc, Journal de Physique [2], vol. 5, p. 116, 1886, and vol. 6, p. 189, 1887; Lenard and Howard, Elektrotechn. Zeitschr. vol. 9, p. 340, 1888; Lenard, 'Wied. Ann. vol. 36, p. 619, 1890. 312 DETEKMINATION OF THE FIELD-INTENSITY whole preparation amounts to less than a millimetre, its resist- ance is of the order of 10 ohms ; both ends of the bismuth wire are soldered to stout copper wires, which pass through a handle of ebonite and terminate in binding screws Jc l and /,- 2 ; this forms at once a suitable fastening and is also convenient for manipulating the flat spiral. The plane of the spiral is placed at right angles to the direction of the field ; the law connecting the increase of resis- tance with the intensity of the field may be represented by an empirical curve for each spiral which is very nearly the same. Such a one is given in fig. 71 ; the abscissas representing the intensity of the field, and the ordinates the ratio g of the resistance R' in the field to that in ordinary conditions R. 1 For each resistance observed the corresponding intensity of "',- 1,'i- v : : ^X t ^ X A ^ ^ ^/ -^ _^ X j ^^ ^ * ^f - -> ^ zooa woo eooo 3000 ioooo -tzooo we FIG. 71 the field may be read directly on the curve. The measurements of the resistances R' and J?, within and without the field, are best made in rapid succession, in order to avoid variations of temperature in the spiral. The latter have of course consider- able influence on R' and R, as well as on the ratio p. According to Lenard, different results are obtained according as the resistance is determined by means of alternating currents or of electrical oscillations and the telephone, or by means of steady currents. 1 This curve is taken from a paper by Bruger (Industries, vol. 12, May 1893). To save space the range of ordinates < p < 1 is omitted. The curves for various spirals agree to within one or two per cent. Hysteresis has not, so far, been observed in this phenomenon. MAGNETO-HYDROSTATIC METHOD 313 The latter method is probably best ; Hartmann and Braun have constructed a field-bridge in which the intensity is read off directly from the position of the sliding contact ; they have also perfected the construction of the bismuth spirals. For measuring field-intensities of the order of 1,000 O.G.S. the method described offers certain advantages. [An extended investigation of the behaviour of bismuth-spirals was recently published by Henderson. 1 He finds that the curve of fig. 71 for any given temperature, continues rectilinear up to fields of 40,000 C.G.S. The higher the temperature, the higher, of course, is the initial ' zero-field ' resistance, but the less steep the ascent of the resistance-curve, so that it actually meets the curves for lower temperatures at a certain field- intensity. This means that in such a field there is no tempera- R R FIG. 72 ture-variation ; plotting the resistance as a function of tempera- ture, between and 80C., Henderson finds a considerable de- crease for a field of 20,000 C.G.S., practically no variation from 10,000 C.G.S., and, of course, the usual increase outside the field. For further details the original paper must be referred to. Quite recently, further light has been thrown on the electric behaviour of bismuth by the work of Sadowsky, 2 and of Dewar .and Fleming. 3 H. du Bois.] 1 J. B. Henderson, Wied. Ann. vol. 53, p. 912, 1894; Phil. Mag. November 1894. 2 Sadowsky, Journ. de Physique (3), vol. 4, p. 186, 1895. 3 Dewar and Fleming, Phil. Mag. (5), vol. 40, p. 303, 1895. 314 DETEKMINATION OF THE FIELD-INTENSITY F. Magneto-hydrostatic Method 203. Principle of the Method. What is called the magneto- hydrostatic method was first used by Quincke in an extended investigation of the properties of paramagnetic and diamagnetic (that is in the sense hitherto used indifferent *) liquids. If these properties be assumed to be known, or if they are eliminated, the method may conversely be used for determining the intensity of the field. A /-tube, the best form of which is that represented in fig. 72, where the section of the narrow tube R R may be dis- regarded in comparison with that of the reservoir A A, contains liquid, the meniscus of which is at first at Z r If a magnetic field is produced here, it is observed that the meniscus rises (or falls) according as the liquid is paramagnetic (or diamagnetic) [ 7] ; owing to its large area, the level in A A does not appreciably vary. If > is the intensity of the field, 3 the magnetisation of the liquid, D its density, a its rise due to magnetism, g the acceleration of gravity, P the pressure which corresponds to the rise, it may be shown that 2 (15) This integral manifestly corresponds to the area enclosed by the curve of magnetisation ( 13), the axis of abscissae, and the ordinate corresponding to the value of its upper limit (compare also 148). In so far as it can be assumed that the liquid has a con- stant susceptibility K (compare the author's paper referred to), 3 = K <; if this is introduced into the above integral, 3 1 Quincke, Wied. Ann. vol. 24, p. 374, 1885. The method, with some modifications, was afterwards also used for gases and solids. 2 Compare du Bois, Wied. Ann. vol. 35, p. 146, 1888. 3 Strictly speaking, K is the difference between the susceptibilities of the liquid and of the gas above it. Equation (16) was deduced by Quincke (loc. cit.~) from Maxwell's general formula for electromagnetic stress. The confir- mation of that equation by Quincke and others forms, therefore, in any case a support for Maxwell's theory, even though it only refers to feebly magnetic substances (p. 166). MAGNETO-HYDKOSTATIC METHOD 315 (16) P = agD = The order of magnitude of these actions is such that, with a field of 40,000 C.G.S., which for the present can scarcely be exceeded ( 175), a (diamagnetic) water meniscus would sink by about 0'5 cm., while a concentrated (paramagnetic) aqueous solution of iron chloride would rise by about 100 times as much that is, by 50 cm. 204. Practical Execution. This action may still further be increased by inclining the tube R R at an angle a with the FIG. 73 horizon ; a displacement of the meniscus b is then obtained which is connected with the vertical ascent a by the equation a = 1} sin a Fig. 73 represents an apparatus constructed by the author (loo. cit.) ; instead of Quincke's Z7-tube of fig 72, there is a glass vessel A A REG 8 by means of which the field between the pole-pieces of an electromagnet can be investigated at any inclination of the measuring tube, since the whole appa- 316 DETEKMINATION OF THE FIELD-INTENSITY ratus can turn about the axis of the field which meets the plane of the figure in E. The displacement of the meniscus is read off by the microscope M fixed to the micrometer screw F. Powerfully paramagnetic concentrated solutions of iron, or manganese salts, are less suited for measurement, as their viscosity is too great, and in time they are apt to undergo chemical change. For practical purposes, one of the best is a semi-concentrated solution of the green rhombic nickel sul- phate (NiS0 4 , 7H 2 0), the density and susceptibility of which are determined once for all in a known field. These do not alter if the liquid is kept in a closed vessel so as to prevent evapora- tion, and consequent changes of concentration. The intensity of the field is found from the formula in which the expression within the bracket may be conveniently brought to a round number, 10,000 for instance (if a is expressed in centimetres). The measurements must always be made at about the ordinary equable temperatures of rooms ; for further details the work cited must be referred to ; since the ascent is proportional to the square of the intensity, the method is only adapted for determining very intense fields, the order of, say, 10,000 C.G.S. for instance. 317 CHAPTER XI EXPERIMENTAL DETERMINATION OF MAGNETISATION OR OF INDUCTION 205. General Remarks. Theoretically, all the methods of determining the intensity discussed in the previous chapter may also with more or less considerable alterations be adapted for the absolute or relative measurement of magnetisation ; we shall therefore once more consider them in the same order from this point of view, which, however, in practice is quite different. And here, again, we lay chief weight on newer methods, especially in so far as the principles which hold for magnetic circuits apply to them ; however, for the sake of completeness the older methods will be briefly mentioned. In making a selection among the many possible methods, we must, as in the deter- mination of field-intensity, be guided by the conditions of the most varied kind, which apply in each case. The object of measurement will in most cases be that of obtaining the normal curve of magnetisation or that of induction. In case the measuring apparatus used does not directly give these curves ( 212, 214), their abscissae <> are, in the first place, to be determined experimentally by the methods of the previous chapter or in many cases calculated and, in the second place, the ordinate 5 or 93 is to be measured. The observations some- times give directly the former vector frequently as the magnetic moment per unit volume and sometimes the latter. When two of the principal vectors , SB and 3 are known, the third may be at once deduced by means of the fundamental expression [equation (13), 11] $ = 471-3 + & The calculation is particularly simple in the earlier stages of the magnetisation, where we may write 93 = 4?r3 ( 11, 59). ItJNIVERSIT' x.. .PF 318 DETEKMINATION OF MAGNETISATION OE OF INDUCTION It is of great importance to allow for.the influence of the shape of the body investigated, since this, as has already been suffi- ciently dwelt on, in many cases far exceeds the influence of the special properties of the metal, the temperature, and other factors ; and even an error in numerically eliminating this factor might partially conceal the properties of the ferromagnetic substance with which we are primarily concerned. This fact cannot be sufficiently insisted on, especially as until quite recently it has been frequently overlooked, and this omission has led to a series of errors and false conclusions. 206. Discussion of the Shape of the Test-piece. That omission, however, is strictly speaking only admissible if the shape of the body to be investigated is that of a closed ring, which is also advisable for theoretical reasons. Such a ring should never be forged from a bar, for even the most skilfully made weld always acts more or less like a joint that is, it has a demagnetising and leaking action ( 151); the ring should rather be turned from a plate. This has, of course, the dis- advantage that the direction of the laminaB or fibres is every- where parallel, and therefore, in general, not peripheral, the homogeneity too is generally doubtful. A closed ring, finally, is only available for measurements by the ballistic method, against the use of which objections may in many cases be raised ( 216, 220), especially when solid undivided material is used. As regards the use of non-endless figures, the ovoid comes first, because this alone acquires uniform magnetisation in a uniform field. For accurate investigations more especially it is desirable to use elongated ellipsoids (m2,50,JV= 0-0181, Table I, p. 41 .) The material used should be as homogeneous as possible ; the turning of an ovoid in a lathe offers no great difficulty when using a standard gauge ; the volume may be determined by the weight of water displaced, and affords a control of the direct measurement of the dimensions and the excentricity calculated from it. By using the demagnetising-factors of Table I, p. 41 (if necessary interpolated by means of the values of 0), the magnetisation curves of the ellipsoid may by shearing ( 17) be conveniently transformed to normal curves. If commercial wire is to be used directly for tests, the mean DISCUSSION OF THE SHAPE OF THE TEST-PIECE 319 demagnetising factor for cylinders Table I, p. 41, must be taken into account. 