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PREFACE 
 
 TO 
 
 THE SECOND EDITION. 
 
 IN this edition many misprints have been corrected, several 
 sections have been revised and improved, and some additions 
 made on the subject of modern measuring instruments. 
 
 In answer to certain criticisms the author would add, firstly, 
 that the book is not intended to be a practical companion to the 
 laboratory, but rather to indicate the amount of theory that it is 
 advisable to master before entering upon technical work ; secondly, 
 that it is assumed that the student can see and examine the 
 instruments referred to, the descriptions being intended merely to 
 enable the learner to understand at once the principle of each 
 instrument that comes under his notice. 
 
 The writer wishes to acknowledge his obligations to Dr. A. H. 
 Fison for help most kindly given with respect to corrections and 
 
 additions for this edition. 
 
 W. L. 
 
 August 1888. 
 
 6 >*< * i" t\ f 
 ?o^y 
 
PREFACE. 
 
 IN this Course the writer aims at giving a sound though elementary 
 knowledge of the modern science of Electricity. With a view to 
 rendering the book suitable for use in public schools, it has been 
 thought better to assume no more mathematical knowledge than is 
 usually possessed by the higher boys in a classical school. 
 
 For reasons of space, no attempt has been made to treat the 
 history of the subject at all completely. Names and dates may be 
 occasionally mentioned ; but for information on this head the 
 reader can consult works in which the historical sequence of dis- 
 covery has been carefully traced. 
 
 In that part of the subject in which an elementary knowledge 
 of Chemistry is demanded, the student is referred to the elementary 
 text books of Roscoe and of others, in which he can easily read up 
 to the desired standard. 
 
 But with respect to such knowledge of systems of units and of 
 mechanical principles as is essential for the proper understanding 
 of the present Course, the writer conceives that the case is different. 
 Believing it to be far from easy for the student to collect from 
 various sources just the kind of mechanical knowledge required 
 for the present purpose, the writer had originally included in the 
 Course all the mechanics needed. He has, however, since been 
 obliged, for reasons of space, to curtail ; and for the most part to 
 indicate merely the exact amount of knowledge required. 
 
 With regard to diagrams, it may be remarked that there have 
 
vni PREFACE 
 
 to be considered two classes of drawings. There are the simple 
 diagrams illustrative of principles, and there are the complete 
 figures representative of actual pieces of apparatus. After some 
 deliberation the writer decided to draw the former specially for 
 the Course ; but the latter have, for obvious reasons, been freely 
 borrowed from other works, wherever, through the courtesy of the 
 publishers, this was permitted. For such use of diagrams the 
 writer is chiefly indebted to Messrs. Longmans, Green, & Co.; 
 and also to Messrs. Elliott, to the publishers of * Nature,' and to 
 others. 
 
 At the end of the book will be found questions and numerical 
 examples ; these are intended both to serve as an exercise, and 
 also to some extent to supplement the teaching of the text. Some 
 few of the numerical exercises have, with the kind permission of 
 the publishers, been borrowed. 
 
 It would be impossible for the writer to acknowledge in detail 
 all that is owed to other books, papers, and lectures. But he 
 would here express his thanks to Professor Tait, Professor D. E. 
 Hughes, and Professor Crookes, for most kindly answering ques- 
 tions and giving information on certain points. Also to Mr. 
 Lupton, of Harrow, for kindly giving permission to use some 
 tables given in his * Numerical Tables.' 
 
 But especially would he acknowledge his obligations to his 
 friends Professor W. H. Heaton, of Nottingham, who most kindly 
 looked over the MSS. and the proofs of the first ten chapters ; 
 and Professor W. N. Stocker, of Cooper's Hill, who, with like kind- 
 ness, looked over the MSS. of the remaining fifteen chapters. To 
 both of these the writer is indebted for many valuable suggestions 
 and criticisms. Also to his friend and colleague, Mr. A. S. Davis, 
 who undertook the work of looking over the proofs of the last 
 fifteen chapters. 
 
 CHELTENHAM COLLEGE : June 1887. 
 
REFERENCES. 
 
 The writer here gives some references to works which the student 
 may consult if he desires to go deeper into the subject. He has 
 thought that it may be useful to some if he gives also references 
 to various matters discussed in * Nature/ volumes of this paper 
 being found in most libraries that are even partly scientific. 
 
 The list that follows is, however, very far from being complete. 
 
 I. Magnetism. 
 
 Lloyd's Magnetism. [Longmans, Green & Co.] 
 
 Airy's Magnetism. [Macmillan.] 
 
 Admiralty Manual of Scientific Enquiry. [Murray.] 
 
 Elementary Manual for the Deviation of the Compass, by 
 
 F. J. Evans. 
 
 Theory of Electricity, by L. Cumming. [Macmillan.] 
 Electricity and Magnetism, by Mascart and Joubert ; trans- 
 lated by E. Atkinson. [De la Rue & Co.] 
 
 II. Theory of Potential. 
 
 Elementary work, not employing the notation of the calculus, 
 Theory of Electricity, by L. Gumming. [See under I.] 
 
 Treatise on Statics, vol. ii., by G. M. Minchin. [Clarendon 
 Press.] 
 
 Mascart and Joubert. [See under I.] 
 
 Elementary Treatise on Electricity, by Clerk Maxwell. [Clar- 
 endon Press.] 
 
 Electricity ; the large work by Clerk Maxwell. [Macmillan.] 
 
 III. Electrolysis. 
 
 Faraday's Experimental Researches. 
 
 Stewart and Gee's Elementary Practical Physics, vol. ii. p. 72, 
 gives a list of references. [Macmillan. J 
 
 Sprague's Electricity [Spon & Co.] gives useful practical 
 details. 
 
 There is to be a discussion on Electrolysis at the B. A. meet- 
 ing for 1 887. Probably therefore much useful information 
 will be given in the report of this discussion. 
 
 IV. Electric Testing, Telegraphy, &c. 
 
 Handbook of Telegraphy, by Culley. [Longmans.] 
 Stewart and Gee's book. [See under III.] 
 Practical Physics, by Shaw and Glazebrook. [Longmans.] 
 Electric Testing, by Kempe. [Spon & Co.] 
 Telegraphy, by Preece and Sivewright. [Longmans.] 
 Practical Electricity, by W. E. Ayrton. [Cassell.] 
 
X REFERENCES 
 
 V. Phenomena in High and Low Vacua. 
 
 A Physical Treatise on Electricity and Magnetism, by Gordon 
 [Sampson Low], gives description and figures of experi- 
 ments by Crookes, Spottiswoode, De la Rue, and others. 
 
 We may refer also to ' Nature,' vols. xx. pp. 174, 199, 228, 250, 
 438; xxii. pp. 101, 125, 149, 153, 174, 196; xxiii. pp. 442, 
 461 ; xxiv. pp. 66, 89, 130, 255,346, 569 ; xxv. pp. 539, 594; 
 xxvii. p. 434 ; xxviii. p. 381 ; xxx. p. 230. 
 
 VI. Dynamos and Motors ; Electric Lighting, &c. 
 
 Electric Transmission of Energy, by Kapp. [Whittaker.] 
 Dynamo-Electric Machinery, by S. P. Thompson. [Spon.] 
 Modern Applications of Electricity, by Hospitalier, translated 
 
 by Julius Maier. [Kegan Paul.] 
 Arc and Glow Lamps, by Julius Maier. [Whittaker.] 
 
 VII. General. 
 
 Clerk Maxwell's book. [See under II.] 
 
 Die Lehre von der Elektricitat, von G. Wiedemann. [Fried- 
 rich Vievveg und Sohn, in Braunschweig.] 
 Cours de Physique, par M. J. Jamin. [Gauthier-Villars, Paris.] 
 Qiuvres de Verdet. [Imprimerie Nationale, Paris.] 
 Mascart and Joubert's book. [See under I.] 
 
 VIII. Miscellaneous Papers in ' Nature.' 
 
 Hydrodynamical Analogies to Electric Phenomena, vol. xxiv. 
 
 p. 362 ; xxv. p. 271 ; xxvi. p. 134. 
 
 Electricity, Medical Use of, vol. xxv. p. 521; xxviii. p. 463. 
 Electric discharge by red-hot balls, and towards a flame, 
 
 vol. xxv. 475, 523. 
 
 Electric Light and Horticulture, vol. xxiv. p. 567. 
 Aurora Borealis, &c., vol. xxii. pp. 33, 146; xxiv. p. 613 ; 
 
 xxv. pp. 53, 319, 368 ; xxvi. p. 571 ; xxvii. pp. 31, 389, 443 ; 
 
 xxviii. pp. 60, 107, 128 ; xxix. p. 409; xxx. p. 80; xxxi. 
 
 p. 372 ; xxxii. pp. 275, 348 ; xxxv. 433. 
 Electric Current Meters, vol. xxviii. p. 162 ; xxx. p. 220 ; 
 
 xxxiv. p. 508. 
 
 * Contact-Electricity* of Metals, vol. xxxv. p. 142. 
 Magnetism; Hughe? Researches in, vol. xxviii. pp. 159, 183; 
 
 xxix. p. 459. 
 
 Earth's Magnetism, vol. xxv. p. 66. 
 Magnetisation, Change in Dimensions of Magnetised Iron, &*., 
 
 vol. xxii. p. 543 ; xxxii. p. 45 ; xxxiii. p. 597. 
 Saturation of Cores of Electromagjiets, vol. xxxiv. p. 159. 
 Magnetic Field produced by rotation of an Insulator, vol. xxii. 
 
 p. 89. 
 
 Floating Magnets, vol. xvii. p. 487. 
 Diamagnetism, vol. xxxiii. pp. 484, 512. 
 
CONTENTS. 
 
 CHAPTER I. 
 GENERAL PHENOMENA OF MAGNETISM. 
 
 SECTION 
 
 i Introductory . 
 
 PAGE 
 
 2. First Phenomena observed 
 3 Polarity 
 
 ; . i 
 
 2 
 
 4. Constitution of a long, thin, Magnet . 
 5. Molecular Theory of Magnetism 
 6. Induction, General Phenomena. 
 7. Use of Keepers .... 
 8. Methods of Magnetisation . 
 
 ,.-..' . . . 2 
 
 . .' '. .'; ... 3 
 .. i ... 7 
 . . . . . 9 
 . . . 10 
 
 CHAPTER II. 
 
 MECHANICAL AND MAGNETIC UNITS. 
 
 1. Introductory. . . . . . . . 12 
 
 2. Fundamental and Derived Units . . . . . . . 12 
 
 3. Velocity and Acceleration . . . . . ' ^. ., " 13 
 
 4. Force . . . . . . . . . 13 
 
 5. Parallelogram of Forces, &c. . . . . . . . . 13 
 
 6. Moments and Couples . . . ...,.,. . . 14 
 
 7. Unit Magnetic Pole . . ....... .14 
 
 8. Magnetic Fields, and Unit Field . . . .... 15 
 
 9. Magnetic Moment of a Needle . .. . . -.-. . 15 
 
 10. Magnetic Moment of Practical, not Ideal, Magnets . . . . 16 
 
 11. Magnetic Curves . . . . . .-.'.. 17 
 
 12. Magnetic Induction takes place along the Lines of Force . 19 
 
 CHAPTER III. 
 MAGNETIC MEASUREMENTS. THE EARTH'S MAGNETISM. 
 
 1. Coulomb's Torsion Balance .... 7" .. 2 
 
 2. Use of Torsion Balance ' at constant angle* . . . . . 21 
 
 3. Method of Oscillations .... 24 
 
xii CONTENTS 
 
 SECTION PAGE 
 
 4. Laws of Magnetism . .'. . . . . 26 
 
 5. Proof of Law II. by Torsion Balance ...... 27 
 
 6. Proof of Law II. by Method of Oscillation . . . . . 27 
 
 7. Measurements as affected by Induction ...... 28 
 
 8. Earth's Magnetism. General Ideas . . . . . . . 28 
 
 9. Compasses . .- . , * V' 29 
 
 10. Modification of Earth's Lines of Force by the Presence of Iron 
 
 Masses . . . . . . . . . . . 30 
 
 11. The Earth's ' Magnetic Elements '. . . . . . .30 
 
 12. Measurement of Declination . . . . . . 31 
 
 13. Resolution of Earth's total Field into two, or three, Components . 34 
 
 14. To find the Inclination, or Dip . . . . . . . . 36 
 
 15. Measurement of the Earth's Magnetic Elements . . . -37 
 
 16. The ' Method of Deflexions ' 38 
 
 17. Magnetometers. Changes in the Earth's Field . . . .40 
 
 .CHAPTER IV. 
 
 THE SIMPLER PHENOMENA OF ELECTROSTATICS. 
 
 1. Introductory . . . . . . . . . . 42 
 
 2. Electrical Excitement ......... 42 
 
 3. Dryness needed ; not High Temperature . . . . . . 43 
 
 4. Attraction and Repulsion Phenomena ...... 43 
 
 5. Conductors and N on- Conductors . . . . . . 44 
 
 6. Electrics and Non-Electrics ........ 45 
 
 7. Two sorts of Electrification . . ' . . . . 45 
 
 8. The two sorts of Electrification are always produced together . . 46 
 
 9. Equal Quantities of the opposite Electrifications are always pro- 
 
 duced simultaneously . . . . . . . . 47 
 
 10. The 'Fluid' Theories of Electricity . . . . .48 
 
 n. The three Laws of Electrostatics -49 
 
 12. Law II. The Force varies as Q x Q' . . . . -49 
 
 13. Law III. The Force varies as . . . . -, . . 51 
 
 14. First Ideas as to Induction ........ 52 
 
 15. First Ideas as to Distribution . * 55 
 
 1 6. Faraday's Ice-pail ; illustrating the Laws of Distribution and of 
 
 Induction ........... 57 
 
 17. ' Electrophorus ... 59 
 
 18. Frictional Electric Machines . . .*- i '-. . .60 
 
 19. Miscellaneous Experiments with the Electrical Machine . . . 63 
 
CONTENTS xiii 
 CHAPTER V. 
 
 INTRODUCTORY CHAPTER ON POTENTIAL. 
 
 SECTION PAGE 
 
 1. Quantity of Electrification . . . ,\ . , V -65 
 
 2. Electrical Level, or Electrical Potential , ..'.." ' -.:" '. . . 66 
 
 3. Measurement of differences of Electrical Level by Work . ,. -67 
 
 4. Elementary Ideas on ' Capacity ' . . . ,'..... 68 
 
 5. Lines of Force, and Equipotential Surfaces . .' ,''''. . 69 
 
 6. Induction ; from a ' Potential ' Point of View . . . . . 70 
 
 7. Necessity of distinguishing Sign of Charge and Sign of Potential . 74 
 
 8. Further on Capacities . ' . f 74, 
 
 CHAPTER VI. 
 ELEMENTARY DISCUSSION OF CONDENSERS. 
 
 1. General Ideas. Apparatus used . . . .... -76 
 
 2. Experiments with the two Condenser Plates . . . . . 78 
 
 3. Discussion of the Terms ' Bound ' and ' Free ' = , . ., . .81 
 
 4. Conditions Affecting the Magnitude of the * Bound ' Charge . . 82 
 
 5. An Isolated Body considered as the Limiting Case of a Condenser . 83 
 
 6. Alternate Discharge . . . . . . . . . . 83 
 
 7. Leyden Jars 85 
 
 8. The Unit Jar 86 
 
 9. Cascade arrangement of Leyden Jars . . ".'.. f . 88 
 
 10. Nature of the Leyden Jar Charge . . v- . . 90 
 
 11. Various Effects of the Discharge .. .-.'; . . . 91 
 
 12. Induction Effects of the Discharge . .. . ' . . . 94 
 
 13. Wheatstone's Spark-board . ....... . . .96 
 
 14. The Condensing Electroscope . . .. r . . . . 98 
 
 15. Various Forms of Electrical Discharge . . . .-.,.. 98 
 
 CHAPTER VII. 
 INDUCTION MACHINES. 
 
 1. Some further Propositions in the Theory of Potential* . ... ioa 
 
 2. Application to Induction Machines . . . . . . .102 
 
 3. Sir W. Thomson's Water-dropping Accumulator . . . . . 102 
 
 4. Varley's Induction Machine . . . .. ....". .103 
 
 5. Sir W. Thomson's Replenisher . . . . ' .. . . . 104 
 
 6. The Voss Machine ' . . . .. ,. . . . 105 
 7- The Holtz Machine . .... .. . .. . . . . 108-' 
 
XIV CONTENTS 
 
 CHAPTER VIII. 
 
 ATMOSPHERIC ELECTRICITY. 
 
 SECTION PAGE 
 
 1. Franklin's, and Other Early Experiments . .'' . . . 113 
 
 2. Lightning Conductors . . . ... > ..113 
 
 3. Return Shocks . ' . V . . . . ,.' . . 115 
 
 4. The High Potential of Thunder-clouds . . .' . . .115 
 
 5. Potential at a Point in the Atmosphere . . .,,. . H7 
 
 6. Methods of Measuring the Potential at a Point in the Atmosphere . 1 18 
 
 7. Results of Observations ...... JI 9 
 
 8. Sheet- Lightning and other Phenomena ... . . 120 
 
 CHAPTER IX. 
 
 SPECIFIC INDUCTIVE CAPACITIES. 
 
 I Definition . . . , . 
 
 * . 121 
 
 2. Variation with Time . , . 
 3. Cavendish's Method 
 4. Faraday's Method .. . .. 
 5. Modern Methods ... . . . 
 
 ., . . . 122 
 . . ... . .122 
 ... ... 124 
 . . .125 
 
 CHAPTER X. 
 
 ELECTROSTATIC POTENTIAL. 
 
 1. Introductory .,.%..-.... 127 
 
 2. Definition and Measurement of Work . .. .. . .127 
 
 3. Dimensions of Work . . . V. . ... . . 128 
 
 4. Energy and Conservation of Energy . . .... .128 
 
 5. Work against a Constant Force . ., . . . . . . 129 
 
 6. Work where the Force varies as the Distance . . . ;' ,129 
 
 7. Work where the Force varies as the Inverse Square of the Distance, 
 
 or as L . ' . . . . . . . . . 129 
 
 8. Potential, and Difference of Potential . ' . . . -131 
 
 9. Application of 7 to the Measurement of Electric-Potential . .131 
 
 10. Equipotential Surfaces . . . . , ' T 33 
 
 11. Lines of Force are Perpendicular to Equipotential Surfaces. . -134 
 
 12. Field-strength; and Rate of Change of Potential . . . . 135 
 
 13. The Mapping Out of Lines of Force ; Simple Case . . .., . . .136 
 
 14. General Case . ,, . . . . ... .. 137 
 
 15. Total Number of marked Lines of Force . . . . . 138 
 
 16. Tubes of Force. ' Fa is Constant ' .... .. . 138 
 
 17. Statement of some further Theorems on Lines of Force ~. . . 139' 
 
 18. The Potential of an ' Isolated ' Body . . . .... .141 
 
LOWER DIVISION 
 
 CONTENTS XV 
 
 19. Potential of an Isolated Sphere . . . . v . . . .141 
 
 20. Capacity of an Isolated Sphere . . . . . . . .142 
 
 21. Distribution; from the Potential Point of View . Y . . .143 
 
 22. Two Spheres of Different Radii .- . , '. . . . 143 
 
 23. Potential, and Density, distinguished . '. '.',... . . .144 
 
 24. Force on a + Unit, acting Perpendicularly to a Conducting Surface . 144 
 
 25. Important Case of a Spherical Condenser . . . '. . . 145 
 
 26. The Plate Condenser v . "' . . 147 
 
 27. Formulae for Capacities, &c. . . . . . ... 148 
 
 28. Energy of Charging and Discharging . . . . . .150 
 
 29. Examples in Energy of Discharge . . . . . ..151 
 
 30. Energy of Discharge in the Cascade Arrangement of Leyden Jars . 152 
 
 31. Electroscopes and Electrometers ....... 153 
 
 32. Electrometers. The < Attracted-Disc ' Form 153 
 
 33. Sir William Thomson's Quadrant Electrometer . . . . . 155 
 
 34. Uses of the Quadrant Electrometer . . . . . .158 
 
 35. Examples in Energy of Discharge, &c. . . . . . . 159 
 
 36. General Consideration of Electrostatic Fields of Force . . .161 
 
 CHAPTER XI. 
 
 THE PHENOMENA OF ELECTRIC CURRENTS. BATTERY-CELLS 
 AND BATTERIES. 
 
 1. Introductory . . . . . . . . . . . 163 
 
 2. Galvani's Experiment . . . . . . . .164 
 
 3. Volta's Experiments and Views . . . . . . . . 165 
 
 4. Volta's Pile ; from Volta's p6int of View . . . . . .167 
 
 5. Volta's Cell, and the Couronne des Tasses ; from Volta's point of 
 
 View 169 
 
 6. The Contact ' and ' Chemical' Theories 170 
 
 7. Theory of the Simple Volta's Cell . . 171 
 
 8. Digression on the Galvanometer . . . . . . . 173 
 
 9. The Solution of Zinc in the Voltaic Cell 173 
 
 10. Polarisation . . . . . . . . . . 175 
 
 11. Constant Batteries 176 
 
 12. Remarks on Cells and on Batteries 180 
 
 CHAPTER XII. 
 
 THE CHEMICAL PHENOMENA ACCOMPANYING THE PASSAGE 
 OF THE CURRENT. 
 
 1. Introductory . . . . '''". . . . . 182 
 
 2. Heating Effects ; a Brief Account ". * . " . . . V .182 
 
 3. Chemical Effects ; General View Y V '. .' . . '. . 183 
 
 a 
 
XVI CONTENTS 
 
 SECTION PAGE 
 
 4. Grothiiss's Hypothesis. Nature of Electrolysis . . ,' . .186 
 
 5. Primary and Secondary Decompositions . . . .. . ' . . . 188 
 
 6. Simultaneous Decompositions . . . .-'. . .189 
 
 7. Faraday's Laws of Electrolysis . . . . . . . . 189 
 
 8. Further on Faraday's Laws of Electrolysis . . . . .190 
 
 9. Electro-chemical Equivalents . . . , . . ..191 
 
 10. Electro-plating . . . . . . , . . . 191 
 
 11. Polarisation of the Electrodes . . .. -, . . . . 194 
 
 12. Secondary, or < Storage,' Cells . . . . . . . 196 
 
 13. Plante's Secondary- Cell v ........ 197 
 
 14. Faure's Accumulator . . ^ , . .. , r . . 200 
 
 CHAPTER XIII. 
 OHM'S LAW. 
 
 I. General Ideas as to the Scope of Ohm's Law , . 
 2. Statement of Ohm's Law . 
 3. Resistance further Discussed . . . - . . 
 4. The Exact Conditions on which Resistance Depends 
 5. Conductivity . . . . 
 6. Application of Ohm's Law in a Simple Case . 
 7. Graphic Representation of Ohm's Law . . . 
 
 . . 201 
 
 . - . .203 
 . . 204 
 . 204 
 . . 206 
 
 '. . .207 
 
 . ; . . 209 
 
 9. Divided Circuits ....... 
 10. 'Shunts' . . ' . " . * . " , * 
 
 . . r . . 211 
 21,1 
 
 II. Fall of Potential through the Circuit . . . 
 12. Kirchhoffs Two Laws . . ' . " . '. . 
 1 1. Maximum Current with a given Battery 
 
 . . 215 
 . . . . 216 
 . 2IQ 
 
 CHAPTER XIV. 
 MEASUREMENT OF RESISTANCES AND OF E.M.F.S. 
 
 1. Preliminary, on the Units Employed . ', . .. V . 221 
 
 2. Resistance Coils, and Resistance Boxes, . . . . . - . . . 222 
 
 3. Wheatstone's Rheostat . . . . . . . . . . . ' .. . . 224 
 
 4. Wheatstone's Bridge ; General Principle . . . - . . . 226 
 
 5. Slide-form of Wheatstone's Bridge ....... 228 
 
 6. Wheatstone's Bridge ; Resistance Box Form ... . . 229 
 
 7. Resistance of a Galvanometer . . . . . . . 229 
 
 8. Resistance of a Battery- Cell . . . . . . . . 230 
 
 9. Measurement of E. M. F. ' . , . . . . .231 
 
 10. Electrometer Methods ; Open Circuit . . . . . 232 
 
 11. Volt-meter Galvanometers . . . . . . . . 233 
 
 12. Method of Opposition . * . ' ....... 234 
 
CONTENTS xvii 
 
 SECTION PAGE 
 
 13. Latimer-Clark's Potentiometer . . ' . . . 234 
 
 14. Table of Resistances in Ohms (or o>) . . . .... 235 
 
 15. Table of E.M.F. sin Volts . . . . . . . .237 
 
 CHAPTER XV. 
 JOULE'S LAW, AND CONSERVATION OF ENERGY. 
 
 1. General Survey. . . . . . .... 238 
 
 2. Units of Heat, Work, and Activity . . . . . 239 
 
 3. Energy of the Electric Current . . . . . . . 240 
 
 4. Joule's Law ........... 241 
 
 5. The Heating of Uniform Wires , . 243 
 
 6. Distribution of Heat in the Circuit . . . . . . 244 
 
 7. Heat Evolved with various Arrangements of n Cells . . . 245 
 
 8. Case of no Back-E.M.F. in the Circuit 246 
 
 9. Case of a Back-E. M. F. e in the Circuit .... . . 246 
 
 10. Numerical Examples . . . . . . . . 247 
 
 11. Failure of a Smee's Cell to Decompose Water . ... 248 
 
 12. Partial Polarisation in the foregoing Case ..... 249 
 
 13. Connection between E.M.F.s and ' Heats of Combination '. . . 249 
 
 CHAPTER XVI. 
 
 THERMO-ELECTRICITY. 
 
 1. Introductory. ... . . . . . . . 252 
 
 2. The Simple Thermo-Cell . 253 
 
 3. The Thermo-Pile 254 
 
 4. Thermo-Electric Series ......... 254 
 
 5. Thermo-Electric Powers ........ 256 
 
 6. The Neutral Point .......... 257 
 
 7. Thermo-Diagrams .......... 259 
 
 8. Peltier Effect ; Observed Facts 263 
 
 9. The Thomson Effect ......... 264 
 
 10. Theory of the Simple Thermo-Cell ....... 265 
 
 n. Theory of the Peltier and Thomson Effects . . .. . 268 
 
 CHAPTER XVII. 
 
 GALVANOMETERS ; WITH A PRELIMINARY ACCOUNT OF 
 THE MAGNETIC ACTIONS OF CURRENTS. 
 
 1. Magnetic Field about a Simple Rectilinear Current . . . . 271 
 
 2. The + and - Directions of the Lines of Force .... 273 
 
 3. Simple Form of Galvanometer . . . . . . . 275 
 
 4. Relation of Strength of Field to Current-Strength . . . 275 
 
xviii CONTENTS 
 
 SECTION PAGE 
 
 5. The Tangent Galvanometer . . . '. . . . . 276 
 
 6. The Sine Galvanometer . . . . ... . . 279 
 
 7. The Multiplying Galvanometer . . . . . . . . 280 
 
 8. Astatic Galvanometer ; Two Needles ...... 280 
 
 9. The Controlling Magnet ' Method 282 
 
 10. Sir W. Thomson's Mirror Galvanometer . ... . . . 283 
 
 11. The Differential Galvanometer . . V 285 
 
 12. The Ballistic Galvanometer . t , . . . . . 286 
 
 13. Sir W. Thomson's ' Graded ' Potential Galvanometer . . . . 289 
 
 14. Sir W. Thomson's Graded Current Galvanometers . 290 
 
 15. Weber's Electro-Dynamometer . . . . . . 291 
 
 16. Some General Observations on Galvanometers ... . 292 
 
 17. Galvanometers for Practical, or Commercial, use . ~ . . . 293 
 
 18. Calibration of Current-meters and Voltmeters .... 293 
 
 CHAPTER XVIII. 
 
 ACTIONS BETWEEN CURRENTS AND MAGNETIC POLES. MAG- 
 NETIC EQUIVALENT OF A CURRENT. ACTION BETWEEN 
 CURRENTS AND CURRENTS. 
 
 1. Action of an Infinite Rectilinear Current on a Magnetic Pole . . 294 
 
 2. Action of an Element of a Current on a Pole ..... 294 
 
 3. The Absolute System of Electro- Magnetic Units 297 
 
 4. Summary of Electro-Magnetic Units (see 3) . . . . . 298 
 
 5. The Dimensions of the Derived Units . . . . . . 299 
 
 6. Field due to a Circular Current . . . . . . . 302 
 
 7. Magnetic Shells ; and the Fields due to Them . . . . . 302 
 
 8. Magnetic Potentials due to Magnetic Shells ..... 304 
 
 9. Magnetic Equivalent of an Electric Circuit . . . . . . 306 
 
 10. This Equivalence is for the External Field only .... 307 
 
 1 1. Principle of Sinuous Currents ........ 307 
 
 12. Reaction of a Pole on an Element of Current ' \ * . . . 308 
 
 13. Action of a Pole upon a Closed Circuit . . . . . . 308 
 
 14. Action of a Pole on an Incomplete Circuit . , . . .310 
 
 15. Action of the Earth's Field on Currents Completely or Partly Mobile 311 
 
 1 6. Actions between Currents ; Ampere's Laws . . . . .311 
 
 17. Continuous Rotations of Currents .-.., . . . .313 
 
 1 8. Ampere's Laws of the Actions between Elements of Currents . .313 
 
 CHAPTER XIX. 
 
 LAWS OF THE MOVEMENTS OF CURRENTS AS DEDUCED FROM 
 THE CONSIDERATION OF MAGNETIC FIELDS AND POTENTIALS. 
 
 1. Magnetic Fields and Potentials . . * ../, . .-, .315 
 
 2. Movements are from Higher to Lower Potentials . . . * 315 
 
CONTENTS xix 
 
 SECTION PAGE 
 
 3. Potentials on Poles and on Circuits - . . . . : . . . 316 
 
 4. General Law of Movement of (Magnetic Shells or of) Electric 
 
 Circuits ;.. .. .. '. \. .. . -..'<' ' . . . 318 
 
 5. The Case of a Uniform Field . . .... . . . 318 
 
 6. The Case of a Field not Uniform ...;. . . . . . .318 
 
 7. The Case of Incomplete Circuits . . . . . . .319 
 
 8. Reconsideration of Ampere's Laws . . ".'-. ' .,'" . . 321 
 
 9. Cases of Continuous Rotation . . . . . . . 322 
 
 10. Potentials Due to Circuits . . . . .' ..... 322 
 
 11. Potentials on Circuits . . . ^ .... . . . 323 
 
 CHAPTER XX. 
 
 SOLENOIDS, ELECTRO-MAGNETS, DIAMAGNETISM, AND 
 ELECTRO-OPTICS. 
 
 1. Cylindrical Magnet built up of Circular Laminae ."... . 324 
 
 2. The Ideal Solenoid . . . . . -,. ' . ' . . . 324 
 
 3. The Practical Solenoid . . . . . . > . . . 324 
 
 4. Ampere's Theory of Magnetism . . . . . .' , - . . 326 
 
 5. Solenoid, and Hollow Cylindrical Magnet, Contrasted . . . 327 
 
 6. Matter Placed in a Uniform Magnetic Field of Force . * '. . 328 
 
 7. Movements of Small Bodies in a Non-Uniform Magnetic Field . 330 
 
 8. The ' Setting ' of a Long Body in a Uniform Magnetic Field . . 330 
 
 9. A Long Body in a Non-Uniform Field . . . ... 332 
 
 10. Solenoid With, and Without, an Iron Core . .-..-. 332 
 
 11. Electro-Magnets .......... 333 
 
 12. Paramagnetic and Diamagnetic Phenomena . . : . , . 334 
 
 13. Pseudo-Diamagnetic Phenomena . . . , . . 335 
 
 14. Relative Magnetism or Diamagnetism . . . . . 335 
 
 15. Is there Absolute Diamagnetism ? . . . ...-'-. . 336 
 
 1 6. Rotation of "the Plane of Polarisation in a Magnetic Field . 337 
 
 17. Other Electro-Optical Phenomena . . . . . . . 340 
 
 18. The Electro- Magnetic Theory of Light . . . . . 340 
 
 CHAPTER XXI. 
 
 ELECTRO-MAGNETIC INDUCTION. 
 
 1. General Account .of Induction Phenomena . . ; , 343 
 
 2. General Reason for ' Induced Currents ' . , ; . . , . 346 
 
 3. More Exact Reasoning, in a Simple Case . . . . . 347 
 
 4. General Expression for Induced E.M.F. -..,. . *'*:'' . 349 
 
 5. Induction where there is no Initial Current .- ." .- . - . 351 
 
 6. Direction of the Induced Currents ; Lenz's Law . ., . . . 352 
 7- Constant Induced Currents -. .. .;. -. ... ... . 355 
 
 8. Changes that Give Induced Currents . . . . . . . 357 
 
XX CONTENTS 
 
 SECTION PAGE 
 
 9. Coefficient of Mutual Induction, or of Mutual Potential . . . 359 
 
 10. Self-Induction. The Extra Current ' . . . . . 360 
 
 11. Induced Currents of Higher Orders . ... . . . 362 
 
 CHAPTER XXII. 
 
 ARAGO'S DISC, RUHMKORFF's COIL, AND OTHER CASES OF 
 INDUCTION. 
 
 1. Induced Currents (' Eddy Currents ') in Solid Metallic Masses moving 
 
 in a Magnetic Field' . . . . . . . . . 363 
 
 2. Arago's Disc and Magnetic Needle ....... 363 
 
 3. Continuous Current Collected from Barlow's Wheel . . . . 364 
 
 4. Induction in the Earth's Field . . . ,. . - r . . 366 
 
 5. Induction Coils ; General Plan ........ 368 
 
 6. Practical Difficulties to be Overcome ...... 369 
 
 7. Ruhmkorffs Coil , '. ..'.. . ' *. ; . . 370 
 
 8. The Part Played by the Condenser .. . . 4 . . .372 
 
 9. Condition of the Secondary Circuit when Closed . . . . 373 
 
 10. Secondary Circuit with Air-Break . . - .'. . . . 374 
 
 11. Electrostatic Condition of the Open Secondary Terminals. The 
 
 Charging of Leyd en Jars *'..* *, 374 
 
 12. Various Phenomena of the Secondary Discharge .... 376 
 
 13. High and Low Vacua . ' V . . . ' . . . 377 
 
 14. Discharge in Low Vacua .' . . * . . . . 377 
 
 15. Discharge in High Vacua . . .. . ..." - . -* . . 379 
 
 CHAPTER XXIII. 
 
 DYNAMO-ELECTRIC MACHINES. 
 
 1. General Nature of a * Dynamo ' . ... , .. . 383 
 
 2. General Account of Induced Currents . ... . . 383 
 
 3. Clark's Machine . . . . . ...... 385 
 
 4. The Simple Commutator . . . ."'.'. . . . 386 
 
 5. Siemens's Armature ......... 387 
 
 6. The Self-Exciting Principle .,'..>,. . . . . 387 
 
 7. ' Continuous Current ' Machines . . ,.,.,,-'. . . 387 
 
 8. The 'Gramme.' Construction of Armature ... . . . 388 
 
 9. The Gramme. The E.M.F.s Induced in the Ring . ' ; . . 390 
 
 10. The Gramme. The Collecting Brushes . . . ' . 392 
 
 11. Curve of Potential Round the Collecting Axis. . . , . 392 
 
 12. The * Lead ' that Occurs when a Current is Running . . 393 
 
 13. Armatures Wound 'for E.M.F.' and 'for Current' . . . . 394 
 
 14. The Siemens- Alteneck Armature ....... 395 
 
 15. The Brush Machine . - V . . 395 
 
CONTENTS xxi 
 
 SECTION PAGE 
 
 1 6. Magneto Machines . . . 395 
 
 17. Separately- Excited Machines ........ 396 
 
 18. Series-Excited Machines * . , . : '.-'' . . . 397 
 
 19. Shunt-Dynamos . . . -. --. . , . . . 398 
 
 20. Other Methods of Winding > '. . % . - , ( ' . '.. . 399 
 
 21. Alternate Current Machines . . - ."'.'. . .399 
 
 22. The Ferranti Alternate Current Machine . . '.'". 401 
 
 CHAPTER XXIV. 
 