1 With a magnetically hard material the dimensional ratio ot the dimensions may be taken lower than with a softer ferromag- netic substance, because the normal curves of the former are in themselves more distant from the axis of ordinates, and there- fore are relatively less influenced by shearing. Apart from special cases, it is scarcely worth while to investigate cylinders whose length is less than twenty times the diameter ; if, on the other hand, m > 500 (JV < 0-00018), the influence of shape may usually be altogether neglected ; still a magnetisation 3 = 1000 C.G.S. produces a demagnetising intensity $< = 0-18 C.G.S. If the demagnetising action of the ends or, what comes to the same, the magnetic reluctance of the surrounding air is dimi- nished by a yoke (see 218), the correction that is, the amount of shearing of the curves will be smaller, and somehow or other has to be determined once for all. 207. Details of the Method. The older methods were mostly intended for investigating the moment of permanent magnets ; as there are then no magnetising coils, the placing of the magnets is far simpler. Arrangements in which it is sus- pended by a fibre or hangs to a balance are scarcely applicable when coils have to be used. In investigating the induced magnetisation of ovoids or cylinders, the magnetising coil must be two or three times as long as the body tested, and must encircle it as closely as possible; the necessity of having a cooling arrangement by means of water between the test-piece and the coil often, how- ever, stands in the way of the last condition. The coil is suitably inserted in a circuit containing a commutator, an apparatus for measuring the current, a storage battery and a variable resistance which had best be a liquid rheostat so that the current can be conveniently and gradually reversed, varied, and measured. In many cases a suitable correction has to be made for the mag- netic action of the coil, which is of the same kind as that of the test-piece ( 211). 1 [This table has since been superseded by the special investigation of Eiborg Mann, Entmagnetisirungsf actor en Jtreiscylindrischer Stale, Disserta- tion, August 1895. H. du Bois.] 320 DETERMINATION OF MAGNETISATION OE OF INDUCTION According to the kind of curve to be determined, the cur- rent is varied by repeated reversals with" a series of increasing or decreasing values of the current ; or it is varied step by step, which gives the ascending or descending branches of the (statical) hysteresis loop, the area of which may be determined either by graphical integration, or by weighing the corresponding areas cut out of paper in the well-known manner. 1 It is in general desirable to demagnetise the test-piece as completely as possible before each series of measurements that is, to destroy the influence of previous magnetic actions. This is best effected by a process which can be described as that of * diminishing reversals.' The test-piece is subjected to the influence of a series of slowly or rapidly succeeding fields, the intensity of which each turn changes in direction, and at the same time gradually decreases to zero. If the initial value was large enough, the test-piece is afterwards demagnetised. In practice more resistance is gradually brought into the circuit, for which purpose a liquid rheostat is best used while the current reverser is being constantly worked ; instead of letting the cur- rent decrease, it is simpler, if possible, to draw the test-piece gradually out of the magnetising coil. 208. Determination of Distribution. The distribution of the vector 3 or 9B within a ferromagnetic body is not, of course, accessible to direct observation ; this is only possible with phenomena external to the body, from which conclusions as to internal distribution can only be drawn under certain limitations. In the first place, the normal component of intensity <$ v may be determined as near the body as possible, but outside the surface. This is identical with the normal component 93,,, or 3V, for external or internal points near the surface, but is not identical with the component of intensity %' v in the latter points. We have (52, 56, 58), a) . . & = =:$; = ,; + 4 *r 3; In most cases the first term on the right may be neglected in comparison with the second ( 11, 59), so that we may put 3>^ = <,,/4 TT. The normal component of magne- tisation to be hereby determined gives, further, directly the 1 The dissipation of energy owing to hysteresis may be directly measured by calorimetrical or watt-metrical methods (note, p. 228), upon which we can- not here enter. MAG-NETOMETKIC METHODS 321 strength of the superficial end-elements which define the action of magnets at a distance ( 27, 46). In the cases in question no further end-elements occur in the interior ( 50, 59). For the experimental determination of the following simpler, but not very trustworthy, methods have hitherto been used : (a) Kepulsion of one end of a long magnetic needle in the torsion balance (Coulomb) ; (6) Oscillations of a short mag- netic needle at right angles to the surface in front of the given point of the surface (Coulomb) ; (c) The same, with a soft iron needle ; (d) Detachment of a small iron ball, or of a suitable piece of iron (Jamin's test-nail). In the last two cases the value $ is measured. The only method free from objection is to use a small exploring coil, which is laid flat on the surface to be investigated, and is then rapidly moved away. 1 In this way the flux of induction $ through the surface of the windings of the exploring coil is measured ; dividing that by 8 gives . An arrangement of Rowland's, which is much used in closed or open magnetic circuits with any given centroid but invariable cross-section, is to let the exploring coil surround the magnet, and then move it suddenly backwards, or to pull it away altogether. The momentary current due to the rapid change of position measures the variation of the flux of induction encircled by it, or the number of induction tubes cut ( 63, 64). If, for instance, we consider the short cylinder represented (fig. 7, p. 31), and place an exploring coil about the middle, it is manifest that the flux of induction from the surface passed over each time is measured by the stepped motion of the coil. The distribution of iron filings, moreover ( 189), in the neighbourhood of the surface gives in such cases an approximate idea of the conditions. The investigations communicated in 88 III and 92 furnish ex- amples of the application of the method. 2 A. Magnetometric Methods 209. Plan of the Experiments. In discussing Gauss's method ( 191) it was mentioned that the moment of the auxiliary 1 This arrangement offers a certain external analogy with the use of ' proof planes ' for investigating electrostatic distribution. 2 See further Mascart and Joubert, Electricity and Magnetism, English translation, vol. 1, 413, 424 ; vol. 2, 1211-1218. Also Chrystal, art. ' Magnetism,' Eneyclopadia Britannica, 9th ed. vol. 15, p. 242, Edinburgh, 1883. Y 822 DETERMINATION OF MAGNETISATION OR OF INDUCTION magnet could be determined at the same time. The method is therefore adapted to determining magnetic moments. In applying it to this purpose it may be considerably simplified by assuming as known the horizontal intensity which directs the magnetometer. It is either determined once for all, or, by means of a local variometer, it is referred to the known value at another place. The instrument in question is adapted for checking the time-variations. The deter- > > East m i na tion of the directive force may also \be avoided by referring the moments to be measured to the known moment of a con- stant permanent magnet ; or, better, if a coil of known area ( 195) it may even be the magnetising coil used is traversed by a current determined in absolute value, the latter forms, as it were, a standard of magnetic moment. The magnet, of volume V and cross- section 8, whose magnetisation 3 is to be determined, is investigated in the first, or. less conveniently, in the second, chief position (fig. 61, p. 294). We find equation [191, equation (5)] FIG. 74 (2) or J = ? Very long magnets may, according to Ewing, 1 be placed due east or west of the magnetometer (fig. 74) in a vertical position, whereby the top is on about the same level as the latter, that is, so as to give a maximum deflection. 2 It may be shown that the following expression is obtained for the magnetisation : D\ % tan a 3 (3) . ;. . SI -fA (2)1 1 The vertical component of the earth's field is best eliminated by means of a special coil; compare Ewing, Magnetic Induction, 40, chapter ii., where all experimental details of the magnetometric method are described. 2 The use of ovoids with this method, especially with regard to the ques- tion of maximum deflection, has quite recently been fully discussed by Naguoka, Zur Aussenrcirkung gleicJiformig magnetisirter Rotations-Ellijjsoide. Wied. Ann. vol. 57, 1896 ; H. du Bois. VIRTUAL LENGTH OF THE MAGNET 323 in which a 3 is again the observed deflection of the magneto- meter, and the signification of D l and D 2 is obvious from fig. 74. 210. Virtual Length of the Magnet. The equations given have all been obtained by assuming that the entire action at a distance proceeds from the two ends of the magnet ( 22). This assumption, however, does not hold good, especially not with short cylinders, and still less with ovoids, the whole sur- face of which may be considered as overspread with end- elements. It may, however, be shown that the actions at distances, in comparison with whose fourth power that of the length of the magnet may be neglected, may be calculated by supposing the end-elements concentrated in two ' virtual ends,' and with the strength (3 S) m ( 19) which prevails in the middle of the magnet. The distance of these ends may be called the virtual length of the magnet L 1 '. We have, obviously, Here L r < L, the geometrical length of the magnet, because in the above fraction the denominator that is, the strength in the middle of the magnet is always greater than its mean strength (3 $), which is defined by the equation L 3S = m With circular cylinders the ratio of the virtual to the geometri- cal length depends, strictly speaking, on the magnetisation and on the ratio of the dimensional ratio. We may, with F. Kohl- rausch, usually write * (5) ' = |L If this value is introduced instead of L into the series which holds for the first chief position [ 191, equation (3)], the com- ponent of intensity p due to the magnet at the distance D., is, with sufficient approximation, 2 m r i L* i i 4 1 F. Kohlrausch, Wied. Ann. vol. 22, p. 414, 1884. The virtual length L' is often called the ideal or reduced length, or is spoken of as the polar distance, Y 2 324 DETERMINATION OF MAGNETISATION OR OF INDUCTION With ovoids uniformly magnetised, we have in all circum- stances, as is capable of strict proof, (7) ... It-\L '"":'.'': -.