 DYNAMOS AND MOTORS. 
 
 1. Magnetos, .or Separately-Excited Machines, as Motors . . . 402 
 
 2. Series-Dynamos as Motors . . . , . , . 403 
 
 3. General Remarks on Dynamos and Motors . * , 404 
 4.. Formulae for Activity, &c. Maximum Activity . ... . . 405 
 
 5. Efficiency ..... ..... . 408 
 
 6. Representative Curves . . . . , .... 408 
 
 7. S. P. Thompson's Diagrams ......... 409 
 
 8. Transmission of ' Power ' from a Distance . . . . . 409 
 
 9. Electric Railways and Tram- Cars ; Telpherage; &c. . . .411 
 
 10. Distribution of Potential in the Circuit of a Dynamo and Motor . 412 
 
 11. Work done per Second upon a Dynamo as Related to the Velocity 
 
 v of Rotation . . . . . '.'.,'. . 415 
 
 CHAPTER XXV. 
 
 VARIOUS APPLICATIONS OF ELECTRICITY ; TELEGRAPHS, TELE- 
 PHONES, MICROPHONES, ELECTRIC LIGHTING. 
 
 1. Introductory . . . . . -.-.-. . . 416 
 
 2. General Principle of Telegraphy . . .. . . . .416 
 
 3. Telegraphic Alphabets . . . . . . ' . ..419 
 
 4. The Needle System of Telegraphy . ' ..... . ... . 420 
 
 5. The Morse System .......... 421 
 
 6. Relays . . . . . . * '. . . . . 422 
 
 7. Transmission through Cables, under Water . , f , . 422 
 
 8. Earth Currents. Condenser System of Working .... 424 
 
 9. Insulation of Wires . . . ; . . 426 
 
 10. Duplex Telegraphy . . . . . . . . 426 
 
 11. Telephones. Introductory . . , , , . 428 
 
 12. The Bell Telephone . ,~. . . ' . ., . 429 
 
 13. Telephones with External Source of Current , . . , . . 431 
 
 14. Microphones . . . . . . , . . * . .431 
 
 15. Properties of Selenium. The Photophone . ..... 432 
 
 16. General Account of Electric Lighting >,.. 433 
 
XX11 
 
 CONTENTS 
 
 SECTION 
 
 17. The Incandescent ' Lamp ' 
 
 18. Arrangement of Incandescent Lamps in Parallel 
 
 19. The Economy of Incandescent Lighting . 
 
 20. The Arc Lamp . . . 
 
 21. The Reverse E.M.F. of the Arc 
 
 22. * Equivalent Resistance ' in an Arc Lamp 
 
 23. Series Arrangement of Arc Lamps 
 
 24. Further Remarks on the Use of Arc Lamps . 
 
 25. Current-meters or Ammeters . . ; 
 
 26. Voltmeters . !. V : . ' '. 
 
 27. ' Damping ' in Ammeters and Voltmeters . 
 
 28. Wattmeteis . ... 
 
 29. Integrating Instruments. Coulomb-meters . 
 
 30. Integrating Instruments. Energy-meters 
 
 31. Testing of Coulomb-meters and Energy-meters . 
 
 32. General Principle of the System . . 
 
 QUESTIONS AND EXAMPLES . ; ; ; . 
 ANSWERS TO QUESTIONS . . . , ; . 
 INDEX 
 
 434 
 436 
 
 . 437 
 438 
 440 
 441 
 442 
 443 
 443 
 446 
 446 
 447 
 448 
 449 
 45 
 450 
 
 xxiii 
 
 xl 
 
 , xlv 
 
 SOME ABBREVIATIONS USED. 
 
 Magnetic pole-strength /u- 
 
 Magnetic moment . . m and /a/ 
 
 Field-strength H and I 
 
 Work and Energy W 
 
 Electric Quantity Q 
 
 Electrostatic Capacity . . . . K 
 Specific-inductive-capacity cr 
 
 Electric and Magnetic density . . . p 
 
 Potential V 
 
 Difference of Potential . AV 
 
 Gold-leaf electroscope . . g.l.e. 
 
 Current-strength C 
 
 Resistance R and r 
 
 Equivalent-resistance R' 
 
 Battery-resistance ..... B 
 Galvanometer-resistance . . . . G 
 
 Shunt-resistance S 
 
 Electromotive Force . E.M.F., E, and e 
 
 /Joule's equivalent J 
 
 [This is the factcr which reduces calories 
 to ergs ; and = 4*175 x io 7 , or 42 x io s 
 V nearly.] 
 
 Note. In cases where the above symbols have other meanings, the context will obviate 
 ambiguity. Thus V may sometimes mean -velocity ', H may mean heat ; and p may mean 
 specific resistance. 
 
. <,,,,, . 
 gijVsw 
 
 i ", O-S'.'.',' 
 
 $jti'& '-' 
 
 ELECTRICITY. 
 
 CHAPTER I. 
 
 GENERAL PHENOMENA OF MAGNETISM. 
 
 i. Introductory. The subjects of magnetism and electricity 
 e in reality not two, but one ; all magnetic, electro-magnetic, 
 voltaic phenomena to use terms with which most of our 
 readers will have some acquaintance belong to one great branch 
 of science for which we have not yet one comprehensive name. 
 Perhaps it might be more logical were we to give at once a general 
 survey of all the above-named classes of phenomena before pro- 
 ceeding to a more detailed discussion of each. But, as in this 
 course no previous knowledge has been assumed, it has been 
 thought better to avoid any chance there might be of confusing 
 the mind of the student by the presentation of a multitude of 
 strange facts, and hence we shall first discuss the main phenomena 
 of magnetism. A study of these will serve as a training to the 
 beginner, and he will incidentally become acquainted with many 
 facts and conceptions that will prove to be of great value in the 
 study of the other branches of our science. 
 
 2. First Phenomena observed. Accustomed as most of 
 us are to the use of a 'magnet,' there is still something very 
 startling in the simple elementary experiment with a ' magnet ' or 
 a piece of lodestone. When a piece of lodestone (magnetic oxide 
 of iron, Fe 3 O 4 , magnetised by the influence of the earth) is held 
 above a piece of iron or steel of not too great weight, the piece 
 of iron or steel will move up to the lodestone, against the force 
 due to gravity. And, more remarkable still, the lodestone can 
 convert pieces of hard steel into permanent ' magnets,' it being 
 
 B 
 
2 u ELECTRICITY en. i. 
 
 merely necessary to make a convenient bar of steel and to stroke 
 it repeaterlly' fr^rr 1 end to end with the lodestone, taking care that 
 the strokes' are alike 'in all respects.' We shall henceforth employ 
 such steel magnets instead of lodestone. 
 
 3. Polarity. If we examine our magnetised bars of steel we 
 shall find that, though the lodestone passed along every portion of 
 them equally, the ends always possess far greater magnetic powers 
 than do the middle portions. Further than this, we find that in 
 each magnet these two ends are different. If we magnetise a bar 
 A B, and another A' B', in such a way that A and A', B and B' 
 are respectively corresponding ends (and we can do this by 
 stroking the one bar from A to B, the other from A' to B' with 
 the same portion of one piece of lodestone), then we find that A 
 repels A', B repels B', but that A attracts B' and A' attracts B. 
 We find also that if such bars be suspended they will take up a 
 definite position with respect to the earth, setting themselves ina 
 direction called magnetic north and south, the similarly treated 
 ends A and A', B and B', lying in similar directions. 
 
 We may say that the ends of the bar have acquired remarkable 
 properties, these properties being, in a sense, of opposite natures 
 at the two ends respectively. It is usual to call the ends where 
 the magnetic properties are displayed poles^ and the bar is said to 
 have acquired magnetic polarity. 
 
 We may here point out a possible source of error. We have seen 
 that like poles repel, and unlike poles attract, one another ; and 
 further that all needles or bars, that are magnetised and suspended, 
 turn so as to point in a northerly and southerly direction. That end 
 of the bar that points towards the geographical north pole of the earth 
 is usually called a north pole, or north-seeking pole ; the other end is 
 called a south pole, or south- seeking pole. But from what we have 
 seen it is evident that the north-seeking pole of the bar must be of 
 opposite polarity to the ' north pole ' of the earth towards which it 
 turns; and so with the south-seeking pole of the bar. 
 
 It is better, therefore, to speak of the ' north pole ' of the earth, 
 but the ' north-seeking ' pole of a magnetised bar. 
 
 4. Constitution of a long thin Magnet. Much light is 
 thrown upon the nature of this Polarity by the following simple 
 experiment : 
 
 A strongly magnetised knitting-needle is taken, and the 
 
CR. i. GENERAL PHENOMENA OF MAGNETISM $ 
 
 polarity of the ends, and neutrality of the middle, tested by use 
 of filings and compass-needle. 
 
 It is then broken in half. 
 
 We now find each half a complete magnetic needle ; two 
 opposite poles having apparently started into existence in the pre- 
 viously neutral middle. 
 
 Each half is then again broken, and so oral 
 
 The original needle, and the condition wherWHr broken, are 
 shown in the accompanying figure. 
 
 It would seem that all these intermediate poles were existing 
 in the original needle, but that they neutralised each other as far 
 as any external action went. 
 
 As far as experiment goes there is nothing against the sup- 
 position that if we had a magnetic bar very thin, in fact, a single 
 row of molecules of steel, we might continue this process of 
 breaking up until we should find each molecule a complete little 
 magnet with poles at its ends (if we may use the term ends with 
 respect to a molecule which may be spherical), and a neutral 
 region at the middle. 
 
 Experiments, (i. ) Magnetising a bar of steel with lodestone, we find that 
 iron filings will cluster chiefly at the ends. 
 
 (ii. ) Take a series of unmagnetised knitting-needles ; testing them by 
 seeing that either end acts the same on the ' north ' end of an ordinary compass- 
 needle. 
 
 Lay them down side by side, and magnetise them successively with a piece 
 of lodestone or with a steel magnet ; marking with gummed paper the ends 
 that must, by the process of magnetisation, be similar. 
 
 It will then be found that (a) similar poles repel, dissimilar attract, one 
 another ; (b} the suspended needles will all set themselves with the marked 
 ends turned in the same direction. 
 
 5. Molecular theory of Magnetism. The above experiment 
 suggested, very early, that probably magnetism is molecular. The 
 view adopted was that when a long thin needle was magnetised, 
 its molecular constitution was somewhat as represented in the 
 accompanying figure. N s is the magnetic needle, supposed to 
 
 B 2 
 
 ' CF CALIFORNIA 
 
4 ELECTRICITY CH. i. 
 
 consist of a single line of molecules. Each of these molecules is 
 a small magnet, and they are arranged so that north-seeking poles 
 are opposed to south-seeking poles all along : there being left at 
 the one end^attorth-seeking pole, and at the other end a south- 
 seeking pole, uncompensated. 
 
 FIG. i. 
 
 It is assumed, as justified by such experiments as that alluded to 
 in 4, that the south-seeking pole of one molecule can neutralise, 
 as far as external manifestation of magnetic properties goes, the 
 opposed north-seeking pole of another molecule, while the north- 
 seeking and south-seeking poles of the same molecule, ''thougrr.yery 
 close to one another, do not so neutralise one another. ^ 
 
 The state of neutrality was formerly considered to be a 
 'higgledy-piggledy' arrangement, or rather absence of arrange- 
 ment, producing on the whole external neutrality. 
 
 Thus the end of a magnetised bar would present to external 
 bodies a whole set of molecular north-seeking or south-seeking 
 poles, while the end of an unmagnetised bar would present a 
 mixed surface of north-seeking and south-seeking poles whose 
 total external action would be nil. 
 
 Professor D. E. Hughes has taken up the molecular theory, 
 and has reduced it to order, much extending it. In fact, he has 
 done so much that the chief questions left unsolved are the funda- 
 mental ones : * What is the polarity of a molecule ? and, Why do 
 similar poles repel and dissimilar attract one another?' The 
 question of diamagnetism also needs some further research. 
 
 The main drift of Professor Hughes's theory, each portion of 
 which he has supported by experiment, is somewhat as follows : 
 
 (i.) Each molecule (possibly each atom) of every substance 
 possesses to a fixed and unalterable degree a property called 
 polarity. The poles of the molecule are probably fixed in that 
 molecule, and can be made to lie in a changed direction only by 
 rotation of the molecule. 
 
 (ii.) The opposite poles of consecutive molecules do, when the 
 molecules are so rotated that these ' poles ' lie one over against the 
 other, neutralise each other with respect to external action. 
 
en. i. GENERAL PHENOMENA OF MAGNETISM 5 
 
 (iii.) In neutrality there is always some symmetrical arrange- 
 ment by which the molecules neutralise one another with respect to 
 any external action ; either by the molecules being paired off with 
 opposite poles together, or by a whole line of molecules neutralis- 
 ing each other's poles in one long chain, or by some equivalent 
 arrangement. 
 
 One of the great features of Hughes's theory W the symmetry 
 of arrangement that is considered to exist under all circumstances. 
 The accompanying figures indicate two possible cases of neutrality. 
 
 Since the molecule, whatever its real form, is practically for mag- 
 netic purposes a short line with poles at the ends, we have here re- 
 presented the molecules as lines ; indicating thereby their magnetic, 
 though not their actual shape. 
 
 ft IHHHHf \ IHHIKHMHi 
 
 7 ^ 
 
 nf Aft 
 
 1 ). 
 
 FIG. iii. 
 
 (iv.) When the bar is magnetised to the greatest possible 
 degree, the arrangement will be such that there is a chain of mole- 
 cules in which the north-seeking pole of the molecule at the one 
 end of a chain, and the south-seeking pole of the molecule at 
 the other end of the same chain, will be left unneutralised ; 
 while between these there is complete neutralisation. And, further, 
 these chains must be arranged side by side so that the free north- 
 seeking poles are all at one end of the bar and the free south- 
 seeking poles at the other. 
 
 In the above two cases this might be effected in fig. ii. by a 
 rotation of the molecules so as to form one line as in fig. i., while 
 in fig. iii. it would be only necessary to break the chain at any point. 
 
 Thus there is a limit to possible strength of magnetism. 
 The utmost that can be done is to arrange the molecules in the 
 
6 ELECTRICITY CH. i. 
 
 most advantageous position. The strength of magnetism will 
 then depend on the ultimate properties of the molecules. 
 
 (v.) Usually, or perhaps in all cases, the setting of the mole- 
 cules so as to give evident magnetism is not completed. The 
 molecules are partly rotated away from their position of what we 
 may call ' short-circuiting ' or neutralisation, and we get all inter- 
 mediate cases^between perfect neutrality and perfect magnetisa- 
 tion. A piece of iron when magnetised to its fullest extent is said 
 to be saturated. 
 
 (vi.) Arrangements of the molecules, in which they do not 
 pair off, or in any other way neutralise one another, and so give 
 zero external action, are unstable. In fact, if only there be given 
 free movement to the molecules, so that they can act on each 
 other without hindrance, they will always ' short-circuit ' with one 
 another and give us zero external magnetic action. 
 
 (vii.) The rearrangement of the molecules so as to give ex- 
 ternal magnetism is resisted both by their action on each other 
 and by the mechanical difficulties in the way of rotation of the 
 molecules. This mechanical rigidity, that opposes the ' magnetisa- 
 tion ' of a bar and also opposes the molecular tendency to a 
 return to the neutral, short-circuited, condition, is called by the 
 inexact name of coercive force. Any action (as heating, ham- 
 mering, and the like) that gives freer play to the molecules will 
 diminish the coercive force of the bar. In general, therefore, it is 
 useful to hammer a bar while it is being magnetised, but bad to 
 do so when it is no longer under the magnetising influence. 
 
 (viii.) The degree of molecular rigidity depends upon the 
 chemical nature of the metal and upon its temper. Thus, some 
 specimens of ' soft iron ' will hardly retain any trace of magnetism 
 when the magnetising influence has been removed, while some 
 specimens of 'hard steel' will remain unaltered for a long 
 time. 
 
 (ix.) Temporary and Residual Magnetism. In general, if we 
 magnetise a bar, it will readily lose part of its magnetism, but will 
 retain the remainder unaltered for a long time. It is supposed that 
 the molecules can, without much difficulty, rotate part of the way 
 back to their original positions of neutrality ; but that at a certain 
 point their freedom of movement is limited by the ' molecular 
 rigidity ' (or coercive force) spoken of above. 
 
CH. i. GENERAL PHENOMENA OF MAGNETISM 7 
 
 It is found that when iron or steel is raised to a white heat, it 
 is not attracted by a magnet ; in fact, it ceases to be magnetic. 
 This phenomenon has not been satisfactorily explained by any 
 theory ; and, until further investigation has been made, we cannot 
 say whether it makes for or against the above molecular theory. 
 
 Experiments to illustrate molectilar theory. (i.) We can magnetise a tube 
 of steel filings, and then render it neutral by shaking it up. As the filings 
 are not free to traverse the tube, but merely turn about where they are, this 
 illustrates the case of neutrality by the pairing off of the molecules. 
 
 (ii. ) Magnetise a strip of watch-spring, and test it. 
 
 If this watch-spring be now made into a complete circuit, we get almost 
 complete external neutrality ; while the evident magnetism is again restored 
 when we break the circuit. 
 
 (iii. ) A poker may be magnetised by being held along the lines of the earth's 
 magnetism (see Chapter III. 8) and in that position hammered. 
 
 (iv.) The magnetism of a steel knitting-needle may be reversed if it is held 
 with south-seeking end downwards along the lines of earth's magnetism, and 
 in that position kept at a bright red heat for some time ; being then allowed 
 to cool in the same position. 
 
 (v. ) It was shown by Joule that a bar when magnetised increases in length. 
 This might well be the case if each pair of molecules (see fig. ii. above) does 
 in fact rotate from a parallel, to an end-on, position. 
 
 (vi. ) A tube is filled with water in which is suspended magnetic oxide of 
 iron in a finely divided condition ; the whole being nearly opaque to light. 
 
 If the tube be magnetised in the direction of its length, it is found that it 
 is now less opaque to light in this direction. This would tend to show that 
 the particles of the magnetic oxide have arranged themselves end-on in the 
 direction of magnetisation. 
 
 6. Induction, General Phenomena. The matter of induc- 
 tion will be clearer to the learner when he has learnt something 
 of ' Fields of Force,' in Chapter II. 
 
 At present we shall only describe the actual facts observed. 
 
 When a piece of iron is placed near the pole of a magnet, the 
 end of the iron that is nearest the pole acquires a polarity oppo- 
 site to that of the said pole ; while the end of the iron that is 
 furthest away acquires a polarity similar to that of the magnet- 
 pole. 
 
 This action of a magnet, in making iron near it also magnetic, 
 is called 'Induction.' 
 
 It is more powerful as the ' inducing ' magnet is more power- 
 ful, and as the iron is nearer to the magnet. 
 
8 
 
 ELECTRICITY 
 
 CH. I. 
 
 FIG. i. 
 
 It is stronger and less permanent as the ' coercive force ' of the 
 iron is small (or the molecular freedom great), weaker and more 
 permanent as the ' coercive force 7 is greater. 
 
 Experiments in Induction. (i. ) NS is a powerful magnet ; and ns, 
 n's , n"s" y &c., are pieces of very soft iron arranged near NS as shown. The 
 
 letters of the accompanying figure 
 indicate the observed polarities of 
 the pieces of soft iron. One may 
 test these polarities either by means 
 of a small compass-needle applied 
 to the ends of the iron pieces, or 
 in some such way as that indicated 
 in the next experiment. 
 
 (ii.) The writer believes that 
 the following experiment is due 
 to Professor Guthrie. 
 
 NS is a powerful permanent 
 magnet and v<r is a bar of soft iron 
 hung above NS, parallel and close 
 to it. Then v<r will be acted upon 
 inductively, and will temporarily 
 become a magnet, its poles being 
 at v and <r. 
 
 If a north seeking pole N' be 
 approached to the end v which lies 
 above the south-seeking pole S, it 
 is found that N' repels v. Hence 
 v is north-seeking, or S has induced 
 at v a polarity opposed to its own. 
 This is, in fact, indicated by the simple fact that vff will, if displaced, return 
 to the position represented in the figure. 
 
 We may say generally that induction takes place through any 
 solid, liquid, or gas ; it making no perceptible difference what 
 substance is interposed between the magnet and the iron, unless 
 we are making very delicate observations indeed. 
 
 There is, however, one very important exception. A sheet of 
 perfectly soft iron, or iron with absolutely zero 'coercive force,' 
 would act as a perfect screen ; and no induction would take place 
 on a piece of iron or steel suspended in a closed vessel made of 
 such soft iron. To a greater or less degree, according to its 'soft- 
 ness,' all iron and steel will act as a screen. 
 
 We have spoken hitherto of iron and steel only. There are, 
 
GENERAL PHENOMENA OF MAGNETISM 
 
 9 
 
 however, other bodies, notably cobalt, nickel, and manganese, which 
 possess to a certain degree magnetic properties. More will be 
 said of these in Chapter XX. ; and it will there be pointed out how 
 feebly magnetic are these bodies when compared with iron or 
 steel. 
 
 7. Use of Keepers. When a bar is magnetised, there are 
 at the one end a set of molecules with their north-seeking poles 
 'free,' and at the other end a corresponding set with their south- 
 seeking poles ' free.' This condition of things is, as we have said 
 before, unstable ; and there is in a magnetised bar a tendency for 
 the molecules to pair off again in such a way as to give neutrality 
 of the nature indicated in 5, fig. ii. This tendency is resisted by 
 molecular rigidity, or coercive force ; but a loss of magnetisation 
 is inevitable if the bar be subjected to molecular disturbance of 
 any kind. 
 
 But if we can, by means of soft iron pieces applied to the 
 magnet, make the molecular circuit complete, we shall obtain a 
 
 FIG. i. 
 
 stable molecular arrangement similar to that 01 5, fig. iii. The 
 whole system will give but little signs of external magnetism until 
 the soft iron pieces are removed. The accompanying figures show 
 how a horseshoe magnet, or a pair of bar magnets, may be pro- 
 vided with soft iron keepers. 
 
 The letters indicate the polarities temporarily induced in these 
 
10 ELECTRICITY CH. i, 
 
 keepers. The molecular arrangements, similar to that of 5, 
 fig. iii., can readily be imagined. 
 
 8. Methods of Magnetisation. In making a magnet we 
 must have several points in view. 
 
 (i.) The material must be one having great molecular rigidity 
 or coercive force; so that the rearrangement of molecules, to 
 which is due the external or evident magnetism, may be retained. 
 Hence we use hard steel. 
 
 (ii.) During the process of magnetisation the molecules should 
 .be given as free play as possible. Hence, hammering, twisting, 
 and other mechanical aids to molecular freedom may be employed 
 if convenient. It is found that repetitions of the magnetising pro- 
 cess, even when each action first undoes the work of the preced- 
 ing action before redoing it, give better final results than a single 
 application of the process. 
 
 (iii.) We must apply as powerful a magnetising influence as we 
 can. The more powerful the 'field' (see Chapter II.) in which 
 we place our bar, the more will the molecules be rotated from 
 their 'short-circuited' position into the position giving most power- 
 ful external magnetism. 
 
 I. Single Touch. Here a magnet is simply drawn from end 
 to end of the bar in the position shown in the figure. This ' strok- 
 ing ' process is repeated many times. 
 
 Here we can imagine the molecules rotating, as the inducing 
 magnet pole moves down the bar. They will always, e.g., turn 
 their south poles towards a north inducing 
 magnet pole. Hence we should predict that 
 finally the end last touched by a north induc- 
 ing pole will be left of south polarity, while 
 that first touched will be left of north polarity. 
 Each time that the 'stroke' is repeated it is 
 W///////////////////M clear that the molecular arrangement due to 
 1 the preceding ' stroke ' will -be undone, or 
 at least seriously disturbed. It seems, there- 
 fore, at first sight somewhat strange that there is so much 
 gained by repetition. Probably the explanation is that this 
 repetition gives greater freedom of movement to the molecules ; 
 and hence the final stroke has greater effect than had the first. 
 
 II. Double Touch. Here two magnets are used ; they are 
 
CH O i. GENERAL PHENOMENA OF MAGNETISM 1 1 
 
 placed, N pole to S pole, in an inclined position, as shown. 
 We begin the stroke in the middle of the bar, and we move 
 the magnets up to one end, back to the other end, and then 
 up to the middle again ; this process being repeated many times. 
 The ends of the bar may also be placed on the opposite poles 
 of two magnets. This increases the effect. 
 
 FIG. ii. 
 
 III. Separate Touch. Or again the two inducing magnets may 
 be drawn away from each other to the two ends respectively, and 
 brought back to the middle by being lifted in a wide curve through 
 the air ; and this process repeated. 
 
 IV. By an Electric Current. But by far the most powerful 
 method is magnetisation by means of an electric current. Of this 
 we shall say much hereafter in Chapter XX. 
 
 
12 ELECTRICITY en. n. 
 
 CHAPTER II. 
 
 MECHANICAL AND MAGNETIC UNITS. 
 
 i. Introductory. For the study of magnetism and elec- 
 tricity it is necessary that the student should have a knowledge of 
 the principles of elementary mechanics, and a clear understanding 
 of what is meant by a system of units. He will then be in a 
 position to understand in their exact sense the terms employed 
 in the discussion of magnetic and electrical measurements. For 
 reasons of space it is impossible to give here any satisfactory 
 account of the mechanical principles involved in our present 
 subject. We shall merely indicate briefly the nature and extent 
 of the knowledge required. 
 
 2. Fundamental and Derived Units. The three funda- 
 mental units to which all other mechanical quantities can be 
 ultimately referred, are the units of length, time, and mass. We 
 shall in what follows employ the centimetre as the unit of length, 
 the second as the unit of time, and the gramme as the unit of 
 mass. The system of units built up upon this foundation is called 
 the ' centimetre-gramme-second system,' or more usually the 
 'C.G.S. system.' 
 
 Examples of derived units will occur in 3 and 4 and later 
 on. Very simple cases of derived units are the square centimetre 
 as the unit of area, and the cubic centimetre as the unit of 
 volume. 
 
 The exact way in which any derived unit involves the funda- 
 mental units constitutes what is called the dimensions of the 
 quantity measured by that derived unit. For the purpose of ex- 
 hibiting the dimensions of any derived unit it is convenient to 
 represent the three fundamentals by the symbols [L], [M], and 
 [T] respectively. Thus the C.G.S. unit of volume has dimensions 
 represented by [L] 3 . 
 
CH. ii. MECHANICAL AND MAGNETIC UNITS 1 3 
 
 3. Velocity and Acceleration. In the same system the unit 
 of velocity will be the velocity of one centimetre per second. If we 
 represent this unit by the symbol [V], we may express the dimen- 
 sions of velocity by the relation [V] = L-J. 
 
 The rate at which velocity changes is called acceleration and in 
 our system unit acceleration will be the adding of unit velocity 
 during unit time. Hence the C.G.S. unit of acceleration is one 
 centimetre per second, per second. 
 
 We may express the ' dimensions ' of acceleration by the 
 relation 
 
 4. Force. The C.G.S. unit of force is called the dyne. It is 
 that which acting on unit mass for unit time gives to it unit 
 velocity. 
 
 Thus we may express symbolically the 'dimensions' offeree 
 by the relation 
 
 The student will have learned, in his study of mechanics, to dis- 
 tinguish carefully between mass and weight. In virtue of gravitation, 
 a mass of one gramme is urged downwards with a certain force ; this 
 force being called the weight of one gramme. Near the earth's surface 
 this force is such that acting for one second it will give to the gramme 
 mass a velocity of about 981 centimetres per second. Hence, near 
 the earth's surface, the weight of our unit mass is a force measured by 
 about 981 dynes. A gramme then is a mass; and we take it for our 
 unit of mass. But the weight of a gramme is a force, and is very far 
 indeed from being unit force in our system. 
 
 Note. In the English system of units we take the foot, second, and fauna 
 as our three fundamental units. 
 
 The unit of force in this system is called the poundal, and is that which 
 acting for one second on one pound mass will give it a velocity of one foot per 
 second. 
 
 5. Parallelogram of Forces, &c. We shall make frequent use 
 of the principle commonly called the * Parallelogram of Forces.' 
 
14 ELECTRICITY 
 
 CH. It. 
 
 It will be assumed that the student is thoroughly conversant 
 with the formulae for the composition of two component forces 
 into one resultant, and for the converse resolution of a single 
 force into two components, in the case in which the directions of 
 the two components are at right angles to one another. 
 
 The case of ' oblique resolution ' will not occur. 
 
 6. Moments and Couples, Let us suppose that we have a 
 body (such, e.g., as a magnetic needle) capable of rotation about 
 an axis. If one or more forces act on the body we shall in 
 general have, as a result, rotation of the body about the axis. 
 
 In our present subject the only cases usually occurring are 
 very simple ; the forces acting in one plane which is perpendicular 
 to the axis. We shall have to consider two cases. 
 
 (i.) A single force acting. Let a single force of F dynes act 
 upon the body, and let the perpendicular between the axis and 
 the direction of the force be p centimetres. Then the ' turning 
 power ' or moment of the force about the axis is, as we know, 
 measured by F x /. This is the moment urging the body to 
 rotate about the axis ; and there is also a force of F dynes 
 tending to move the axis bodily in the direction of the force. 
 
 If, for example, the body be a horizontal bar, suspended from 
 its middle point by a thread, and we push one end of the bar, we 
 shall find that the bar tends to rotate and also to move bodily in 
 the direction of our push. 
 
 (ii.) A couple acting. Now consider the case where two equal, 
 parallel, and opposite forces act upon the body, each force being 
 F dynes, and the distance between them being / centimetres. 
 The moment of this couple is, as we know, measured by F x p. 
 In this case we have simply a tendency to rotation, and no 
 pressure on the axis. 
 
 When the product F x p is numerically equal to unity (F 
 being measured in dynes and p in centimetres] then we have what 
 we call unit moment in the C.G.S. system. 
 
 7. Unit Magnetic Pole. Returning now to magnets and 
 magnetic poles, we observe that similar poles of various pairs of 
 magnets repel one another with forces of various magnitudes. 
 Thus we naturally say that poles may be of different strengths. 
 
 We should then endeavour to express the * strengths ' of mag- 
 
CH. ii. MECHANICAL AND MAGNETIC UNITS 1 5 
 
 netic poles in terms of some convenient unit-pole ; and further 
 we should choose our unit strength of pole such as to give the 
 simplest relation to force of repulsion expressed in dynes, and 
 distance apart expressed in centimetres. 
 
 We have seen that a single pole never occurs alone. But in 
 an indefinitely long straight thin magnet, perfectly magnetised, 
 we have the poles exactly at the ends and indefinitely far apart ; 
 or, in other words, we can easily conceive of, and approximately 
 obtain, a pole isolated from another pole. We can then in our 
 definitions speak of 'a pole.' We choose to call the north- 
 seeking pole +, and the south-seeking pole . 
 
 It is natural to define unit-pole as that which at a distance of 
 i cm. from a similar pole repels it with a force of \ dyne. 
 
 So a pole of two units would be defined as that which at a dis- 
 tance of i cm. from unit-pole repelled it with 2 dynes ; or at a dis- 
 tance of i cm. from a similar pole repelled it with 2x2 = 4 dynes. 
 
 We shall designate the numerical value of the strength of a 
 pole by fj. 
 
 8. Magnetic Fields, and Unit Field. Any region where a 
 pole (e.g. a + unit-pole, which we may conveniently employ as a 
 test-pole) would be urged by magnetic force is called a magnetic 
 field. 
 
 In so far as such force is due to any particular magnetic body, 
 we speak of the field due to that body, 
 
 The direction in which our + unit-pole at any point is urged 
 is called the line of force at that point. 
 
 If a + unit-pole is urged with i dyne, the field at that point 
 is defined to have unit strength. 
 
 If with 5 dynes, then the field has a strength 5. 
 
 In general we shall designate the numerical value of the 
 strength of a field by the letters H or I. 
 
 Thus a pole of strength in a field of strength H is urged by 
 a force of n x H dynes If a field is uniform it means that the 
 strength is the same at all places in it, the lines of force being 
 moreover, as a necessary consequence, parallel to one another ; 
 as will be seen in Chapter X. 13. 
 
 9. Magnetic Moment of a Needle, Let us consider a 
 magnetic needle in a uniform field. Since the poles, n and s, are 
 equal and opposite, and since the lines of force are in a uniform 
 
 UNlVERCiT ' OP 
 
1 6 ELECTRICITY CH. n. 
 
 field parallel, it follows that the forces acting on the poles will 
 be equal and opposite ; or that a magnetic needle in a uniform field 
 is acted on by a pure couple. 
 
 We have seen that each force will be p H dynes. Hence 
 if the needle lie at right angles to the lines of force, and if its 
 
 length be /cm., the couple 
 acting on the needle will 
 be measured by H/x/ 
 
 units of couple. 
 
 The part ^ / depends 
 on the needle alone ; and 
 into whatever field we 
 
 4L introduce the needle, so 
 
 long as it remains mole- 
 cularly unaltered, this yu / remains constant. 
 
 We give to this quantity /z / the name of the magnetic moment 
 of the needle. We call it m. When /z/=i, both /u and / being 
 measured in C.G.S. units as given in 2 and 7, then the needle 
 is said to be of unit moment. 
 
 A needle of unit moment in unit field, placed perpendicu- 
 larly to the lines of force, will be acted on with unit couple. 
 
 It will be useful for the student to prove for himself that when 
 a needle of pole-strength //, and length /, is placed in a field H so 
 as to make an angle 6 with the lines of force, then the couple 
 acting on the needle is 
 
 H p. I sin 0, or H m sin 6. 
 
 If we make up a cube of needles by first laying similar needles 
 side by side so as to form a layer, and then piling layer on layer, we 
 can see that the following statement will be true ; the magnetic moment 
 of a uniformly magnetised bar is proportional to the volume of the bar. 
 
 The student should puzzle this out by the ' building up ' method 
 just suggested. 
 
 10. Magnetic Moment of Practical, not Ideal, Magnets, 
 We have considered our magnet as consisting of two poles occu- 
 pying points, separated by a distance called 'the length of the 
 needle.' But the actual magnet may be considered to consist of 
 an indefinite number of such pairs of poles, the poles of each 
 pair being of equal strength and of opposite sign ; thus we may 
 
CH. ii. MECHANICAL AND MAGNETIC UNITS I/ 
 
 have at the two ends the poles ^ and /^ corresponding to one 
 another and separated by a length / 1 ; then ^ 2 an d /* 2 w ith distance 
 / 2 between, and so on. The poles p l and f* t &c. near the ends 
 will be the strongest ; in fact, the strength of the pairs of poles 
 ought, in a good magnet, to fall off very rapidly as we leave the 
 ends and approach the centre. 
 
 So we have the moments /u^, /i 2 / 2 , & c - And the magnetic 
 moment of the whole needle will be the sum of these, or will be 
 
 + m + m + &c. 
 
 where the first terms, ; 1 + m 2 + &c. are the largest. 
 
 . The complexity of this result is due to the difference between 
 our ideal magnetisation and the imperfect manner in which we 
 must needs perform it. The magnetisation is not effected uni- 
 formly all over and all through the bar and hence there is not 
 the complete internal neutralisation, with evident magnetism only 
 at the ends, which would result from uniform magnetisation. 
 