-"" If this again is introduced into equation (3), 191, we obtain / ,fi 2 m fl i. 2 L * 4 l L * ^ == H + By calculations based on the theory of gravitation a strictly accurate finite expression may moreover be deduced for the action of an ovoid at points at any distance on the pro- longation of their axis of rotation ; it is as follows : 1 3n 2 2 ' here denotes the excentricity of the meridian ellipse (.29), n the ratio of the distance D, to half the axis of rotation ; that is, = A / 1 -z and If a series is preferred, it may be obtained by dividing and developing the logarithm, and runs as follows : 6 E* 9 4 4 G 15 s ' ^ 5 n^ 7 n^ 3 The true value given by this series within the brackets is slightly larger than that which would correspond to the bracketed expression in equation (8). 211. Von Helmholtz's Method. Compensating Coil. If we have three permanent magnets 1, 2, 3, with the virtual lengths L/, L 2 ', and Z/ 3 ', and the moments 3^, 2tf 2 , 3^ 3 , the latter, accord- ing to Helmholtz, may be determined by hanging the magnets in pairs to a balance, free from iron. Thus, for instance, 1 may be hung vertically at one end, and 2 at the other end parallel 1 Roessler, Untcrsuchimgen iiber die Magnetisirung des E%seii$, pp. 27-31 ; Inaugural Dissertation, Zurich, 1892. [It is also possible to strictly calculate that action for points not situated on the axis, by the aid of unifocal ovoids ; see F. Neumann, Vorlesungen iiber Magnetisimis, p. 81. Leipzig, 1885. H. du Bois.] HELMHOLTZ'S METHOD 325 to the beam. After reversing cue magnet, let the weight required to restore equilibrium be M ; if ND is the distance of the knife edges, g the acceleration of gravity, then 1 " 2 D 2 3 D 2 In like manner (2ft 3 2,) and (3# 2 2# 3 ) are determined, and then, for instance, we have (12) 9ft, = A / V &c. We have mentioned the method in this connection, since it also depends on the measurement of actions at a distance, though it can scarcely be called a magnetometric one. Where permanent magnets are not the subject of investiga- tion, but, as is much more frequently the case, a ferromagnetic substance magnetised by the field of a coil, the action of the coil, and of the conducting wires on the galvanometer, must generally be taken into account. Their action, which is pro- portional to the current in the coil, may be separately deter- mined and afterwards subtracted from the total action. It is more convenient to neutralise it by a second adjustable compen- sating coil, the action of which is equal but opposite. In coils which are to produce intense fields, such a compensation can scarcely be accurately and permanently obtained, as their action at a distance far exceeds that of the enclosed body. It is in any such case advisable (1) to use a magnetometric system of as small extent as possible, (2) to keep down the temperature of the coils, (3) to twist the two leads together especially before the current reverser, so that they do not enclose any appreciable area. If, however, the compensation is still not complete, the residual differences must be separately determined, and allowed for. 212. Searle's Curve Tracer. By the instrument to be described it is possible to measure simultaneously the intensity 1 Von Helmholtz, JBerliner Berichte, p. 405, 1883; Koepsel, Wied. Ann. vol. 31, p. 250, 1887 ; F. Kohlrausch, Leitfaden, 7th ed. p. 248, Leipzig, 1892. 326 DETERMINATION OF MAGNETISATION OR OF INDUCTION and the magnetic moment induced by it, as well as to trace automatically the corresponding curve of magnetisation. 1 An aluminium fork ABDE is suspended by a silk fibre KA (fig. 75) ; a horizontal magnetic needle s'n r is fixed to the fork, and tends to set in the magnetic meridian /b'A 7 "; its motion is damped by a mica vane G'. Between D and E a second silk fibre is stretched which supports a mirror F, on the back of which a vertical magnetic needle sn is cemented in the usual manner. As the plane of the mirror is in the meridian, it is evidently only the vertical terrestrial component (and par- tially also gravity) which tends to set the mirror vertically; the damping of its vibrations is effected by the horizontal mica vane G, which hangs at a small distance above the fixed plate P. At a little distance east from F, and therefore above the plane of the figure, the upper end of a vertical ferromagnetic body to be investigated is placed as represented in fig. 74, p. 322 ; this there- fore will produce at F, a component at right angles to the plane of the figure which tends to turn the needle, together with the mirror, about DE. The deflec- tion affords a measure of the magnetic moment 2)t to be determined ; the direct action of the magnetising coil is neu- tralised by the current traversing a second compensating coil placed somewhere west of F ; moreover it flows along a third coil which is placed east of the upper needle, s'n', and thus im- parts to the entire system a deflection about the vertical, which is proportional to the current in the coil, and therefore also to the intensity of the field <. A spot of light may be reflected from the mirror, the displacement of which in two directions at right angles to each other, as with the well-known Lissajous' figures, is proportional to the values to be measured, 3)1 (vertical ordinate) 1 Searle, Proc. Phil. Soc., Cambridge, vol. 7, p. 330, 1892. Fm. 75 EICKEMEYEE'S DIFFERENTIAL MAGNETOMETER 327 or $ (horizontal abscissa). Hence, the spot of light, as the current increases or decreases, traces the desired curve of mag- netisation or hysteresis loop. If the absolute measure of the co- ordinate scale is specially determined, the indications of this curve tracer may also be expressed in absolute measure ; accord- ing to the inventor's statement, this simple instrument is chiefly adapted for purposes of demonstration. 213. Eickemeyer's Differential Magnetometer. 1 This prac- tical appliance, which is intended for an approximate com- parison of the magnetic reluctances of two specimens of iron, has a magnetic circuit, a diagram of which is represented in fig. 76 ; s l F l n l and s 2 F 2 n 2 are two equal, heavy, S-shaped blocks of Swedish iron : x and y are the two iron bars to be compared. The flux of induction is in the direction of the arrows F 2 n 2 x s l F l n l ys 2 F 2 . The coil, a single horizontal turn of which is indicated by (7, encircles the two middle cheeks of the iron blocks. A magnetometer needle, not represented in the figure, swings within the vertical field of the coil. This will set vertically, for reasons of symmetry, when the magnetic reluctances x and y are equal ; but if x is greater, it will swing to one side, and if y is greater to the other, owing to leakage, which is now no longer symmetrically distributed, and which produces a horizontal component at the point where the needle is. By means of a set of standard bars of a definite kind of iron, arranged like ' a set of weights, the unknown reluctance x may be approximately equalised in a readily intelligible manner. Besides this zero method, several others, mostly imitations of Wheatstone's bridge, have been described by Edison and others, as to which hardly any informa- tion has been published ( 215). In so far as magnetic shunts are concerned, we may again refer to the discussion 123. FIG. 76 Steinmetz, EleUrotechn, Zeitschr. vol. 12, p. 381, 1891. 328 DETERMINATION OF MAGNETISATION OR OF INDUCTION B. Mectrodynamic Methods 214. Swing's Curve Tracer. This instrument, which is represented in outline in fig. 77 and in perspective in fig. 78, 1 was invented at the same time as that of Searle, and, like it, has a mirror reading. The perpendicular to the mirror, or, in other words, the reflected ray of light, can again move in two planes at right angles to each other. The deflection in the horizontal plane is proportional to the intensity < to be measured ; that in the vertical to the induction 93 in the specimen bar. The deflections are, however, produced in quite FIG. 77 a different manner, which resembles the method of measurement described in 192. The wires A A and BB conveying the current are stretched in the magnetic field by adjustable weights like strings, and when the latter is excited, they experience a sag proportional to the intensity, which is suitably transferred to the mirror E by means of cross wire. Two specimen pieces DD of the material are required, which 1 Ewing, Mektrotechn. Zeitsclir. vol. 13, pp. 516, 712, 1892; and vol. 14, p. 451, 1893. Also Ewing and Klaassen, ' Magnetic Qualities of Iron,' Phil. Trans, vol. 184, p. 985, 1893. EWING'S CUEVE TEACEE 329 are either solid or are built up of wire, ribbon, or sheet. It is desirable that they should have the rectangular cross-section (about 2-4 x 1-2 cm.) of the two normal specimens, about 45 cm. in length, supplied with the instrument. The specimens are clamped in the two pole-pieces, and likewise in a yoke which connects them at the other end, as seen in fig. 78, where also the coils are represented which magnetise the specimens. The variable current, proportional to the abscissae, passing through them also passes through the stretched wire BB. This moves in a constant vertical field, which is produced by the split iron tube C. The horizontal sag of that wire is therefore a measure for <%. The constant current of a few amperes which, in order to magnetise the iron tube, is led through its coil, now also FIG. 78 traverses the wire A A , which is stretched in the horizontal field in the slit between the pole-pieces of the specimens. Its vertical elongation, therefore measures the intensity of this field that is, the induction S5 in the specimen bars. The amplitude of the deflection of the mirror may be regu- lated both by the strength of the constant current in C and in A A', as well as by the lever effect of the weights which stretch the wires. The motion is quite dead-beat. The succession of positions of the projected point of light on the screen is registered photographically or by hand. The scale of co- ordinates may be reduced to absolute measure for the abscissas <% by calculation from the ampere-turns of the magnetising coils, and for the ordinate 9B, if necessary, by means of exploring 330 DETERMINATION OF MAGNETISATION OR OF INDUCTION coils round the specimen bars. Fig. 79 represents curves obtained with such an apparatus. Owing to the magnetic FIG. 79 reluctance of the yoke, the pole-pieces, and the air-gap, these are to be read off from the slanting line 22, instead of from the axis of ordinates xx. 1 FIG. 80 If the variations of the magnetising current are sufficiently rapid within about -^ second the projected spot of light 1 Ewing has, moreover, described an ingenious kinematic device by which the correction in question is made automatically. The motion of the mirror corre- sponding to the ordinates is so arranged that the perpendicular to the mirror APPARATUS OF KOEPSEL AND KENNELLY 331 FIG. 81 appears to trace a continuous line, which then directly represents the induction-curve. The instrument thus forms an apparatus for demonstration as ingenious as it is instructive. The con- dition above mentioned is satisfied by Ewing by means of a specially constructed rotating liquid rheostat, with or without a commutator. 1 Fig. 80 shows another form of the curve tracer, with a fixed magnetic circuit KK instead of the test bars, as best suited for purposes of demonstration. 215. Apparatus of Koepsel and of Kennelly. The measure- ment of the field by means of a coil directed by torsion springs is the principle on which is based an apparatus for investigating iron, de- scribed by Koepsel. 2 It is represented in fig. 81 in section, and in fig. 82 in plan. The magnetising coils S S, traversed by a current up to five amperes, produce in the intermediate space a field parallel to their axis. Parallel also to the latter is the plane of the windings of a measuring coil, fastened above and below to a torsion spring, which, at the same time, con- veys the current. The auxiliary current through this is adjusted to a constant value of a deci-ampere, for instance. When the coils 8 S are excited, the measuring coil will tend to set parallel to them. The angle of torsion which brings it back to zero is a measure of the field of the coil. If a specimen bar is placed in each magnetising coil, the angle of torsion required is far greater; it is, approximately at least, proportional to the induction in the test bars. In the arrangement of the magnetic circuit de- scribed, the shape of the specimen is difficult to allow for ; sweeps a plane inclined to the vertical ; hence, when there is no current in the wire, the spot of light does not trace the axis of ordinates, but the line zz, from which then its horizontal deviations take place. 1 Electrician, vol. 30, p. 65, 1892. 2 Koepsel, Verhandl. phys. Gesellsch., Berlin, p. 115, 1890 ; EleUrotecJm. Zeitschrift, vol. 13, p. 560, 1892. FlG. 82 332 DETERMINATION OF MAGNETISATION OR OF INDUCTION D Koepsel l has therefore recently modified the apparatus in the following manner : A magnetising spiral contains a single test-piece, which is enclosed in a yoke ( 218). The measuring coil is wound on an iron cylinder, which rotates in a suitable cavity in the yoke like an armature. Its deflection is not compensated by torsion, but is read off SSje A on a scale by an index. This gives directly the flux of induction in abso- lute measure; whereby, however, a definite value of the auxiliary current is assumed, and must be separately adjusted or measured. An arrangement by Kennelly may, in conclusion, be men- tioned, by which an approximate comparison of the magnetic reluctances of two specimens of iron is possible, as in the differential magnetometer described in 213. Both specimens are placed in AF or FC respectively (fig. 83). If the reluctances are equalised, there will, from symmetry, be no induction in the central iron cross-piece FD. The criterion of this is the fixity of a copper disc D, traversed by a current in a radial direction, which can rotate in a suitable slit about the unifilar suspension OP, which conveys a current. in FIG. 83 C. Induction Methods 216. The Ballistic Method, notwithstanding its many dis- advantages, is still undoubtedly the most important for experi- mental physics chiefly on account of its universal applicability, and of the unlimited range within which it can be utilised by raising or lowering the sensitiveness. It is less suitable for practical measurements, owing to the considerable disturbance to which it is liable from external influences. Its application to the measurement of the intensity of a field has been discussed in detail ( 195-197), and the way in which 1 Koepsel, Elektrotechn. Zeitsclir. vol. 15, p. 214, 1894. ISTHMUS METHOD 333 it is used for determining magnetic distributions has also been pointed out. In Chapter V we have elucidated by an example its application to the investigation of toroids, so that we may here dispense with further general discussion. 1 The present method is the only one which can be used with completely closed magnetic circuits. But as in this case it is not possible to pull away the exploring coil, the flux of induction actually existing at any time can never be ascertained, let alone measured, but its sudden variation can be determined. The ballistic method fails when the time required for such a varia- tion exceeds about a second ; in that case it is not possible to make the period of the galvanometer sufficiently long in comparison. With large electromagnets, and with high self- induction, this order of magnitude is very soon reached ( 170). 2 This forms a chief objection to the ballistic method ; in other respects it leaves little to be desired, especially for laboratory purposes. 217. Isthmus Method. It only remains to mention the particular modification in which the ballistic method may be em- ployed for special purposes. For measuring magnetisation at high intensities up to 4? = 25,000 C.G.S., EwingandLow 3 introduced what is known as the Isthmus method. The ferromagnetic sub- stance 'to be examined for example, a piece of iron is turned on a lathe into the shape of an ordinary bobbin ; its ends are either plane, or cylindrical about an axis at right angles to the principal axis. In the former case the iron can be suddenly withdrawn from the magnetic circuit of the electro- magnet used in this method ; a diagram of the latter form is represented in fig. 84. By means of a handle the whole iron bobbin can be turned about an axis at right angles to the plane 1 Further details are found in Ewing, Magnetic Induction, chapters iii. and iv. 2 In cores of soft iron thicker than say 1 cm., of high magnetic permeability and electrical conductivity, eddy currents play a considerable part ( 187). They seem, indeed, to diminish the self-induction, and therefore accelerate variations of current in the magnetising spiral (note, p. 282) ; but apparently, by their screening action, they produce a considerable time-lag of the vector S3 in respect of <$, the amount of which is difficult to check or bring into calculation. 3 Ewing and Low, Proc. Roy. Soc. vol. 42, p. 200, 1887; Phil. Trans, vol. 180 A, p. 221, 1889. 334 DETERMINATION OF MAGNETISATION OR OF INDUCTION of the figure, within the pole-pieces in which a corresponding cavity is bored, so that the direction of the field in respect of the bobbin appears suddenly reversed (see fig. 55, p. 260). A secondary coil is closely wound about the narrow neck of the bobbin (the isthmus), by means of which therefore the flux of induction through the neck can be ballisti- cally measured, and, on di- vision by the value of the section, gives the induction. At a somewhat greater dis- tance from the neck a second secondary coil is wound, so that between them is a narrow annular, indifferent space. The difference of the ballistic throws observed with each of the secondary coils, divided by the section of the space, is obviously a measure of the field- intensity within this latter; 1 and, from the tangential continuity of the vector in question, this may be regarded as identical with the intensity < in the neck itself. In this way both the ordinates of the induction curves and also their abscissae are determined. The latter can be increased far more than is possible with ordinary coils ; by rightly shaping the conical part of the iron bobbin, and by using a powerful electromagnet, both the intensity and the uniformity of <>, and therefore also of SB, within the neck are most favourably influenced, as discussed before ( 174, 175). With such a coil of annealed Lowmoor wrought iron, in which the section of the neck was 1 ,500 times less than that of the pole-pieces, Ewing and Low obtained the following values for the magnetic vectors : 24,500, 9B = 45,350, 3 = 1660 1 If the two secondary coils whose number of turns may be equal are inserted so as to oppose each other, this field -intensity may be directly measured. In this way the correction for the thickness of the wire is also obtained. See also Ewing, Magnetic Induction, &c. chapter vii. YOKE METHOD 335 218. Yoke Method. It is scarcely possible in practice that the ferromagnetic substance examined shall always have a specially prescribed form. Several methods have therefore been devised to use the material, which commercially is usually supplied in the shape of bars of various section or as wire, with the ballistic method without much trouble. J. Hopkinson 1 first endeavoured to diminish the considerable self-demagnetisation of a bar-shaped specimen by enclosing it in a simple or double dosed yoke (fig. 85), A 4, of large cross-section, and of the softest iron. The magneti- sation curve is now much less sheared to the right than would be the case without such a closed yoke ( 17, 206). The direc- trix for a given closed yoke may be determined by means of a normal specimen, the magnetisation curve of which is known. FIG The induction in the test-piece within the yoke was deter- mined ballistically by J. Hopkinson. In his original experi- ments the specimen bar consisted of two parts, CG (fig. 85), separated in the middle. By means of a handle the part on the right could be suddenly pulled out so far that, by means of a spring, the secondary coil D sprang out of the field between the primary coils B B. The momentary current then measures the flux of induction which had existed in the place in question. The surface of separation between the two halves is obviously in an unfavourable place, where the irregularities due to it ( 151) directly influence the secondary coil. The two halves may, of course, be left in position, and, the secondary coil being fixed, step-by-step measurements may be made in the usual manner. Hopkinson's arrangement does good service, especially if a 1 J. Hopkinson, PML Trans, vol. 176, p. 455, 1885. 336 DETERMINATION OF MAGNETISATION OR OF INDUCTION suitable shearing of the induction curve is not omitted, and the specimen bar is not of highly permeable soft iron like the yoke ; for the magnetic reluctance of a ' harder ' kind of iron or steel is, of course, far greater in comparison with that of the yoke. 219. Various Forms of Closed Yoke. The arrangement is more favourable when, as proposed by Ewing, 1 two test bars are FIG. 86 used, and their ends connected by two massive yokes (fig. 86, p. 336). The secondary coils are best arranged about the middle of each bar. One of the yokes is so arranged that when it is pulled off, the secondary coils come away with it. FIG. 87 In the apparatus represented in fig. 85 the test bars are inserted in the perforations in the yoke in which case they must fit well, so as to avoid air-spaces ; hence a very accurate form of section is necessary. In this respect, the simple connection at the ends, as in fig. 86, is more practical. The rods, strips, or 1 Ewing-, Magnetic Induction, 60, 161. CASE OF GEE AT SELF-INDUCTION 337 bundles of wire, of any cross-section, need only be of a definite length, and with their ends cut off straight and smooth. The magnetic pull itself produces a close contact, which can be still further improved by external pressure as by screws, for example ( 152). An apparatus for practical purposes depending on the method in question is that described by Corsepius l under the name of Siderognost (at,Bijpds t iron), in which a simple U-shaped yoke is used. Fig. 87 represents the smaller form of this apparatus, which, after what has been stated, needs no further explanation. A simple arrangement has recently been described by Behn- Eschenburg, 2 by means of which rapid comparative measure- ments of various kinds of iron may be made (fig. 88). The FIG. 88 test-piece P is fixed to the posterior bar of a closed yoke 5jjB 3 JB 2 5 4 ; between P and the front bar is an iron plate A carrying the secondary coil A, which, by means of a lever H, can be suddenly withdrawn, as in Hopkinson's arrangement. 