 To sum up all these terms, m { + ;;z 2 + &c, about each of 
 which too little would be accurately known, were impracticable. 
 
 We have, however, other methods of finding ;;/. We may (e.g.) 
 suspend the needle in a uniform field of known strength H, and 
 measure by mechanical means the moment (in terms of unit 
 couple or unit moment) required to keep the needle perpen- 
 dicular to the lines of force. Let this moment be found to be S. 
 
 Then we have 
 
 S = H m (see 9). 
 
 whence m = . 
 H 
 
 In fact, we measure the magnetic moment m as a whole by 
 observing the action of a known field upon it. 
 
 ii. Magnetic Curves. Consider a bar magnet NS, and a 
 point P in its field, and suppose a + unit pole to be placed there. 
 
 This will be urged from N in the line N P ; urged to S (by a 
 weaker force if it be nearer to N than to S) in the line S P. 
 
 If we draw the parallelogram of forces we shall get a resultant 
 force P R lying (in the case given in figure) nearer N P than S P 
 in direction. 
 
 As the pole P moves, the component and resultant forces 
 change continuously. And if P is always allowed to follow the 
 
 c 
 
18 
 
 ELECTRICITY 
 
 CH. II. 
 
 direction of the force urging it, then it will trace out a curve of 
 
 force, called, in the subject of magnetism, a magnetic curve. 
 
 The curve is such that at any point in it the tangent gives the 
 
 direction of the force at this point in the field. Now if we put 
 
 into the field a needle so 
 small that we can consider its 
 two ends to be practically in 
 the same part of the field, 
 then the needle will be acted 
 upon by a couple, and it will 
 turn until it lies in a straight 
 line with the lines of force. 
 In no other position but this 
 could there be equilibrium for 
 the needle. 
 Hence, a ' small needle ' will always point out the direction of 
 
 the lines of force at the place it occupies. 
 
 FIG. i. 
 
 
 #$ZR 
 
 "fe^x 
 v \>^l^' 
 
 ^^r^-'-- 
 
 ~~ ^fyf' ' '/j'f ' ' I ' v'V^ 7 '"' 
 
 FIG. ii. 
 
 Now iron filings do, by induction, become small magnetic 
 needles when they are placed in a magnetic field. Hence, if we 
 
-CH. II. 
 
 MECHANICAL AND MAGNETIC UNITS 
 
 place a magnet under a sheet of glass and scatter over the glass 
 iron filings, tapping the glass to allow of arrangement, we shall 
 find traced out for us many of these lines of force that lie in the 
 plane of the glass. 
 
 The accompanying figures show us the general aspect of the 
 curves thus obtained. 
 
 In fig. ii. we have respectively the cases of a bar-magnet ; a 
 horseshoe-magnet ; two bar-magnets with unlike poles opposed ; 
 and the same with similar poles opposed. 
 
 12. Magnetic Induction takes place along the Lines of 
 Force. It is along these lines of force that we must consider our 
 molecules, regarded as small magnetic needles, urged to direct 
 themselves. 
 
 Hence, if these molecules were perfectly free to move, any 
 mass of iron would by the rearrangement of its molecules become 
 magnetised along the lines of force of the inducing field. 
 
 But as the molecules are subject internally to mechanical con- 
 straint as well as to each other's action, the resulting magnetisation 
 is determined by the combined influences, internal and external. 
 
 If the bar or needle lies so that its greatest length is along the 
 lines of force, and if the form of the bar is symmetrical about this 
 
 axis, then the magnetisation should be whether weak or strong 
 along the lines of force. 
 
 But if the bar be unsymmetrically situated with respect to the 
 lines of force, the resulting magnetisation may be quite unsym- 
 metrical and be oblique to the lines of force. 
 
 c 2 
 
20 
 
 ELECTRICITY 
 
 CH. III. 
 
 CHAPTER III. 
 
 MAGNETIC MEASUREMENTS. THE EARTH'S MAGNETISM. 
 
 i. Coulomb's Torsion Balance. In considering the subject 
 of magnetic measurements we shall first describe the torsion 
 balance. This instrument deserves notice as the earliest by which 
 exact magnetic and electrostatic measurements were obtained, and 
 
 some study of it will be in- 
 structive. In practice it has, 
 however, been now super- 
 seded by other instruments. 
 The figure represents one 
 form of the instrument. A 
 rectangular or cylindrical 
 glass case is provided, either 
 with a graduated scale round 
 the sides, as here shown, or, 
 what is better, with a plane 
 mirror at the bottom, on 
 which is marked a circle 
 graduated in degrees. 
 
 In what follows we shall 
 
 = assume that the latter method 
 
 of graduation has been 
 adopted, and that the centre 
 
 of this graduated circle is called O. Above the mirror is sus- 
 pended (when the instrument is used in magnetism) a magnetic 
 needle a b, in such a way that its axis of suspension is immediately 
 above the centre O of the graduated circle. 
 
 By looking down from above we can tell over what degree the 
 needle is lying. By moving the eye until the needle and its image 
 coincide, we avoid errors due to the fact that the apparent position 
 of the needle over the scale varies according to our point of view. 
 
CH. in. MAGNETIC MEASUREMENTS 21 
 
 The needle is suspended by a fine wire or thread of glass, so 
 hung that its prolongation would pass through O, the centre of the 
 graduated circle. 
 
 This thread is fixed at its upper end to a brass piece e d (called 
 a torsion head} that caps the glass tube shown. This piece is so 
 constructed that the thread can be twisted without being displaced 
 laterally, and the angle of twist can be measured. 
 
 A sight of the instrument will make clear how this is contrived. 
 
 Now to twist a wire or thread of glass without displacing it 
 laterally requires a simple couple. 
 
 It can be proved experimentally that as long as the thread or 
 wire is not permanently altered by a twist, the couple required 
 to twist it is directly proportional to the angle of twist. 
 
 Or if we twist the wire through 10, 20, 30, 170, &c., we are 
 exerting couples proportional to 10, 20, 30, 170, &c. We cannot 
 measure the couples in absolute C.G.S. units unless we know more 
 about the length, radius, and material of the thread. But we can, 
 without this knowledge, compare couples. If the top of the wire 
 be twisted through ft one way, and the bottom through a the 
 other way, then the total angle of twist will be + a. 
 
 2. Use of Torsion Balance * at constant angle.' We can 
 use the torsion balance to compare the strength of magnetic 
 fields and of magnetic poles ; or 
 even to measure them in C.G.S. units 
 if we are acquainted with the 'con- 
 stants ' of the instrument we are 
 using. 
 
 In the accompanying diagram 
 we are supposed to be looking down 
 upon the whole instrument. The 
 small circle represents the graduated 
 circle of the torsion head ; and, in 
 the case given, the torsion wire has 
 been twisted either through an angle a, or through (n x 360 
 4- a), where n is some whole number. 
 
 The large circle is the graduated circle on the mirror ; and, 
 in the case given, the needle has been deflected through ah angle 
 B from zero. 
 
 The opposite directions of twist of the needle and of the 
 
22 ELECTRICITY CH. m. 
 
 torsion head respectively, that must exist where we are dealing 
 with repulsions, are indicated by the opposite directions of gradua- 
 tion. In this figure the needle n s answers to the needle a b of 
 i. Supposing now that we wish to compare the strength of two 
 magnetic poles. 
 
 We start with needle and torsion index both at o. There is 
 a hole in the glass top to the instrument in such a position that 
 we can lower the one or the other pole of the magnets we are 
 comparing into the place occupied by the pole of the needle n s 
 when this lies over o. The needle is swung to one side, and the 
 first pole, which we will call /u t , lowered. As we always take care 
 to lower a pole similar to that pole of the needle whose place at 
 o it takes, we have the needle deflected until the moment of 
 the force of repulsion about the axis of suspension of the needle 
 just balances the moment of the couple due to the twisting of the 
 wire. 
 
 As a rule we shall find it advisable to turn the torsion head, 
 and so to twist the wire, until k the needle is forced back to some 
 angle 6 of deflexion from zero, this angle being much less than 
 the original angle of deflexion. One reason for this is that if we 
 cause ft to be small, we may neglect the restoring couple due to the 
 earth's field. 
 
 Let us suppose that the torsion head has been turned through 
 (n { x 360 + a^ . Then the total twist on the wire will be 
 (n l x 360 + aj + 0) ; and the couple that it exerts will be mea- 
 sured by k (, x 360 + G! + 0), where k is some constant that 
 depends upon the nature and dimensions of the wire. 
 
 Now this couple is balanced by the moment about O due to 
 the action of the pole MI upon the needle ns ; and this moment 
 will be measured by the product \i\h. In this product ^ represents 
 the pole strength of the first magnet ; while h is some quantity 
 depending upon the magnetic moment of the needle n s and upon 
 the angle 6 of deflexion, and so will be constant while is con- 
 stant and while n s remains unaltered. 
 
 Since the two are in equilibrium, we have 
 
 Hi h = k (! x 360 + oj + 6) 
 or >!!=(! x 360 + a, +6). 
 
CH. in. MAGNETIC MEASUREMENTS 23 
 
 Repeating the process with the second magnet pole p. 2 , arid 
 turning the torsion head through such a number of degrees that 
 remains constant, we have 
 
 A<a=;j(2 x 360+ a 2 +0). 
 Finally, by division we obtain the result that 
 
 \ _i x 360 + a i + 0. 
 p 2 n 2 x 360 + a 2 + tf 
 
 Notes, (i. ) The reader should notice that we here compare primarily the 
 two moments acting upon the needle n s in the two cases respectively. 
 
 But if the magnetic moment of ns, and also the angle Q, remain unaltered, 
 it follows that we are really comparing the two field-strengths due to the poles 
 /i, and fj. 2 respectively. 
 
 And finally, since these field -strengths must be proportional to the pole- 
 strengths yii, and fjL 2 respectively when the conditions of distance and situation 
 remain constant, it follows that we are comparing /*, and ^ 2 , as was desired. 
 
 (ii. ) We assume that the magnetic moment of the needle n s is constant. 
 As pointed out at the end of 3, this is not true absolutely ; and under certain 
 conditions the error, due to the alteration in this magnetic moment, may be 
 great. 
 
 This method may also be employed to observe the distribution 
 of evident magnetism along a bar-magnet ; since we may, without 
 otherwise altering the process, lower the bar-magnet so that the 
 different portions of it may be successively opposite to the pole n 
 of the needle. 
 
 It will be found that the evident magnetism is distributed 
 along a good bar-magnet somewhat as here represented by the 
 
 FIG. ii. 
 
 diagram. In this the height of any ordinate such as a a' repre 
 sents the strength of polarity at the point a in the magnet. 
 
24 ELECTRICITY CH. in. 
 
 If we find the centres of gravity of the two areas included between 
 the axis of the bar-magnet, the ordinates at its two ends, and the 
 curve; and if through these centres of gravity we draw lines perpen- 
 dicular to the axis of the magnet, meeting it at P and P', these points 
 P and P' may be called \\\e principal poles of the bar. If we consider 
 any external point that is so remote from the bar that straight lines 
 drawn from it to the extremities of the latter are practically parallel, 
 then the action at this point will be the same as if the polarities of the 
 bar were concentrated at the two points P and P' respectively. But 
 for points that are relatively near to the magnet, the action is not so 
 simple. 
 
 In using the torsion balance it must not be forgotten that the 
 needle n s is under the influence of the earth's field. This obliges 
 us to place the instrument in such a position that the needle, 
 when under the earth's action only, stands at the zero mark, or o. 
 
 When the needle is displaced through an angle the earth's 
 field gives a restoring couple that is proportional to sin 6 ; and 
 this is added to the torsion couple. 
 
 If be made small, we may often neglect this couple in com- 
 parison with the large couple due to torsion ; since the smaller 6 
 is, the smaller is the earth's couple, and the larger is the torsion 
 couple. Or we may find experimentally what angle of torsion is 
 required to displace the needle through 6 under the earth's action 
 only. If the angle of torsion required is /3, then we must add /3 
 to the angle through which the torsion head was turned in the 
 experiment. 
 
 3. Method of Oscillations. The figure represents a uniform 
 magnetic field of strength H, and a needle n s of magnetic 
 
 moment m placed in this 
 field. 
 
 If disturbed from this 
 
 -Q ~- position of rest, the needle 
 
 * will oscillate more or less 
 
 quickly ; the number of 
 
 oscillations per second 
 
 depending upon the field- 
 strength H, the magnetic moment m, and mass and shape of the 
 needle. 
 
 Now it can be shown that in all such cases of oscillation 
 (whether of magnetic needles or of pendula or of elastic rods) 
 
CH. in. MAGNETIC MEASUREMENTS 25 
 
 the time of a single oscillation is independent of the angle of oscil- 
 lation, provided that this is very small. If this angle do not exceed 
 a few degrees we may consider that this law is true. Hence we 
 can speak of ' the rate of oscillation,' without specifying the extent 
 of oscillation in degrees. In the case we are considering it can 
 be shown by mechanics that for the same needle 
 
 The product H m is proportional to the square of the number of 
 oscillations per second. This gives us a means of comparing, or of 
 measuring, two field-strengths ; for we may oscillate in the two 
 respectively a needle of constant magnetic moment m, counting 
 the number of oscillations per second. We may also compare two 
 pole-strengths ; since, at constant distance, the field- strengths 
 will be proportional to these. 
 
 In this method we must reckon in the earth's action ; since 
 we cannot here make it relatively so small as to be negligible. 
 
 We so arrange matters that the lines of force of the field that 
 we are considering act in the same direction as those due to the 
 earth ; i.e. so that the two fields, acting upon the needle, are 
 simply added. 
 
 Let the needle under the earths action only, or in the field 7z, 
 oscillate O times per second; under the earths action + the first 
 jield, or in the field h + H,, let the number be O, ; and under 
 the earths action + the second field, or in the field h + H 2 , let the 
 number be O 2 . 
 
 Then by the mechanical law quoted above we have 
 (mh = k.0 * ...... (i.) 
 
 m (^ + H 1 ) = .O, 2 . (ii.) 
 jn (h + H 2 ) = /.O 2 2 . . . (iii.) 
 
 where k is some constant involving the mass and dimensions of 
 the needle. 
 
 (Subtracting (i.) and (ii.) we have ;;/ H, = k (O, 2 O 2 ) 
 (and subtracting (i.) and (iii.) we have m H 2 = k (O 2 2 O 2 ). 
 
 From this we obtain by division that 
 
 H 2 2 2 - O/ 
 
 This gives us also the ratio of the two pole-strengths, if the 
 distance be constant. 
 
26 ELECTRICITY CH. in. 
 
 Experiment, Such experiments are best made with a short massive needle ; 
 the oscillations being counted for a sufficient length of time to give accurately 
 the number per second. 
 
 Notes. (i.) The needle must be so short that there is no practical difference 
 in the field-strengths at its two ends. 
 
 (ii. ) The reader must notice that we assume that m is constant in fields of 
 different strengths. This is unfortunately not true ; as m will, as a rule, 
 change when the field -strength changes. But the error is not great pro- 
 vided that the fields are so weak that the magnetism of the needle is found 
 not to be permanently affected. 
 
 Comparison of the magnetic moments of two needles. This 
 method enables us to compare the magnetic moments of two 
 needles. For if we oscillate them in the same field H, e.g. that 
 due to the earth, we have 
 
 for the two respectively. From this it follows that 
 m _k . O t 2 
 
 Here, the constants k and k' depend upon the masses and 
 dimensions of the two needles respectively ; and may be readily 
 calculated, provided that the needles are of simple forms. If they 
 are identical in mass and in dimensions, we have simply that 
 
 m_ (V 
 m' <V' 
 
 4. Laws of Magnetism. There are two fundamental laws in 
 magnetism. 
 
 I. Like poles repel, unlike poles attract, one another. 
 This simple observed fact needs no comment. 
 
 II. The force between two poles varies inversely as the square 
 of the distance between them. 
 
 It may also be stated that the force between two poles /x and 
 n' is proportional to the product JJL x //. But this is hardly a sepa- 
 rate law ; since the magnitude of ^ and // would be found by 
 measuring forces of repulsion ; and hence we should be reason- 
 ing in a circle were we to give this as a third independent law. 
 
 The second law is very important, and requires experimental 
 proof. 
 
CH. in. MAGNETIC MEASUREMENTS 2/ 
 
 5. Proof of Law II. by Torsion Balance. Here we keep 
 the same pole acting on the needle-pole #, but vary the distance. 
 (See 2, fig. i.) Now if we only keep the angle small enough, 
 the following statements will be approximately true. 
 
 (i.) When the needle is deflected through 0, 0, ^0, &c., 
 the arm of the repulsive moment acting upon n s remains constant, 
 the force only changing. 
 
 (ii.) If the pole n be at a certain distance from the repelling 
 pole when the deflexion is 0, then at ^0 it is at half this distance, 
 at jjO it is at one-third of this distance, and so on. 
 
 The method is simple enough. 
 
 Let the needle be deflected through 0, and let the torsion 
 angle be (n^ X 360 + i). Then the total angle of torsion is 
 (n, x 360 + QI + 0). 
 
 Now let us force the needle up to ^0 by a larger angle of 
 torsion (n 2 x 360 + 2 ) . Then the total angle of torsion is 
 (n 2 x 360"+ a 2 + 10)." (See i and 2.) 
 
 Hence 
 
 Force at distance 6 n\ x 360 + i + 
 
 Force at distance ^0 n* x 360 + a 2 + \G 
 
 If Law II. be true we shall find that this fraction comes out to 
 bej. 
 
 So again for an angle of deflexion ^0, we should find that the 
 ratio comes out to be -J- ; and so on. 
 
 We here suppose that the earth's action has been allowed for, 
 or is relatively insignificant. 
 
 6. Proof of Law II. by Method of Oscillation. Using a 
 ' short ' needle we can count the oscillations at distances 10 cm., 
 20 cm., 30 cm., &c., from a given pole. The magnet whose pole 
 we use should here as in all cases where we wish to consider 
 the action of one pole only be so long that for all these dis- 
 tances the other pole has a negligible action on the vibrating 
 needle. 
 
 Eliminating the earth's action as before, we get the ratio of the 
 field- strengths at 10 cm., 20 cm., 30 cm., &c. If the law be true, 
 then we should have 
 
 H 10 : H 20 : H 30 : c. = i : : i : c. 
 
2 8 ELECTRICITY ; ' , CH. in. 
 
 7. Measurements as aiFected by Induction. The same 
 needle will give quite different results under the same external 
 conditions if its own magnetism alters. 
 
 Hence, in all the above, where a needle is used for two or 
 more measurements, and where it is assumed that there is no 
 change in its magnetic moment, we must take care that the fields 
 are not strong enough to alter its magnetism permanently. 
 
 This can be tested by vibrating it under the earth's action only 
 before and after the series of experiments, counting the oscillations 
 per second. 
 
 Even then there will be error due to the temporary alteration 
 of the magnetic moment. 
 
 8. Earth's Magnetism. General Ideas. We have spoken 
 of the ' earth's action,' implying that the earth acts as a magnet. 
 
 It does so act ; but from the nature of the causes of its mag- 
 netic action we cannot give a simple account of the position of its 
 poles or distribution of its magnetism. 
 
 The lines of force viewed as a whole (see Chapter II. n) 
 would be seen to be wavy ; and it would probably be seen that 
 they converged to more than one north and south pole. 
 
 But it will give some idea if we say that the earth's action is 
 nearly that which there would be if the whole earth were neutral, 
 and if there were buried at the centre a powerful magnet about 
 half as long as the earth's axis, whose position lay about 20 from 
 the earth's axis, and varied from year to year and century to century. 
 
 This would give us lines of force lying horizontal over the 
 equatorial regions, dipping more and more as we go north or 
 south, and plunging perpendicularly into the earth at the points 
 cut by the prolongation of the axis of our buried magnet. 
 
 Any magnetic needle used will be so very small with respect to 
 the earth, that the lines of force will be with respect to it practi- 
 cally parallel, or the earth's field practically uniform. In fact, we 
 could not detect any difference between the earth's field in different 
 parts of the same room. 
 
 The earth 's action therefore on a needle will be a pure couple, and 
 will simply direct it. 
 
 If it were not so we ought to find besides the couple a hori- 
 zontal force, or a vertical force, or both. 
 
 But experiment shows that no such force exists. 
 
CH. in. MAGNETIC MEASUREMENTS 29 
 
 Experiments. (i. ) A needle floating on a cork shows absence of any 
 horizontal force, by not moving in any direction. 
 
 (ii. ) A needle weighed before and after magnetisation shows absence of a 
 vertical force by absence of change in weight. 
 
 Note. In what follows we shall often speak of the single force acting on 
 one pole of a magnet needle, instead of the couple that acts on the needle as a 
 whole. The reader will see that this is for convenience, and does not in any 
 Way contradict the above. 
 
 9. Compasses. 
 
 Definition of magnetic axis. A B represents a magnetic needle, 
 perfectly symmetrical ; its ends A B and its point of suspension O lying 
 in a straight line. 
 
 If A B be also symmetrically magnetised, then it will come to rest 
 with the geometric axis A O B lying along the lines of force acting 
 upon it. 
 
 But if the needle be unsymmetrically magnetised, then, in the 
 position of rest, some other line as n O s will lie along the lines of 
 force. This line through O, 
 that lies along the lines of 
 force when the needle is at 
 rest, is called the magnetic 
 axis of the needle. 
 
 It ought to coincide with the geometric axis A O B ; and in a well- 
 made lozenge-shaped needle this will be the case. 
 
 When an ordinary compass-needle comes to rest, the direction 
 of its magnetic axis shows us the direction of the horizontal com- 
 ponent of the earth's lines of force at the place where the compass 
 is used. (For horizontal component, and resolution , see further on.) 
 
 This will be all that we can learn. Unless we have further in- 
 formation as to the declination (see n) at the place in ques- 
 tion, we cannot tell the direction of the geographic north and 
 south. 
 
 The mariner's compass differs from trie ordinary compass in 
 that the whole card turns on a pivot ; there being several needles, 
 parallel to each other, attached to the card underneath. There is 
 a fixed mark on the side of the compass-box, and one reads off 
 what point on the card lies against this mark ; whereas in the 
 other compass one reads off the point on the fixed card over 
 which the needle has come to rest. 
 
 The compass-box is, moreover, provided with two sets of 
 
3O ELECTRICITY CH, in. 
 
 pivots at right angles to each other, so that the card may remain 
 horizontal in spite of the rolling of the ship. 
 
 10. Modification of Earth's Lines of Force by the Presence 
 of Iron Masses. Any masses of iron or steel will modify the 
 earth's field of force in their vicinity. 
 
 This disturbance of the field is of especial importance in ships, 
 since there it may bear a large proportion to the whole field. 
 
 To render the magnetism of an iron ship as symmetrical as 
 possible, it should lie along the lines of force while it is being 
 subjected to the prolonged hammering during its construction. 
 
 ii. The Earth's 'Magnetic Elements.' If we take any 
 point on the earth's surface there will be through this point 
 a line of force. This will lie more or less horizontally in equa- 
 torial regions and vertically in polar regions. In our latitude it 
 will lie obliquely, dipping down towards the north. That vertical 
 plane, through the point, that contains the line of force, is called 
 the plane of the magnetic meridian ; just as the vertical plane, that 
 passes through the geographical north and south points, is called 
 the plane of the geographical meridian. 
 
 The earth's field can be resolved into a vertical and a hori- 
 zontal component in this plane, as we shall see further in 13. 
 It is the horizontal component that acts on the ordinary compass, 
 so that the magnetic axis of the needle will come to rest in this line. 
 
 Hence, the plane of the magnetic meridian can also be defined 
 as that vertical plane that contains the magnetic axis of a com- 
 pass-needle at the place in question. 
 
 At any place we know all about the earth's field when- we 
 know the three particulars given below. These are called the 
 earths three magnetic elements at the place. 
 
 (i.) The declination is the angle between the planes of the 
 magnetic and geographic meridian. It is called west or east 
 declination, as the needle points to the west or east of the geo- 
 graphic north respectively. 
 
 (ii.) The inclination is the angle between the direction of the 
 lines of force and the horizontal plane. It is also called the dip. 
 
 (iii.) The intensity is the field-strength measured in C.G.S. 
 units, as explained in Chapter II. 8. 
 
 Besides the 'elements' there are other terms that need ex- 
 planation. 
 
CH. in. MAGNETIC MEASUREMENTS 3! 
 
 Variations are changes taking place (from hour to hour, day 
 to day, year to year, and age to age) in any of the elements. 
 Most of these changes, and probably all, are periodic. 
 
 Magnetic storms are ' irregular ' disturbances, or those which 
 as yet we cannot predict. The most powerful are those accom- 
 panying marked and sudden alterations in the sun, such as a 
 sudden appearance or disappearance of sun-spots. 
 
 Magnetic equator is an irregular line round the earth's equa- 
 torial regions at all points along which there is no inclination, the 
 lines of force being horizontal. 
 
 Iso-dinic lines answer to the parallels of latitude. They are 
 irregular lines, only very roughly parallel to the magnetic equator, 
 connecting points of equal inclinations. 
 
 Iso-gonic lines answer to meridians of longitude. They also are 
 wavy and irregular. They converge towards points where the 
 lines of force are vertical ; points which would be called ' the 
 earth's magnetic north and south poles,' only that there appear 
 to be more than one of each, and each of these is perhaps not a 
 point but an area. 
 
 12. Measurement of Decimation . The finding of the angle 
 between the planes of the magnetic and geographic meridian is 
 effected by means of an instrument called a declinometer ; this 
 being provided with means by which we can find both meridians. 
 
 Near the base of the instrument is a compass-box with a very 
 carefully graduated card, over the centre of which is suspended a 
 compass-needle. This compass-needle is of lozenge shape, and 
 has at each end a fine cross scratched. It is suspended in a sort 
 of stirrup, so that it can be turned over if required. Thus, we 
 can use it with either surface uppermost. 
 
 On a vertical axis, coinciding with the axis of suspension of 
 the needle, and passing through the centre of the graduated card, 
 turns a telescope. This telescope is provided with an object- 
 glass so made that both distant and near objects can be viewed 
 without altering the focus of the instrument. It is capable of 
 being inclined to the horizontal plane as well as of turning about 
 a vertical axis. 
 
 There is a fixed horizontal graduated circle and an index 
 movable with the telescope, to tell us through what angle we turn 
 the telescope about its vertical axis. 
 
ELECTRICITY 
 
 CH. in. 
 
 The general method, without detail, is as follows. 
 
 With the telescope observe some distant object whose geo- 
 graphical bearing is known, e.g. some star not far from the 
 horizon. Read off the position of the index on the graduated 
 circle ; and, from knowledge of the bearing of the object, note 
 down at what division on the graduated circle the index would 
 stand if the telescope were directed to the geographical north. 
 Next set the telescope parallel to the magnetic axis of the needle, 
 
 FIG. i. 
 
 which, if the instrument be perfectly made, will be done by turn- 
 ing the telescope round till the end of the needle is viewed. 
 Read the position of the index. The difference between this 
 reading and the last will be the angle between the magnetic axis 
 of the needle and the geographical north and south line, or it will 
 be the declination. 
 
 Corrections. (i.) If the vertical axis about which the telescope turns 
 does not pass through the point of suspension of the needle, then the 
 
MAGNETIC MEASUREMENTS 
 
 33 
 
 telescope, when the whole instrument is viewed from above, would be 
 seen to lie off to one side of the needle. The consequence of this 
 would be that when the telescope was directed so as to point at one 
 end of the needle, it would be inclined to the needle and not parallel 
 to it. 
 
 In fig. ii. (which represents this state of things as viewed directly 
 from above, or is a horizontal projection of the needle and optical axis 
 of the telescope) O and O' are projections of the 
 vertical axes of rotation of needle and telescope re- 
 spectively. When we view one end n of the needle, 
 the direction O' n of the telescope is not parallel to 
 the axis n s of the needle. But if we first view , 
 then view j, and bisect the angle between O' n and 
 O' s produced back through O', then the direction 
 O' n is that parallel to n s. 
 
 Thus we first regard the geographical north along 
 O' N ; then, by taking two readings and striking an 
 average, we find the line O' '#' parallel to sn\ and 
 so arrive at the declination N O' n f . We will call 
 this angle 8. 
 
 (ii.) In the above, sn is the axis of figure (or 
 geometric axis) of the needle. It may not be the 
 true magnetic axis. 
 
 If it be not we proceed as follows. 
 
 If it only be remembered that it is the magnetic axis of the needle 
 that preserves a constant direction, it will be seen that if the needle be 
 
 FIG. iii. 
 
 turned over in its stirrup, it will now lie as much to the one side of 
 magnetic north and south as it before did to the other. 
 
 Hence, the average position of the needle will give us the true 
 direction n m of its magnetic axis (see fig. iii.) 
 
 D 
 
34 
 
 ELECTRICITY 
 
 So that if the process given in correction (i.) be applied over again 
 with the needle turned over in its stirrup, then the average value of 
 d, as obtained from thesey^r readings, will be the true declination. 
 
 [There are also other corrections not given here.] 
 
 13. Resolution of Earth's total Field into two, or three, 
 Components. Let us first suppose that we are viewing the plane 
 of the magnetic meridian at a place, from a position off to the 
 west of this place- 
 In the diagram, VON represents the plane of the magnetic 
 meridian ; O R represents in direction and magnitude the earth's 
 total field ; which, as we know, lies in 
 
 M H ^o this plane. We can resolve this into two 
 
 components ; O H, or H, horizontal ; O V, 
 or V, vertical. In the case of an ordinary 
 compass-needle, which can only move in 
 a horizontal plane, the vertical component 
 V is lost, and only H acts on the needle 
 to direct it. 
 
 But let us consider a needle swinging 
 in a vertical plane, and supported on a 
 horizontal axis. The figure in 14 shows 
 
 such a needle. We can turn the instrument round so that the 
 vertical plane in which the needle is free to swing is in any 
 position ; from one in which it coincides with the plane of the 
 magnetic meridian, to that in which it lies magnetic west and east. 
 In the first position the whole force R acts to direct the 
 needle ; and the needle, if properly balanced, is directed along 
 the lines of force. 
 
 In the second position mentioned, only the vertical com- 
 ponent V can act on the needle to direct it ; the 'horizontal com- 
 ponent being directly resisted by the axis. 
 
 In intermediate positions, in which its plane of swing makes 
 an angle 6 with the plane of the magnetic meridian, we have the 
 whole of V acting, and part of H. 
 
 If R is the whole force, and a the angle of dip, then we 
 have 
 
 [H = R cos a. 
 (V = Rsina. 
 
 FIG. i. 
 
CH. III. 
 
 MAGNETIC MEASUREMENTS 
 
 35 
 
 Now in the next figure it is supposed that we are looking 
 down from above. 
 
 POP' is the projection of the vertical plane in which the 
 needle swings ; O N is the magnetic north and south line ; E O W 
 
 H r 
 
 is the magnetic east and west line ; O H is seen in its full length ; 
 O V has sunk into the point O ; and O R is not given, since it 
 would appear in a foreshortened condition. The angle PON 
 is 0. 
 
 Resolving H into two components, of which the one OH' 
 acts upon the needle, while the other O A' is directly resisted by 
 the pivots, we have acting upon the dip-needle 
 in this position 
 
 (OH 7 , or H.cosfl 
 
 J or R . cos a . cos 6 . . . horizontally. 
 
 (OV, or R sin a ...... vertically. 
 
 In our third figure we give a side view ot 
 the vertical plane in which we suppose our dip- 
 needle to be swinging ; this vertical plane being 
 at an angle from the plane of the magnetic meridian. 
 
 / Horizontal component = H' = R cos a . cos 0. 
 Vertical component = V = R sin a. 
 
 (Total force) 2 = (R') 2 = R 2 cos 
 
 tan ft = ? 
 
 a . cos 
 
 + R 2 
 
 sin a 
 
 COS a . COS 0. 
 
 sn a 
 
 D 2 
 
I .'.$! v- M.'V 
 
 36 ELECTRICITY } CH. m. 
 
 ,{;',! I '!] fi ?' * v f * ^ 
 
 Here /3 is the angle made by the needle with the horizontal 
 plane, and R 7 is the resultant field acting upon the needle in the 
 given position of its plane of swing. 
 
 14. To find the Inclination, or Dip. We will first give a 
 general description of the method of finding the dip ; and later 
 will give more detail. 
 
 We will suppose that a steel needle has been accurately 
 balanced on a horizontal axis, so that it swings in a vertical 
 plane. While unmagnetised it will, if truly balanced, take up 
 any position in this plane indifferently. 
 
 Now suppose that this needle is accurately magnetised so that 
 its magnetic axis coincides with its axis of figure. The needle 
 will now set in the direction of the resultant field R' of the last 
 section ; this direction depending upon the angle which its 
 plane of swing makes with the plane of the magnetic meridian, 
 as well as upon the angle of dip a at the locality in question. If 
 be zero, then the needle takes up the direction of dip, or it 
 makes with the horizontal plane the angle a ; for in this case R' 
 is the same as R. 
 
 If 6 be 90, or if the plane of swing stand magnetic east and 
 west, the needle stands in a vertical position ; since in this case 
 we have only the vertical component V acting upon it. Hence 
 we can find the angle a of dip, if we have a vertical graduated 
 circle fixed near to the needle. We first obtain the direction of 
 magnetic east and west, by turning the plane of swing round until 
 the needle stands vertically. We then turn this plane through 90, 
 so that it may coincide with the plane of the magnetic meridian. 
 The angle now made by the needle with the horizontal will be the 
 required angle a of dip. 
 
 Corrections. In practice, however, it is necessary to make several 
 corrections owing to unavoidable imperfections in the instruments. 
 
 (i.) For error in the magnetic axis of the needle. We wish to find 
 the position in which the magnetic axis of the needle is vertical, since 
 it is this that gives us true magnetic east and west. Hence, we must 
 turn the plane until the needle is vertical ; must then take the needle out 
 of its pivots and replace it in a reversed position ; must, if necessary, 
 turn the plane slightly until the needle is again vertical ; and finally, 
 must give to the plane the mean of these two positions. This will be 
 the true magnetic east and west. 
 
 (ii.) Rrror of balance. It is almost impossible to balance the 
 
CH. III. 
 
 MAGNETIC MEASUREMENTS 
 
 37 
 
 needle so truly that, when unmagnetised, it will take up any position 
 indifferently. Such a defect can be corrected for by repeating the 
 above processes with the needle reversed in magnetism. 
 Thus we have in allfour measurements to make. 
 
 [There are also other corrections not given here ] 
 
 ' Goolden's and Casellds Dip-circle.' 1 The accompanying figure 
 represents a very convenient form of apparatus by means of which 
 the dip can, for teaching purposes, be 
 found with sufficient accuracy. 
 
 Full instructions as to its use are 
 supplied with the instrument, and 
 need not be given here. 
 
 For ordinary purposes we can 
 assume that error (ii.) has been cor- 
 rected by the makers. If we follow 
 carefully the instructions given, it 
 will be found that they amount simply 
 to 
 
 (i.) Showing us how to set the plane 
 of swing, and its axis, in a vertical 
 position. 
 
 (ii.) Showing us how to correct for 
 error (i.) given earlier ; the error viz. 
 respecting the magnetic axis of the 
 needle. 
 
 There is a very simple device by means of which we can turn 
 the plane of swing through 180 or 90 from any initial position. 
 
 15. Measurement of the Earth's Magnetic Elements. We 
 have described in 12 and 14 how two out of the three magnetic 
 elements (see n) at any place can be measured. It remains only 
 to show how the intensity R of the earth's field can be determined. 
 