220. Case of Great Self-induction. It was stated in 216 that the ballistic method, in its ordinary form, fails when the self-induction is too high. This defect may be partially remedied by bringing into the circuit of the magnetising coil a considerable non-inductive resistance, by which variations of current, and therewith magnetic variations, are less re- 1 Corsepius, Untersuckungen zur Konstruktion tnagnetischer Maschinen, pp. 46-61, Berlin, 1891. 2 Behn-Eschenburg, Elektroteclw. Zeitsclir. vol. 14, p. 330, 1893. Z 338 DETERMINATION OF MAGNETISATION OE OF INDUCTION tarded. 1 A corresponding higher electromotive force is then, of course, necessary. On the other hand, this very fact may be taken advantage of. By determining the self- inductance, by any suitable method, certain conclusions can be drawn as to the properties of the ferromagnetic bodies. For, from 153, equation (8), for a closed magnetic circuit we have, in our usual notation, as intro- duced there Let a variation S< of the intensity, and S95 of the induction, correspond to a small finite variation S I of the current, during which the self-inductance may be regarded as constant. Then, from the above equation, (14) . ; S9 . c 4 TT ri 2 8 nS If the self-inductance is determined each time for a series of such small variations, it is obvious that, by a suitable inte- gration, we may obtain the relation between 95 and $. For (15) . = _^UcZ$=_l Ud/ 4 TT n 2 8 J n 8 J o o Swinburne and Bourne 2 have described a practical method, which may be regarded as coming under this head, in which the material to be investigated is in the form of wire. They wind the wire in a mould of known dimensions, so as to form a ring, which is then wound uniformly with a primary and secondary coil. In order to dispense with the ballistic measurement of the current in the secondary, it is compensated by an equal and opposite one, so that the galvanometer is only used as a 1 Compare p. 265, note 1. See also J. Hopkinson, Original Papers on Dynamic Machinery, p. 198, New York, 1893. 2 Swinburne and Bourne, Phil. Mag. [5], vol. 24, p. 85, 1887 ; The Elec- trician, vol. 25, p. 648, 1890. METHODS OF HOPKINSON AND OF GEAY 339 zero-indicating instrument. For this purpose the primary coil of the ring is connected up in the same circuit with that of an ' induction-box.' l The secondary coils of the latter are, on the other hand, in the same circuit with that of the ring, but in such a manner that the two induced impulses act in opposition to each other. Secondary coils are now inserted in the induction-box until compensation is obtained. Its mutual inductance is thus equal to that of the two coils of the ring. This mutual inductance, multiplied by the ratio of the number of turns [ 176, equation (39)], gives the unknown self-inductance of the primary coil. As regards the details of Swinburne and Bourne's method, we may refer to the paper quoted. 221. Methods of J. and B. Hopkinson and of T. Gray. The following method has been applied by Drs. J. and B. Hop- kinson 2 for determining the hysteresis with rapidly successive cyclical processes. Iron or steel wire of J mm. diameter was used, and wound into a ring; on this wire n (= 200) turns of copper wire were coiled. Through this ' induction coil ' (as explained in 153) an alternating current was passed. The current 7, as well as the impressed electromotive force Z e , were, by means of a rotating contact-maker and an electrometer pro- vided with a condenser, determined as periodic, functions of the time, and represented graphically by an (/, T)-curve and an (JE e , T)-curve. By multiplying the ordinates of the former with the resistance of the coil, and subtracting from the ordinates of the latter, a third curve is obtained, which, by equation (7), 153, (16) . . > = Ee -IR represented this differential quotient as a function of time. Integration gives, in the first place (11 ), and therefore, also, 23 = (n)fnS as a function of T; and then, by reference to the corresponding ordinates of the (7, 2 7 )-curve, as a function of I or of % = 4tirn I/L ; that is, the unknown curve of induction is 1 Such an induction box is the analogue of a resistance box. It contains two sets of coils, whose mutual inductance may, within certain limits, be regulated, either step by step or continuously. See Graetz, Wied. Ann. vol. 50, p. 769, 1893. 2 J. and B. Hopkinson, Tlie Electrician, vol. 29, p. 510, 1892. z 2 340 DETERMINATION OF MAGNETISATION OR OF INDUCTION ultimately obtained. Its area in the usual manner represents the dissipation of energy due to hysteresis. Besides from periodic current curves, either by the above method, or by an inversion of Sumpner's method ( 156), the curve of induction may, as T. Gray l has done, be also deduced from the curve representing the rise of the current on making. As this method, which is of theoretical interest, has been dis- cussed and graphically explained, we shall be content with a few statements in reference to what is rather a circuitous method of experimenting. Measurements and graphs of the time-course of rapidly varying currents have been obtained by Blaserna 2 with ro- tating contact arrangements, and by Helmholtz by means of his well-known contact-pendulum. 3 In subsequent investiga- tions these methods, more or less modified, have been used. For determining his current curves T. Gray at first used a method like the one given by Hopkinson. With the very high time ratio of his electromagnet (compare fig. 41, p. 243) the slow changes of current could be followed with a rapidly vibra- ting, but strongly damped, reflecting galvanometer. The mirror reflected a spot of light on a chronograph drum rotating about a horizontal axis, and covered with co-ordinate paper. By following the motion of the spot of light with a pencil, curves were obtained of which we have represented in fig. 40 (p. 242) and 41 (p. 243) some observed with a closed magnetic circuit, together with the induction-curves deduced from them in the manner there described. D. Magneto-optic Methods 222. Kerr's Phenomenon. The intensity of a field may be measured optically ( 199) ; in like manner we may determine magnetisation by means of Kerr's magneto-optic phenomenon ( 10). Linearly polarised monochromatic light is allowed to fall (normally in preference) on the reflecting magnet, and the reflected light is then investigated by means of a special 1 T. Gray, Phil. Trans, vol. 184 A, p. 531, 1893. Compare also Evershed's -criticisms, The Electrician, vol. 32, p. 316, 1894. 2 Blaserna, Giornale di Scienze Naturali ed Economiche, vol. 6, p. 38, 1870. 3 Helmholtz, Wixsensch. Abhandl. vol. 1, p. 629, Leipzig, 1882. KEEK'S PHENOMENON 341 polariser and analyser. A simple rotation s of the plane of polarisation, without any disturbing ellipticity, is then observed which is proportional to the normal component of magnetisa- tion, as has been mentioned in introducing that vector ( 11) ; hence we put / 1 7\ T^c* (i/j . . . = KJ V K is a negative or positive factor of proportionality, and is known as Kerrs constant ; for a given metal it depends on the wave- length, but scarcely at all on the temperature; the absolute values are known for the ferromagnetic metals. Fig. 89 represents an experimental arrangement used by the author. 1 The reflecting polished plate M to be investigated is fixed to one massive pole-piece, P, of a powerful electro- Ojilisck FIG. 89 magnet ; the other pole-piece is bored so as to allow incident and transmitted light to pass. In addition to the magnetisation to be deduced from equation (17), the corresponding abscissa of the curve of magnetisation has also to be determined. For 1 Du Bois, Phil Mag. [5], vol. 29, p. 293, 1890; Wied. Seibl. vol. 14, p. 1156, 1890. Compare also Wied. Ann. vol. 39, p. 25, 1890 ; and vol. 46, p. 545, fig. 1, 1892. Since, from equation (17), the rotation is proportional to the normal component of magnetisation, it would theoretically be possible to determine the distribution'of 3i> for instance, by means of a very thin proof mirror ' (see 208). 342 DETERMINATION OF MAGNETISATION OK OF INDUCTION this purpose the field intensity J$ t , normal to the mirror and immediately in front of it, is measured, by means of standard glass plate G, silvered at the back 8 (compare 200), On account of the normal continuity of the total induction the value of this vector within the plate, that is 98/, is equal to that of & (compare note, p. 83). We obtain in this manner (18) . 3 = funct. (SB!) = funct. ($; + 4 TT 3) that is, a direct relation between the two vectors 3 and SB/. Since, however, it is not usual to represent this relation graphically, but ordinarily to introduce the single independent variable <% t as abscissa, the curve of magnetisation, that is, (19) . ... 3 = funct. ($) may be conveniently obtained by shearing backward from the axis of ordinates to a directrix, the equation of which is (20) ... $ = -47r3 223. Kundt's Phenomenon. The rotation of the plane of polarisation when light passes at right angles through thin transparent ferromagnetic layers, transversely magnetised, which is known as Kundt's phenomenon, may theoretically also be used for measuring purposes. The rotation s here observed is (21) . . . e = 3d in which d is the thickness of the layer, ty a factor called Kundt's constant. 1 The relation expressed by equation (18) is thus also directly obtained ; this also follows from the fact that the curve of magnetisation of a plate is the limiting curve, which corresponds to the greatest possible demagnetising factor N 4 TT ( 30, 33). These magneto-optical methods there- fore are suited only for investigating the saturation-stage ( 13) of magnetisation ; for in the lower stages the curve of magneti- sation differs so little from the straight line symmetrical with the directrix [equation (20)], the equation of which is (22) . . . $= + 47r3 1 Kundt, Wied. Ann. vol. 23, p. 228, 1884; and vol. 27, p. 191, 1886. Compare also du Bois, Wied. Ann. vol. 31, p. 966, 1887. HALL'S PHENOMENON 343 that the differences are quite within the range of errors of obser- vation. This case is the most striking example of the manner in which all special properties of the ferromagnetic substance are concealed by its shape. Although the rotation with transmitted light may in certain circumstances be almost ten times that on reflection, it cannot for various reasons be so accurately determined. In the former method we are restricted to thin electrolytic layers of metal, and cannot investigate any given mass of material ; in conse- quence of this the method is in general less to be recommended than that depending on Kerr's phenomenon. E. Hall's Phenomenon. Bismuth Spiral 224. Hall's Phenomenon. Bismuth Spiral. It has recently been shown by Kundt l that the angle of deflection fi of the equipotential lines (compare fig. 69, p. 310), in thin ferro- magnetic layers transversely magnetised, is proportional to the rotation s of the plane of polarisation of the transmitted light. He concludes from this, that that angle, or, in other words, the potential of difference between the two ' Hall's electrodes,' must also from ( 223) be proportional to the magnetisation, and hence be suited for measuring it. This mode of measurement, which is perhaps capable of development, has been just as little used for determining field-intensity as the corresponding phenomenon in non-magnetic metals (201). The method might specially be suited for investigating the stage of saturation, for we are dealing with the limiting .curve corresponding to the value N = 4 -IT discussed in the previous paragraph. It is to be ob- served that the measurement of the difference of potentials of the Hall electrodes is far more accurate and convenient than that of the rotation of the plane of polarisation on transmission. 