 Now by 13 it is clear that, if we have measured the inclina- 
 tion a, we can find R by measuring the horizontal component H 
 of the earth's total field. For we have the simple relation that 
 H = R cos . a. This is convenient ; since, for practical reasons, it 
 is easier to measure the strength of a horizontal field than of one 
 inclined at any angle to the horizontal. 
 
 The general method of finding H is somewhat as follows. 
 
 A steel bar is made, of simple rectangular shape ; such that its 
 
ELECTRICITY 
 
 CH. III. 
 
 mechanical properties, as depending upon its mass and its dimen- 
 sions, can be readily calculated. This is very carefully magnetised, 
 and is then suspended by a thread of fibre of negligible torsional 
 elasticity, so' that there is complete freedom of vibration in a 
 magnetic field. 
 
 It is then vibrated in a horizontal plane under the influence of 
 the earth's action only. As explained in 3, this enables us to 
 determine, in absolute C.G.S. units if we so desire, the value of the 
 product H m ; where H is the horizontal component of the earth's 
 field, and m is the magnetic moment of the bar. But we cannot 
 in this way separate H from m. 
 
 TT 
 
 Next we find the value of the quantity . To effect this, we 
 
 compare the actions of the earth's field and of the magnetic bar 
 respectively upon a small magnetic needle ; as it were setting the 
 field H against a field that can be shown to be proportional to m. 
 We shall in 16 describe more exactly how this is done. 
 We thus have 
 
 )(i.) H m a (where a is known). 
 TT 
 (ii.) b (where b is known). 
 
 Hence, by multiplying, we obtain the 
 value of H; and so also of R, since R 
 = H . sec a. 
 
 16. The ' Method of Deflexions,' If 
 a small needle O of magnetic moment ni 
 be under the horizontal component H of 
 the earth's field only, it will lie along the 
 lines of this field ; and if it be deflected 
 to an angle 3 from this direction, it will be 
 acted upon by a restoring couple measured 
 by H . m' . sin I (see Chapter II. 12). 
 
 Now if the needle be thus deflected 
 by a field T, due to a magnetic bar N S 
 ^so placed that its lines of force run at 
 ^ right angles to the earth's lines, the de- 
 flecting couple due to this magnetic bar 
 will be measured by T. m' . cose. When there is equilibrium it 
 must be that these couples are equal ; and thus 
 
CH. in. MAGNETIC MEASUREMENTS 39 
 
 ' ' ' H =tan? - " 
 
 (Compare with the theory of the 'Tangent galvanometer,' 
 in Chapter XVII. 5.) 
 
 We thus can find the ratio of T to H, without any knowledge 
 of the magnetic moment m' of our small needle. For this moment 
 ;;/' 'cancels out' when the ratio between the two fields is con- 
 sidered. When the bar and small needle are suitably situated 
 with respect to one another, it can be shown that the field T is 
 proportional to the magnetic moment m of the bar. 
 
 In what follows we shall investigate the relations that exist 
 between the angle S of deflexion, the two fields H and T, the 
 magnetic moment m of the bar, and the distance of this latter from 
 the small needle when these two are suitably arranged. And we 
 shall show how we may experimentally determine the value of 
 
 TT 
 
 - ; a result of which the importance is indicated in 15. 
 
 ' Total action' of a magnetic bar upon a small magnetic needle; 
 
 TT 
 
 and the determination of. In the figure the small needle is placed 
 m 
 
 at O ; the word ' small ' having the usual relative meaning, and implying 
 that we may consider its poles to be in a field that does not differ 
 sensibly from the field at its centre O. The line O H represents in 
 direction and in magnitude the earth's horizontal field H. The mag- 
 netic bar N S is so placed that the line H O passes through its centre 
 C, and is perpendicular to its axis. The pole-strength of N S is /a, and 
 its length is 20, so that its magnetic moment m is 2/z. The distances 
 N O and S O from the centre of the small needle to the poles of the 
 magnetic bar are each measured by r. 
 
 If we mark out on S O and on N O produced, the equal distances 
 O .$ and O n representing the two components of the field due to the 
 two poles respectively, then the diagonal O T of the completed 
 parallelogram will represent on the same scale the resultant field- 
 strength T at the point O due to the magnetic bar ; the lines of force 
 thus being perpendicular to the earth's lines. 
 
 The small needle will take up a position lying along the resultant 
 of O T and O H, so that if the angle of deflexion from the magnetic 
 
 meridian be 5, then we have the relation tan $ = - . 
 
 11 
 
 Now the lines O n and O s each represent the components of the 
 
40 ELECTRICITY CH. in. 
 
 field due to one pole, i.e. represent ^-. And by similar figures we 
 
 have the proportion _ = ; or T = . ^L. Since we may write 
 On r ' r r' z 
 
 in for 2<Zyu, we have 
 
 *'' i, H COtS 
 
 r , whence = ; 
 
 T m r * 
 
 H 
 
 and these results give us all that we desired. 
 
 It is clear that the same method enables us to eliminate H and to 
 find m, if we wish to do so. 
 
 And the latter part of the experiment, or the ' method of deflexion,' 
 enables us to compare the magnetic moments of two magnetic bars 
 such as N S, and that without any knowledge of their relative masses. 
 We must, however, know approximately the positions of their ' poles,' 
 otherwise we could not substitute a measured value for r. 
 
 17. Magnetometers, Changes in the Earth's Field. When 
 a magnetic bar is so constructed and suspended that by means of it 
 we can determine the absolute value of H (and hence of R) at any 
 place, we have the essential part of that important instrument, the 
 Magnetometer. 
 
 For the construction and use of this class of instrument the 
 reader must consult more technical works. He will there see how 
 a magnetometer can be also caused to record automatically, in a 
 continuous manner, the daily and hourly changes in the earth's 
 field. 
 
 We will here merely indicate in a general manner how such 
 changes could be observed. 
 
 (i.) Variations in 
 H. Let O N repre- 
 sent the direction of 
 the lines of force in 
 the horizontal com- 
 ponent of the earth's 
 
 w field, and let n s be a 
 
 magnetic needle that 
 
 has been deflected into a position lying due magnetic east and west, 
 by means of torsion. 
 
CH. in. MAGNETIC MEASUREMENTS 41 
 
 In this position the couple due to H is first balanced by the 
 torsion couple. 
 
 Any change in H will evidently destroy the equilibrium, and 
 the position of the needle will change. Very slight movements 
 can be detected if we employ a beam of light reflected from a 
 mirror attached to the needle. (See Chapter XVII. 10.) 
 
 (ii.) Variations in V. Any needle must be balanced after 
 magnetisation, if it is to maintain a horizontal position. In such 
 a position we then have on the one hand a gravity couple, if the 
 centre of gravity is not directly under the pivot ; and on the other 
 hand a magnetic couple, due to the vertical component of the 
 earth's field. These two couples are in equilibrium. 
 
 Any change in the vertical component V of the earth's field 
 will disturb this equilibrium, and the consequent movements of 
 the needle (in a vertical plane) can be observed by means of the 
 reflected beam method. 
 
42 ELECTRICITY CH. iv. 
 
 CHAPTER IV. 
 
 THE SIMPLER PHENOMENA OF ELECTROSTATICS. 
 
 i. Introductory. We think it best to give the reader a 
 general view of the phenomena of statical electricity before dis- 
 cussion of the theory of electricity as based on ideas of potential 
 and fields of force. 
 
 So we propose, in this preliminary view, to use language that 
 may not be very scientific, but at any rate will not be misleading 
 to a learner previously warned. 
 
 The reader must carefully note in what follows that in electrostatics 
 as in magnetism all the phenomena are dual in nature. 
 
 In magnetism there are two polarities that, together, neutralise 
 each other's external effect ; apart, they give a field of magnetic force ; 
 the two ends of all the lines of force terminating against matter of + 
 and polarity respectively. 
 
 So, in electrostatics, we shall find always two sorts of electrifica- 
 tions that, together, neutralise each other's external effect ; apart, they 
 give us a field of electric force, the two ends of all the lines of force 
 terminating against + and electrifications respectively. 
 
 In magnetism this polarity is molecular, and we can never isolate 
 the one polarity. In electrostatics we shall see that the case is some- 
 what different ; but yet every + electrification has corresponding to it 
 a electrification of just the magnitude to neutralise it. 
 
 2. Electrical Excitement. As early as 640 B.C. it was noticed 
 that when certain bodies, such as amber, were rubbed, they 
 became possessed of many peculiar properties. Thus they could 
 ' attract ' to them light bodies such as bits of paper or pith-balls ; 
 when held near the face they gave rise to a peculiar sensation, 
 similar to that produced by coming into contact with cobwebs ; 
 and, under certain conditions, luminous phenomena could be 
 observed. 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 43 
 
 Experiments. (i. ) We hang up a large pith-ball by means of a fine wire 
 If now we bring near a glass rod rubbed with silk, or an ebonite rod rubbed 
 with flannel, the pith-ball will be attracted to the rod. Many experiments of 
 this sort can be tried. 
 
 (ii. ) If we approach the excited rod to the face the peculiar sensation, 
 referred to above, is at once felt. 
 
 (in.) If we rub the rod in the dark, luminous effects will' be at once noticed. 
 
 Note. In order to succeed in the above experiments we should warm all 
 the apparatus used, so as to insure its perfect dryness. 
 
 3. Dryness needed ; not High Temperature. It was found 
 that in order to make these experiments succeed it was better to 
 warm the rubbers and the rubbed at a fire. 
 
 But a little further experiment will show that it is not high 
 temperature, but the dryness of surface obtained, that we need. 
 
 Experiments. (i. ) If we heat the bodies in damp steam we fail to get a 
 result, though the bodies are hot. 
 
 (ii.) If we dry the bodies by leaving them for some time under a glass 
 case in presence of some desiccating body, the experiments will succeed though 
 the bodies are cold. 
 
 4. Attraction and Repulsion Phenomena, We find that after 
 a body has been in contact with an excited body, and if the 
 conditions are such that it itself becomes and remains excited by 
 contact, then it will be repelled by the body from which it received 
 its excitement. 
 
 But if the excitement it has received be removed from it, the 
 original attractive action will be again seen. 
 
 Experiments. (i.) If we hang our pith-ball by a dry silk thread, we find 
 that it becomes and remains excited after contact with the excited rod. In 
 this condition it will be repelled by the rod. But if we hang it by a metallic 
 wire, or if we keep touching with the hand, it will not retain its excitement, 
 and it will always be attracted by the rod. 
 
 (ii. ) The two leaves of a gold-leaf electroscope repel one another when 
 charged. 
 
 Note on the Gold-leaf Electroscope. This instrument is a very sensitive 
 indicator of electrical excitement, and it may be constructed in various ways. 
 In the ordinary form there are two strips of gold leaf, n n, suspended from a 
 brass wire which passes through the neck of a glass vessel B. This wire 
 terminates in a knob or plate C. Inside the vessel is usually placed some 
 desiccating substance, such as calcium chloride. Two strips of tin-foil are 
 often gummed on to the inside of the vessel so as to be opposite to the gold- 
 
44 
 
 ELECTRICITY 
 
 leaves ; and these strips should be in metallic connection with the earth, or 
 with the table upon which the instrument stands. The glass of the vessel B 
 should be dry, and the upper part may be coated with shellac-varnish. The 
 
 reasons for these details of construction 
 will be understood later on. 
 
 Test for electrification, We shall 
 also understand by what follows later 
 that the only certain test for electrifi- 
 cation is the divergence of the leaves. 
 If the leaves be charged and be di- 
 vergent, it is found that the approach 
 of a neutral (or unexcited) body to the 
 knob C will cause a greater or less 
 convergence of the leaves. Hence, if 
 in any case we obtain such convergence 
 of the leaves by the approach of a body, 
 we should compare the amount of this 
 convergence with that produced by the 
 approach of the same, or a similar body 
 when unexcited. It is only if the 
 
 former convergence is markedly greater in amount than the latter that we 
 can say that it indicated electrical excitement in the body approached. 
 
 5. Conductors and Non-Conductors, Such threads as do 
 not allow the pith-ball, after excitement by contact with the 
 excited rod, to remain excited, were said to conduct away what 
 was formerly called ' the electric fluid.' Other such experiments 
 were made, and it was found that certain bodies kept any electrical 
 excitement isolated, while along others the excitement leaked away 
 to the earth, or would pass to and excite other isolated bodies. 
 
 When a body showed the one or the other property to a 
 marked degree, it was called a non-conductor or conductor respect- 
 ively. In reality all bodies are to a greater or less degree con- 
 ductors, as in the case of heat, but non-conducting properties are 
 more strongly marked for electricity than for heat. 
 
 Experiments. (i.) By experiments with the suspended pith-ball we find 
 dry silk, glass fibres, threads of sealing-wax^ &c. to be non-conductors ; while 
 damp silk, cotton and wire are conductors. 
 
 (ii.) Excite an ebonite rod, and throw over it threads or wires of various 
 sorts in turn. In each case let the thread or wire, as it hangs from the excited 
 rod, come into contact with the knob of a gold-leaf electroscope. 
 
 The 'conducting' threads will allow a charge to pass to the gold-leaves 
 from the rod, and the leaves will remain permanently divergent. 
 
 The 'non-conducting' threads will not allow a charge to pass. 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 45 
 
 Note, Table of some bodies remarkable for good or bad conduction respect- 
 ively. 
 
 Conductors. Nan- conductors. 
 
 Metals. 
 
 Well-burnt charcoal. 
 
 Graphite. 
 
 Acids. 
 
 Saline solutions. 
 
 Moist bodies. 
 
 Flame. 
 
 Linen. 
 
 Cotton. 
 
 Ebonite. 
 
 Resins, shellac, amber, &c. 
 
 Caoutchouc 
 
 Dry gases. 
 
 Sulphur. 
 
 Glass. 
 
 Silk. 
 
 Wax. 
 
 Many dry bodies. 
 
 6. Electrics and Non-Electrics (so-called). If various 
 bodies be held in the hand and rubbed with other bodies, some 
 give signs of excitement and some do not. 
 
 This rough method of experiment gave rise to a division of 
 all bodies into electrics (from which it was easy to obtain electric 
 phenomena), and non-electrics (from which it was not). 
 
 This division was premature. 
 
 It was soon found that there was no reason for supposing but 
 that all bodies rubbed together gave electric phenomena ; at least, 
 if they were not absolutely identical in material, surface, tempera- 
 ture, &c. But conducting bodies allow the electrification to pass 
 away at once, while non-conducting do not. Hence electrics and 
 non-electrics mean non-conductors and conductors respectively. 
 
 Experiment. A brass rod held in the hand and rubbed with silk gives no 
 signs of excitement. But, if fixed at the end of an insulating ebonite rod, it 
 will now remain excited after the rubbing. 
 
 7. Two sorts of Electrification. It can be shown that there 
 are two sorts of electrification such that similarly electrified bodies 
 repel one another, and dis- similarly electrified attract one another. 
 
 The reader will of course be at once reminded of the two 
 polarities in magnetism. 
 
 Experiments {see 2, note). (i. ) Suspend a gilt pith-ball by dry silk 
 thread. Having left it excited and repelled by an ebonite rod rubbed with 
 flannel or catskin, bring up to it a glass rod rubbed with silk. It will be 
 attracted. 
 
 Various experiments of this nature may be tried. 
 
 (ii. ) Let a gold-leaf electroscope be charged. Then the approach of the 
 excited ebonite will cause the leaves to diverge or converge according to the 
 
46 ELECTRICITY CH. iv. 
 
 nature of the charge of the leaves. Note which is the case, and then try the 
 effect on the leaves of the approach of the excited glass rod, or of other 
 excited bodies (see end of note to 4). 
 
 (iii. ) Make a simple wire frame and suspend it by dry silk, so that excited 
 rods may easily be placed in it and hang insulated. 
 
 Excite various rods and try their action on each other. 
 
 Thus, if an ebonite rod rubbed with flannel be so hung, and another simi- 
 larly rubbed be approached, they will repel each other. But if we approach 
 a glass rod rubbed with silk, or another ebonite rod rubbed with silk on which 
 has been spread some electric amalgam, we have attraction. (Such electrical 
 amalgam is usually an amalgam of mercury and tin ; this is reduced to a 
 powder, and is then made into a paste with lard. ) 
 
 It is usual to call ' + ' that electrification that we get on 
 smooth glass rubbed with silk ; and ' ' that which we get on 
 ebonite rubbed with flannel. 
 
 Sometimes the former is called vitreous, and the latter resinous, 
 electrification (or electricity). 
 
 8. The two sorts of Electrification are always produced 
 together. Further experiment will show us that rubber and 
 rubbed are always oppositely electrified ; and that this simultaneous 
 occurrence is the only thing that one can, without previous investi- 
 gation, predict for certain when one body is rubbed with another. 
 
 Thus, ebonite may be made either -4-ly or ly excited ; 
 but it is certain that in either case the rubber will be oppositely 
 excited. 
 
 Experiments (see 2, note). (i. ) Charge an electroscope, and then hold 
 near it the rubber and the rubbed in turn. The leaves will always diverge 
 more in the one case, and fall together somewhat in the other ; showing the 
 rubber and rubbed to be oppositely electrified. 
 
 (ii. ) In the simple wire frame, mentioned in 7, hang an excited rod. It 
 will always be attracted by the rubber that has excited it, but repelled by 
 another similarly excited rod. 
 
 The following pairs of substances can be tried as rubber and rubbed. 
 
 Ebonite and catskin, ebonite and amalgamated silk, smooth glass and silk, 
 roiigh glass and flannel, sealing-wax and flannel, sealing-wax and giin-cotton, 
 gun-cotton and cotton-wool, flannel and silk, two silk handkerchiefs of different 
 makes, brown paper and india-rubber. 
 
 It is found that we can arrange bodies in a certain order, in- 
 variable under ordinary conditions, such that when two of the 
 bodies are rubbed together, that which is higher on the list be- 
 comes + , and that which is lower becomes . 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 4/ 
 Such a list is here given ; it is taken from Ganot 
 
 Catskin. 
 
 Flannel. 
 
 Ivory. 
 
 Rock-crystal. 
 
 Glass. 
 
 Cotton. 
 
 Silk. 
 
 Wood. 
 
 Metals. 
 
 Caoutchouc. 
 
 Sealing-wax. 
 
 Resin. 
 
 Sulphur. 
 
 Gutta-percha, 
 
 Gun-cotton. 
 
 Note. It is sometimes not easy to obtain signs of electrification on bodies, 
 as flannel, gun-cotton, &c. , which are not good insulators. In such cases we 
 may fasten them to the ends of thin ebonite rods. Thus, if we so fasten a 
 piece of gun-cotton and of cotton-wool to the ends of two ebonite rods, and 
 then rub them together, we shall find them to be strongly excited with opposite 
 electrifications ; but if we hold the two in the hand and then rub them, we 
 shall get little or no such signs. 
 
 9. Equal quantities of the opposite Electrifications are 
 always produced simultaneously. This is a fact of great and 
 fundamental significance ; and is indissolubly bound up with the 
 essentially dual nature of electrostatic phenomena. 
 
 In every case of electrical excitement there is a separation of 
 a state of equilibrium into what may be expressively, though per- 
 haps not very exactly, termed an overbalance and an underbalance 
 respectively. 
 
 What * + and electrifications ' mean we do not yet know. 
 But one thing at least is certain ; that the total electrifications 
 produced in any way from a state of neutrality will, if collected 
 together on one conductor, give zero electrification. Or, we never 
 have any + or electrification produced without an equal 
 quantity of or + electrification (respectively) being simul- 
 taneously produced. The sense in which the word ' equal ' is 
 used has just been explained. 
 
 Experiments. - (i. ) The following experiment is well-known. 
 
 A flannel cap is made, so as to fit on to the end of an ebonite rod, and the 
 whole is carefully discharged in a Bunsen's flame. Then, by means of a long 
 and dry silk thread attached to the cap, this latter is twisted round upon the 
 rod. 
 
 Still it will be found that the whole has no effect upon the leaves of an 
 electroscope. 
 
 But if the cap be drawn off the rod, still insulated by means of the silk 
 thread, it will be found that the two are oppositely excited. 
 
4 3 
 
 ELECTRICITY 
 
 CH. IV. 
 
 Here then we have opposite electrifications produced, whose equality was 
 Shown by the fact that the who/e system had zero action on the electroscope. 
 
 (ii. ) In 16 we shall see that when a charged conductor is lowered into a 
 hollow vessel such as A, one that is nearly closed and is open only towards the 
 comparatively remote ceiling, the effect upon this vessel will be the same as if 
 it had received the entire charge of the body. This will be the case whether 
 the charged body be a good or a bad conductor, whether it touch the vessel 
 or hang insulated within it. 
 
 In the figure we have such an insulated vessel A, and with it is connected 
 an electroscope E. Hence, in all cases of electrical excitement we may 
 
 introduce the rubber and rubbed into 
 the vessel A simultaneously. If this 
 be done with the rod and cap of the 
 last experiment, it w T ill be found that 
 they give together a zero action, while 
 
 A separately they cause the leaves to 
 
 diverge equally with opposite electrifi- 
 cation. 
 
 ' Notes. (i. ) On discharging bodies. 
 
 + All bodies may be readily discharged 
 
 by passing them through the flame of a 
 Bunsen's burner. The completeness 
 of discharge should be tested by their 
 having zero action on an electroscope. 
 
 (ii. ) On testing the charge for ' sign. ' 
 When the leaves are divergent with 
 any charge, a rod of ebonite excited 
 
 with flannel may be approached. If the leaves diverge still more, their 
 charge was similar to that of the rod, or negative ; but if they fall together, their 
 charge was positive. The reason for this will be explained later ; that it is 
 the case can easily be proved experimentally by the student. 
 
 10. The 'Fluid' Theories of Electricity. The mobility of 
 electrification (or electricity), its ready passage along conductors, 
 and certain other characteristics early suggested the name electric 
 fluid. But the absence of weight, the attractions and repulsions 
 of the two sorts of electrification, and other phenomena, such as 
 the extraordinary speed of movement, arguing absence of mass, 
 all show us that if we are to hold previous ideas associated with 
 the word fluid we must regard the term 'electric fluid' merely as 
 a rough analogy, if indeed it should be used at all. 
 
 Believing that too little is known as to the nature of electrifi- 
 cation, and of the phenomena of electric strains and movements, 
 to make any term at present in the language really a good one, we 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 49 
 
 shall, for lack of better terms, speak of ' + and electrifications 
 or electricities? 
 
 The expressions ' + ' and ' ' retain their algebraic mean- 
 ing. Equal and opposite electrifications, when imparted to the 
 same conductor, give zero electrification ; while, if the quantities 
 be unequal, we have left simply the balance of the one or the 
 other as the case may be. 
 
 When it is assumed that there are two fluids possessing, in a 
 sense that has been made clear, ' opposite ' properties, we are said 
 to be employing the two-fluid theory of electricity. 
 
 When it is assumed that there is but one fluid of which an 
 excess in one place and a defect in another give rise to the 
 phenomena of + and electrical charges respectively, we are 
 said to employ the one-fluid theory. The reader is again warned 
 that in the present state of knowledge it is unadvisable to lay 
 much stress upon either view. 
 
 ii. The Three Laws of Electrostatics. With respect to the 
 attractions and repulsions of electrified conductors, it is found that 
 three main laws hold. These are as follows. 
 
 Law i. Like electricities repel, and unlike attract^ one another. 
 
 This means that if two bodies are charged with electrifications 
 of like or of unlike sign respectively, there will be observed 
 between them ' repulsion ' or * attraction ' respectively. (We here 
 use the usual terms without asserting that they are accurate.) 
 
 12. Law II. The Force varies as QxCl'. This means that 
 if we have two quantitiesVf electricity, Q and Q' respectively, on 
 two conductors, and if the conditions as to distance remain con- 
 stant, then the force in dynes that exists between the conductors is 
 proportional to QxQ', being repulsive if the algebraic product 
 be +, attractive if the product be . . 
 
 We must here suppose the bodies to be in the middle of a 
 very ' large ' room ; and the conductors should be two ' small ' 
 spheres, so that we may consider the distance between Q and Q' 
 to be the distance between the centres of these spheres. 
 
 Note. The word large means very great compared with the distance 
 between the spheres ; the word small means very small compared with the 
 same. 
 
 Measurement of Quantity ; unit Quantity. It was said that 
 in magnetism (see Chapter III. 4) we had no independent law 
 
 RSiTY OF CALIFORNIA 
 
ELECTRICITY 
 
 CH. IV. 
 
 that force varied as ^ /u', since we measured the product ;< /*' by the 
 force, and would be arguing in a circle if we made of this an 
 independent law. 
 
 In electricity the case is somewhat different. We do indeed 
 define unit quantity as that which at unit distance from an exactly 
 similar quantity repels it with unit force, thus measuring our unit 
 by force. But we can find multiples or submultiples of this unit 
 without reference to force. 
 
 Thus we can empty a series of bodies, each carrying unit 
 charge, into an insulated hollow body ; the charge will go wholly 
 to the outside surface, thus charging one and the same surface 
 with any number of units desired. 
 
 Or again, we can charge a sphere with unit quantity, and then, 
 hanging it in the middle of a large room, bring into contact with it 
 one, two, three, &c., exactly similar spheres. 
 
 Theory and experiment will show that the unit is subdivided 
 into J, ^, ^, &c,, provided that the spheres are arranged with perfect 
 symmetry. 
 
 (i.) Proof of Law II. by torsion balance. Referring to the 
 figure of the torsion balance we have to imagine only the follow- 
 ing modifications. 
 
 Instead of the needle we have a 
 light arm of straw, A B, carrying a 
 small gilt sphere at each end. 
 
 When the straw arm is at zero each 
 of these equal spheres rests against 
 another equal sphere at C and D, these 
 latter being connected with each other, 
 but otherwise insulated. This arrange- 
 ment of two spheres &c. is to insure 
 its being a true couple that acts on 
 the needle. A like arrangement would 
 have been desirable in magnetism, bufwe could not have found 
 two exactly equal poles to act on the two poles of the needle, 
 whereas we can readily get equal quantities of electricity on the 
 gilt balls. If C is now charged, the charge will be shared by all 
 four balls alike if they have been made accurately equal spheres, 
 and the arm A B will be deflected. We bring it back by torsion 
 to some conveniently small angle, and measure the total angle of 
 
 130 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 51 
 
 torsion (see Chapter III. 2), and so the torsion couple. We 
 may suppose that we have a sphere S t charged, so large that the 
 quantity on it is not appreciably altered by several times charging 
 the torsion balance from it. Let us charge the balls of the torsion 
 balance from the sphere S^ and measure the torsion as above. 
 Then discharge the whole. 
 
 Next make S t share its charge with an equal sphere S 2 , and 
 charge the balls again from either S L or S 2 . 
 
 A little thought, and reference to the assumption as to the size 
 of B! made above, will show us that each ball now has one-half the 
 charge that it had before. We shall now find that for the same 
 deflexion the torsion couple required to balance the electric 
 couple is one-quarter what it was before ; that is, the force between 
 the balls is one-quarter the former force. 
 
 Now since the quantity on each ball is one-half what it was, this 
 result confirms the law that ' the force varies as Q x Q'. 
 
 13. Law III. The Force varies as - 2 . The meaning of 
 
 this law has been already explained in Chapter III. 
 
 In order to test this law by direct experiment it is clear that 
 we must deal with charged spheres of very small size as compared 
 with the distance between them. Otherwise, we could not assign 
 a distinct meaning to the distance r between the two quantities of 
 electricity. 
 
 All direct experiment is only approximate, and serves to con- 
 firm a law for which there is much indirect evidence. 
 
 The law theoretically refers to quantities of electricity collected 
 2& points, just as do the corresponding laws in magnetism and in 
 gravitation. This gives a definite meaning to r. 
 
 Assuming that our methods of experiment and of reasoning 
 have proved the law, the integral calculus enables us to deal with 
 the action on each other of conductors on which there is a 
 known distribution. 
 
 (i.) Torsion balance method. If the student will read Chapter 
 III. 5, and has learnt the modifications in the torsion balance 
 required for electricity as given in the last section, he will need no 
 further explanation of this method. 
 
 (ii.) Indirect^ but exact^ proof. One of the easiest matters to 
 
52 ELECTRICITY CH. iv. 
 
 prove experimentally is that if a hollow vessel A B be charged, 
 there is no force due to this charge acting on a small charge P in- 
 side ; or there is no field of electrical force 
 inside a hollow, charged, conductor. 
 
 Faraday constructed a small insulated 
 room. He went inside it with delicate 
 electroscopic apparatus, and then the 
 whole was highly charged. No electric 
 field inside, due to this charge, could be 
 detected. 
 
 Now it can be proved mathematically 
 that no possible law of force between the 
 
 small charge P and the different portions of electricity on the 
 vessel A B, except the law of inverse squares, could account for 
 this resultant zero force on P. 
 
 Formula expressing the three laws. We can express all the 
 three laws by the one formula 
 
 F _QxQ' 
 
 ** 
 
 Here we pay proper attention to the ' rule of signs ' in the pro- 
 duct Q x Q', and consider the force to be repulsive if the product 
 is +, attractive if the product is . The force F is in dynes if r 
 is measured in centimetres, and if Q and Q' are measured in the 
 absolute units given in 1 2. 
 
 14. First Ideas as to Induction, As stated in i, we shall 
 at present use the somewhat unscientific ' two-fluid ' language in 
 this first sketch of the main phenomena of electrostatics. 
 
 Instruments used. The gold-leaf electroscope has already been 
 described. The insulated conductors need no comment. 
 
 But the proof-plane needs some explanation. It consists of a 
 small metal (or gilt paper) disc at the end of a thin insulating rod. 
 The theory of its use involves the following considerations. We 
 shall see that electric charges reside only on the surfaces of con- 
 ductors. Consequently, if the small disc of the proof-plane be 
 applied to a conductor so as for the time to form part of its 
 surface, the electricity that was on the portion of the conductor 
 covered by the disc will now be found on the disc. When the 
 proof- plane is removed (care being taken to remove it so that 
 every part of it leaves the conductor at one and the same 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 
 
 53 
 
 moment) the charge that it carries away can be taken as a sample 
 both in sign, and in degree of concentration, of the charge existing 
 on that part of the conductor to which the proof-plane was 
 applied. 
 
 The main facts of induction will be understood from the 
 experiments that follow, and the figures represent roughly to the 
 eye the condition of things in each case. 
 
 Note. Insulating stands. A very good, but somewhat expensive, form of 
 insulating stand is that due to Professor Clifton. It can be obtained at 
 Powell's glass-works, or through electrical instrument makers. Professor 
 S. P. Thompson has devised a simple and very effective form, that can be made 
 in any laboratory. A description of it may be seen in Nat^^re, vol. xxix. 
 page 385- 
 
 Well-polished ebonite rods are also good, but in delicate experiments care 
 must be taken to discharge them frequently in a Bunsen's flame. They may 
 be cleaned with paraffin oil, this latter having been rendered anhydrous by 
 means of a few lumps of sodium kept in the bottle. 
 
 Experiments. (i.) A B is a large rod of ebonite excited with catskin and 
 placed in a stand. C D is an insulated cylinder, previously discharged, placed 
 
 FIG. i. 
 
 near A B. It will be found that the state of equilibrium of C D is altered. 
 Whereas it was totally uncharged, we now find on it a distribution somewhat 
 as given below. 
 
 Applying the proof-plane to different parts on its surface in succession, and 
 converging the charges so obtained to an electroscope, we can show that 
 the end C opposite to the charged A B is strongly + charged. The 
 charge is feebler as we approach the centre of the cylinder ; about the centre 
 there will be a region where no charge is to be detected ; while, as we approach 
 
54 
 
 ELECTRICITY 
 
 CH. IV. 
 
 the end D, we find a stronger and stronger charge. The proof-plane must 
 be discharged in a Bunsen's flame between each two trials. 
 
 Note. When the charge has been conveyed to the gold-leaf electroscope 
 by the proof-plane, its strength can be roughly seen by observing the amount 
 of divergence of the leaves, while its sign can be recognised by the approach 
 of an excited ebonite rod. 
 
 (ii. ) While the conductor C D is in the presence of the charged body A B, 
 let us touch C D at any point. This will connect it with, and so make it form 
 part of, the earth. 
 
 We shall find that all the ' repelled ' electricity ' goes to earth,' leaving 
 the body charged + . 
 
 We should have expected this, since the portions of the earth near A B 
 would probably, judging from experiment (i. ), be +ly charged, while the 
 would go as far away from A B as possible. But still it seems at first a little 
 
 surprising that the charge should 
 
 _ + fifl escape as readily if we touch the end C 
 
 I as ^ we toucn ^e end D. This will be 
 I explained in Chapter V. 6 (d'\ Here 
 | we may only say that the presence of 
 the + at the end C renders it as easy 
 I for the to pass to earth by the 
 I route near A B as by the other route. 
 
 lit! II (iii.) The divided cylinders. We 
 
 FIG. iii. may construct our cylinder of two 
 
 halves separately insulated. 
 
 If we allow the whole, when forming one cylinder, to be acted upon induc- 
 tively as in fig. i., and then separate the two halves, we shall find the half 
 nearer A B to be +ly charged, the half further from A B to be ly charged, 
 (iv. ) Charging the gold-leaf electroscope by induction. The second experi- 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 55 
 
 ment shows us how we may charge a conductor by induction ; and this method 
 is almost invariably employed when we desire to charge the gold-leaf electro- 
 scope. 
 
 An excited rod is held near the knob of the electroscope, so that the leaves 
 diverge with a charge of a similar sign to that of the rod. We then discharge 
 the electroscope of this ' repelled ' electricity by touching the knob, so that the 
 leaves again collapse. If we remove the hand and then withdraw the excited 
 rod, the leaves will diverge with a charge whose sign is opposite to that of the 
 inducing rod. 
 
 15. First Ideas as to Distribution. In considering the dis- 
 tribution of an electric charge on a conductor, we assume that we 
 are dealing with simple continuous conductors, or those in which 
 all the parts are connected by conducting matter. Such cases as 
 that of an insulated ball hung inside a vessel will be considered 
 later under the head of condensers. They are cases of systems of 
 conductors and not of simple conductors. 
 
 We shall find the following general facts to hold, it being taken 
 for granted that, unless the contrary is stated, the conductor in 
 question is in the middle of a large room (see n, note). 
 
 (i.) The charge is on the surface of the body only. In the 
 case of closed, or nearly closed, hollow conductors, the charge will 
 in general be on the outside surface only. 
 
 (ii.) The distribution depends upon the shape of the body. 
 The density of charge is greatest on all salient points ; all project- 
 ing corners, or prominent regions of great curvature, having a far 
 greater density than have the other regions. 
 
 We here give some facts with respect to the distribution on 
 bodies of certain forms. 
 
 \ Note. By density of charge we mean quantity of electricity per unit area. 
 ' We shall say more concerning this in Chapter X. 
 
 On a sphere the distribution is uniform. 
 
 On a ellipsoid of revolution the densities of the charges at the 
 extremities of the axes are proportional to the lengths of the 
 /TjD axes. 
 
 On a cube Riess found that the density at the middle points 
 of the edges is about two and a half times as great as at the 
 middle of a face, while at the corners it is more than four times as 
 great. 
 
 On a circular disc the density is greatest round the edge. As 
 
56 ELECTRICITY CH. IT. 
 
 we move from the edge the density falls off rapidly, and for a con- 
 siderable region near the centre of either face of the disc the 
 density is nearly uniform. 
 
 More exactly, Coulomb found the following distribution on a disc of 
 10 inches diameter. 
 
 At the centre . , the density was rooo 
 
 1 inch from the centre . rooi 
 
 2 inches M . , . ' 1*005 
 
 3 inches * '"- 1*170 
 
 4 inches . .. 1-520 
 4-5 inches . 2-07 
 
 5 inches, or at the edge . 2*90 
 
 On a cylinder of an elongated form the density is much greater 
 at the ends than at the middle. 
 