225. Bruger's Apparatus. The method of measuring a field by means of a bismuth spiral, described in 202, has been used by Bruger as the basis of a magnetic apparatus, the most recent form of which is represented in fig. 90. 2 A handle-shaped yoke of circular section encloses the test-bar, which ought to 1 Kundt, Wied. Ann. vol. 49, p. 257, 1893. 2 The original form of the apparatus was shown at the Frankfort Electro- technical Congress in 1891. (See BericTite der Sections- Sitz, p. 87, Frankfort, 1892.) 344 DETEEMINATION OF MAGNETISATION OK OF INDUCTION have the same section as the yoke. An air-gap is left (on the right in the figure), within which the field can be measured by means of a bismuth spiral. The width of the slit is determined each time by the micrometric arrangement represented. The FIG. 90. | nat. size corresponding magnetomotive force, expressed direct in ampere- turns, is denoted by M' e (p. 196, note 1) : (23) . . . M;=0-8d Let the total value of the ampere-turns be 3//; then for the yoke there remains Mj and for the test-bar M' p ; so that (24) . . M' p a- M; = Ml - M' If 8 is the constant cross-section, then, independently of leakage, the flux of induction is = <$ S. Hence, we can plot M' p as a function of , which is the object of the apparatus. Before doing this, MJ must be determined once for all as a function of by means of a standard test-bar of the same kind of iron as the yoke. Leakage may be allowed for by taking the bismuth spiral of somewhat larger diameter than the slit. Compare what has been said on the behaviour of bismuth spirals in 202. F. Traction Methods 226. Thompson's Permeameter. We mentioned in 107 the experiments of Shelford Bidwell on toroids divided diametrically, SILVANUS THOMPSON'S PERMEAMETER 345 as well as with divided bars. This method of using the tractive force to measure the induction or the magnetisation, forms the basis of a class of apparatus to which reference has been made. Silvanus Thompson has developed the latter arrangement mentioned, so as to form an apparatus for practical purposes, which he has called a permeameter. 1 This is represented in fig. 91 . The specimen to be tested passes through a hole in the top of the yoke, fitting very closely, and through a coil, so that its lower end, faced true, rests on a portion of the yoke, which is Spring balance Screw Clamp . . \Big block of iron ,-Coil Conducting wires, for the current FIG. 91 also faced true. The force required to pull off the bar is deter- mined by a spring balance. As the coil remains at rest when the bar is detached, Thompson takes into account only the first member of equation (13), 104 and therefore puts (25) . . ft = 3S= 27r3 2 # in which S is the section of the bar, $ the total tractive force. If the latter is expressed in kilogrammes-weight, and the above quantities are distinguished, as in 103, by being written g^ 1 Silv. Thompson, Dynamo- electric Machinery, 4th edition, p. 138, 1892. 346 DETERMINATION OF MAGNETISATION OR OF INDUCTION and 3u we obtain, with sufficient approximation, the following formula : (26) 3 = The spring balance may, of course, be so graduated that the value of 3 may be at once read off. All the sources of error ( 107) which we have already dis- cussed in detail when speaking of experiments on lifting power, of course also influence measurements with the permeameter. The joint is, moreover, in an unfavourable place, as the lines of induc- n~ I 2030 40 Sn TFJ-H-H-I.U.H 90 JOO ~ rr r~l 130 FIG. 92. scale tion present irregularities in passing from the narrow bar into the yoke. Ewing has, accordingly, proposed that the joint should be in the middle of the bar. 227. Magnetic Balance. In order to get rid of these drawbacks, the author has designed a magnetic balance, in which there is no actual separation of two ferromagnetic substances in contact. An outline section of the apparatus is given in fig. 92, while fig. 93 gives a perspective view. The test-bar T is clamped automatically ( 219) between the wrought -iron cheeks V l and F 2 , after having been cut exactly to 15 cm., and, if necessary, turned to a diameter of 1*128 cm. This represents a cross-section of 1,000 sq. cm., which need not be strictly adhered to, but which is so far convenient that it dispenses with 1 See G. Kapp, the Electrician, vol. 32, p. 498, 1894. MAGNETIC BALANCE . 347 all calculation. The coil C furnishes fields up to 300 C.G.S. units. It is 4-7T cm. in length, 1 and has 100 turns, so that the field < inside it is at once obtained if the current (in amperes) is multiplied by 10. Above the cheeks, and at a slight distance, is a yoke YY, which also forms the beam of the balance. Its knife-edge rests excentrically on the apparatus. Equilibrium is restored by the lead block P. The magnetic attractions on each side, which are equal by reason of symmetry, produce, nevertheless, unequal couples, owing to the unequal leverage. The resultant, as was shown by FIG. 93. scale special ballistic experiments, is proportional, within the range of the experiments (0 << 16,000), to the square of the flux of induction in the middle of the test-bar. This shows that with the corresponding comparatively weak magnetisation of the cheeks and of the yoke the leakage is unaltered (compare 109, 110). On account of the comparatively low fields pro- duced, 3> and 95 may, further, be taken as proportional (compare 11, 59). The yoke is attracted downwards on the left, and this action is compensated by sliding weights W im or W 4 . The instrument is so adjusted that, by multiplying the reading of the scale by 10 (= yTOO) or 2 (= y^4) respectively, the 1 It is, in fact, somewhat shorter (12'2 cm.), so as thereby to compensate in a simple manner for the influence of the ends. 348 DETERMINATION OF MAGNETISATION OR OF INDUCTION absolute value of the magnetisation is obtained, if the section of the test-piece is 1 sq. cm. ; otherwise, after plotting the curves, the ordinates must be divided by the cross-section. It is imma- terial, in this respect, whether the section is circular, square, or of any other form, or whether the specimen consists of a bundle of wires, ribbons, or strips ; it should, if possible, not differ by more than 10 per cent, from 1 sq. cm. Since the magnetic equilibrium is by the nature of the case always unstable, no fixed position can be read off as on the pointers of a balance ; but by trial a position of the sliding weight is found at which the yoke is just detached from the adjustment screw ; with a little practice this is effected with more than sufficient accuracy. The yoke oscillates like a Morse key with very little play (Ol 0-2 mm.) over the screw I, and the stop A. By its tilting the reluctance of the whole magnetic circuit theoretically alters to a slight extent, as the motion of the yoke is subject to the principle of least magnetic reluctance ( 158). The necessity of this follows, moreover, from the elementary consideration, that when the yoke is tilted through a given angle, the width of the left air-gap decreases [increases] more than that of the right increases [decreases] ; these varia- tions of reluctance are, however, so small as not appreciably to alter the value of the flux of induction. The tilting, to the left for instance, produces the consequence that the air-gap on the left becomes narrower ; the leakage, and therefore the efficient cross-section for the transmission of the lines of induction, is less, and accordingly, with a constant flux 01 induction, the total tractive force becomes greater there ( 109) ; with the air-gap on the right the reverse is the case. This explains in the first case the unstable equilibrium, and secondly the necessity that the detachment must take place with a definite invariable width of both air-gaps. The latter amount to about 0'3 mm. ; the instrument is so designed that variations in this are excluded as much as possible. In graduating it by means of a standard bar, 1 the adjusting screw I on the left is so adjusted 1 [This is the method of graduation as carried out by the Physikalisch- technische Reichsanstalt ; the standard bar was once for all compared with an ovoid of the same material ; see Zeitsclirift fur Instrumenten-Kunde, vol. 15, p. 330, 1895 H. du Bois.] USE OF THE BALANCE 349 that the reading gives the correct values of magnetisation, and is then provided with a sealed cap. 228. Use of the Balance. In order to demagnetise all parts of the magnetic circuit as completely as possible, the yoke may be slowly raised by a lever arrangement, while the commutator in the circuit is rapidly worked ( 207) ; when the instrument is not in use the yoke is stopped at the highest position. If this method is not sufficient to destroy the effects of the previous magnetic history of the yoke, which is of great importance, a special demagnetising coil must be used. The demagnetisation N \ XT* '/u ~*"-v 1 % ^ X \l \ rrw % ^ w V \\ *r>o d 1 ^ N \ \ --=-*- - 01 t/ ?00 V \ y, ,-o ~^ '-' s NX V; ' ' ^ y \\N \ ; ^j r **OQ \\ 1 A 1 ,\ l> a 3 jl \ \ \ jl - n $ B r ^ $ \ \ \\ i ^ 5 \ \ v | \\ , n ' ^ *. o 3 Z / < ; / ? 2 A > t 5 y to ? FIG. 94 of the specimen may, from 207, be effected by pushing the coil towards the observer, and drawing the specimen slowly out. while the reversals are being continued. According to Table I, p. 41, the demagnetising factor N= 0-12 responds to a dimensional ratio m = 15 ; in the magnetic balance N is about equal to 0'02, corresponding to the ratio 45 ; this is therefore apparently trebled by applying the yoke to close the circuit. The same value of N would be 350 DETERMINATION OF MAGNETISATION OR OF INDUCTION obtained by bending the 15 cm. bar into a toroid, until its ends formed a slit J mm. in width. N is moreover not constant ; in plotting the curves it is therefore better, instead of starting from straight directrices ( 17), to plot the curves from suitable lines of demagnetisation (fig. 21, p. 131). In graduating the appa- ratus three of these are determined once for all, one for magneti- sation increasing from zero, and two for the ascending or descending branches of hysteristic loops of cyclic magnetising processes. By their use we automatically correct the influence of the magnetic reluctance of the cheeks V l and F 2 , of the yoke Y F, of the air-gaps with their leakage, as well as of the small resistance on the joint between the test and the instruments. This is to be reduced to as small an extent as possible by previously making the ends of the test-bar as straight and smooth as may be. These ' instrumental lines of demagnetisa- tion' may be drawn on a transparent sheet of horn or gelatine, which in plotting the curve is laid upon the left upper or the right lower quadrant of the scale paper. In fig. 94 they are represented full ; the dash-and-dotted line corresponds to the value N = 0'02. The dotted curves of magnetisation were obtained for a specimen of malleable cast iron by means of the balance. 