 In the case of a cylinder of circular section whose diameter was 
 2 inches, length 30 inches, and whose ends were hemispherical, Cou- 
 lomb obtained the following results. 
 
 ( At the middle . . . the density was I'oo 
 
 1 2 inches from the ends . 1-25 
 
 1 i inch from the ends . 1-80 
 
 v At the ends , 2-30 
 
 It is to be noticed that this is somewhat the form given usually to 
 the ' prime-conductors ' of electrical machines. 
 
 On a cone the density is greatest at the apex. 
 
 When the cone is of a very small angle, and the point is very 
 sharp a common needle is an extreme case of this kind the 
 density at the point becomes very great indeed. As a consequence 
 of this, though why it is a consequence it is not so easy to explain 
 as would at first sight appear, the air no longer serves to separate 
 the charge residing on the point from the charge induced upon 
 the walls or other surroundings. It is found that a stream of 
 electrified air proceeds from the point towards the induced charge 
 of opposite sign that exists on the walls, &c. ; and the conductor 
 is thus discharged. Thus a sharp projecting point will almost 
 completely discharge any conductor to which it is fixed. In a 
 similar manner, but by inductive action, an earth-connected 
 pointed conductor will nearly discharge a charged body, towards 
 which it is presented ; for there will proceed from the point to the 
 
CH. IT. SIMPLER PHENOMENA OF ELECTROSTATICS 57 
 
 body a stream of air charged with the electricity of opposite sign 
 that is induced on the point by the charged body. 
 
 If there be a sharp point inside a closed or nearly closed 
 vessel, there will be no charge upon it, and hence the above 
 action will not occur. Let us, for example, take a tin vessel (such 
 as a common pint measure), and let us fix inside it a needle, solder- 
 ing it in an upright position to the bottom of the vessel. If the 
 needle's point be well within the vessel, we may charge this latter, 
 and shall find that the needle does not discharge it. But if 
 the needle be long enough to rise out of the vessel, it will dis- 
 charge it. 
 
 .Experiments. (i. ) By the help of a proof- plane we can verify the above 
 statements as to the general character of the distribution on insulated con- 
 ductors of various shapes. 
 
 (ii. ) Insulate a somewhat deep and narrow pail, and charge it. Examining 
 it with the proof-plane we shall find that approximately all the charge is on the 
 outside surface. (As the vessel is not quite closed there will be some charge 
 on the inside surface, especially near the mouth of the vessel.) 
 
 This is approximately true even of a cylinder of wire gauze, or a butterfly- 
 net. A well-known bit of apparatus is ' Faraday's net ' ; it was used to show 
 the above. 
 
 (iii.) Connect a small insulated pail with the electroscope, having placed 
 in the pail a quantity of brass chain to whose end is attached a silk thread. 
 Now charge the pail, and note the extent of divergence of the leaves. Next 
 raise a considerable length of chain into the air by means of the silk thread, 
 leaving the lower end in the pail. The leaves will fall somewhat, showing 
 the charge to be less concentrated than it was. Here we have increased the 
 external surface of the whole conductor without changing its mass ; and the 
 result shows that the charge has spread. 
 
 1 6. Faraday's Ice-pail; illustrating the Laws of Distri- 
 bution and of Induction. The experiments that follow illustrate 
 to some extent the laws of distribution. But their chief value is 
 that they throw much light on Induction. 
 
 We shall find that, among other things, the following facts are 
 true. 
 
 (a) Every charge induces on surrounding surfaces a charge equal 
 to its own in magnitude, but of opposite sign. If some of the 
 surroundings be nearer to the charged body than are others, then 
 the greater part of this induced charge will in general be on the 
 former. 
 
 (b) A partly closed hollow conductor ceases to act as if 
 
58 ELECTRICITY CH. iv. 
 
 entirely closed when there is introduced into it (though not in con- 
 tact with it) a conductor connected with the earth. The opening 
 in the vessel now becomes of great importance ; for we have 
 introduced an earth-connected conductor. The whole surface of 
 the vessel, inside as well as outside, is now opposite to part of the 
 earth ; and the charge will be distributed over the whole surface. 
 Referring to the statement, which is in general true, that ' When a 
 hollow conductor is charged, the charge resides upon the outside 
 surface only,' we must in the present case consider that the only 
 true inside is the interior of the mass of the metal. This case 
 will be examined in experiment (vi.) below. 
 
 (f) The manner in which a charge distributes itself upon the 
 outside surface of a hollow conductor is quite independent of the 
 positions of charged or uncharged bodies situated entirely within 
 the vessel. It depends solely upon the form of the conductor, 
 and the situation and nature of surrounding bodies. 
 
 This is a particular case of the general fact that a conductor 
 acts as a screen between two electrified systems (see Chapter X. 
 
 17(3)). 
 
 Apparatus used. We use a tall narrow ice-pail placed on an insu- 
 lating stand, and connected with an electroscope ; also a brass ball, 
 hung from a silk thread, to which a charge can be given from a small 
 electrophorus (see 17). The ball is kept insulated unless the con- 
 trary is stated. 
 
 Experiments. (i.) We can show that the ball is totally discharged if 
 allowed to touch the inside of the pail, while if allowed to touch the outside it 
 only shares its charge with the pail. (See figure to 9, experiment (ii. )). 
 
 (ii. ) Charge the ball + and lower it insulated into the pail. The leaves 
 diverge with a + charge repelled. Now let the ball touch the pail so as to 
 be totally discharged. The leaves do not stir. This shows that the + charge 
 repelled by induction equals in amount the inducing + charge on the ball. 
 Hence also the induced - charge on the inside of the pail must be equal to 
 the inducing + charge, since the pail as a whole has zero charge. 
 
 (iii.) While the ball hangs inside insulated, touch the pail outside with the 
 finger. The leaves collapse, the repelled + charge having been removed. 
 Now re-insulate the pail and remove the ball, when the leaves will diverge 
 just as much as before, but with a charge. This illustrates the above equality 
 of inducing + , and induced - , charges respectively. 
 
 (iv.) Or again, as follows. Remove the repelled + charge, as in (iii.), and 
 again insulate the pail. Then allow the ball to touch the pail. We get zero 
 charge, showing that the inducing + and induced have just cancelled each 
 other. 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 
 
 59 
 
 (v.) It can be shown by careful experiment with a proof-plane and a very 
 delicate electrometer that (a) the distribution, induced inside, moves about 
 as we move the ball ; (b~) while the repelled + outside is independent of the 
 position of the ball when this is once well inside the vessel. 
 
 (vi. ) Charge the pail and lower into it the insulated ball uncharged. No 
 change in the gold-leaves is observed. Now connect the ball with earth while 
 yet insulated from the pail. The leaves drop a little, showing that part of the 
 charge has (as can be directly proved with a proof-plane) gone to what we 
 have hitherto called the inside of the pail, but what is now a part of the pail's 
 surface opposite to the earth. The charge on the pail induces on the walls, 
 ceiling, floor, and ball (all of which are portions of the earth) a charge equal 
 and opposite to its own, and a portion of this is now on the ball. 
 
 This portion can be isolated by again insulating the ball and removing it 
 from the pail. 
 
 (vii. ) We may hang several pails, insulated from each other, one inside the 
 other. 
 
 Experiments similar to the above may be tried. It will be a useful exercise 
 to the student to puzzle out for himself what will happen in each case, remem- 
 bering the two principles that 
 
 (a) A charge induces an equal and opposite charge on surroundings. 
 
 (b) Each pail serves as a ' screen ' (see Chapter X. 17) between the charge 
 on its outer surface and any charge inside it. 
 
 17. Electrophorus. We shall here describe, in an elemen- 
 tary manner, a simple contrivance for obtaining charges of elec- 
 tricity by induction. 
 
 A B is a circular cake of resin, 
 or ebonite, or any of the good 
 ' electric ' compounds that can be 
 made by melting together various 
 electrics ; (e.g. the following sub- 
 stances may be melted together in 
 the proportions by weight here given: 
 5 parts shellac, 5 parts gum mastic, 
 2 parts Venetian turpentine, and 
 i part marine ghie. This recipe is 
 given by Professor Guthrie). 
 
 This is fixed in a shallow pan C D, 
 made of tinned iron or other metal. 
 
 E F is a circular disc of brass or tinned iron of smaller diameter 
 than the cake, having well-rounded edges, and being provided 
 with a knob F. This is fixed on to an insulating handle I. 
 
 To use the instrument we proceed as follows. 
 
60 ELECTRICITY CH. iv. 
 
 (i.) The upper surface of A B is strongly excited by catskin, 
 and acquires a strong charge. This induces on surrounding 
 objects an equal + charge. Now while E F and other bodies are 
 remote from A B, nearly the whole of this induced + charge 
 will be on the ' sole ' C D ; a repelled charge passing away from 
 the sole to earth. 
 
 The presence of C D helps to keep the upper surface of A B 
 from getting gradually discharged. 
 
 (ii.) When E F is lowered on to the cake A B it is found that it 
 behaves as if very close to A B but not in contact with it. In fact, 
 the plate rests upon a few projections of the badly-conducting 
 cake, and so is acted on inductively only. 
 
 (iii.) The repelled charge can be removed from E F by 
 touching the knob F ; and there remains on E F a + charge 
 opposite to the charge on A B . In this position of things, the 
 plate E F being so much nearer to the excited surface than is the 
 ' sole 'CD, we have on E F nearly all the induced + charge ; the 
 sole C D returning to a nearly neutral state by what may be de- 
 scribed either as + electricity passing from C D to earth, or as 
 electricity returning to C D from the earth. 
 
 (iv.) The plate E F can now be moved by its insulating handle, 
 and its + charge utilised. By the time that E F is again remote from 
 A B, the ' sole ' has returned to the condition in which it was in (i.). 
 
 The reader should draw for himself figures representing the 
 condition of the charges, &c, in all these stages of the use of the 
 electrophorus. 
 
 Experiment. By the use of a substantial insulating stand, on which the 
 electrophorus is placed before excitement, the changes in the condition of the 
 sole, &c. , as given above, can be easily demonstrated. A proof-plane of large 
 size will carry off enough of the charges to affect inductively a previously 
 charged electroscope. The fact that in (i.) electricity would pass from the 
 sole to earth, in (ii. ) and (iii. ) + electricity would do so, and in (iv. ) again 
 electricity would pass away all this can be shown well. 
 
 Of course we must put the sole to earth at the stages (i.), (iii.), and (iv. ), 
 after showing what nature of charge is ready to pass away. 
 
 1 8. Frictional Electric Machines. 
 
 I. The glass cylinder machine. The principle of frictional 
 electrical machines is best understood from a description of one 
 of the simpler forms. As seen in the figure the essential parts are 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 6l 
 
 as follows. A glass cylinder rotates on a horizontal axis. This is 
 pressed on one side by a rubber made of leather stuffed with 
 horsehair and faced with silk ; on the surface of the silk is spread 
 a coating of ' electric amalgam ' made plastic by mixing it in a 
 
 FIG i. 
 
 powdered form with lard. A silk flap passes over the cylinder 
 from the rubber, descending on the other side to nearly half-way 
 down. 
 
 The rubber can be insulated when required. On the other 
 side to the rubber, a large insulated conductor presents a row of 
 fine points to the cylinder. These points nearly touch the glass, 
 and are just below the lower edge of the silk flap. 
 
 Excepting for these points, the insulated conductor is well 
 rounded off. 
 
 When we turn the cylinder, the surface of the glass acquires a 
 + charge, while the rubber acquires an equal charge. 
 
 The +ly excited glass, its charge kept from leakage by the 
 flap of silk, next comes opposite to the conductor with its row of 
 sharp points. On the conductor it acts inductively, repelling + 
 and attracting charge. This electricity is discharged, by con- 
 vection currents of charged air, from the points on to the glass ; 
 so that this latter passes on in its original unexcited condition. 
 The + charge is left on the prime conductor, to be utilised. 
 
 Use of putting the rubber to earlh.^No. have seen that the 
 
62 
 
 ELECTRICITY 
 
 glass surface returns to the rubber in a neutral, or nearly neutral, 
 condition. But the rubber was left charged . Hence, if the 
 original separation of + or were to go on, then after some turns 
 the rubber would become highly charged, and the tendency to 
 part with some of its charge to the glass would at some point in 
 the process at last balance the tendency to excite the glass +ly. 
 Things would come to a standstill. 
 
 But if we put the rubber to earth, we shall get the initial con- 
 dition of the rubber maintained by a steady passage of electricity 
 to earth. Later on we shall treat of this matter from a potential 
 point of view. 
 
 So, if the prime conductor be put to earth, we can use the 
 rubber as a source of electricity. If the rubber and conductor 
 
CH. iv. SIMPLER PHENOMENA OF ELECTROSTATICS 
 
 both be insulated, but they be nearly connected by conducting 
 rods, when we turn the cylinder we shall see a series of sparks 
 bridge over the gap left ; the + charge of the conductor and the 
 charge of the rubber continually neutralising each other. 
 
 II. The common plate machine. In this there is nothing es- 
 sentially different from the cylinder machine. A glance at the 
 figure will explain all. 
 
 There are generally two rubbers ; and in this form of machine 
 they cannot well be insulated if required ; so the machine cannot 
 be used as a source of both + and electricity. Instead of glass, 
 ebonite plates may be used, the rubbers being of amalgamated 
 silk. 
 
 Note. With ebonite two precautions must be taken ; (a) never to heat the 
 plate, only to warm it gently ; (b] after use to clean the plate with paraffin oil 
 from which all water has been removed by the introduction of a lump or two 
 of sodium into the bottle. 
 
 III. Winter's plate machine. -In this the rubber and the points 
 of the prime conductor are more widely separated ; and the prime 
 conductor can therefore acquire a higher level (or potential) of 
 charge without discharge over the glass to the rubber. The 
 rubber can be insulated or not, as required. A curious feature is 
 an addition to the prime conductor in the shape of a large ring of 
 brass enclosed in baked wood. This ring 
 
 increases the ' capacity ' (see Chapter V. 
 4 &c.) of the prime conductor. 
 
 The machine gives very long sparks 
 for its size, as compared with an ordi- 
 nary plate machine. 
 
 19. Miscellaneous Experiments 
 with the Electrical Machine. 
 
 Experiments. (i. ) Illustrating the action of 
 points. Besides the simple experiment con- 
 sisting in discharging the conductor either by 
 presenting to it an earth-connected point, or 
 by fixing the point on to it, experiments can 
 be easily performed in which the rush of elec- 
 trified air from the points is made use of. We may blow out a candle by 
 presenting it to the point. 
 
 Or again, we may by means of the 'electric vane,' shown in the figure, 
 make evident the repulsion between the charged particles of air and the points 
 
64 ELECTRICITY 
 
 CH. IV. 
 
 whence they derive their charge. The air will stream one way, the vane rotate 
 in a contrary sense. 
 
 (ii. ) 7^he insulating stool. A person standing on a stool supported by glass 
 legs can be electrified by his placing the hand on the prime conductor while 
 the machine is worked. He can then ' give sparks ' to other persons, light gas 
 with sparks, &c. ; and his hair will, if fine in quality, show a tendency to 
 stand on end. 
 
CH. V. 
 
 CHAPTER V. 
 
 INTRODUCTORY CHAPTER ON POTENTIAL. 
 (For further, see Chapter X.) 
 
 i. Quantity of Electrification, We have spoken of + and 
 electrification, or electricity, as something measurable and 
 divisible, that is as a quantity. Formerly it was spoken of also as 
 a fluid ; a view that was fairly justified by certain observed phe- 
 nomena, but not by others. 
 
 This conjunction of the words ' quantity ' and * fluid ' leads us 
 to consider in what respect there is, and in what respect there is 
 not, a resemblance between electricity on the one hand, and such 
 fluids as water and gases on the other. 
 
 (a) Points of resemblance. Electricity can be measured as a 
 quantity, as can water and gases. 
 
 In any system of vessels between which there is proper con- 
 nection, there can be statical equilibrium of water only when all 
 is at the same level, or of a gas only when all is at the same 
 pressure. So on any electrical conductor there can be electrical 
 equilibrium only when there is throughout the same electrical 
 level or electrical potential. 
 
 In the former case the water or gas will move from places of 
 higher level or of greater pressure to places of lower level or less 
 pressure respectively, until the state of equilibrium is arrived at. 
 So will there be a flow of + electricity from places of higher to 
 places of lower potential, or of electricity in the opposite direc- 
 tion, until a uniform potential is arrived at. 
 
 Water, and gases, and electricity, are alike unalterable in quan- 
 tity ; only in the case of electricity we must remember that it is 
 the algebraic sum of any + and quantities that is constant, 
 equal and opposite charges giving a zero sum. 
 
 Note. In the above we have neglected the compressibility of water, and 
 the action of gravity on gases, as relatively insignificant. 
 
 F 
 
66 ELECTRICITY 
 
 CH. V. 
 
 (b) Points of difference. Water and gases all possess mass ; 
 indeed, from a mechanical point of view they are merely masses 
 endowed with certain mechanical properties, their chemical com- 
 position signifying nothing. 
 
 Electricity on the other hand has, most probably, no mass. 
 There is, -in the case of electricity, no gravitation-force, no 
 mechanical inertia, no mechanical kinetic energy. In a word, 
 electricity is not matter, while all fluids are. 
 
 If we are to call electricity ' a thing ' at all, it must be justified 
 by the precedent of calling energy ' a thing ' ; and we merely 
 express by the term the unalterability of electricity as a quantity. 
 
 In the case of water and gases we have little or nothing 
 answering to the duality of electricity ; nothing like che attractions 
 and repulsions between unlike and like charges respectively ; nor 
 is there anything like the variation of the intensities of these forces 
 with the nature of the medium in which the charged bodies are 
 situate (see Chapter X. 25, end). 
 
 To return to the subject of our section. 
 
 Electricity is measured as a quantity; and the unit is that quantity 
 which at unit distance (i.e. i cm.) from an equal quantity repels it 
 with unit force, or with one dyne ; the medium between the two 
 quantities being assumed to be air. 
 
 2. Electrical Level, or Electrical Potential In the con- 
 sideration of the statical equilibrium of bodies of water, or the 
 work to be got out of any water supply, we are concerned mainly 
 with level, differences of level, and quantity of water. 
 
 We usually take ' the sea-level ' as our arbitrary zero of level, 
 and reckon the level of any body of water as so many feet + 
 or ; the feet being measured vertically up from, or down from, 
 this arbitrary zen> respectively. 
 
 Now in electricity we have, roughly analogous to the above, 
 the question of electrical potential,. difference of electrical potential, 
 and quantity of electricity. 
 
 We take as a zero of level (whether it is quite arbitrary or not 
 we do not here discuss) the electrical potential of the earth ; and 
 we reckon potentials as + or when above or below our zero 
 respectively. 
 
 In gravity levels the test of the same level and different levels is 
 
CH. v. INTRODUCTORY CHAPTER ON POTENTIAL 6/ 
 
 to see whether such a fluid as water will, or will not, flow from the 
 one spot to the other when free to move. We call that the 
 higher level from which the water flows. In electrical levels 
 (or potentials) two points are at the same potential or different 
 potentials, according as + electricity is not, or is, urged from 
 the one to the other point when free to move. That is called 
 the higher potential from which the + electricity is urged. 
 
 Thus in gravity a spot has a level if water will run from the 
 sea-level to it ; in electricity a point is at potential if + elec- 
 tricity will flow from the earth to it. 
 
 In the dual nature of electricity, and in the opposite move- 
 ments of + and electricity, we perceive the limitations of our 
 rough analogy. 
 
 3. Measurement of Differences of Electrical Lsvel by 
 Work. In Chapter X. we shall discuss this matter of measure- 
 ment of potential more fully. Here we wish only to show in 
 what way the measurement is made. 
 
 The measurement of gravity levels may be made by means ot 
 a tape, caused to hang vertically by being weighted at the end ; 
 or by some equivalent method in which the action of gravitation 
 gives us either a vertical line to measure along, or a horizontal 
 plane from which the vertical can be deduced. In all such 
 cases the measurement is made directly in feet. 
 
 But there is another way that is theoretically possible. 
 
 We shall in Chapter X. consider the meaning and measure- 
 ment of work. Here we will only say that in lifting weights from 
 a lower to a higher level we do work ; that the work done is pro- 
 portional to the weight raised, and also to the vertical height 
 through which it is raised ; and that it is independent of the 
 route, whether direct or roundabout, by which the weight is raised 
 from the lower to the higher level. 
 
 Now if we could in some way accurately register work done, 
 then, by carrying some standard weight from one place to another, 
 we could, by the amount of work done, estimate the difference of 
 level of the two places. There has, however, been elaborated no 
 exact method of this sort, because the former method is so very 
 simple and direct. 
 
 In electricity, on the contrary, there is known no method 
 analogous to the direct plumb-line measurement. The only 
 
 F 2 
 
68 ELECTRICITY 
 
 CH. v. 
 
 conceivable method of measuring the difference of electrical level 
 between two points is by measuring the work done in moving a 
 + unit of electricity from the one point to the other. 
 
 If we assume that the reader understands in what unit work 
 is measured, we can say that we have now shown how to give 
 an exact numerical meaning to that symbol ' V ' which is used 
 to express the potential of a point with respect to the earth as 
 zero. 
 
 Notes. (i. ) We are said to do + work when we move against the lines of 
 force, or expend energy ; we are said to do work when we move down the 
 lines of force, or have energy expended on us. 
 
 (ii.) The reader will understand from the above section, that the electrical 
 potential of a point in space (or of a body) is + , zero, or , according as we 
 do + , zero, or work respectively in moving a + unit of electricity from the 
 earth to the point in space (or body). The potential is higher as this work is 
 greater. 
 
 The + unit of electricity was defined in i of this chapter. 
 Any work done in moving it from one place to another is due to 
 the repulsive and attractive actions of other electrical quantities 
 on this + unit. [As regards sign, see Chapter IV. 7.] 
 
 4. Elementary Ideas on ' Capacity.' A certain quantity of 
 water will fill a certain vessel to a definite level, this level depend- 
 ing on the quantity of water and on the dimensions of the vessel. 
 The more the water, and the less the horizontal section of the 
 vessel, the higher the level to which the latter will be filled. 
 
 In electricity, when we charge a conductor with a quantity Q 
 of electricity, we raise the conductor to an electrical potential V. 
 This potential will be greater or less according to a property of 
 the conductor which we may naturally call, by analogy, its electrical 
 capacity. We call the capacity greater or smaller according as the 
 potential V, to which a given charge Q of electricity will raise it, 
 is smaller or greater respectively. Now we have already indicated 
 how Q and V are defined and measured. Since capacity (K) is a 
 new term we may define it, and express it numerically, by the 
 relation 
 
 which is equivalent to saying tfie capacity of a conductor is directly 
 proportional to the quantity of electricity required to raise it to a 
 
CH. v. INTRODUCTORY CHAPTER ON POTENTIAL 69 
 
 certain potential, and inversely proportional to the potential to which 
 it is raised by a certain quantity. 
 
 Of the units in which V and K are measured we shall treat 
 more fully in Chapter X. We see, however, from the above 
 definition of K, that 
 
 A conductor has unit capacity when unit quantity raises it to 
 unit potential, or to unit electrical level above the earth. 
 
 5. Lines of Force, and Equipotential Surfaces, We have 
 in Chapter III. sufficiently explained what are meant by lines of 
 force. The reader can easily, mutatis mutandis, define for himself 
 lines of force in a gravitation field of force, or in an electrical field 
 of force. 
 
 Thus in an electrical field a line of force is a line along which 
 a particle charged with + electricity, and free from all other forces, 
 would move under the influence of electrical forces ; it being 
 assumed that the movement is indefinitely slow, so that there is 
 not any ' centrifugal ' desertion of the line of force when this is 
 curved. 
 
 It is along these lines of force that electrical level is measured; 
 just as it is along the lines of gravitation force (i.e. vertical lines) 
 that gravitation levels are measured with the plumb-line. 
 
 It is in movement of electricity along the lines of electrical 
 force, as in movement of masses along the lines of gravitation 
 force, that we do work. 
 
 Surfaces over which we can move our + unit of electricity 
 without doing work are called equipotential surfaces. 
 
 In the case of gravitation these are clearly horizontal surfaces ; 
 a mass moved over these surfaces always cuts at right angles the 
 lines of force of gravitation, and hence no work is done, or 
 these horizontal surfaces are equipotential. 
 
 In electrostatics let us consider the simple case of a single 
 isolated charged particle. Since a + unit would be urged straight 
 from it or straight towards it, according as the charge on it is 4- 
 or , it follows that the lines of force are straight lines radiating 
 from the particle as centre. 
 
 What are the equipotential surfaces, i.e. level surfaces, or surfaces 
 of no work ? They must be those over which our + unit always 
 cuts the lines of force at right angles, thus doing no work. 
 
 That is, they must be spheres having the particle as centre 
 
70 ELECTRICITY CH v. 
 
 and the lines of force as radii. In more complicated systems of 
 electrical charges we may have lines of force curved about in any 
 way whatever. But still the equipotential surfaces, over which 
 our -f unit moves without doing any work, will be those that cut 
 at right angles the lines of force. 
 
 From what we have said the reader will see that wherever are 
 lines of force, there is rise or fall of potential ; and conversely. 
 
 A region where are no lines of force must be all at one 
 potential ; since there can be no work done in moving our + 
 unit against zero force, and therefore there can be no differences 
 of electrical level. And conversely, where a region is all at one 
 potential, there are no lines of force. 
 
 6. Induction ; from a ' Potential ' Point of View. We will 
 now consider briefly what light the above considerations throw 
 upon the matter of electrostatic induction. 
 
 Definition of a conductor. We must first, however, define 
 what we mean by the often-used term conductors. We mean 
 bodies on which electrical charges are perfectly free to move, 
 bodies which can resist no electrical stress due to differences of 
 electrical potential. 
 
 By our definitions then it follows that when any conductor is 
 left until the electrical charge has had time to arrange itself, it 
 will be all at one potential ; for were two parts at different poten- 
 tials, the electrification would readjust itself until the two places 
 were at the same potential. This readjustment is so rapid that 
 we can consider it to be practically instantaneous. 
 
 The reader may object that one might conceivably have a con- 
 ductor of which two parts remain at different potentials for the reason 
 that there is no electrical charge to be readjusted. The answer to 
 this is that experiment shows us the universal presence of unlimited 
 quantities of + and electrifications in all bodies, these being equal 
 in amount if the body be in what we call an uncharged condition. 
 If two parts of the conductor be at different potentials, then there is 
 unlimited + electricity ready to flow from the higher level to the lower, 
 and unlimited electricity ready to flow from the lower to the higher 
 level ; and a flow takes place until all is at one potential. 
 
 Hydrostatic analogy to electrostatic inductions. We may with 
 advantage discuss, in connection with the matter of electrical 
 induction, a hydrostatic analogy. We shall consider the sea-level 
 
CH. v. INTRODUCTORY CHAPTER ON POTENTIAL /I 
 
 as analogous to the electrical level or potential of the earth ; a hill- 
 side as analogous to a region through which there is a fall of 'electric 
 potential, or what we may term an electric hill : and a trough of 
 water as (very roughly indeed) analogous to an electrical conductor. 
 In a still rougher sense we may sometimes take a tower as analo- 
 gous to a + electrical charge \ and a 7^// as analogous to a elec- 
 trical charge. We may now note the following facts. 
 
 (a) The water in the trough will always be at one level, or 
 water will have no more tendency to flow in one direction than in 
 the other, in whatever position the trough be placed. 
 
 (b) If the trough be placed upon a hill-side, the whole will be 
 above the level of the sea ; and water would flow from any part of 
 the trough down to the sea, if connected therewith by a pipe. 
 
 (c) Further, the water will rise at one end and will sink at the 
 other ; still remaining level. It will be shallower at the end that 
 lies uphill, and deeper at the end that lies downhill. 
 
 (d) If instead of a trough we have a trench of unlimited depth, 
 and connected this with the sea-level, water will flow out until all 
 is at the sea-level. The water will then be at the sea-level, but 
 will be deep below the surface of the hill. 
 
 (t') If the hill-side now sank, with the trench still cut in it, to 
 the sea-level, the water in the trench would be now below the sea- 
 level ; and water would flow from the sea into the trench until the 
 level was again raised to sea-level. 
 
 (/) Let there be a trough of water whose surface is at the sea- 
 level. Now let this trough be raised up on to a hill-side and there 
 be put in a sloping position. The water will dispose itself as in 
 (c) above. Now let a partition be put in the middle of the trough, 
 parallel to the ends, and let the whole be replaced in its original 
 position. It is obvious that in the one half we shall now have the 
 surface above, and in the other half the surface below, the sea-level. 
 On removing the partition the original state of things is restored, 
 and the surface is once more all at sea-level. 
 
 (#) Let us build a tower on the sea bottom so that its top 
 reaches the sea-level. If the sea-bottom were to rise up, the top 
 of the tower would now be above sea-level. 
 
 (h) Let us dig a well on the hill so that its bottom reaches 
 sea-level. If the hill sinks to sea- level, the bottom of the well will 
 be below this. 
 
/2 ELECTRICITY CH. v. 
 
 We will now consider the facts in electrical potential, or elec- 
 trical level, that are analogous to the above. Both by the word- 
 ing of the statements and by the correspondence of the lettering 
 (a\ b', c, &c.), the reader will be able to see how the two sets of 
 facts answer the one to the other respectively. We must, however, 
 .again repeat that, though such analogies are interesting, too great 
 stress must not be laid upon them ; above all we must not argue 
 from analogy. 
 
 Let us suppose that there is a room whose walls &c. are all 
 connected with the earth, and are therefore at the same electrical 
 level, or potential, as the earth ; this we take for our arbitrary zero 
 of potential. Further, let there be a +ly charged body A sus- 
 pended in this room. 
 
 There will be a fall of potential, or an electrical hill, from this 
 body A, down to the walls, &c. Such was the case, if only we 
 change the sign of the charge, in Chapter IV. 14. 
 
 If any insulated uncharged conductor B be placed between the 
 walls and the body A, this body B will also be at a + potential ; it will 
 take work to bring a + unit of electricity from the earth up to B, 
 in consequence of the presence of A. So that A has given rise to 
 an electrical hill running down to the walls ; and all insulated un- 
 charged bodies placed, as it were, on this hill, will be at a + poten- 
 tial ; though of course at a lower potential than is A, since it will 
 take less work to bring a + unit of electricity up to B than up to A. 
 
 The corresponding electrical facts are as follows. 
 
 (a) A conductor such as B will always be at one electrical 
 potential over all, wherever it be placed. 
 
 (b 1 ) When the insulated conductor B is placed between A and 
 the walls, i.e. on an electrical hill, the whole is above the electri- 
 cal potential of the earth ; and -f electricity would flow from any 
 part of this conductor to the earth, if it be connected therewith by 
 a wire. 
 
 (c'} Further, we shall find electricity at the end nearer to A, 
 and -4- electricity at the end nearer to the walls. This readjust- 
 ment, or separation of + and electricities, will be just such as 
 will leave B all at one potential ; or will be such that a + unit of 
 electricity will move on B in any direction indifferently, and this 
 in spite of the presence of A (see Chapter IV. 14, Experiment (i )). 
 
 (d 1 ) If we connect B to earth by a wire, + electricity will 
 
cir. v. INTRODUCTORY CHAPTER ON POTENTIAL 73 
 
 flow out of B until it is all at zero potential. It will then be charged 
 with electricity, though all at zero potential. Since B is all at 
 one potential, this flow of electricity to earth will take place from 
 any point on it indifferently (see Chapter IV. 14, Experiment (ii.)). 
 
 (e f ) If A be now discharged, or reduced to the level of the 
 earth's potential, our electrical hill disappears. It is now found 
 that B is at a potential ; and + electricity will flow from the 
 earth into it until it be again at the earth's potential. 
 
 (/') Let an uncharged insulated cylinder, capable of division 
 about its centre \see Chapter IV. 14, Experiment (iii.)), be placed 
 between A and the walls. It will now exhibit a + charge at one 
 end, and a charge at the other. If the two halves be separated 
 and A be removed or discharged, we shall find these halves to be 
 above and below zero potential respectively. On again connect- 
 ing the two halves the whole resumes its original condition of 
 zero potential. 
 
 (g') When a body is in a region of potential let us give it 
 such a + charge as will raise it to zero potential. If the region 
 rises in potential, the charge given to the body will cause it to be 
 now above zero potential. 
 
 (/i 1 ) So when a body is in a region of 4- potential let us give 
 it such a charge of electricity as will reduce it to zero potential. 
 If the region now sink to zero potential, the body with its 
 charge will be below zero potential. 
 
 From what we have said above it is clear that all cases of 
 'induction can be regarded as electrical redistributions that take 
 i)lace on conductors when placed in regions of varying (or unequal} 
 potential : these redistributions being such as to maintain the con- 
 ductor at the same potential all over. 
 
 Note. Charging the gold-leaf electroscope by induction. In Chapter IV. 
 14 (iv.) we indicated how we could charge the gold-leaf electroscope induc- 
 tively. It is now time to explain more exactly what occurs. 
 
 Let the electroscope be uncharged, and let a + ly excited rod be approached. 
 The electroscope will be in the field between the rod and the earth, and it 
 will be at a + potential. There will, therefore, be a field of force between 
 the leaves and the wall ; and + electricity will be ready to flow from any 
 part of the instrument to earth. The leaves will, therefore, move from each 
 other towards the walls, or will ' diverge with + electricity. ' If now we 
 touch the instrument with the hand, we reduce it to zero potential, and the 
 leaves collapse ; it now has a charge sufficient to keep it at zero potential 
 
74 ELECTRICITY CH. v. 
 
 in the presence of the + charged rod. If the hand be removed, and then the 
 rod, the instrument will be at a potential in virtue of its - charge, there 
 being now no region of + potential to be counteracted ; and the leaves will 
 'diverge with electricity.' The approach of a charged rod will cause 
 its potential to be still greater, and the leaves will diverge more. But the 
 approach of a + charged rod will again cause the region about the electro- 
 scope to rise in potential. Hence, according to the strength of charge of the 
 + ly excited rod and the nearness of its approach, the electroscope will become 
 less , or zero, or even + in potential ; and its leaves will diverge less, 
 collapse, or diverge with +,' in the three cases respectively. 
 
 7. Necessity of distinguishing Sign of Charge and Sign 
 of Potential, Where we have an isblated charged body, there the 
 potential of the body is + or according as the charge is + or 
 in sign. For we do + or work respectively in moving our 
 test + unit of electricity from the earth to the body. But if we 
 have more than one charged body we cannot predict the sign of 
 the potential of a body from its charge. Thus if we have a pail 
 charged +, a ball hung inside it may be at a + potential even if 
 it be itself charged ; for, owing to the presence of the pail, it 
 may require + work to bring our test + unit of electricity from 
 the earth up to the ball. 
 
 So again, in Chapter IV. 14, Experiment (ii.), the conductor 
 C D was at zero potential although it had a + charge on it. 
 
 8. Further on Capacities. When a body is charged with a 
 quantity + Q, there is a complementary quantity Q on the 
 surrounding surfaces. 
 
 If these surroundings are so far off that this charge Q exerts 
 no perceptible force on our test-charge of a ' + unit ' as we carry 
 it from the earth up to the body, then the work done on our 
 ' + unit ' as we so carry it will be independent of the surroundings ; 
 it will depend only on the charge Q of the body, and on the shape 
 and size of the latter. And the potential of the body, measured 
 by this work, will also depend only on its shape, size, and charge. 
 
 In such a case the body is said to be isolated, and it has a 
 definite capacity. 
 