1 229. Magneto-hydrostatic Methods. As the magnetic rise of liquids in C/-tubes is also ruled by the general laws of electro- magnetic stress (note 3, p. 314), it may in this place be men- tioned how the phenomenon in question may be used for deter- mining magnetisation. From the general equation (15) 2Q3, for the pressure P we obtain by differentiation (27) ... 3 = \j "<,/ 1 For further details as to the construction and use of the instruments reference must be made to the following publications : du Bois, Her. Sekt. Sitz. EleTitroteclin. Kong., Frankfort, 1891, p. 77; Elektrotechn. Zeitschrift, vol. 13, p. 579, 1892 ; ZeitscJirift fur Imtrumenten-Kunde, vol. 12, p. 404 1892. MAG-NETO-HYDEOSTATIC METHODS 351 If, therefore, the pressure P is determined as a function of <, the magnetisation may be obtained by graphical differentiation, in which of course the magnetic intensity is now assumed to be known. For weak para- or diamagnetic liquids of constant susceptibility this process reduces to a very simple method of determining that number by the equation (16) 203, which applies to this case, into which, however, we need not enter. As regards liquid ferromagnetic bodies, we need only consider amalgams of iron, cobalt, and nickel. Quincke has made measure- ments with the former which have been discussed by the author with the aid of the above equation. 1 This method has not, hitherto, been any further developed. 1 Quincke, Wied. Ann. vol. 24, p. 374, 1888; du Bois, Wied. Ann. vol. 35, p. 156, 1888. The magnetic behaviour of the amalgams in question can scarcely be considered as sufficiently investigated. INDEX OF NAMES The references are to the Articles ABDANK-ABAKANOWICZ, 164 Ampere, 27, 64 Arons, 199 D'Arsonval, 193 Ascoli and Lari, 25 Ayrton, Mather and Sumpner, 196 Ayrton and Perry, 123, 133 Ayrton and Sumpner, 148 BARLOW, 70 Barus (see Strouhal) Beer, 71 von Beetz, 120 Behn-Eschenburg, 219 Bernoulli!, 110 Bidwell, Shelford, 104, 107, 226 Biot and Savart, 5, 20, 45 Bjerknes, 113 Blakesley, 185 Blaserna, 221 du Bois, H., 25, 33, 55, 58, 70, 75, 82, 104, 108, 133, 139, 142, 146, 168, 169, 194, 199, 203, 222, 227, 229 du Bois, H., and Rubens, 196 du Bois-Reymond, A., 193 Boltzmann, 93 Bosanquet, 108, 116, 119 Bourne (see Swinburne) Braun (see Hartmann) Bruger, 163, 202, 225 CAMACHO, 166 Carhart, 130 Chattock, 27 Chrystal, 120, 208 Clausius, 40, 49, 138 Cohn, 4, 147 Corsepius, 118, 130, 219 Coulomb, 18, 20, 21, 24, 27, 46, 76, 109, 208 Culmann, 89 Gumming, 111 Czermakand Hausmaninger, 152, 171, 174 DEPREZ, Marcel, 126 Deprez and d'Arsonval, 193 Dove, 111 Dub, 105, 111, 163 EDISON, 125, 129, 132, 140, 145, 213 Edser and Stansfield, 193 Eickemeyer, 213 Elphinstone, Lord, 115 Esson, 130 von Ettingshausen, 93 Euler, 27, 111, 113, 114 Evershed, 221 Evershed and Vignoles, 148, 164 Ewing, 8, 13, 17, 25, 27, 56, 107, 134, 147, 148, 151, 187, 192, 209, 214, 217, 219, 226 Ewing and Klaasen, 214 Ewing and Low, 56, 151, 152, 167, 174, 217 FARADAY, 2, 7, 64, 65, 112, 114, 124, 144, 160, 177, 180, 183 Fechner, 120 von Feilitzsch, 163, 169 Ferraris, 182, 185 Fick, 124 Field and Walker, 164, 194 Finzi (see Gerosa) Fizeau, 178 AA 354 INDEX OF NAMES Fleming, J. A., 155, 158, 177, 180, 184, 185, 187 Forbes, G., 130, 132, 144 Forbes and Timmis, 166 Foucault, 143, 153 Fourier, 113, 118, 124 Fritzsche, 60, 144 Frolich, 0., 33, 118, 121, 125, 137, 138 GAUSS, 21, 190, 209 Gerosa and Finzi, 8 Graetz, 220 Grassmann, 3 Gray, A., 192 Gray, T., 156, 221 Green, 35, 70 Grotrian, 146, 169 Guillemin, 166 Guthrie, 113 HAECKER, 110 Hall, 201, 224 Hamilton, 3 Hankel, 163 Hartmann and Braun, 202 Hausmaninger (see Czermak) Heaviside, 3, 45 von Hefner- Alteneck, 128 Hellesen, 193 von Helmholtz, H., 40, 64, 155, 156. 211, 221 Henderson, 202 Henry, J., 153, 177 Hering, 130, 189 Hertz, H., 178 Heydweiller, 188, 195 Hibbert, 197 Himstedt, 193, 195 vom Hofe, 75 Holborn, 150 Honig {see Warburg) Hopkinson, B., 148, 221 Hopkinson, E., 96, 109, 122, 125, 128, 129, 130 Hopkinson, J., 96, 109, 122, 125, 126, 128, 130, 136, 140, 145, 148, 149, 180, 186, 187, 218, 220 Hopkinson and Wilson, 136, 187 Hospitalier, 8 Howard (see Lenard) Hughes, 164 JAMIN, 150, 208 Janet, 60 Jones, E. T. 108 Joubert (see Mascart) Joule, 67, -105, 111, 166 KAPP, Gisbert, 118, 137, 185 Kelvin, Lord (Sir W. Thomson), 19, 27. 36, 54, 64, 67, 113, 124, 160, 192, 193, 197 Kennelly, 118, 121, 215 Kerr, J., 10, 222 Kirchhoff, 27, 31, 54, 57, 59, 66, 70, 72, 73, 74, 83, 90, 95, 123 Kittler. E., 125, 130, 142, 143, 146, 185 Klaasen (see Ewing) Knott, 154 Koepsel, 199, 211, 215 Kohlrausch, 188, 189, 191, 193, 195, 210 Kundt, 201, 223, 224 Kunz, W., 148 LAHMEYER, 130 Lament, 105, 138, 150 Lang, K., 118 Laplace, 41, 48, 50 Lari (see Ascoli) Leduc, 169, 171, 194, 202 Lehmann, H., 83, 85, 88, 99, 123, 133, 171 Lejeune-Dirichlet, 40, 49 Lenard and Howard, 202 Lenz, 64 Lindeck, 189 Lippich, 200 Lippmann, 194 Lissajous, 212 Low (see Ewing) MASCART and Joubert, 5, 44, 51, 71, 102, 124, 180, 188, 194, 208 Mather {see Ayrton) Maxwell, Clerk, 3, 10, 27, 34, 35, 40, 51, 54, 65, 95, 101, 104, 108, 112, 114, 120, 124, 164, 165, 180, 203 Merritt (see Ryan) du Moncel, 105, 179 Mues, L., 75, 93 Miiller, J., 118, 137 Murphy, 71 NAGAOKA, 13, 209 dal Negro, 105 Neumann, C., 171 Neumann, F., 27, 54, 70, 210 Newall (see J. J. Thomson) Nickles, 105, 166 INDEX OF NAMES 355 OERBECK, 31, 92, 183 Oerstedt, 64 Ohm, 117, 118, 120, 124, 155, 180, 184 Ostrogradsky, 35 PACCINOTTI, 144 Perry (see also Ayrton), 164 Pisati, 118, 123 Poisscn, 27, 50, 54, 70, 111 Pouillet, 120 QUINCKB, G-., 108, 203, 229 EADPOED, 166 Rayleigh, Lord, 17, 199 Kiborg Mann, 206 Richarz, 27 Riecke, 71 Righi, 202 Ritchie, 111 de la Rive, 111 Roberts, 166 Roessler, 210 Rornershausen, 166 Rowland, 74, 115, 145, 208 Rubens (see du Bois) Ruhmkorff, 167, 171, 173, 179 Ryan and Merritt, 184 SADOWSKY, 202 Sanford, 120 Savart (see Biot) Schellbach, 113' Schlomilch, 38 Searle, 189, 212 Siemens brothers, 136 von Siemens, Werner, 105, 106, 117 Silliman, 177 Stansfield (see Edser) Stefan, 101, 174 Steinmetz, C. P., 118, 148, 213 Stenger, 171, 193 Stoletow, 74 Strouhal and Barus, 150 Sturgeon, 111, 165 Sumpner (see also Ayrton), 156, 221 Swinburne and Bourne, 220 TAIT, 3 Tanakadate, 25, 148 Thompson, Silv., 18, 104,'IOS, 110/115, 125, 127, 130, 135, 137, 142, 143, 158, 166, 169, 170, 185, 226 Thomson, J. J., 187 Thomson, J. J., and Newall. 151 Thomson, Sir W. (see Kelvin) Threlfall, 108 Timmis (see Forbes) Toepler, 190 Trotter, 130 UPPENBORN, 185 VEBDET, 199 Vignoles (see Evershed) Vincent, 115 Volta, 112, 123 WALKER {see Field) von Waltenhofen, 25, 105, 118, 137, 163 Warburg, E., 8, 147, 148 Warburg, E., and Honig, 147, 148 Wassmuth, 71, 82, 106 Weber, L., 71 Weber, W., 27, 32, 197 Wedding, W., 130 Wheatstone, 213 Wiedemann, G., 27, 70, 105, 120, 134, 137, 163, 166. 177 Wilson, 136, 187 i ZlPERNOWSKY, 179 A A 2 INDEX OF SUBJECTS The references are to the Articles ABSOLUTE (C.G.S.) system, 4, 11, 14, 45, 101, 106 Action, attractive, of coils on iron cores, 163 Adhesion, 107, 108 Ageing of a magnet, artificial, 150 Air gap, 8 ; equivalent, 151 Air-reluctance, 132, 133, 206 Ampere turns, 114, 126, 135, 141, 165, 181, 214 Analogy of magnetic with other cir- cuits, 111-118 Anchor ring, 9 Approximation, practical rule of, 11, 59, 66, 77 Armature, of a dynamo, 60, 125, 133, 140, 187 ; of a magnet, 150, 166 ; reactions, 135, 136 ; ring, 143 Attraction, magnetic, 101, 167, 169, BALANCE, magnetic, 227 Ballistic, galvanometer, 2, 9, 84, 130, 196 ; method of determining induc- tion, 130, 170, 189, 195, 206, 216, 220 Bar magnet, 10, 19, 22, 67, 77, 150 Bell magnet, 166 Bid well's experiments on magnetic lifting force, 107, 108 Bifilar suspension, 190, 193 Biot and Savart's law, 5 Bismuth spirals, used for measuring a field, 202, 224 Box, induction, 220 Branched magnetic circuits, 123, 145, 186 Brake, magnetic, 165, 166 Break sparks, 178 Brushes, 127, 134 CELLS, 41 Centroid, 6, 9, 15, G5, 100, 168, 180 Characteristic of a dynamo, 126, 127 : magnetic, 96, 131, 136, 180, 181 Circuits, magnetic, 15, 125, 145 Clubfooted electromagnets, 166 Coil exploring, 2, 7, 19, 130, 132, 189, 195, 208 ; choking, 155, 157 Coils, attractive action on spheres, 162 ; on iron cores, 163 Commutator, 126, 135, 142, 214 Condenser, 102, 133, 178, 196, 221 Conductivity, magnetic, 112; analogy of, with thermal, 111, 117, 124 Conductor, magnetic field of circular, 6 Cones, experiments with truncated, 175 Conservation of the flux of induction, 61, 65, 96 Contact, magnetic, 107 Continuity, 35, 37, 58, 78, 96, 102. 217, 222 Convergence of a vector, 49, 59, 73, 79,95 Co-ordinates, transformation of, 13, 17 Core transformers, 185, 186 Correction for leakage, 99 Current, intensity, 44: alternating, 140, 145, 155, 157, 180, 221 Curve, hyperbolic of magnetisation, 33 ; normal of magnetisation, 85, 87 ; tracer, Swing's, Searle's, 214 Curves, shearing of, 13, 17, 32, 54, 88, 149; transformation of, 98 Cut, action of a, 16 Cycles, magnetic, 147, 154, 221, 228 Cylinder, magnetisation of , 24, 30, 61, 70, 163, 206 INDEX OF SUBJECTS 357 DAMPING, measurement of a field by, 198 Damping-ratio, 190 Dead, coil, 182 ; iron bar, 166 Declinometer, 189 Deflection, observations of, 191 Demagnetisation, lines of, 88, 228 Demagnetising, action, of a bar, 24, of leakage, 95 ; factor of circular cylinder, 25 ; intensity, 55 Density of unit shells, 39 ; of tubes, 64 Detachment, experiments on, 107, 22(5 Diacritical point, 138 Diagrams, transformer, 184 Diamagnetic substances, 7, 108, 203, 229 Di-electrical polarisation, 113, 117, 124, 133 Differential magnetometer, 213, 215 Diffusion, 124 Dimension, 14, 101 Dimensional ratio of magnets, 24 Directrix, 13, 87, 149 Disc armature, 143 Discontinuity, surfaces of, 34, 52, 56, 62 Dissipation of energy, 126, 141, 148, 157, 182, 207, 221 Distance, apparent action at a, 10 ; ! centres of action at a, 27, 50, 73, | 79,95 Distribution, lamellar, 38, 39 ; lamellar solenoidal, 41, 42 Divergence of the lines of induction, 77 Drum armature, 143 Dynamo machines, 60, 100, 115, 126, 127, 158, 170, 181, 186, 193 Dynamos, magnetic circuit of, 100, * EDDY currents, 134, 143, 153, 179, 182, 187,216 Electricity, conduction of, 112, 116, 117, 124, 133, 201 Electrodynamic action, 1, 104, 108, 192, 214 Electromagnetism, mechanisms de- pending on, 159 Electromagnets, Ruhmkorff's, 167; j du Bois, 169; with large lifting powers, 165 Electromotive forces, induced, 64 Electromotor, 125, 158, 182 Elementary particles, magnetic, 27 Ellipsoid, magnetisation of, 28, 69 Empty running of a transformer, 181, 184 End-elements, 26, 79 Endlessness, 16, 32, 54 Ends of a magnet, action at a dis- tance of, 18, 19, 22 ; attraction and repulsion between, 21 ; action of external fields on pairs of, 23 Energy, dissipation of, 126, 141, 148, 157, 182, 207, 221 Equipotential surfaces, 39, 57, 160 Equivalent air-gap, 151 Evanescent magnetisation, 149 Ewing's curve tracer, 214, and Low's isthmus method, 217 Exploring coil, 2, 7, 19, 130, 132, 189, 195, 208 Extra current, 177 FACES of a slit, 103, 144, 165, 218, 226 Factor, mean demagnetising of a bar, 24 ; of a field, 28 Faraday's lines of force, 65 Ferromagnetic, substances, 8 ; body conveying a current, 60 Field, magnetic, 1. 5, 188 ; demagnetis- ing factor of, 28 ; distribution of a, 188; experimental determination of the intensity of a, 188 ; measurement of a, by damping, 198 ; by bismuth spirals, 202; represented by unit tubes, 63 ; uniform, 28 Field-magnets, 115, 125, 140, 168 Filtration, 113, 124 Flat, pole, 171, 173 ; ring, 74 Fluid, magnetic, 18, 27, 49, 102, 111 Flux of induction, 119 ; conservation of the, 61, 65, 96 ; standard, 197 Force, magnetic, 4 ; lines of, 4, 65, 112, 114-118 ; measurement of a dynamical, 192 Frequency of alternating currents, 145 Friction, magnetic, 166 GALVANOMETER, ballistic, 2, 9, 130, 133,196; Lippmann's, 194; Leduc, 194 ; standardisation of, 84 Gauss's magnetometric method, 190 HALL'S phenomenon, 201, 224 2> I } Henry, 1 53 History, magnetic, 8, 228 Hoop, magnetisation of, 74, 106 Hopkinson's synthetic method, 96 Horseshoe magnets, 150, 166 Hydrokinetics, 111, 113 Hyperbolic curves, 33, 80, 90, 138 358 INDEX OF SUBJECTS Hysteresis, 8; dissipation of energy by, 148 ; influence of, on transform- ers, 182 ; loop, 147, 154 ; systems free from, 164 IMPEDANCE, 155 Inclinometer, 189 Indicator diagram, 127, 184 Inductance, mutual, 177 Inductance, self, 153 Induction, box, 220; coils, 153, 167, 168, 169; currents, 61 Induction, curves of, 13, 96, 137, 156, 205, 217. 221 ; lines of, 61, 62, 77, 90, 123, 125, 132, 134, 141, 226, 227 ; in ferromagnetic media, 51, 58 ; properties of the resultant, 58; magnetic, 7, 10 ; mutual, 177 ; experimental determination of the, 205 ; tubes of, 61, 77, 112 Inductors, 176 Integrals, surface, 35 Intensity, magnetic, 4 ; lines of, 4 Interferric, 8 Interspace of a dynamo, 125, 144 Inverse functions, 96 Ion, 27 Iron, apparatus for investigating, 215, ! 225 ; filings, use of, 4, 111, 151, 189 ; i shell, 163, 166 Irrotational motion, 39, 113 Isthmus of an electromagnet, 174 ; j method of Ewing and Low, 217 Isodvnamic surface, 160 JOINTS, magnetic reluctance in, 25, 144, 151, 165, 169, 170 KERB'S phenomenon, 222 Kilowatt, 148 Kundt's phenomenon, 223 LAG, magnetic, 134, 148, 157 Lamellar distribution, 39, 47, 56, 66, 113 ; magnets, 150 Lamellar-solenoidal distribution, 41, 47, 54, 66 Laplacian distribution, 41 Leakage, 123 ; coefficient of, 78, 88 ; j correction for, 99 ; distribution of, i 88 ; empirical formula for, 90 ; \ experimental determination of, 130 ; investigation of, in an electro- magnet, 173; influence of , in trans- formers, 183 ; self-compensating effect of, 95 Lifting force, theoretical, of a divided toroid, 103 ; magnetic, 105-108 Limbs of a fieJd magnet, 125 Line-integrals, 40, 54, 72, 95, 122, 124, 135; of the demagnetising intensity, 81, 119 Lines, of force, 4, 65; of intensity, 4 ; of magnetisation, 26 ; law of action between, 20 ; of induction, refrac- tion of the, 62 ; divergence of, 77 ; of demagnetisation, 88 Load-ratio of a magnet, 110, 165 Local, coils, 92, 118 ; variometer, 189, 209 Loop, normal, 149 MAGNET, virtual length of a, 210 Magnetic, balance, 227 ; intensity, 56, distribution of, 47, due to mag- netisation and to external causes, 53 ; cycles, 147 ; circuit, modern conceptions of, 119 ; circuits, dia- grams of various, 146 ; field, dis- tribution of, 189, represented by unit tubes, 63 Magnetisation, 11; curves of, 13; determination of distribution of, 208 ; of an ellipsoid, 69 ; experi- mental determination of, 205 ; hyperbolic curves of, 33 ; lines of, 26 ; non-uniform, of a ring, 91 ; problem of, 26 ; properties of, 57 ; peripheral, of a solid of revolution, 72 ; tubes of, 50, 79 ; uniform, 68 Magnetism, effective, 33 ; free, 50 Magneto-hydrostatic method, 203, 229 Magnetometer, differential, 190, 213 Magnetometric methods, 190, 209 Magneto-optical methods, 171, 199; of determining a field, 222 ; pheno- mena, 10, 33, 167 Magnets, rigid, outlines of the theory of, 34 Make spark, 178 Measurement, absolute system of, 4, 11, 14, 45, 101, 106, 170 Mechanisms, electromagnetic, 159, 170; polarised, 164 Mitis metal, 142 Molecular currents, 27 Moment, magnetic, 6 ; of a bar, 22 NEGATIVE end of a magnet, 1!>, 26 Neutral zone in a dynamo, 127, 134 North pole of a magnet, 19 INDEX OF SUBJECTS 359 OHM'S law, 117, 120 Osmotic pressure, 124 Ovoid, 29, 31, 70, 206, 210 PARAMAGNETIC substances, 7, 108 Peripheral distribution of magnetisa- tion, 73 Permanent magnet, 150, 164, 190, 197, 206, 209, 211 Permeability, 113, 119, 124 Permeameter, S. Thompson's, 226 Permeance, 119, 133 Pierced-disc armatures, 143, 144 Planes, law of action between, 20 Plate, demagnetising factor of, 30 ; poles, 173 Points, law of action between, 20 Polarisation, magnetic rotation of the plane of, 199 Polarised mechanisms, 164 Pole, magnetic, 18 ; pieces of an electromagnet, 125, 128, 133, 136, 142, 167, 170, 214, conical, 174 Poly phase alternating current, 140, 186 j Porous substances, 113 Positive end of a magnet, 19, 26 Potential, magnetic, in the field outside conductors, 45 ; of a permanent magnet, 48 ; analogy with gravita- tional potential, 49 Predetermination of a magnetic cir- cuit, 126 Pressure, magnetic, 101 Pressure, measurement of a hydro- static, 194 ; di-electric, 124, 133 Prism, ferromagnetic, 25 Pull, magnetic, 101 QUADRANT, 153 Quaternions, elementary conceptions of, 3, 34, 41 RATIO, dimensional, 24 Refraction of the lines of induction, 62, 78, 90, 123 Reluctance, 119, 121, 131, 132; in joints, 151 ; principle of least, 158 Reluctivity, 14, 119 Repulsion between the ends of a magnet, 21, 208 Residual magnetisation, 149 Resistance, ohmic, 155, 184 Retentiveness, or retentivity, 8, 46, 104, 149, 150, 207 Reversals, curves of, 13, 85, 154, 170 Revolution, ellipsoid of, 29, 70 ; solid of, 72 Rigid magnets, 34, 67 Ring, anchor, 9 ; armature, 143 ; non- uniformly magnetised, 91 Rotary currents, 186 SAFETY, factor of, against demagneti- sation, 151 Saturation of magnets, 11; influence of, on transformers, 182 ; law of, 57, 76, 90, 95, 123 Scalar quantities, 3 Screening action, 187, 216 Secohm, 153 Secondary generator, see Transformer Self -inductance, 153 Self-induction, 177; coefficient of, 153, 176, 216 ; influence of variable, 156 Sensitiveness of a galvanometer, 196 Shearing, 13, 17, 32, 88, 98, 149, 154, 206, 218, 222 Shell, iron, 197 ; transformers, 185 ; unit, 39 Short circuit, magnetic, 130, 141 Shunts, magnetic, 123, 164, 213 Siderognost, 219 Sine-function, 155, 157, 180; curve, 142 Sinusoidal electromotive forces, 157 Slit, 16, 75, 86, 102, 109, 170 Solenoid, 37 Solenoidal distribution, 36, 37 South fluid, 27 ; pole, 19 Sphere, demagnetisation of, 30 ; in a magnetic field, 160 ; action of circu- lar conductors on, 161 Spheroid, 29, 70 Spring balance, 226 Standard, magnetic, 197, 200, 209, 222; permanent field, 150, 197 Standardisation of the ballistic gal- vanometer, 84 Starting a dynamo, 142 Strain, 101 Stray field, 130 Strength, magnetic, of a bar, 19, 49 Stress, electromagnetic, 101 Surf ace, integrals, 35, 61 ; isodynamic, 160 Susceptibility, 14 ; differential, 154 Synthetic method, Hopkinsons', 96 Syphon recorder, 193 TANGENT-LAW of refraction, 62 Temperature gradient, 118, 124 360 INDEX OF SUBJECTS Tension, magnetic, 101 ; Maxwell's law of, 109 ; resultant, in air-gap, 102 Test-piece, 206, 214, 225, 226; nail, Jamin's, 208 Throttling, 109, 165, 168 Thrust, magnetic, 101 Time variations of magnetic condi- ', tions, 153 ; integral, 64, 153, 178 Toothed -ring armatures, 144 Toroid, 9 ; ferromagnetic, 10, 75 ; with j several radial slits, 81, 83 Torque, 190 ; measurement of a, 193 Torsion, 190, 193, 215 Transformers, 177, 180, 181, 185; dia- grams, 184 ; magnetic circuit of, 186 ; influence of hysteresis on, 182 Tubes of induction, 61 , 1 J 2, 208 ; unit, 63, 64, 95 UNIT shell, 39; tube, 37, 62, 95 U-tube method, 175, 204 VACUUM, 1, 7, 14 Variometer ,-189 Vector, 3; distribution, 34, 43, 61, 168, 208; equation, 11, 51, 57; lines, 36, 37; potential, 44; shell, 39 ; tube, 37, 41, 61, 63 Velocity, potential, 113 Virtual length of a magnet, 191, 210 Volume density, 49 Vortices, 27 WATT, 148 Winding, area of, 4, 64, 126, 193, 195, 209, 211 YOKE, of an electromagnet, 125, 128, 136, 166, 167, 214, 227: method. 218, 219 ZERO, retardation of, 157, 182 Zone, neutral, 127, 135 NOMENCLATUEE A, Power ..... D, Distance . . . . D, Density ..... d, Thickness or width . E, Electromotive force FH, Hopkinson's function J, Electrical current (in decamperes) /', Electrical current (in amperes) I e , Steady current J, Impedance .... K, Moment of inertia . K, Kerr's constant L, Length ..... M, Magneto -motive force N, Demagnetising factor JV, Frequency .... n, Number of windings P, Hydrostatic pressure Q, Quantity of electricity jR, Ohmic resistance . r, Radius ..... S, Area, Section .... T, Time F, Volume . . . . F, Permeance .... X, Reluctivity .... I 7 , Inductive resistance 2, Auxiliary vectors 1 The nomenclature and notation of conceptions of frequent occurrence are here arranged in order, and they are not therefore entered in the Index of Subjects. The dimension refers to the absolute system of units ( 5) ; the number of the article is that in which the special term first occurs or is defined. In many cases, (, m, n, denote direction-cosines ( 35) ; x, y, z, j/, T, suffixes indicating direction ( 34, 53) ; e, i, t, suffixes indicating the source. With reference to the accents on letters, see 51, 97. Bars over letters stand for mean values ( 24, 75). L. If. T. Article 2 1 3 148 1 22 -3 1 203 1 75 f 1 2 | 64 96 * * - 1 5 00 * $ 1 OO 153 1 1 155 2 1 23 * -i 1 222 1 5 * i 1 | 119 24 1 155 5 1 1 2 194 * * 2 1 1 ! 2 1 5 2 2 1 64 3 22 1 119 1 119 1 1 155 1 44 362 NOMENCLATURE 33, Magnetic induction .... @> Current flow . . . ... e, Excentricity ..... , Mechanical force .... 5, Vector quantity .... , Flux of induction .... 4?, Magnetic intensity .... >c> Coercive intensity' . . . ,~- . . 3, Magnetisation. . . , . " . .e> Evanescent magnetisation .R, Eesidual magnetisation . . $, Moment (of a torque) t, Eetentivity 9ft, Magnetic moment . . .. , m, Dimensional ratio .... m, Axial ratio . . . . . 91, Normal . . . n, Function, sign of . . p> Transformation-ratio . .' . r, Convergence of the magnetisation . u, Energy per unit volume . , . 3> Resultant tension . . . a > Angle . . .... f, Gravitation potential . . . e , Rotation of the plane of polarisation 0> Time -ratio . . . . . *, Magnetic susceptibility . A, Self- inductance 7 M, Magnetic permeability . . v, Leakage coefficient . H, Mutual inductance . . '.-; , Magnetic reluctivity r, Period . . . . . T, Magnetic potential . <, Potential ...... X, Difference of phase . . . $, Kundt's constant . . . , o>, Verdet's constant L. M. T. Article -4 I 1 10 i I 1 44 29 i 1 2 21 - 34 1 ^ 1 61 i 1 1 4 i 1 1 149 -3 I 1 11 i 1. 1 149 i. 1 149 2 1 2 23 149 | 1 6, 22 24 29 1 34 80 181 -1 -1 49 1 1 2 148 1 1 2 102 5 1 69 199 1 154 14 1 153 14 78 1 177 14 1 23 ^ 1 45 39 155 1 i 1 223 -i L 1 199 Spotlistroode & Co. Printers, New-street Square, London i UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 25Mar'50CS 28 OCT101955LU| 56C1 , NQV - : :EC'D LD REC' FEB MAV 15 '69 --t DEC 15 1979 IS. DEC LD 21-100m-9,'48(B399Bl6)476 Q J UNIVERSITY OF CALIFORNIA LIBRARY