 But if the surroundings be not so far removed as to make the 
 above condition hold, then the work done on our ' -f unit ' will 
 depend in part on the position of the surroundings. In such a 
 case the potential of the body depends not only on its shape and 
 size and on the charge given to it, but also on the position of 
 
CH. v. INTRODUCTORY CHAPTER ON POTENTIAL 75 
 
 surrounding surfaces. The body is then not isolated ; and we 
 must consider the whole system of body and surroundings. 
 
 Note. A little consideration will show that if a body previously 'isolated' 
 be brought near (e.g.} to the walls of the room, most of the complementary 
 Q will now be on the part of the walls near to the body. This will have the 
 effect of lessening the work required to bring our + unit up to the body. In 
 other words, the nearer approach to the walls will increase the body's capacity- 
 
ELECTRICITY 
 
 CH. VI. 
 
 CHAPTER VI. 
 
 ELEMENTARY DISCUSSION OF CONDENSERS. 
 (For further, see Chapter X.) 
 
 i. General Ideas, Apparatus used. In this chapter we 
 consider the alteration of the capacity of a conductor as the dis- 
 tances and positions of surrounding conductors alter. Or, more 
 accurately, we consider the electrical capacity of the system of 
 conductors taken as a whole. 
 
 We shall use the following apparatus. 
 
 FIG. i. 
 
 (i. ) The sliding condenser plates. (ALpinufs condenser. ) A and 
 B are two discs of metal, having well-rounded edges. They are 
 
FIG. ii. 
 
 CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 77 
 
 insulated by being fixed on to elbow-bent rods of glass. These 
 rods are set in stands that run, guided by rails, along a board 
 three or four feet in length. 
 
 (ii.) A form of Peltier electrometer. Insulated inside a glass 
 shade, the method of insulation being indicated in the accompany- 
 ing figure, is a frame of wire abed. Balanced on a steel point 
 is a magnetic needle n s, provided at the 
 ends with gilt pith- balls. 
 
 The apparatus is so placed that the 
 needle, as directed by the earth's field, rests 
 with the pith-balls just lightly touching the 
 sides of the brass frame. 
 
 When the knob K is connected with a 
 conductor at a + or potential, the frame 
 and needle all become of a + or potential 
 with respect to the earth ; there is an electric 
 field, and the needle is urged away from the 
 frame towards the walls. 
 
 Or, in ordinary two-fluid language, if we 
 connect K with a conductor that we have charged 4- or , we 
 shall have the needle urged away from the similarly charged frame 
 towards the walls of the room ; and it finally rests in such a 
 position that the couple due to the electrostatic field balances 
 the couple due to the earth's magnetic field. 
 
 (iii.) We shall also use a charged ebonite rod with which 
 to test the sign of the charge that may be at any time causing the 
 needle of the electrometer to be deflected ; this testing having 
 been explained earlier. 
 
 We may add that the electrometer can be made more or less 
 sensitive by weakening or strengthening the magnetic field acting 
 on the needle, by means of a controlling magnet, as explained in 
 Chapter XVII. 9. 
 
 (iv.) Source of electricity. For charging the plates we shall 
 use an electrophorus or frictional machine ; and shall assume that 
 it is so regular in its action that it is of a constant potential. That 
 is, the machine will continue to give a charge to a conductor, 
 cease to do so, or receive a charge back from the conductor, 
 according as this latter is below, up to, or above, this fixed poten- 
 tial of the machine respectively. 
 
?8 ELECTRICITY en. vi. 
 
 The analogy would be a water-supply kept at a certain fixed 
 level. 
 
 2, Experiments with the two Condenser Plates. In the 
 following experiments the two plates A and B are connected with 
 the electrometers E A and E B respectively, and are insulated, unless 
 the contrary is stated. 
 
 Experiment. .(i.) Let B be at some distance, say -5 metre, from A, and 
 let both plates be initially uncharged. Now charge A with (say) + electricity, 
 until it will receive no further charge from the source. It will be found that 
 
 FIG. i. 
 
 the needle E A is deflected with + electricity. At the same time the usual 
 proof-plane test will show that on the face of B turned towards A there is 
 electricity ; while E B is deflected with + electricity. 
 
 In two-fluid language we should say that the * charged body 
 A induces a charge on the face of B turned towards it, repelling 
 the complementary + to the further side of B and to E B (see 
 Chapter IV. 14). 
 
 In potential language we should say that there is on B and 
 E B such a redistribution of the total charge (whose algebraic sum 
 is zero) as to leave this system at one potential. Since the whole 
 system is between a + charged body A and the earth, it must be 
 at a + potential, since it would take work to bring our + unit 
 from earth up to B. Hence + electricity tends to run from all 
 parts of the system to the earth ; and hence the needle of E B is 
 urged down an electric field between the brass frame and the 
 walls. 
 
 Experiment. (ii. ) Put B to earth, by touching any part of it. It will be seen 
 that E B returns to zero deflexion, while EA is deflected less than it was before, 
 and further that A will now receive more charge from our source. 
 
 In two-fluid language we can say that we have allowed the 
 repelled + to pass away. In consequence of this, A has further 
 inductive action on B j so that there is more on B than 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 79 
 
 before, and more + passes to earth. Corresponding to this larger 
 
 charge on B, there is drawn over to the surface of A that faces 
 B more of the + charge on A ; and, as a consequence, E A is less 
 deflected, and the whole will receive further charge. 
 
 (The reader may here look at the diagram given at end of this 
 section. It represents roughly the effect that the presence of B 
 has on the distribution of the charge on A.) 
 
 In potential language we may say that we have put B and E B to 
 zero potential. There is therefore no field of force between the 
 brass frame of E B and the walls, and hence the needle has no 
 couple deflecting it. 
 
 Now, there was in (i.) on the side of B turned towards A a 
 certain charge. . This by itself would have made B of a poten- 
 tial ; but, owing to the presence of the + charge on A, B was yet 
 at a + potential. In (ii.) we have B at zero potential, since it is 
 to earth. It seems clear, then, that there must now be a larger 
 charge on B than there was before, since it is as close to the + 
 charge of A as it was before and is yet at zero potential. 
 
 For a similar reason to that indicated in Chapter V. 8, note^ 
 the presence of this larger charge near A will lessen the work 
 required to bring a ' + unit ' up to A. In other words the poten- 
 tial of A will be in this case (ii.) lower than it was in case (i.). 
 
 Experiment. (iii. ) Re-insulate B, and approach it nearer to A. We shall 
 find that E B again diverges with 4- , while E A falls a little in its deflexion. 
 
 And further, as we should expect from the falling off of the deflexion of E A , 
 we can give A more charge from our source. 
 
 In two-fluid language we have a further separation of -f and 
 
 on B, the + being repelled to the regions (such as E B ) remote 
 from A. There is also in A a further concentration of the charge 
 on the face turned toward B, and consequently a weakening of 
 the charge in E B . We should thus expect to be able to give more 
 charge to the whole, A and E A . 
 
 In potential language we have moved B higher up the ' elec- 
 trical hill ' that exists between A and the earth, or into a region of 
 higher potential than before. If, before, its charge of electricity 
 was such as to keep it just at zero potential, this charge will be 
 insufficient to keep it at zero now that it is nearer to the + charged 
 A. It will therefore again become of a 4- potential, or + electricity 
 
8O ELECTRICITY CH. vr. 
 
 will be ready to flow from it to the earth. The electrometer 
 needle is again in an electric field, and is again deflected. 
 
 At the same time the potential of A will be lowered for a 
 reason similar to that given in case (ii.). 
 
 Experiment. (iv.) Next put B to earth. We find that EB is no more 
 deflected, E A falls slightly. 
 
 The discussion of this will be similar to that of Experiment (ii.). 
 
 Experiment. (v. ) Again insulate B and remove it further from A. The 
 needle EB is deflected with this time, while E A is deflected more strongly 
 with + . 
 
 In two-fluid language the removal of A and B from each other 
 causes some of the + and electricities, concentrated on the op- 
 posed faces of A and B respectively, to pass away from these faces 
 to the other portions of the conductors, and so to E A and E B . 
 
 In potential "language the potential of A will be raised, for a 
 reason similar to that given in cases (ii.) and (iii.). There will be 
 a stronger field between the wire frame of the electroscope and 
 the walls, and the needle will be more deflected. 
 
 With respect to B, the charge that kept it at zero potential 
 when in the nearer presence of A, is too much for this purpose 
 when A is removed to a greater distance ; it renders B in sign 
 of potential ; and the charged needle of the electrometer is in 
 an electric field running down from the walls to the frame, and 
 therefore moves up this field away from the brass frame. 
 
 It is important to note that by charging A when B is near, 
 and by then removing B, we can get in A a charge of a much 
 higher electrical potential, or level, than that of the source from 
 which we charged A initially. 
 
 Experiment. (vi.) When A Bare very close and B is to earth, the charge 
 that can be given to A [roughly measured by counting the number of sparks 
 given] will be seen to be very large indeed as compared with the charge that 
 can be given to A when isolated. 
 
 The proof-plane, if it can be used, will show that this charge is mainly on 
 the face of A turned towards B ; while on the opposed face of B is an opposite 
 charge of equal magnitude (see 4). 
 
 If B be pushed still nearer, a brilliant blue spark will be seen to span the 
 gap between A and B, and a strident ' crack ' will be heard. 
 
 If a person holding B touch A, this discharge will pass through him and 
 will give him a shock. Such an experiment must, however, be tried with 
 caution. 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 8[ 
 
 The accompanying figures show roughly the cases of A charged 
 when isolated, and of A and B acting as a condenser, respectively. 
 The crowding of the signs + and 
 is meant to indicate the con- 
 centration of charge. 
 
 3. Discussion of the Terms 
 'Bound' and 'Free.' We have 
 seen that when a body is charged 
 with a certain quantity, say + Q, 
 of electricity, there is induced 
 
 : , . , . , FIG. ii. FIG. iii. 
 
 on surrounding objects an equal 
 
 quantity of opposite sign, or Q (see Chapter IV. 16). The 
 distribution of this quantity + Q on the conductor depends on its 
 form and on the relative positions of surrounding bodies. 
 
 It is found to be part of the essential nature of * electricity ' 
 that a + or charge does not exist alone ; it is always the one 
 or other side of an electrical field. 
 
 Now the charge on the plate A (see figure at end of 2) in- 
 duces on the whole an equal and opposite charge on the walls, &c., 
 and the greater part of this will be on that surface of B which is 
 opposed to A ; since B is by far the nearest part of the earth that 
 is presented to A. 
 
 If we now cut off B from the earth and insulate it, without 
 otherwise disturbing the system, and then put A to earth, what 
 will happen ? 
 
 To begin with we may say roughly, but as we shall see in 6 
 not quite exactly, there will pass away from A that part of the 
 charge that can, by so passing away, get at and neutralise the cor- 
 responding part of the induced charge. If B were still to earth, 
 then all the charge would pass from A, and the two sides of the 
 field (the equal and opposite + and charges) would close in 
 and leave no field or charge ; the + and would entirely ' neu- 
 tralise ' each other. But B has, we suppose, been insulated again 
 before A is put to earth. Hence, to speak roughly, that part of the 
 charge on A which answers to the charge induced on B, will have 
 no tendency to pass away, since it cannot thereby get at the 
 charge on B, which is the opposite side of its field. But if we 
 stood on B, insulated from the earth, and then touched A, we 
 should withdraw from A all that part of the charge answering to 
 
 G 
 
82 ELECTRICITY CH. vi. 
 
 the charge induced on B ; while on A would remain that part 
 of its charge which has its corresponding induced charge on the 
 walls and ceilings, &c., from which we are now insulated. When 
 B is insulated, therefore, we can divide the charge on A into two 
 portions ; one part can be withdrawn by a person in connection 
 with the walls, ceiling, &c, and unconnected with B, but cannot 
 be withdrawn by a person connected with B and insulated from 
 the walls, &c. ; the other part can be withdrawn by a person con- 
 nected with B and unconnected with walls, c., but not by a person 
 insulated from B. The former is free with respect to any person 
 connected with the walls, &c., the latter is bound with respect to 
 such a person. ,-^r 
 
 This is the origin of the terms bound and free ; the reader 
 will see in how relative a sense they must be understood. 
 
 4. Conditions affecting the Magnitude of the ' Bound 
 Charge. When B is very close to A we may neglect the * free ' 
 charge on A as relatively insignificant (see 3). The ' charge ' 
 of the condensing system, composed of A and B, will then mean 
 the ' bound charge ' residing on that face of A that is turned to- 
 ward B ; the inseparable accompaniment of an equal and opposite 
 charge on B will be understood, but not generally mentioned. 
 
 The magnitude of the charge can be experimentally shown to 
 depend on several conditions. 
 
 (i.) On the size of the plates A and B. If these are equal in 
 area, and if the distance between them be very small as compared 
 with the diameter of either, then the charge will be (approximately) 
 directly proportional to the area of the plates. 
 
 (ii.) On the distance between the plates A and B. The nearer 
 the plates, the greater the charge. And, with the condition just 
 given, the charge is inversely proportional to the distance between 
 the plates. 
 
 (iii.) On the difference of potential between the plates A and B. 
 We have not yet explained exactly in what units we measured 
 electrical level or potential ; but we have sufficiently indicated the 
 nature of the unit employed. For further, the reader must wait 
 for Chapter X. 9, 19, 25, &c. W T e may, hovvever, here state 
 that the charge is directly proportional to the difference of the 
 potentials of A and B. 
 
 (iv.) On the nature of the dielectric (see note). If the charge is 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 83^ 
 
 of a certain magnitude Q when the dielectric is air, then cceteris 
 paribus it will be <r.Q when the dielectric is some other substance. 
 This n is (as we shall see in Chapter IX.) the specific inductive 
 capacity of the particular dielectric. If then Q represent the 
 numerical value of the bound charge, V^ and V 2 the potentials of 
 the plates A and B, and K be a quantity including the area of 
 the plates, the distance between them, and the specific inductive 
 capacity of the dielectric, and called the capacity of the condenser, 
 then we shall have the formula ....... ..... 
 
 or Q = K . V, ........... 
 
 when the outer coating is put to earth and so is at zero potential. 
 
 Note. ' 'Dielectric, ,' Any non-conducting medium, be it solid, liquid, or 
 gas, that intervenes between two conductors, is called a dielectric. It is across 
 dielectrics that electrostatic attractions and repulsions are exerted ; and it is in 
 dielectrics that fields of electrostatic force exist. The usual dielectric is air ; 
 but in certain instruments glass, or ebonite, is the dielectric. 
 
 5. An Isolated Body considered as the Limiting Case of a 
 Condenser. If the plate B be at zero potential, then the formula 
 becomes 
 
 Q=K. V 
 
 where V is the potential of A. 
 
 Now, as stated in Chapter V. 8, it can be shown (see 
 Chapter X. 19) that when B and all other surrounding bodies 
 are at a distance from A very great as compared with the dimen- 
 sions of A, then any further change in their positions and dis- 
 tances will make no difference in the capacity of A ; that is, 
 under such conditions the value of the capacity K depends only 
 on the size and shape of A. 
 
 We thus arrive at our old formula for isolated bodies, viz. : 
 
 Q = K.V;orV = |. 
 
 6. Alternate Discharge. Let us have two equal plates A 
 and B very close together, B being put to earth, and A being 
 charged to its fullest extent ; and let us call the charge on A, 
 unity. 
 
 The total induced charge answering to the charge -f i on A 
 will be i j and of this the larger part by far will be on B. Let 
 
 UMivcT" or C4L ,1 
 
ELECTRICITY 
 
 CH. VI. 
 
 this larger part be called m, where m is some large proper fraction 
 
 such as 22. ; the other being on the walls, &c. Then as lone 
 100 100 
 
 as A is untouched we can say that a charge + i on A ' binds ' a 
 charge m on B, in the sense that we cannot withdraw it from 
 B as long as A is insulated. 
 
 In the same way, since the plates are equal and similarly 
 situated, a charge i on B would * bind ' a charge + m on A. 
 Or. more generally, a charge Q on either would bind a charge 
 
 ;// Q on the other. 
 
 In the case we have supposed, the charge + i on A binds 
 
 m on B. This charge m on B can bind only + m x m, 01 
 + m 2 , on A. 
 
 Hence, if we insulate B and touch A, putting it to earth, we 
 withdraw an amount + (i m 2 ) from A, and leave on it + m* 
 
 bound by the m on B. The reader may put e.g. - ' for ;;/. 
 
 Insulating A and touching B, we withdraw from B what A 
 does not bind ; that is, since the + m* left on A will bind 
 
 mx m*, or m 3 , on B, we leave ; 3 on B and withdraw 
 m m 3 from B, and so on. 
 
 Each time that one plate is put to earth, part of the charge on 
 the other plate becomes 'free.' 
 
 This process of alternately putting either plate to earth is 
 called the alternate discharge. 
 
 We give here a table. 
 
 Withdrawn from 
 A + 
 
 On A + Binds On B - 
 
 Withdrawn from 
 B - 
 
 I 111? 
 
 ^ 
 
 > m ' - 
 
 
 
 m' <^ 
 
 
 m nt 3 
 
 m* - m* 
 
 
 ^> m* 
 
 
 
 ' "" < 
 
 
 m* m 5 
 
 
 m* J^ 
 
 S* 
 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 85 
 
 Note. As regards the exact meaning of m, the reader will understand the 
 following explanation better when he has read Chapter X. 
 
 Let us consider the plate A only. There is on the inside surface next to B 
 a bound charge Q ; and on the outside surface, opposite to the relatively distant 
 walls, a relatively insignificant charge Q'. The charge Q raises the inner 
 surface, and the charge Q' raises the outer surface, to a potential V A , which is 
 necessarily the same for the two surfaces. If then K is the capacity of the 
 inner surface, and K' the capacity of the outer surface, we have the relation 
 
 V A - ^ = ^ We may call K the ' bound capacity ' of A, or the capacity 
 K. .K. 
 
 of the condensing system, its magnitude depending on the shape and size of A 
 and also on the position &c. of B. We may call K' the ' free capacity ' of A 
 or rather of its outer surface ; its magnitude depends on the shape and size of 
 this outer surface, but on nothing else provided that the walls &c. are so remote 
 that the outer surface of A may be considered to be ' isolated ' with respect to them. 
 From the relation given above we easily obtain the result that 
 
 / 
 
 Remembering then that Q + Q' is the total charge on A, and that the 
 
 K * 
 
 charge on B equals Q in magnitude, we see that in - -. 
 
 K + Iv 
 
 Experiments* (i.) With the plates A and B, each connected with a Peltier 
 electrometer, the alternate discharge can be shown ; though the arrangement 
 is not one adapted to give indication for more than a few discharges. As we 
 touch A we receive a spark, and EB, previously undeflected because at zero 
 potential, will be deflected with electricity, B and E B being now at a 
 potential, while E A will fall to zero deflection. If we now touch B, E B falls 
 to zero and E A is deflected with + , and so on. 
 
 (ii. ) A Leyden jar (see next section) is mounted on an insulating stand. 
 While it is being charged the outer coating is put to earth, and then the whole 
 is left insulated. 
 
 We can, by alternately touching the knob connected with the inside, and 
 the outside coating, draw off sparks of + and electricity in turn?. The 
 signs of these alternate charges can readily be tested as in Chapter IV. 
 
 7. Leyden Jars. When it is required merely to store a large 
 charge at the potential of the source, and when there is no need 
 to alter the distance between the plates, the condenser usually 
 takes the form of a jar of glass having inside and outside coatings 
 of tin-foil. A knob gives connection with the inside of the jar ; 
 this knob should, if possible, pass up from the inside without any 
 connection with the neck of the jar, in order to insure good 
 insulation. 
 
 Since the charge, cczteris paribu s, is proportional to the differ- 
 
86 
 
 ELECTRICITY 
 
 ence of potential of outside and inside, it should be our object 
 to make this difference as large as possible. As a rule the highest 
 available potential is that of our source, and with this we connect 
 the knob ; the lowest available is the zero potential of the earth, 
 
 and so we put the outside coating to earth. 
 
 A larger charge would be obtained if we could 
 
 put the outside to a potential. 
 
 When the outer coating is to earth, the 
 
 charge of the jar is determined by the formula 
 Q=K.V 
 
 where K is the capacity of the jar, and V is 
 the potential of the inner coating. 
 
 Glass is used for several reasons. It is 
 easy to make glass jars ; glass has a high 
 specific inductive capacity (see Chapter IX.); 
 and the jar can be made thin, or the distance 
 between the plates small and therefore the 
 capacity of the jar large, without a discharge 
 taking place through the glass. 
 
 For the calculation of K, as well as for the units in which V 
 is measured, the reader is referred to Chapter X. 
 
 8. The Unit Jar. If we can fix the capacity of a jar and the 
 difference of potential between its two coatings, then we have 
 fixed the charge in that jar. 
 
 A unit jar is a small Leyden jar of some convenient shape ; 
 
 usually so made as to admit of 
 easy cleaning and drying, and 
 to insulate well. 
 
 Connected with the outside 
 is a knob B, and connected 
 with the inside is a knob A, 
 whose distance from B can be 
 regulated by sliding the piece 
 A C along the graduated rod 
 DE. 
 
 It is proved experimentally 
 that the distance between A 
 
 and B across which the jar will just discharge itself is directly 
 proportional to the difference of potential between A and B. 
 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS S/ 
 
 (It depends also on the curvature of the knobs, and on the medium 
 between A and B ; but we suppose these conditions to remain 
 constant.) 
 
 Hence, if we keep the distance between A and B fixed, the 
 difference of potential between A and B at the moment when the 
 jar discharges itself in a spark across the gap A B will also be 
 fixed. Let us now put the outer coating of the jar to earth, put A 
 at a fixed distance (that we will call i) from B, and charge the jar 
 by placing the knob D near the prime conductor of a machine ; 
 and let us suppose that it takes a difference of potential mea- 
 sured by v\ for a spark to pass from A to B. 
 
 Then on charging the jar we shall shortly perceive a brilliant 
 discharge to pass from A to B ; this discharge leaving the jar 
 uncharged. At the moment of this discharge there was on the 
 inner coating a charge + Q, determined by the relation 
 
 QIK(V A -V*) 
 
 or, by hypothesis, Q, =K v } . 
 
 There was, on the inner surface of the outer coating, a charge 
 Q! ; and there had passed to earth a quantity + Q t . 
 
 Hence, every time that the jar discharges itself, the + Q, and 
 the Q], that are opposed to each other, neutralise each other; 
 while there has passed to earth a quantity + Q,. 
 
 If we put A and B at a distance apart twice as great as before, 
 or at a distance 2, then we have Q 2 = K . v 2 K . 2 v\ = 2 Q,. 
 
 Let us now see how the unit jar is used. 
 
 We connect the outside coating of our small unit jar with the 
 inside of a large Leyden jar whose outer coating is to earth. 
 
 If we now work the machine, then each time that the unit jar 
 discharges itself we know that a certain quantity + Q x of elec- 
 tricity has passed away from the outside of the unit jar into the 
 large jar. And so, by counting the sparks of discharge of the 
 unit jar, we are able to give the large jar a charge represented by 
 
 Qi, 2 Qi> 3Qi,&c. 
 
 Comments and notes. (i. ) We ought to have the unit jar so small as com- 
 pared with the jar to be charged, and the knobs A and B at such a distance 
 apart, that a considerable number of discharges of the unit jar may take place 
 by the time that we have given the required charge to the large jar. For we 
 cannot stop charging the moment that a spark passes, and therefore there will 
 always be a fraction of + Q, in excess given to the large jar. 
 
88 ELECTRICITY CH. vi. 
 
 (ii. ) We have said that, at each discharge of the unit jar, a quantity -f Q 
 has passed into the large jar. This is not accurately true. The outside of the 
 unit jar is a small conductor at a considerable distance from surrounding bodies, 
 and the inside of a large jar is a relatively large conductor having the outer 
 coating of the jar very near it and opposed to it. These two form one con- 
 ductor at one potential, since they are connected ; but the capacity of the first 
 is negligibly small as compared with that of the second. The charge + Q, 
 distributes itself between these two portions of the conductor in the ratio of 
 their capacities. Hence, we may say that very nearly the whole charge, that 
 would have passed to earth if the outside of the unit jar had been to earth, 
 will have passed into the large jar. 
 
 (iii.) It requires a certain difference of potential v for a spark to pass 
 between A and B. If then A is connected with the prime conductor of the 
 machine, and B with the inside of a large jar, we cannot charge this latter up 
 to the full potential of the prime conductor, but to a potential less than this 
 by the amount v. 
 
 9. Cascade arrangement of Leyden Jars. P, Q, and R are 
 
 three Leyden ars mounted on insulated stands, arranged 'in 
 
 cascade ' as indicated in the figure. These jars must be equal in 
 capacity. To insure this we may take one jar of many that are 
 presumably equal, and then by comparison choose two or more 
 others equal to it. If we give a certain charge to our jar chosen, 
 and measure the potential of the inside (when the outside is to 
 earth), this potential ought to fall to one-half when the knob is 
 connected with that of an uncharged equal jar. This method is 
 employed in Chapter IX. 4. 
 
 But for rough experimental purposes it will be sufficient to 
 choose the jars as follows. Let an electrical machine be worked 
 for some time so that it has got into a constant condition. Then 
 charge the jars in question with a unit jar interposed, as shown in 
 the last section. If the jars refuse further charge after the same 
 number of discharges of the unit jar have taken place, then their 
 capacity will be approximately equal. 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 89 
 
 Having arranged our three (or more) jars as shown, let us 
 work the machine. 
 
 When no more charge will pass, then the inside of the first jar 
 will be at a potential V t ; which will be that of the prime con- 
 ductor, or something less, according as the knob is or is not in 
 contact with the prime conductor. The outside of the first jar and 
 the inside of the second jar (in contact with the former) will be at 
 some lower potential V 2 . The outside of the second jar and the 
 inside of the third will be at a still lower potential V 3 ; and the 
 outside of the third will be at zero potential, which we may call 
 V or zero-V, as we please. 
 
 Now, as shown in the last section, if there be in the first jar 
 a ' bound ' charge + Q, then there has passed into the second jar 
 also a quantity + Q, and from the outside of this into the third 
 jar also a quantity + Q, if we neglect the trivial ' free ' charges on 
 the outside of the first jar and on the knob of the second jar, and 
 so on. 
 
 That is, in the cascade arrangement we have of necessity equal 
 charges in the jar. But by Chapter VI. 4 (explained further in 
 Chapter X.) we have 
 
 ( for first jar Q = K . (V l - V 2 ) 
 { for second jar Q = K . (V 2 - V 3 ) 
 1 for third jar Q = K . (V 3 - V ). 
 
 So that, since Q and K are the same for each jar, we see that 
 V, - V 2 = V 2 - V 3 = V 3 - V = (V, - V ). 
 
 Hence the total fall of potential that is possible, viz. the fall 
 measured by Vj V (or by V b since we take V as our zero), 
 has been broken up into three equal falls. 
 
 (This is somewhat like the breaking up of a large waterfall 
 into three smaller ones.) 
 
 Now let us consider the charge Q' that is given to one of these 
 jars when its outside coating is to earth and its inside coating is at 
 the potential Vj of the prime conductor ; i.e. its charge, when it 
 has been charged in the usual way. By our formula we shall have 
 
 Q' = K . (V, - V ). 
 But by what has preceded we see that this equals 3Q. Hence 
 
9O ELECTRICITY CH. vi. 
 
 We see then that the sum of all the charges given to the jars, 
 when these are arranged in cascade, equals the charge given to a 
 single jar when this is treated in the usual manner. 
 
 The above reasoning can readily be extended to any number 
 ;z jars. .For further on the 'cascade arrangement' the reader is 
 referred to Chapter X. 30. 
 
 10. Nature of the Leyden Jar Charge. It is now time to 
 investigate, as far as we are able, what is meant by the expression 
 'an electric charge.' All that we have hitherto observed indicates 
 that neither ' + charges,' nor ' charges,' can exist by them- 
 selves ; but that there must always co-exist two equal charges of 
 opposite sign. In fact, it would seem that in all electrostatic 
 phenomena there must be a + charge and an equal charge, 
 separated by some insulating medium called a dielectric. It is pro- 
 bable that in the case of two conductors thus separated by a 
 dielectric it is the surfaces of the dielectric in immediate contact 
 with the conductors that are in the condition that we have called 
 ' + charged ' and ' charged ' respectively ; though it is usual, 
 and perhaps more convenient, to speak of the conductors them- 
 selves as so charged. 
 
 Various experiments tend to show that the dielectric is, under 
 these conditions, subjected to a stress ; and to this it yields to a 
 greater or less degree, becoming deformed or strained. 
 
 The conductors appear to play somewhat the following part. 
 They mark out the portion of the dielectric that can be converted 
 into an electric field and can be put under a stress ; and they 
 allow this stress to be imposed or taken away with rapidity. If 
 the stress be continued for a sufficient time, it is found that all 
 solid dielectrics become strained (or distorted) to an appreciable 
 distance from the conducting surfaces ; and that then the strain 
 cannot at once be removed, or the ' condenser ' cannot be at once 
 discharged. Some of the stress remains and must be allowed to 
 be relieved gradually. This gives rise to the ' residual charges ' 
 discussed below. 
 
 Experiments.- (i.) Leyden jar with mcveable coatings. A is a Leyden jar 
 so constructed as to be separable into three portions ; B the glass jar, D the 
 inside coating, and C the outside coating. 
 
 The jar is put together, charged as usual, and then placed on an insulating 
 stand. The inside is then removed, a small charge coming away with it, and 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 
 
 a small charge being ' set free ' on the outside coating. Then the glass is 
 lifted out of its outer coating and set down on the table, and finally the outer 
 coating may be handled and removed. In all this we notice but very slight 
 discharges on touching D and C respectively. 
 
 FIG. i. 
 
 On carefully putting the whole together again (on the insulating stand), a 
 discharging rod (fig. ii. ) will show us that the whole charge of the condenser has 
 been (approximately) unaffected by the above process. Hence, since the coatings 
 B and C were separately put to earth by handling, the charge of the condenser 
 must have resided in the glass. We could not, how- 
 ever, discharge the glass by itself ; it was necessary to 
 have the metal coatings as distributors. 
 
 (ii.) Residual charge. The penetration of the 
 charge, from the surfaces inwards, is shown by the 
 following. 
 
 If a Leyden jar be discharged and then left for 
 a time, a second small discharge can be obtained, 
 and so on. 
 
 That the interior of the glass is actually strained 
 or distorted is shown by the fact that its optical pro- 
 perties, when it is under the electrical stress, undergo 
 an alteration similar to or identical with that produced 
 by mechanical stress. 
 
 When the electrical stress is excessive, the material may give way and a 
 hole may be made ; through this a discharge takes place, since the air which 
 then separates the two charges is far weaker to withstand the stress than is glass. 
 
 11. Various Effects of the Discharge, The Leyden jar gives 
 us a means of observing the effects of electrical discharge. There 
 is nothing essentially different between the discharge between the 
 prime conductor and the floor and walls, where was collected the 
 induced charge of opposite sign, and the discharge between the 
 inner and outer coatings of the Leyden jar. Only the latter 
 
ELECTRICITY 
 
 CH. VI. 
 
 gives us a means of storing up quantities, ready for discharge, 
 enormously greater than we should get did we discharge the prime 
 conductor only. 
 
 I. Mechanical effects. 
 
 Piercing card, &^c. In several of these experiments we make use 
 of a discharging table. Its construction is evident from the figure. 
 One rod, A, is connected with the outside C of the jar, or battery of 
 jars, and the other, B, can be connected with the inside D of the jar 
 
 FIG. i. 
 
 or battery by means of an insulated discharging rod E. The battery 
 is charged, a piece of card placed between the points of the rods A 
 and B, and then connection made. 
 
 If the pierced card be examined, it will be seen that the hole has 
 been apparently formed by an explosion from inside, due to the sudden 
 expansion of air, or to sudden vaporisation of water in the inside of 
 the card. Whatever the cause, it may help the reader to remember 
 that ' the passage of a spark ' is an expression hardly authorised by 
 our knowledge ; the discharge may take place from + to , or from 
 - to + , or may be a series of alternate discharges, or may be of some 
 dual nature not represented by any of the above suppositions. 
 
 II. Magnetic effects. 
 
 Experiment. (i.) In fig. ii., A B is a wire along which the discharge can 
 be effected. If we, for convenience, agree to say that by the ' direction of a 
 discharge ' we mean that direction in which the + charge would move in order to 
 meet the charge, then in fig. ii. the discharge is from B to A. This expres- 
 sion, ' direction of discharge,' is not here assumed to have a real physical 
 meaning, but is employed in order to save confusion. 
 
 n s is a piece of steel placed at right angles to the wire B A, and under- 
 neath it. After discharge it will be found that the steel is magnetised in the 
 way shown. That is, if we swim with the + electricity and face the steel bar, 
 the induced north pole will be found at our left hand. 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 
 
 93 
 
 Experiments of this sort, and similar ones made with steel filings on glass, 
 show us that round a wire bearing a discharge are lines of magnetic force ; 
 they are circles round the wire as axis, and their + and direction is clearly 
 given by the italicised rule above. (Compare also Chapter XVII. I and 2.) 
 
 FIG. ii. 
 
 FIG. iii. 
 
 The reader can verify this rule when he has determined the sign of the 
 charge on the prime conductor of his machine. 
 
 (ii. ) We may (see fig. iii.) intensify the effect by coiling a wire round a 
 glass tube in which lies the steel needle. The above rule will be found true, 
 and it will be seen to be equivalent to saying that if we face one end of the 
 spiral coiled round the tube, the needle will have been made n or s at that end, 
 according as the discharge so viewed is against or ^v^th movement of the hands 
 of a watch. A reference to Chapter XVII. 2 will make the rule clear. 
 
 This experiment is one of the many that link together the electrostatical and 
 electrodynajnical divisions of our subject. 
 
 III. Heating effects. 
 
 Experiment. (i. ) The brilliant light of the discharge, and the fact that 
 discharges over the dust of various metals scattered over a glass surface give 
 the well-known spectrum of each metal respectively, indicate that in the dis- 
 charge there is an elevation of temperature sufficient to volatilise metals and 
 other bodies, and to raise the vapours to brilliant incandescence. 
 
 By experiments easily contrived we may reduce a platinum wire to fused 
 drops ; or may volatilise gold-leaf between two well dried, and therefore in- 
 sulating, cards. 
 
 Experiment. (ii.) Ignition of gunpowder. If we place gunpowder on the 
 small table of the discharger, between the points of the discharging rods, we find 
 that the discharge merely scatters it. 
 
 But if we interpose between A and C a substance, such as wet string, which 
 
94 ELECTRICITY CH. vi. 
 
 conducts worse than does a metal, we find that the discharge will ignite the 
 powder. We conclude that this retards the discharge and gives sufficient 
 heating effects without such violent mechanical disturbance. 
 
 IV. Chemical effects. 
 
 The subject of chemical decompositions may be left until we come to it 
 under the head of ' Phenomena of electric currents,' in Chapters XL and XII. 
 
 It will suffice here to say that if we keep to the convention given above, as 
 to which we shall call the direction of discharge, then the chemical phenomena 
 given later on in Chapter XL could be repeated with similar though less 
 striking results if we substituted a series of discharges for the continuous 
 currents employed in that chapter, 
 
 12. Induction Effects of the Discharge. 
 
 Note. This section may be omitted until the student has read Chapter XXL 
 
 We will first give a general account of the phenomena that are 
 the subject of this section, and will then describe how these 
 phenomena may be observed. 
 
 If a discharge be effected along a wire B A, it is found that at 
 the moment of discharge there passes along a wire C D, placed 
 parallel to B A and near to it, an in- 
 duced discharge. 
 
 Further examination, in which we 
 are aided by results of experiments with 
 the more manageable currents given by 
 voltaic batteries, shows us that the phe- 
 nomenon is not a simple one. 
 
 It is found that in C D are induced 
 two rushes of electricity ; one answering 
 to a discharge of + electricity from D 
 to C, one in the contrary direction. 
 
 It would seem that these two induced 
 rushes of electricity are equal in total 
 amount of electricity \ but that the rush from C to D, or that 
 whose direction is the same as that of the inducing discharge 
 from B to A, is the more violent in character. 
 
 If we interpose a sufficient gap in the circuit C D E, we find- 
 that the direct induced discharge (or that passing from C to D) 
 will leap over the gap in the form of a spark ; while the inverse 
 induced discharge (or that passing from D to C) will not pass, 
 but will be suppressed. 
 
ELEMENTARY DISCUSSION OF CONDENSERS 
 
 95 
 
 Comparing with the results obtained in Chapter XXL, where 
 we employ electrical currents over whose increase and decrease we 
 have complete control, it would seem that as the discharge from 
 B to A begins, there is induced a rush in the contrary direction 
 from D to C ; while as the discharge from B to A fades away, 
 there is induced a rush from to D in the same direction. 
 
 It would seem also that the direct induced rush from C to D 
 is the more violent of the two, for the reason that the cessation of 
 the inducing discharge in B A is more abrupt than its commence- 
 ment. The inducing current is called the primary ; the induced 
 is called the secondary current. 
 
 Experiments. In the figure, X represents a screen of card or of glass, on 
 the further side of which is fixed a ' primary ' spiral of wire through which a 
 Leyden jar discharge can be passed. On the nearer side of the screen is fixed 
 
 FIG. ii. 
 
 another, 'secondary,' spiral of wire, parallel to the other but separated from it 
 by the glass or card. The ends of this last wire are connected with the ends 
 of a wire coiled round a tube, C D. 
 
 Then, by placing steel needles inside the tube C D, we can [by the results 
 obtained in the last section] examine the direction of the current induced in 
 the secondary spiral. Thus, if the end P of the needle were found to be a 
 north pole, it would mean that the current passed in the general direction from 
 
t>6 
 
 ELECTRICITY 
 
 CH. VI. 
 
 P to Q ; since then, if one swam with the current and looked at the needle, 
 the north pole would be to the left (see the. rule given in last section). 
 
 We shall find that when there is no break in the secondary circuit there is 
 a weak magnetising action making the end P the north pole. This implies a 
 total magnetising action as of a current in the opposite direction to that in the 
 primary circuit. Without further experiment this would indicate merely a 
 weak inverse current induced in C D. Other experiments show us that as 
 regards quantity of current in C D there is really a zero total, though the 
 inverse current is the more effective as regards magnetising powers. But if we 
 make a small break in the secondary circuit, such as will just allow a bright 
 spark to pass, we find a relatively powerful magnetising effect, making Q a 
 north pole. This indicates that now there is in CD a powerfully induced 
 direct current. 
 
 Other experiments, which we do not describe here, are needed to bear out 
 fully our statements as given above. 
 
 13, Wheatstone's Spark-board. This was a piece of ap- 
 paratus designed to investigate whether the spark of -discharge 
 had any appreciable duration, whether the discharge passed from 
 one coating to the other or from both simultaneously, and with 
 what velocity it traversed such conductors as copper bell-wire. 
 The apparatus is sketched in the accompanying diagram, in which 
 the reader is supposed to be viewing it from above. 
 
 FIG. i 
 
 FIG. ii. 
 
 In fig. i., L is a Leyden jar. The discharge takes place 
 through the circuit YabcdefQ ; so that sparks appear at the 
 three breaks ab,cd, and ef. 
 
 Between b and c, d and e, are coils of wire of the same known 
 thickness and length, say 1,000 metres of common copper bell- 
 
CH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 97 
 
 wire. Between a and P, f and Q, there snould be wire of the 
 same thickness and length respectively. In fig. ii., m is a small 
 plane mirror revolving on a horizontal axis that lies parallel to the 
 line of the three sparks a b, c d, and ef. 
 
 Above it is a horizontal screen of ground glass, marked out in 
 degrees in a manner indicated later on. 
 
 The mirror can be made to revolve rapidly, its rate of revolu- 
 tion being indicated by clockwork or by the musical note produced 
 by its striking a fine metal wire. 
 
 If the discharges take place very rapidly, then a person looking 
 down from above will see some of the sparks reflected in the 
 mirror as it passes through the position proper for reflection of the 
 sparks to the eye ; others of the sparks will not be seen, owing to 
 the mirror being unsuitably situated at the moment of their 
 occurrence. 
 
 The rotation of the mirror will of course distort, by ' drawing 
 out ' in the direction of the motion any spark that lasts an 
 appreciable time ; i.e. that lasts while the mirror has moved 
 appreciably. 
 
 Now it is found that, for moderate rates of revolution of the 
 mirror, we see on the screen X Y merely three points of light, 
 
 FlG. iii. 
 
 FIG. iv. 
 
 lying parallel to the sparks of which they are the images, as 
 represented in fig. iii. 
 
 But when the mirror revolves with sufficient rapidity, we see 
 these points ' drawn out ' into three lines of equal length, covering 
 a number of degrees on the glass screen that is definite for each 
 definite rate of revolution of the mirror, as indicated in fig. iv. 
 
 Further, we see that the bright lines answering to the sparks 
 
 H 
 
98 ELECTRICITY CH. vi. 
 
 ab and efhzve their ends lying in a direction parallel to the line 
 of sparks, while the bright line answering to the spark cd is 
 displaced as shown. 
 
 This shows (i.) that the duration of the spark is such that the 
 mirror turns through an appreciable angle while it is reflecting ; 
 and (ii.) that, whereas the sparks ab and ef occur simultaneously, 
 the spark c d occurs later than either. 
 
 The fact that the spark has duration was perhaps obvious 
 beforehand; we cannot conceive of 'no duration.' But this 
 apparatus gives us a measure of -.the duration of a spark under 
 any particular conditions of distance between a and b, and nature 
 of the intervening gas. 
 
 But the fact that the lines answering to the sparks a b and ef 
 are still similarly situate shows that these occur simultaneously, 
 and that the discharge is of the dual nature indicated before, and 
 that when we speak of its * passing from one coating of the 
 jar to the other ' we are but using a convenient convention. 
 
 14. The Condensing Electroscope. Before concluding this 
 chapter on condensers, we must describe the ' condensing electro- 
 scope,' a piece of apparatus in which the principle of experiment 
 (v.) of 2 in this chapter is made use of. If the reader will turn 
 to the figure to Chapter XI. 3, he will there see the apparatus in 
 question represented. 
 
 On the plate N of a gold-leaf electroscope is placed another 
 plate M, separated from the first by a thin insulating film of shellac. 
 
 When M is put to earth, the plate N considered by itself will 
 have a capacity very large as compared with its capacity when 
 'isolated.' Hence, if it and the gold leaves are raised to a very 
 low potential when M is to earth, the charge that it has received 
 will suffice to raise it to a potential many times greater when M is 
 removed. Thus the leaves, which did not stir when at some 
 small difference v from zero, may diverge widely when this has 
 been multiplied to (say) 500 v. 
 
 We can, in fact, multiply any potential v by the fraction, 
 
 capacity of N, when M is on it, and is to earth 
 capacity of N when isolated 
 
 15. Various Forms of Electrical Discharge. We have many 
 times alluded to the passage of a ' spark ' or ' disruptive electrical 
 
<SH. vi. ELEMENTARY DISCUSSION OF CONDENSERS 99 
 
 discharge.' It may not be out of place to enumerate a few of the 
 forms in which this discharge may take place. 
 
 I. The ordinary spark. When we approach the finger, or any 
 other conductor, to the prime conductor of a machine, or to any 
 other conductor that is of a high potential but has not a very 
 great capacity, the discharge usually takes place in the form of a 
 jagged spark of no very great luminosity. If the opposed surfaces 
 be of very gentle curvature, as e.g. two spheres of large radius, the 
 spark will be straighter and more brilliant. In both cases there 
 will be heard a sharp crack or other report. 
 
 II. The Leyden jar discharge. In the Leyden jar we have 
 stored up a quantity of electricity that is very great as compared 
 with that stored on any ordinary isolated conductor. In Leyden 
 jar discharges the spark is usually very brilliant and dense in 
 appearance, and much straighter than in case (i.) above. Moreover, 
 the sound is very strident. 
 
 III. The brush discharge. If any projecting part of the prime 
 conductor of a machine be of very sharp curvature, or still more 
 if it be pointed, the discharge from this part will be in the form 
 of a beautiful faintly luminous brush, and this discharge will be 
 nearly or quite silent. 
 
 IV. Discharge ' in vacuo.' 1 When the discharge takes place 
 through a tube or other vessel in which the air or other gas is 
 very rare, the whole tube is filled with a faint luminosity, and 
 there is no sound. 
 
IOO ELECTRICITY CH. vn. 
 
 CHAPTER VII. 
 
 INDUCTION MACHINES. 
 
 i. Some further Propositions in the Theory of Potential. 
 
 Still pursuing our system of introducing more and more of the 
 theory of potential as we require it, we shall here discuss certain 
 points that we must make clear before we can rightly understand 
 the theory of induction machines. 
 
 By induction machines we mean those pieces of apparatus by 
 means of which we can obtain continuous supplies of + and - elec- 
 tricity without any friction or chemical action ; the essentials being 
 (i) an initial supply of electricity that can act inductively on con- 
 ductors placed near to it, and (2) mechanical work to effect such 
 movements of the parts of the apparatus as shall give us continuous 
 action. The common electrophorus is the simplest form of such a 
 machine ; and the reader can see how it differs in principle from a 
 frictional machine, though in both cases mechanical work is the source 
 of energy. 
 
 (i.) There is no field of force inside a simple closed conductor. 
 If we have a simple closed conductor, that is, a closed vessel con- 
 taining no insulated charged bodies, it can be shown experiment- 
 ally that there is no field of force inside it, or that it, and all the 
 space inside it, are at one potential. 
 
 (ii.) Concerning partially closed vessels. If we experiment with 
 a partially closed vessel, such as a tin pail whose height is greater 
 than the diameter of the mouth, and if it be placed so that those 
 external objects which are opposite to the opening are relatively 
 remote, then we shall find that there is no appreciable field of 
 force in the inside of the vessel. 
 
 (iii.) Insulated uncharged conductors inside a vessel. If into the 
 inside of a charged vessel there be introduced an uncharged con- 
 ductor, there is nothing to alter the fact of the potential being the 
 
CH. VII. 
 
 INDUCTION 
 
 FIG. 
 
 same throughout. The insulated conductor is in a region of no 
 field of force, and hence there is no inductive action between it 
 and the charged vessel. The conductor is at the same potential 
 as this vessel and the space inside it. 
 
 (iv.) An insulated uncharged conductor partially inside a vessel. 
 Let A be an insulated vessel charged + , and let B be an insu- 
 lated uncharged conductor partly inside and partly 
 outside the vessel. 
 
 The conductor B must be at one potential ; 
 and yet the one end is more remote from the 
 vessel than is the other end. 
 
 As in Chapter IV. 14 (i.), and in Chapter VI. 
 2 (i.), there will be a redistribution of electricity A 
 on B, there being a charge on the end inside A, 
 and the complementary + charge on the end outside 
 A ; the whole of B being at a potential below that of 
 A but above that of the earth. There will be also inside A a portion 
 of its + charge equal to the charge induced on the lower end of B. 
 
 All this can be shown experimentally. It is what we should 
 expect from Chapter VI. 2 (i.). 
 
 (v.) An earth-connected body inside a vessel. If the body B be 
 put to earth, it will be at zero potential. As in Chapter VI. 2 
 (ii.), we shall find that it now has a charge greater in amount 
 than the charge in the last case. 
 
 (vi.) Two vessels at different potentials ; a conductor inside each 
 of them. If we have two vessels A and B at potentials + V and V, 
 there being an insulated con- 
 ductor (initially uncharged) in- 
 side each of them, and if these 
 conductors be joined by a con- 
 ducting wire, it is not difficult to 
 predict what will take place. 
 
 The conductor C will, before 
 joining, be at + V, and D will 
 be at V. Hence, when they are brought to the same poten- 
 tial by joining, there will be a. discharge between C and D ; C will 
 become charged ly, and D will become charged +ly. 
 
 Further, if we now disconnect C and D and lift them still in- 
 sulated from the vessels, and drop C into B and D into A, they 
 
 FIG. ii. 
 
 FIG. iii 
 
t. Y t , ELECTRICITY 
 
 will give up the above-named charges, and the difference of poten- 
 tial between A and B will be still further increased. 
 
 2. Application to Induction Machines. The above shows 
 us that we have a means 
 
 (i.) Of getting an unlimited number of electrical discharges 
 (viz. between C and D), :, .... - . . . . 
 
 (ii.) Of indefinitely increasing the difference of potential be- 
 tween two (hollow) conductors, , 
 
 at the expense of mechanical work. For we have merely to repeat 
 indefinitely the cycle of operations indicated above ; and we shall 
 thus get a series of discharges between C and D, each more 
 vigorous than the last, and shall be increasing the difference of 
 potential between A and B. 
 
 It remains to discuss some convenient arrangements designed 
 to repeat continuously and rapidly such a cycle of operations. 
 
 Note. In what follows we 
 shall use the symbol A V to 
 denote difference of potential. 
 
 3. Sir W. Thomson's 
 Water-dropping Accumu- 
 lator. In the arrangement 
 to be described in this 
 section there are two points 
 to be noticed. 
 
 First: we make no use 
 of the discharges spoken of 
 above ; we only aim at in- 
 creasing the AV between 
 two (hollow) conductors 
 that are initially at a small 
 AV. 
 
 Secondly, instead of two 
 vessels we have four, con- 
 nected in pairs so as really 
 to form two. 
 
 The accompanying 
 figure is a sectional sketch of the apparatus in question. The two 
 pairs of vessels, A A' and B B' respectively, are at a certain A V as 
 indicated by the + and signs. 
 
CH. vn. INDUCTION MACHINES IOJ 
 
 In A and B are the ends C and D of water-pipes, leading from 
 a supply that is at the zero V of the earth. So that C and D 
 will be charged inductively with and + electricities respectively ; 
 the orifices being so small that the water breaks away in detached 
 drops, so that these drops are continually carrying away and 
 4- charges respectively. They are caught by funnels in the in- 
 terior of B' and A' respectively ; and so they give up their charges 
 to these vessels, before dropping away to waste. The reader will 
 see that it is essential for the water to fall in drops. 
 
 Hence B' and B are being continually charged with , and 
 A' and A with + electricity ; and therefore the A V between A 
 and B will continually increase. 
 
 It is important to notice that this increase of A V, both in this 
 apparatus and in the Holtz, Voss, and other machines, pro- 
 ceeds on compound interest principles. For as the A V between 
 A and B increases, so will the respective inductive actions of A 
 and B on C and D increase ; and so will the drops carry away 
 charges of increasing magnitude. 
 
 The reader should also notice that we cannot avail ourselves 
 of the discharge between C and D spoken of in 2 (i.) ; for, since 
 C and D are permanently connected, the rearrangement of elec- 
 tricities between C and D is continuous and not disruptive. 
 
 The question will occur, ' Whence do we derive our 
 energy ? ' The answer is as follows. 
 
 The raised water supply represents a certain quantity of 
 mechanical potential-energy ; and, as the water drDps to a lower 
 level, we should get heat developed equivalent to the potential- 
 energy that is lost. But, owing to electrical attractions, the 
 water drops are pulled upwards, and so fall more gently than 
 they otherwise would ; thus giving out less heat. The electrical 
 potential-energy gained is the equivalent of the heat thus lost. 
 In fact, we may say that the energy used is part of that expended 
 in the original raising of water to the level of our supply (for 
 ' Energy,' see Chapter X.). 
 
 4. Varley's Induction Machine, In this piece of apparatus 
 we have two hollow vessels, and metal carriers instead of drops of 
 water. 
 
 In the figure the insulated hollow vessels ee, e t e t , are repre- 
 sented with their front halves removed, so as to expose to view the 
 
a 04 
 
 ELECTRICITY 
 
 CH. TIT. 
 
 carriers, &c. 
 
 In reality they are flat hollow vessels through which 
 
 the carriers c c pass. 
 
 These carriers are mounted on an insulating' disc of ebonite. 
 
 When each carrier has just entered an armature (the hollow 
 
 pieces e e and e t e t are called 
 armatures), it touches a 
 projection g or g t , and so 
 gives up any charge it has 
 to that armature. When 
 halfway through the arma- 
 ture, it becomes momen- 
 tarily connected with the 
 diametrically opposite car- 
 rier by means of a metal 
 piece h h r This metal 
 piece is insulated, and its 
 ends h and h, pass through 
 holes in the armatures, 
 without touching them. 
 Action, Initially the 
 we may suppose one 
 
 armatures must be at. different potentials ; 
 to be at + v and the other at v. 
 
 When a pair of carriers are connected by the conductor h h t 
 they will be reduced to the same potential, and therefore will be- 
 come charged ly and +ly respectively. 
 
 As the disc turns, these carriers are again insulated ; and so 
 they carry away the charges acquired, and deliver them up to the 
 other armatures respectively by contact with the knobs g t and g. 
 
 Thus the A V of the two armatures will continually increase. 
 When the A V is large enough we may introduce a break into the 
 conductor hh, ; we shall then see the electrical readjustment be- 
 tween each pair of opposite carriers take place disruptively, a 
 spark leaping over the break made. 
 
 5. Sir W. Thomson's Replenisher. This is a small piece of 
 apparatus used to keep up the potential of the Leyden jar with 
 which the needle of the quadrant electrometer is connected. We 
 here desire a slow action, and do not aim at powerful action. 
 
 The figure shows that the armatures e e t are hardly to be called 
 * hollow vessels ' ; they are merely metal plates bent into a 
 
as., vii. 
 
 INDUCTION MACHINES 
 
 105 
 
 cylindrical form. They do not therefore act so powerfully on the 
 carriers C C /5 since these are never enclosed by them ; nor do the 
 carriers give up all their charge to the armature when in contact 
 with it. The action, therefore, is less vigorous, but sufficient for 
 the purpose. After 4 it will 
 be sufficient to point out one 
 or two details. 
 
 (i.) One armature is per- 
 manently put to earth. 
 
 (ii.) The conductor that 
 connects the two carriers 
 when they are inside the two 
 armatures is insulated. It is 
 continuous ; a break being 
 only for the purpose of ob- 
 taining phenomena of dis- 
 charge, which are not. wanted 
 here. 
 
 The reader should, by 
 reasoning out the action, discover for himself why h h, may be 
 uninsulated when e and e, are insulated; while it must be insu- 
 lated if either e or e t be put to earth. 
 
 6. The Voss Machine. With regard to the more complex 
 induction machines, the Voss, the Holtz, and others, it may be 
 said that no mere verbal description is enough to enable a student 
 to understand the action any such description should be read by 
 him with the machine actually before him. 
 
 Further, the theory of these machines is by no means in a 
 satisfactory condition. This is due mainly to two causes. In the 
 first place, we no longer have hollow armatures into which the 
 carriers pass, and concerning which the theory is simple ; the 
 armatures are now plane pieces of tin-foil or of paper, and the 
 carriers merely come opposite to these. And, in the second place, 
 there is little doubt but that the extremely rapid movements in 
 these machines give rise to actions concerning which our informa- 
 tion is imperfect ; actions depending upon the relation between 
 the velocity of these movements on the one hand, and the rate at 
 which electrical charges spread on the badly-conducting surface 
 of the varnished glass on the other. These difficulties will be 
 
io6 
 
 ELECTRICITY 
 
 better understood when the following description has been read 
 and the machine has been seen. 
 
 In the Voss, as (see 7, fig. i.) in the Holtz, there are two 
 glass plates mounted on the same axis and very close to one 
 another ; one of these being fixed, the other being capable of 
 very rapid rotation upon the axis. On the former are fixed the 
 sheets of tin-foil or of paper that serve for armatures, and the 
 varnished surface of the latter (supplemented in the case of the 
 Voss by metallic carriers fixed on to the glass) takes the place 
 of the carriers whose functions were explained in 3 and 4. 
 
 Since it is not possible to represent in any one simple diagram 
 the action of the real machine with its double plate, we here give 
 a representative diagram by means of which the actual machine 
 may be understood. It is supposed that, instead of two plates, we 
 have two co-axial glass cylinders, the outer one being fixed, and 
 
 the inner one revolving as 
 shown by the arrows. The 
 diagram shows such an ar- 
 rangement in section, the 
 two cylinders being repre- 
 sented by two thin circles of 
 glass. Hence, the reader 
 must imagine that all the 
 lines (whether long as A, or 
 short as c\\ with the excep- 
 tion of the metal pieces M N, 
 m n, B, and B', are but ' end- 
 on ' views of strips that in reality run down the whole length of 
 the cylinders. A and A', the armatures, are sheets of tin-foil ; 
 while c\, c. 2 , &c., are metallic strips fixed on to the glass. M N 
 and m n are stout brass rods ; and B and B' may be made of 
 thick brass wire. 
 
 For the present we will neglect the piece M N ; and will show 
 how, when the inner cylinder is rotated in the direction indicated, 
 any initial AV between the armatures A and A' will be soon 
 increased up to the limit set by the conditions of insulation in the 
 machine. 
 
 Suppose that initially the armatures A and A' have some 
 
CH. vii. INDUCTION MACHINES TO? 
 
 slight A V ; they will act inductively upon the carriers that come 
 opposite to them. Now, at one point in their course, as the 
 inner cylinder revolves, each pair of carriers as c\ \ are connected 
 with one another through the metallic piece m n, whose extremities 
 are provided with fine wire brushes. In consequence of this, c 
 will pass on charged , and c\ will pass on charged +. 
 
 When the two carriers come opposite to the other armatures, 
 A' and A respectively, they make contact with these through the 
 metal pieces B' and B ; these pieces reach round the ends of the 
 cylinders, and have wire brushes at their extremities. The question 
 now occurs, will these carriers charge the armatures to greater + 
 and potentials as did the carriers in 3 and 4 ? The difficulty 
 here arises from the fact that the carriers do not now pass into the 
 interiors of the armatures, and so do not give up all their charge. 
 
 W T e may perhaps reason somewhat as follows. The carrier <:,, 
 while still uncharged, is brought very near to the armature A ; 
 and we may assume that it is thereby raised to at any rate more 
 than half the potential of A. That is, it is raised to some poten- 
 tial m V, where m is greater than ^. It is then brought to zero-V 
 by connection with the opposite carrier. It must therefore have 
 received a charge that would, in the absence of A, have lowered 
 it to ;;/ V. Next it comes into the same position with respect 
 to A' ; and would therefore, if uncharged, have been lowered to a 
 potential m V. Being charged, however, it will acquire on the 
 whole a potential of m V m V, or 2 m V. But we have 
 supposed ;;/ to be greater than \. Hence 2 m V is numerically 
 greater than V ; and therefore + electricity will flow from 
 armature A ; to carrier t, f or electricity in the other direction ; 
 and hence, finally, the potential of A' becomes lower than before, 
 or its V is numerically increased. 
 
 A like action will meanwhile have taken place with respect to 
 c\ and A. 
 
 So far then we have shown how the armatures are raised to, 
 and maintained at, a great A V. 
 
 Now consider the brass pieces M and N, which are represented 
 as furnished with combs and as having an air- break p q between 
 them. (It must here be noted that the * combs ' with which M N 
 and m n are provided should in reality run parallel to the axis of 
 Ihe cylinder. They are drawn as in the diagram so that they may 
 
I0 8 ELECTRICITY CH. vn. 
 
 be seen ; whereas strictly they should be drawn ' end-on.') Both 
 the metal carriers, and the varnished surface of the glass which 
 serves as a continuous series of insulated carriers, will act induc- 
 tively upon the brass combs ; + electricity from M, and electri- 
 city from N, will pass convectively from the points of the combs on 
 to the carriers and glass surfaces, tending to reduce the regions that 
 lie opposite to the two combs to a smaller A V ; while the ' re- 
 pelled ' and + charges will give a stream of sparks across the 
 air-space / q. 
 
 If there be no break at/0-, and if the combs have very sharp 
 points, the opposite pairs of carriers and of glass surface will be 
 reduced to almost the same potential ; and the piece m n will be 
 idle, M N taking its place. If there be a break at / q it requires a 
 certain A V to give a discharge across this break ; and the opposite 
 pairs of carriers cannot certainly be reduced to anything less than 
 this A V ; so that m n will now act as above described. The limit 
 to the striking distance p q is given by the A V of the armatures ; 
 the A V between M and N being always less than this. When the 
 distance p q is too great, M and N are idle. 
 
 The action, or idleness, of the pieces m n and M N can readily be 
 detected by working the machine in the dark ; the presence or absence 
 of glows and brush-discharges giving the required information. 
 
 The combs with which the piece m n is provided thus play 
 with respect to the glass surface the same part as the brushes play 
 with respect to the metal carriers. 
 
 7. The Holtz Machine. The Holtz machine was invented 
 before the Voss ; but, as its theory is less simple, we have chosen 
 to discuss it later. 
 
 The main differences between this machine and those that we 
 have discussed above are that here the armatures are charged 
 inductively, and not directly, by the carriers ; and that we have 
 as * pairs of carriers ' no metal pieces, but simply the diametrically 
 opposed portions of the varnished glass surface. 
 
 Fig. i. gives a general view of the simplest form of Holtz ; 
 and this figure will serve, mutatis mutandis, to give an idea of the 
 Voss. The larger and hindermost plate A A is fixed. In it are 
 cut two windows F and F' ; and along the edges of these windows 
 are pasted the armatures/ and/', consisting of two sheets of tin- 
 
CH. VII. 
 
 INDUCTION MACHINES 
 
 IO9 
 
 foil or of varnished paper. From the armatures project two 
 pointed tongues of paper, n and n' ; these touch lightly the back 
 surface of the smaller revolving plate B B (which is the nearer to 
 us in the figure), the points of contact being opposite to the 
 windows F F' cut in the fixed plate. The front plate B B re- 
 volves as indicated by the arrows. 
 
 On the nearer side of the front plate B B, and opposite to the 
 armatures//', are the combs O O' of the conductors C C'. These 
 
 FIG. 
 
 conductors are put into contact by means of the pieces K and K', 
 until the action resulting from the revolution of B B has increased 
 the small initial A V of the armatures to a sufficient magnitude. 
 When this has been effected, an air-space may be introduced at 
 rr' , when a discharge will take place across. 
 
 As in all induction machines, this discharge will be nearly 
 continuous, and will consist of a luminous brush accompanied by 
 small sparks ; unless, by connection with the condensers H H' 
 
IIO ELECTRICITY CH. vn 
 
 or by some other means, the capacity of the prime conductors be 
 greatly increased. When this is done, the character of the dis- 
 charge alters ; it now takes place at intervals in the form of a dense 
 and brilliant spark. 
 
 In fig. ii. we give a representative diagram. The lettering 
 of this, similar to that of the figure to 6, differs from that of 
 fig. i. above; but the reader will find no difficulty in recognising 
 
 the various parts. Here we 
 have the sections of two co-axial 
 cylinders, of long strips of tin- 
 foil which form the armatures, 
 and of long open strips which 
 form the windows cut in the 
 outer fixed cylinder. As in 
 the case of the Voss, the 
 combs should run parallel to 
 the axis of the cylinder, i.e. to' 
 the edges of the windows and 
 armatures ; they should have 
 FIG. ii. been viewed end -on instead 
 
 of as here drawn. 
 
 As in the other machines discussed above, there is given 
 initially to A A.' some slight A V. The conductors are then put 
 into contact, no air-space being left at/ q ; and the inner cylinder 
 is turned as indicated by the arrows. The opposite pairs of what 
 we may style 'glass- surface carriers ' are thus charged inductively, 
 as usual. 
 
 Let us consider the charged glass surface that comes 
 opposite to B'. In virtue of its charge, and of the presence of 
 the charged armature A', this portion of glass surface will 
 acquire a negative potential greater in magnitude than that of A' 
 and B'. There will therefore be a field of force running from the 
 armature A' towards the 'carrier,' since this is at a negative poten- 
 tial of greater magnitude. Hence we shall have + electricity 
 passing off the tongue B' on to the back surface of the inner cylin- 
 der ; while the armature A 7 is consequently left at a negative 
 potential numerically greater than before. 
 
 When our 'carrier' glass surface comes opposite to the comb 
 M, it will be charged + as usual ; and the + charge that was 
 
CH. Til. 
 
 INDUCTION MACHINES 
 
 III 
 
 'bound' at the back of the cylinder will become 'free.' When 
 the same portion comes in turn opposite to B, this tongue will be 
 charged to a higher + potential, in part directly by this free + 
 charge on the back of the inner cylinder, and in part inductively 
 by the + charge on the inner surface of the cylinder, in the 
 manner described above. 
 
 When the A V between A and A' is great enough, an air- 
 space may be introduced at / q and sparks will now strike 
 across. 
 
 But it is to be noticed that if this air-space be great, requiring 
 a large A V to drive a discharge across it, then the charges on the 
 glass surfaces will fail to be 
 neutralised and reversed as 
 they pass the combs ; and 
 therefore, as can be seen 
 from the theory of action 
 given above, the machine will 
 become less efficient, or will 
 even cease to act, just when 
 we make the greatest demands 
 upon it. 
 
 To obviate this defect, the 
 armatures may be extended, 
 and a continuous metal piece 
 m n, furnished with combs at the ends, may be introduced, as 
 shown in fig. iii. This form of Holtz is more like the Voss. 
 The piece m n with its combs will now insure the reversal of 
 charge upon the glass surfaces ; and, if the gap / q be great, but 
 not too great for the insulation of the machine, the A V of the 
 armatures A A' will increase until a spark can strike across p q. 
 
 Notes on the Voss and Holtz. (i.) It may be remarked that in both the 
 above machines it is only a small portion of the revolving surface that is 
 employed to raise the armatures to, and maintain them at, the high A V re- 
 quired. In the Voss this part is played by the metallic carriers ; these in the 
 actual machine are small metallic discs which bulge out at their centres suffi- 
 ciently to make contact with the brushes at m and n. In the Holtz the same 
 function is performed by that ring of glass surface which lies in the immediate 
 neighbourhood of the circle traced out on the revolving plate by the sharp- 
 pointed tongues. 
 
 FIG. iii. 
 
1 1 2 ELECTRICITY CH. vn. 
 
 (ii. ) We have not in 6 and 7 pointed out the compound interest action 
 of the Voss and Holtz, this having been already discussed in 3. 
 
 (iii. ) The reader is again warned that possibly the action of these machines, 
 especially as regards the feeding of the armatures, depends in some way at 
 present unknown upon the velocity of rotation. Hence the above discussion, 
 which ignores this velocity as affecting the nature of the action, may be defec- 
 tive. 
 
CH. VIII. 113 
 
 CHAPTER VIII. 
 
 ATMOSPHERIC ELECTRICITY. 
 
 i. Franklin's, and Other Early Experiments, In a 'course' 
 such as the present, it is impossible to give a history of the suc- 
 cessive experiments and discoveries by which little by little our 
 present knowledge has been attained. 
 
 But we cannot pass over without a brief notice the classical 
 ' kite ' experiment of Franklin. 
 
 Having successfully imitated in miniature the phenomena of 
 thunder and lightning with his powerful Leyden jar batteries, 
 Franklin wished to establish the common nature of the large and 
 the small phenomena by obtaining from the thunder-cloud an 
 ordinary electric spark. He had noticed the action of pointed 
 conductors, and considered that the discharging of an insulated 
 conductor by the presenting to it an earth-connected pointed con- 
 ductor was due to the point * drawing ' the charge from the body 
 (see Chapter IV. 15). So he conceived the idea of flying a 
 kite, provided with metallic points, near to or in a thunder-cloud. 
 
 Such a kite was flown. The end of the string was tied to a 
 key, and the key was fastened by an insulating silk cord to a tree. 
 Apparently Franklin had not considered sufficiently the part the 
 string had to play ; for it was when dry a bad conductor, and no 
 results were at first obtained. When, however, the string had been 
 wetted and so rendered more conducting by an accidental fall of 
 rain, abundant sparks could be drawn from the key. 
 
 It is clear that if the string had been put to earth, the thunder- 
 cloud would have been slowly discharged by a current of oppo- 
 sitely charged air proceeding towards it from the points of the kite. 
 
 2. Lightning Conductors. When a cloud charged with + 
 or electricity comes over any portion of the earth, there is in- 
 duced opposite to it a or + charge respectively, of a magnitude 
 
 i 
 
114 ELECTRICITY CH. vm. 
 
 equal to that of the cloud's charge. The stress thus produced 
 will, if of sufficient intensity, be relieved by a disruptive dis- 
 charge ; a ' flash of lightning ' will pass, and the equal and opposite 
 charges will disappear. 
 
 But if on any portion of "the earth there be erected one or 
 more metallic points well connected with earth there will be a slow 
 discharge of the convection-current nature, with the effect ii? 
 general that the A V between the cloud and the earth does not 
 rise high enough for a disruptive discharge to take place. 
 
 During this quiet discharge there will in the dark be observed 
 (as in the smaller experiments performed with electrical machines) 
 a glow or a brush of pale blue light on the points. This pheno- 
 menon, when it occurs on the extremities of a ship's masts, has 
 usually been called * St. Elmo's fire.' 
 
 If the conductor be not well connected with earth, its presence 
 may be a cause of danger, as determining the locality of the now 
 necessary disruptive discharge. The same may be said of a con- 
 ductor whose point is not sufficiently sharp. 
 
 For a good and effective conductor the following points should 
 be attended to. 
 
 (i.) The point should be sharp and of some metal not liable 
 to be corroded. 
 
 (ii.) The lower end should be put well to earth, e.g. be led to 
 a stream or pond, or into a pit filled with metallic refuse or coke, 
 or into a marshy piece of ground ; it being remembered that dry 
 soil does not conduct well. 
 
 (iii.) All detached metallic masses should be connected with 
 the conductor. Otherwise we may get disruptive discharges 
 where there are gaps in the connection. 
 
 (iv.) The conductivity of the conductor should be sufficiently 
 great to obviate any chance of fusion. 
 
 (v.) The height should bear a certain relation to the area to 
 be protected. 
 
 The reader will observe that a ' lightning conductor' aims in 
 reality at preventing discharges of a sudden nature, rather than 
 at receiving these discharges. 
 
 Some trees, such as pines, act as natural dischargers, the 
 numerous points of the leaves acting as so many imperfect 
 lightning conductors. 
 
CH. viii. ATMOSPHERIC ELECTRICITY 1 1 5 
 
 Wire network used as a protector. The inside of a hollow 
 conductor is (as will be further explained in Chapter X.) free from 
 any action due to charged bodies situated outside. Hence, it has 
 been proposed to protect buildings by covering them with an 
 earth-connected system of wires, and so making them approxi- 
 mately hollow conductors. If, however, a disruptive discharge 
 did take place, such wires (which would naturally not be very 
 thick) might be fused, and further, since in a discharge the con- 
 ditions are no longer statical, such a discharge might, even with 
 the wires intact, pass partly through the body of the building. 
 
 3. Return Shocks. Any abrupt change of electrical con- 
 dition makes itself felt physiologically as a shock. Hence the 
 sudden readjustment consequent on a lightning discharge may 
 produce fatal results in the vicinity of the actual discharge. 
 
 Experiments. (i.) Place near the prime conductor of a machine another 
 conductor imperfectly connected with earth. As the prime conductor is gradu- 
 ally charged we shall perceive small sparks passing over the gaps left, the 
 other conductor being acted upon inductively. If we now discharge the prime 
 conductor suddenly, we shall perceive a much stronger spark pass over each 
 gap as the second conductor suddenly loses its induced charge. 
 
 (ii. ) If a person hold his face near a charged conductor he will experience 
 a slight shock when this latter is discharged. Or if a person stand on the 
 insulating stool and hold the prime conductor he will receive a shock when 
 this is discharged, although the spark is not taken from ' him directly. 
 
 It is for the above reason that rule (iii.) of 2 should be 
 observed. 
 
 4. The High Potential of Thunder-clouds, The differences 
 of potential as directly caused by natural actions are very small. 
 They are mainly due to the friction of the winds over the earth, to 
 evaporation, and to the sun's action in producing differences of 
 temperature. The first two causes are the more important. 
 
 Experiments. (i. ) Direct a strong current of air by means of a pair of 
 bellows on to the plate of a delicate gold-leaf electroscope. The leaves will 
 diverge. 
 
 (ii. ) Heat a platinum capsule and place it on the plate of the electroscope. 
 Now drop into it some aqueous solution of copper sulphate. A divergence of 
 the leaves indicates that the vessel is charged . It is necessary, as a rule, 
 to use a condensing electroscope, the platinum vessel being on the upper 
 plate, and the lower plate being temporarily put to earth during the evapora- 
 tion. Pure water gives no results. 
 
 I 2 
 
Il6 ELECTRICITY CH. vm. 
 
 It must, however, be stated that many consider the electrical separation to 
 be due to friction between the heated vapour and the metallic vessel. 
 
 The question naturally arises, * How can we account for differ* 
 ences of potential so great as to give us flashes of lightning that 
 are sometimes perhaps several miles in length ? ' 
 
 Note. The writer is inclined to think that the difference of potential 
 required to produce flashes of lightning (whose length was roughly calculated 
 from the duration of the peal of thunder or by some other means) has been 
 over-estimated. The calculation of the difference of potential required is based 
 on experiments connecting the difference of potential between two brass spheres 
 with the length of spark that will pass between them. In the latter case the 
 spark is a truly disruptive discharge, there being nothing to help it on its way 
 (so to speak) except a little dust in the air. But, in the former case, who can 
 be sure of the nature of those discharges whose great length has been roughly 
 calculated ? They may be discharges only partly disruptive, and partly passing 
 from particle to particle of water. 
 
 We here give the theory that explains how a high potential may 
 be attained. 
 
 The reader will understand it better when he has read 
 Chapter X. 
 
 When visible cloud is first formed from true vapour, the water 
 is in very minute spheres. As these coalesce the cloud gets 
 denser and denser, and it is possible that in a heavy thunder- 
 cloud each sphere may contain thousands or millions of the very 
 small spheres first formed. 
 
 Now each original sphere we may suppose to have acquired a 
 charge from the air and from the vapour from which it was 
 formed, this charge being due to one of the causes named. 
 
 If the charge of each small sphere was Q, and its radius was 
 r, then its potential (as we shall see in Chapter X.) was measured 
 
 by ......... (i) 
 
 Now, let n such drops coalesce into one larger drop. To find 
 the radius R of this drop we have merely to state that its mass 
 equals the sum of the n small masses. 
 
 Hence, if d be the density of water, we have 
 
 = n 
 
viii. ATMOSPHERIC ELECTRICITY 
 
 Hence the new potential will be (from Chapter X.) 
 
 Comparing (i) and (2) we see that 
 
 __Q__ 
 
 New potential _ r \J n _ n __ I 
 Old potential Q ~~ n* ~ 
 
 r 
 
 Thus, if n were 1,000,000, we have 
 
 New potential = 10,000 x (old potential). 
 
 Besides this we shall in general have the charge tending to 
 pass to the outside of a cloud as a whole, and this would still 
 further decrease the capacity and increase the potential. 
 
 When the flash passes, the drops will, as a rule, more readily co- 
 alesce, there being no longer the electrostatic ' repulsion ' between 
 them. This may be one cause of the sudden fall of rain observed 
 to follow a powerful flash. 
 
 5. Potential at a Point in the Atmosphere. Observations 
 on the phenomena of thunder-storms are occasional ; and are, as a 
 rule, of necessity very inexact. 
 
 There are, however, daily observations of a far more exact 
 nature that are made, viz. observations of potential that exist 
 between the earth and points in the atmosphere at different 
 heights above the earth. 
 
 (a) Before discussing this matter we will, even at the risk of 
 repeating somewhat, examine what is meant by the potential at a 
 point in the atmosphere. 
 
 We have, in Chapter V. 2 and 3, given some explanation of 
 what is meant by 'potential' and how it is measured. The reader 
 must wait until he has read Chapter X. to understand fully what is 
 given in a general way in Chapter V. ; he can then turn back to the 
 present chapter and read it again. But the passages referred to 
 are sufficient to show that we measure the potential at a point in 
 the atmosphere by seeing how much work it takes to raise + unit 
 electricity from our arbitrary zero of potential, the earth, up to the 
 point in question. 
 
 (b) A small conductor, elevated above the earth, is at the 
 
IlS ELECTRICITY CH. vm. 
 
 potential of the region where it is situated when it has no charge 
 on it ; since it would require the same work to bring the + unit 
 up to the body from the earth as it would to bring the 4- unit up to 
 the same place before the conductor was put there. 
 
 (c) If an earth- connected conductor be situated at a region A 
 in the atmosphere that is at a different potential from the earth, a 
 charge of opposite sign to this potential will be induced on the 
 conductor, as explained in earlier chapters (see Chapter VI. 2 
 (ii.), &c). If this charge can escape, as it will off a point, or still 
 better off a flame, this convectional escape can only cease when 
 the region is reduced to the zero potential of the earth. The charge 
 induced on the conductor will indicate by its magnitude the poten- 
 tial of the region at A. 
 
 (d) If a conductor be situated at A, and be connected with an 
 insulated conductor situated elsewhere, e.g., nearer the surface of 
 the earth where there is a different potential, induction will take 
 place somewhat as in the last case, a charge being induced of 
 opposite sign to the potential at A. 
 
 If this charge can escape, as by means of a flame, this convec- 
 tional escape will take place until the whole conductor is at the 
 potential of the region at A ; for there cannot be equilibrium until 
 this is the case. But it must be noticed that, if the conductor be 
 large enough, this escape may sensibly alter the original potential 
 of the region A. 
 
 Notes. The above reasoning is of a general nature. It can however be 
 shown, by reasoning somewhat beyond this Course, that 
 
 (i. ) When there is a charge on the surface of a conductor, this means that 
 the conductor is at a different potential from the region surrounding it ; and 
 conversely. 
 
 (ii. ) That any such charge is urged normally outward from the conductor. 
 
 (iii.) That when there is no such charge, this means that the conductor 
 is at the same potential as the region surrounding it ; and conversely. 
 
 6. Methods of Measuring the Potential at a Point in the 
 Atmosphere. 
 
 (i.) One method, formerly used, was based upon (c) of the last 
 section. 
 
 A ball was elevated into the region in question ; was tempo- 
 rarily connected with earth ; was again insulated ; was lowered ; 
 
CH. viii. ATMOSPHERIC ELECTRICITY lip 
 
 and finally the charge induced (of opposite sign to the potential 
 in question) was examined 
 
 (ii.) But the best method is as follows. 
 
 E represents a quadrant electrometer (see Chapter X. 33, for 
 full description and complete figure). One pair of quadrants is 
 put to earth, and from the other pair is raised a wire ending in a 
 flame or slow-match at the point A, whose potential is required. 
 Thus we have an insulated conductor consisting of the slow- 
 match, wire, and pair of quadrants. As explained in (d) of the last 
 section, there will be a convectional discharge by means of the slow- 
 
 Earth. 
 
 match until the end of this latter is uncharged, and is, therefore, at 
 the potential of the region A (see last section (b)). When this is 
 the case, then the wire and the pair of quadrants with which it is 
 connected will all be at the potential of the region A. The needle 
 of the electrometer will then indicate by its deflexion the differ- 
 ence of this potential from the zero potential of the other pair 
 of quadrants. 
 
 (iii.) Or we may employ the dropping of water to carry off the 
 induced charge, and so to reduce the conductor to the potential 
 of the region where the water breaks into drops. 
 
 If with the pair of quadrants there be connected an insulated 
 tank, and if a pipe from this reach to A, and if there the water 
 fall away in drops, we shall attain the desired end just as well as 
 we did by the use of a flame or slow-match. 
 
 7. Results of Observations. Some results arrived at by 
 such observations are here given. In ordinary circumstances the 
 potential of the atmosphere under a cloudless sky is + ; and it 
 
I2O ELECTRICITY CH. vm. 
 
 increases as we move higher from the earth. There are variations 
 during the day, and these are fairly regular for all cloudless days. 
 
 Clouds may be charged to a + or potential ; and hence, 
 when clouds pass, the points in the atmosphere which lie between 
 them and the earth will also have a + or potential respectively. 
 
 Whatever the potential of the atmosphere, there will be found 
 on the exposed surface of the earth a charge of opposite sign, the 
 potential of the earth being of course zero. 
 
 8. Sheet-Lightning and other Phenomena, With respect to 
 the Aurora Borealis, references to papers on this subject will be 
 found in the Preface, p. viii. 
 
 Sheet -lightning may, in some cases, be a true brush or glow 
 discharge. .In other cases this name is wrongly given to the visible 
 reflection of a spark discharge that itself occurs out of sight. 
 
 Forked-lightning. The lightning-spark is nearly always zigzag 
 in its course. It is supposed that a greater resistance is created 
 in the direct line of a spark, and thus the discharge is diverted 
 along the line of least resistance to one side or the other. In very 
 rare cases the discharge is straight ; so that we conclude the 
 crookedness of the usual path to be due in some way to the air 
 through which it passes. 
 
 Globe-lightning. There is one form of discharge of which the 
 explanation is at present very imperfect. It is now sufficiently well 
 established that sometimes a globe form of discharge is seen. This 
 appears like a fiery globe moving slowly, or even at times resting 
 stationary. Its movements are sometimes very erratic. It appears 
 at times to follow conductors ; it is said to break through non-con- 
 ducting structures such as walls ; and all accounts agree that it 
 ultimately disappears with a powerful detonation. Plante has pro- 
 duced on a small scale similar appearances by aid of a voltaic 
 battery, which gave him large quantities of electricity and a high 
 difference of potential ; but it is not certain that there is a true 
 connection between the two phenomena. 
 
 Some believe this ' globe ' form to be a kind of Leyden jar 
 highly charged ; but this explanation does not appear to make 
 matters clearer. 
 
 No doubt in some cases the phenomenon was really a heated 
 aerolite ; but in other cases its nature seems undoubtedly to have 
 been that of an electric discharge. 
 
CH. IX. 121 
 
 
 CHAPTER IX. 
 
 SPECIFIC INDUCTIVE CAPACITIES. 
 
 i. Definition. We stated in Chapter VI. 4 (iv.) that the 
 charge of a condenser depended, cateris paribus, on the nature 
 of the substance that was between the two plates ; or, in usual 
 language, on the nature of the dielectric. 
 
 We propose in this chapter to discuss at some length the 
 relative efficiencies of different substances as dielectrics. 
 
 If we have two condensers of exactly similar construction, one 
 having air as the dielectric and the other having some other sub- 
 stance A as its dielectric, we shall find that for any given dif- 
 ference of potential between the plates the two condensers have 
 unequal charges. Further, we shall find that the ratio between 
 the two charges is constant, whatever the form of the condensers 
 and difference of potential between the plates, provided that all 
 conditions are the same for the two condensers. 
 
 Thus we have for the substance A a constant number repre- 
 senting its efficiency as a dielectric as compared with air. We 
 call this number the specific inductive capacity of the substance A, 
 and usually denote it by the symbol a. 
 
 Or we define as follows. 
 
 Capacity of the condenser when 
 The specific inductive capacity = A is the dielectric 
 
 of a substance A " Capacity of the condenser when 
 
 air is the dielectric 
 
 We here give a table of the specific inductive capacities of a 
 few substances. 
 
 These results must, however, be regarded as approximate only. 
 
 Air .... roo 
 
 Glass . . . .1-90 
 Sulphur . . . 2-58 
 Shellac . . .274 
 
 Paraffin . . .1-98 
 India-rubber . .2*22 
 Gutta-percha . . 2-46 
 
122 
 
 ELECTRICITY 
 
 The expressions inductive power, and dielectric constant, are used 
 as synonymous with specific inductive capacity. 
 
 2. Variation with Time, It was soon found that for solid 
 dielectrics the time for which the experiment lasted had a marked 
 influence on the value obtained for a. 
 
 If, in Chapter VI. 2 (i.), we interpose a plate of ebonite I 
 between the two condenser plates A and B, we have initially a 
 certain inductive action of the plate A on B ; this being indicated 
 by the divergence of the needle of the electrometer E B . 
 
 But we shall observe that this action gradually gets less, as 
 the plate I gradually yields to the electric strain. There is a slow 
 separation of electricities on I similar to the practically instan- 
 taneous separation that occurs in a conducting plate. In fact, 
 with time the ebonite plate I acts more or less as a conducting 
 plate ; and then, like a conducting plate, it screens B from the 
 action of A to a greater or less degree (see Chapter X. 17). 
 
 Or again we can, by waiting, charge a Leyden jar from a source 
 of constant potential with a larger charge than it would receive in 
 the first instance ; the specific inductive capacity of the glass appearing 
 to increase with time. As explained in Chapter VI. 10, there is 
 . a penetration of the 
 
 charge, and we find 
 residual charges. 
 
 The true specific in- 
 ductive capacity would 
 be that obtained be- 
 fore any such action 
 on the dielectric had 
 taken place. 
 
 3. Cavendish's Me- 
 thod. Cavendish de- 
 termined, with a degree 
 of accuracy that is sur- 
 prising when we con- 
 Earth. FIG. i. Earth. sider the apparatus 
 
 that was at his dis- 
 posal, the value of a- for various dielectrics. 
 
 His general method was as follows. He prepared a series of 
 condensers constructed with glass and tin-foil. These he employed 
 as arbitrary standards, with which he compared condensers in which 
 
 
CH. TX. SPECIFIC INDUCTIVE CAPACITIES 123 
 
 various other dielectrics were used. In this way he obtained the 
 inductive capacity of these materials as compared with that of the 
 standard glass. Then he determined the ratio borne by the inductive 
 capacity of this glass to that of air, and so obtained finally the specific 
 inductive capacity (see definition) of the dielectrics employed. 
 
 In fig. i. D represents in section the condenser in which the dielec- 
 tric was the substance whose specific inductive capacity was desired. 
 G represents one or more standard condensers. The upper plates of 
 both were charged from the same source, and therefore to the same 
 potential ; and the under plates were put to earth or were at zero 
 potential. If the charges upon the upper plates were + a and + b 
 respectively, then those upon the lower plates would be (very nearly) 
 a and - b respectively ; the charges on the upper plates exceeding 
 those on the lower by negligible 'free ' charges. 
 
 When the condensers were charged, matters were arranged as in 
 fig. ii. It is not difficult to see that if a and b were equal, then the 
 charges + a and 
 
 - b would (very 
 
 nearly) neutralise 
 
 one another. The 
 
 wire connecting 
 
 these plates com- 
 
 municated with a 
 
 delicate electro- 
 
 scope : and thus 
 
 it could be seen Earth ' FlG ' iL Earth " 
 
 whether there was neutralisation or no. Standard condensers were 
 
 grouped together until such neutralisation was observed. When this was 
 
 the case the capacity of G was equal to that of D. For, by our formula, 
 
 Y. (V-V ) 
 
 where K' and K were the capacities of the two condensers D and G 
 respectively. And since (V-V ) was the same for both, it followed 
 that a could only equal b if K' = K. Then, from a knowledge of 
 the dimensions of the two condensers, Cavendish determined the ratio 
 of the two inductive capacities, or the ratio that would have existed 
 between the charges a and b had the condensers been similar in all 
 respects save in the nature of the dielectric 
 
 In somewhat the same manner he compared the capacities of these 
 standards with that of a large sphere ' isolated ' (see Chapter V. 8 
 &c.) in the middle of the room. From this comparison he was able to 
 calculate what would have been the capacities of the standards had 
 
I2 4 
 
 ELECTRICITY 
 
 air been their dielectric, and thus he finally found the ratio that the 
 charge a would have borne to that of an exactly similar condenser 
 having air as the dielectric. 
 
 4. Faraday's Method. (a) The figures represent, complete 
 and in section, the apparatus used by Faraday ; this being a con- 
 denser of a convenient and symmetrical form. The outer coating 
 consisted of a hollow brass sphere P Q, that was always put to earth. 
 The inner coating was a brass globe C. This could be charged 
 by means of the knob B connected with it. A represents a layer 
 of shellac serving to insulate the wire connecting B and C. 
 
 FIG. i. FIG. ii. 
 
 Further, the space between P Q and C was air-tight ; by 
 means of a tap any gas could be introduced, and P Q could be 
 pulled apart so as to admit of the introduction of various dielec- 
 trics into the space between C and P Q. 
 
 (b) The potentials of B at different times could be compared (not 
 measured in absolute units) by touching B with an insulated ball 
 of a certain size, communicating this charge to a torsion balance 
 (see Chapter IV. 12 (i.) ), and observing the torsions needed in the 
 different cases to keep the needle deflected at constant angle. 
 
CH. ix. SPECIFIC INDUCTIVE CAPACITIES 125 
 
 Note. As this method of comparing potentials is obsolete, we merely stait 
 that they can be so compared ; we do not go further into the matter. 
 
 (c) For his experiments it was necessary to have two such con- 
 densers that were of exactly equal capacities when filled with air. 
 
 To test whether this were the case, he charged the one and 
 measured the potential of its knob B. He then connected this 
 knob B with the knob B' of the uncharged condenser ; thus 
 causing the charge to be divided between the two condensers. 
 He then measured the potential of B again. If the jars were of 
 equal capacity the new potential should be half the old. In fact, 
 in mathematical symbols we have 
 
 v = ^ . . . in the first case, 
 Js_ 
 
 and v 1 = -^ = v . . . in the second case, 
 
 if the capacities are equal (see Chapter VI. 4). 
 
 (d) Having provided himself with two similar condensers, he 
 filled one with the substance whose specific inductive capacity was 
 required. (For convenience he only half filled it, and calculated 
 accordingly.) He then charged the other, or air-condenser, and 
 observed the potential v of its inside coating. The knob B was 
 then connected with the knob of the condenser which had 
 been filled with the dielectric in question and had been left un- 
 charged. The new potential v' of the compound condenser was 
 then measured. As before stated, the outside coatings were to 
 earth. Let K be the capacity of the air condenser, and a K that 
 of the similar condenser filled with the dielectric in question, 
 where n- is the required specific inductive capacity. 
 
 Then we have 
 
 ,,-Q.v- Q 
 
 K' 
 
 Whence 
 
 K 
 
 5. Modern Methods, For the reason given in 2, methods 
 have been devised to obviate the errors due to penetration of 
 charge. These methods employ rapidly alternating + and 
 charges, this reversal of charge occurring even as often as 12,000 
 times per second in some of Gordon's experiments. The condensers 
 
126 ELECTRICITY CH. ix. 
 
 thus never remain charged for more than a fraction of a second j 
 and the charge has, so to speak, no time to penetrate into the 
 body of the dielectric. 
 
 Let us suppose that, when this reversal of charge takes place 
 with a certain rapidity, the value of c- is measured ; and let us 
 further suppose that, on increasing the rapidity of reversal, no 
 change is made in this value of rr obtained. When this is the case 
 it may fairly be said that errors due to penetration of charge have 
 been obviated. 
 
 Gordon's, experiments. Mr. Gordon has made a series of ela- 
 borate experiments on the specific inductive capacities of various 
 solid dielectrics, in which the principle of rapid reversal of charge 
 was employed. 
 
 As it is not possible to give in a brief space a satisfactory 
 account of the apparatus and methods employed, the reader is 
 referred to Gordon's ' Physical Treatise on Electricity and Mag- 
 netism,' where he will find a full description given. It may, how- 
 ever, be mentioned that there are two considerations which may 
 modify considerably the value of the results obtained ; the first 
 objection is of undoubted weight, the second is possibly without 
 foundation. 
 
 (i.) In calculations concerning the circular brass plates which 
 formed the plates of the condensers, no allowance was made for 
 the variations in distribution which occur within a considerable 
 distance of their edges. These condenser-plates should have been 
 provided with guard-rings (see Chapter X 32^. 
 
 (ii.) Further, it may reasonably be asked whether, with these 
 rapid reversals of charge, the conditions are truly statical. 
 Whether, in fact, we are obtaining what we desired, viz. the true 
 electrostatical specific inductive capacity of the dielectric. It is 
 assumed that these extraordinarily rapid swingings to and fro of 
 electrical charges leave the theory of the condenser unaltered. If 
 this be really the case, it must be because the rapidity with which 
 statical electric equilibrium is attained immeasurably transcends 
 the rapidity of the reversals. 
 
<:H. x. 127 
 
 CHAPTER X. 
 
 ELECTROSTATIC POTENTIAL. 
 
 i. Introductory. In the present chapter we purpose deal- 
 ing more fully with the conceptions of electrostatic potential, fields 
 and lines of electric force, and work of electric charge and discharge. 
 
 We recommend the student to read again carefully the follow- 
 ing portions of the earlier chapters of this course. 
 
 (i.) Chapter II. 8, &c. Here he should find no difficulty in 
 making for himself all such necessary changes in the wording as 
 the substitution of ' a + unit of electricity ' for * a + magnetic pole 
 of unit strength/ &c. 
 
 (ii.) The whole of Chapter V. 
 
 (iii.) In Chapter VL, all that relates to the formulae connecting 
 K, V, and Q. 
 
 (iv.) Chapter VII. i ; and Chapter VIII. 5 and 6. 
 
 He will then be in a better position to understand on what 
 points he needs more exact knowledge and fuller discussion. 
 
 2. Definition and Measurement of Work. In addition to 
 the mechanical principles and terms briefly mentioned in Chap- 
 ter II., we must now assume a knowledge of the principle of work. 
 
 The student should clearly understand 
 
 (i.) What * work ' means ; and its essential difference from 
 ' force/ 
 
 (ii.) What ' positive ' and ' negative ' work mean respectively. 
 
 (iii.) In what sense, and under what general conditions, we 
 can say that * when a body is displaced in a field of force from a 
 position A to a position B the work done is independent of the 
 route followed between these two positions.' 
 
 In the C.G.S. system the unit of work is called the ' erg,' and 
 is the work done when a force of one dyne is overcome through a 
 distance of one centimetre along the lines of force. 
 
128 ELECTRICITY CK. x. 
 
 3. Dimensions of Work. We see that the unit of work [W], 
 or the erg, involves the fundamental units in the following way. 
 
 L\V] = [F] x [L] = [M] x [A] x [L] = [M] x B. x [L] ; 
 
 of finally . . . [W] = . 
 
 4. Energy and Conservation of Energy. (a) If we have a 
 supply of heat we can by a properly constructed engine do work 
 and use up (or cause to disappear) some of the heat. 
 
 (b} If a cannon-ball be moving, we can cause it to do work 
 such as the raising of weights, until it comes to rest. 
 
 (c) If we have a body of water at a higher level, we can cause 
 it to do work by means of a water-mill until it is all at the lower 
 level. 
 
 (d) If an elastic be stretched, we can make it do work until it 
 has returned to its unstretched condition. 
 
 (e) If two substances have chemical affinity for one another, 
 we can let them combine and give out heat, and can then cause 
 this heat to be used up in doing work, this going on until the 
 chemical affinity is * satisfied.' 
 
 In each of the above cases we had a condition of things such 
 that we could get work done (measured in ergs, foot-pounds-weight, 
 or in any other unit involving the essential product of force over- 
 come into distance through which it is overcome] at the expense of 
 losing this advantageous condition of things. 
 
 Now when a body, or a system of bodies, is in such a con- 
 dition that it can do work, then it or they are said to possess 
 energy. In fact, energy may be sufficiently accurately defined as 
 capacity for doing work, and it is measured by the work that can 
 be done, or it is measured in ergs. 
 
 Where the energy is possessed (as in cases (a) and (b) above) 
 in virtue of mass and velocity combined, it is called kinetic 
 energy. Where it is possessed (as in the other cases) in virtue of 
 position, it is called potential energy. In case (e) we may call it 
 chemical potential energy. 
 
 One of the greatest generalisations of modern times, the most 
 powerful weapon we possess in discussing the problems of physical 
 science, is the great law of conservation of energy. This law, 
 
CH. x. ELECTROSTATIC POTENTIAL 129 
 
 abundantly proved by direct experiment and corroborated by 
 masses of more indirect evidence, means that energy is indestruct- 
 ible. This means that in a self-contained system (or a system 
 not acting upon, nor acted upon by any other system) the total 
 amount of energy is constant, though it may undergo transforma- 
 tion, as, e.g., from the form of mechanical-energy into the form of 
 heat -energy. 
 
 The student should study carefully the principle of 'conserva- 
 tion of energy ' as given in books on mechanics, and should make 
 himself familiar with cases of transformations of energy. 
 
 5. Work against a Constant Force, Where the field of 
 force acting on a particle is uniform, the calculation of work done 
 is simple enough. 
 
 It is merely necessary to know the force in dynes and the dis- 
 tance between the initial and final positions of the particle, as 
 measured along the lines of force, in centimetres. Then the pro- 
 duct gives us the work in ergs. 
 
 All work done against gravitation on the surface of the earth 
 comes practically under this head ; since, for distances small 
 as compared with the radius of the earth, we may consider the 
 field to be constant. 
 
 6. Work where the Force varies as the Distance. Another case of 
 work may here be considered, though it is of no direct interest in our 
 present subject. 
 
 When a piece of india-rubber is stretched from the length it has 
 when under no tension into some new length, we overcome a gradu- 
 ally increasing force. This force is proportional to the extension. If 
 then we start with zero force and end with a force of F dynes, the 
 extension being b centimetres, it is not difficult to see that 
 
 the work done = ( F x b} ergs. 
 In this case we may call ^ F the average force overcome. 
 
 7. Work, where the Force varies as the Inverse Square 
 of the Distance, or as - 2 . In our present subject we are con- 
 cerned mainly with forces that act towards or from centres of force, 
 the magnitude of the force, due to any such centre, varying in- 
 versely as the square of the distance from that centre. A single 
 magnetic pole, or a single sphere charged with electricity, are 
 simple cases of such centres of force. 
 
 K 
 
tance 
 
 130 ELECTRICITY OH. x. 
 
 We will first discuss this important case without making any 
 reference to the nature of the force, whether it be electrical, 
 magnetic, or of any other description ; and then we will apply 
 the general results obtained to electro- 
 statics. 
 
 Let O represent a centre of force, and 
 let us suppose that there is a particle that 
 is repelled from this centre with a force 
 
 that varies as ... 
 
 (distance)"* 
 
 If at i centimetre the force be measured by F dynes, then at dis- 
 
 F F 
 
 r A it will be > ^, and at distance r B it will be - 2 , dynes ; 
 
 v A) v R) 
 
 r A and r B being measured in centimetres. 
 
 What work then will be done in moving the particle from A to 
 B ? This is evidently no simple matter of arithmetic, for we cannot 
 by any arithmetical means find the average force as we did in 6. 
 
 We might work out the problem here without going into any 
 advanced mathematics. But, as the calculation is one which 
 belongs essentially to the integral calculus, and is a very easy 
 matter to settle with the aid of this powerful mathematical 
 weapon, we prefer in an elementary Course to give merely the 
 result. 
 
 This result is that 
 
 Work done in moving the _ -p / I I \ .^ 
 particle from A to B ' V B r A ) * 
 
 If A be so remote from O that the force at A is quite inap- 
 preciable as compared with that at B, or so remote that z ^ and 
 
 VA) 
 
 *- are practically zero as compared with JL- and -L-, then 
 
 ( r jj VK) VB) 
 
 we may consider A to be at an infinite distance from O. We 
 thus obtain from our formula the result that 
 
 Work done in bringing the particle F 
 from an infinite distance up to B r B * ' 
 
 The reader is here warned to remember 
 (i.) That F is the force acting on the particle when at unit 
 distance from O. 
 
CH. x. ELECTROSTATIC POTENTIAL 13! 
 
 (ii.) That r refers to the distance from O up to which the 
 particle is brought, and is not the distance through which it is 
 moved. 
 
 8. Potential, and Difference of Potential, If the reader will 
 turn back to Chapter V. he will there see explained the general 
 meaning of potential, and the measurement of the same with 
 respect to some chosen zero. The explanation was, however, 
 incomplete, because we had not then considered the question of 
 Work. 
 
 Referring to the diagram given in 7, we now understand that 
 there is between A and B a certain difference of potential ; and that 
 it is measured by the work done on some sort of unit particle in 
 bringing this from A to B. 
 
 If the field of force be one due to gravitation, and be con- 
 sidered to be practically uniform and acting with a force of 981 
 dynes on i gramme mass, or with i dyne on | T gramme, then we 
 do one erg work in raising our mass of -g-| T gramme through i centi- 
 metre vertically. Or we could theoretically measure in centimetres 
 the difference of gravitation potential QV level between two points by 
 finding the work done in ergs in raising a particle of F | T gramme 
 mass from one point to the other. 
 
 If the field of force be electrical, then we take as our unit 
 particle a + unit of electricity ; and we measure the difference 
 of electric -potential between the points A and B by the work 
 that is done on this test-particle in moving it from A to B. 
 
 If the work is found to be negative, we say that the final posi- 
 tion has a lower potential than the initial position. 
 
 Note. The reader must observe that all fields of force can be considered 
 from the potential point of view, and that we may speak of gravitation, 
 potential as well as of magnetic- or electric-potential. 
 
 9. Application of 7 to ths Measurement of Electric-Poten- 
 tial. Referring to the diagram given in 7, let us now consider 
 the case of electrostatic forces. 
 
 Let us take as our test-particle (see above) a + unit of elec- 
 tricity ; and let there be situated at O a quantity of electricity 
 measured by Q units, this giving us an electrical field of force. 
 
 By the laws of electrostatics, summed up at the end of 13, 
 Chapter IV., the force exerted upon our + unit by a quantity of 
 
 K 2 
 
132 ELECTRICITY CH. x. 
 
 Q units will at unit distance be measured by Q dynes ; and, at 
 any other distance r from the centre O, by -^ dynes. 
 
 LetV A and V B represent the numerical value of the potentials 
 at A and B respectively ; these being measured by the number of 
 ergs work done in bringing our + unit from infinity up to the 
 points A and B respectively. Then the formulae given in 7 tell 
 us that 
 
 The absolute potential V A v of the _ O 
 
 point A r A * 
 
 The absolute potential V B of the __ O 
 point B r s 
 
 The difference of potential, V B V A , _ Q / I i \ 
 between the points A and B \r* r A 
 
 An example or two will make the meaning clearer, 
 (i.) Let there be + 24 units of electricity at the point O ; let 
 r^ be 4 cms., and let r A be 6 cms. 
 
 Then at unit distance (or i cm.) from O our + unit will 
 
 be repelled with a force of 24 dynes ; at B with a force of 2i 
 
 4 
 
 = * dynes ; and at A with a force of ^ = - dynes. 
 
 O 
 
 The potential of B will be greater than that of A by an amount 
 measured by 
 
 V B -V A=24 (<->)=, 
 
 Or it will take 2 ergs work to move our + unit from A up to B. 
 
 The absolute potential of B will be measured by ~ = 6 ; and 
 
 4 
 
 ? i" that of A by 24 = 4 ; or it will 
 
 o 6 
 
 A take 6 ergs to move our + unit 
 
 from infinite distance up to B, 
 and 4 ergs to move it from in- 
 +10 finity up to A. 
 
 (ii.) Let there be several 
 
 quantities of electricity, as + 20 at O, 16 at P, + 10 at Q. 
 These will give rise to a somewhat complex field of force. But 
 still the results arrived at in 7 and in the present section enable 
 
CH, x. ELECTROSTATIC POTENTIAL 133 
 
 us to find easily the potentials of any points as B and A in the 
 field, and the difference of potentials. 
 
 For the total work due to this system of electric charges will be 
 the algebraic sum of the works due to each charge separately. 
 
 Thus the absolute potential of B will be measured by 
 
 of Ab + - Hence 
 
 the difference of potential, V B V A , is easily found by subtraction. 
 
 (iii.) Where there are a whole series of quantities q^ q^ q z . . . 
 
 situated at distances r^ r. 2 , r z . . . respectively from a point 
 
 B, then the potential of B as due to these quantities will be 
 
 ^J. + i + fl -f . . . ; which is conveniently represented by 
 r \ r i r i 
 
 2 $. Here, of course, attention must be paid to the signs of the 
 
 r -& 
 quantities ^, &c. 
 
 And V B - V A = S-?_- S -?. 
 >B r A 
 
 (iv.) Where the quantities are distributed continuously over a 
 
 conductor, or over any surface, this adding up of -^ + . . . &LC. 
 
 r \ 
 becomes in general a matter for the application of the integral 
 
 calculus. 
 
 Note. It may occur to the reader that if r be zero, or if we touch the 
 
 charge Q, then the potential becomes infinite, since V oo . In answer to 
 
 o 
 
 this we may remark that it is physically impossible to be at zero distance from 
 any finite mass, or electrical or magnetic quantity. We can be at zero distance 
 only from a geometric point. Hence, we cannot have Q finite and r zero 
 simultaneously. 
 
 10. Equipotential Surfaces. 
 
 (i.) If we so move our + unit of electricity as not to move up 
 or down the lines of force, then we do no work against the 
 electrical forces. Hence, by definition, all the positions thus 
 arrived at must have the same potential. 
 
 Let us consider the case of a simple field of force due to a 
 single charge of Q units situated at the point O (see 7, figure). 
 If we move our + unit over any sphere that has O as centre, we 
 
I 34 ELECTRICITY en. x. 
 
 do no work ; and hence every point on such a sphere has the 
 same potential. 
 
 Thus if we consider the sphere that his O as centre and that 
 passes through the point A, it-is clear that all points on this sphere 
 have the same potential as A ; a potential that is measured by 
 
 -i ergs. 
 
 It is necessary to do work to the amount of Q ( [ ) 
 
 >1s ^"A/ 
 
 ergs to move our + unit of electricity from any point on the 
 sphere passing through A to any point on that passing through B. 
 
 (ii.) Equipotential surfaces in general. Where there are many 
 centres of force, there both the lines of force and the equipotential 
 surfaces will be complex, and will not have the simple forms that 
 they had in the last case. 
 
 But in all cases certain general results hold good. The equi- 
 potential surfaces are always perpendicular to the lines of force 
 (see n); they may be mapped out into a series of 'marked' 
 surfaces such that we do one erg work in moving our + unit 
 between two consecutive surfaces ; these marked surfaces will be 
 closer together where the field is stronger, further apart where 
 the field is weaker, and at a constant distance where the field 
 is uniform. 
 
 ii. Lines of Force are Perpendicular to Equipotential 
 Surfaces. This has been already proved implicitly ; or rather it 
 is a consequence of the very definition of an ' equipotential sur- 
 face.' But it may be well to give a formal demonstration. 
 
 Let P F be a line of force ; and let A P B be a section of the equi- 
 potential surface that passes through P ; and let P T be a section of 
 T the tangent-plane to this surface, at the 
 
 point P. 
 
 ^ B Then shall P F be perpendicular to 
 
 the tangent P T. For if possible let it 
 lie as P F' ; making with P T an angle 
 less than 90. We can now resolve the 
 force P F into two components, along 
 the tangent P T and perpendicular to it 
 respectively. For the former component we have P F cos F' P T ; and 
 this is not zero, since F' P T is not 90. Since then there is a tan- 
 gential component, we should do work in moving our + unit along 
 
CH. x. ELECTROSTATIC POTENTIAL 135 
 
 the surface near P. But this is contrary to hypothesis, since A P B 
 is an equi potential surface. Therefore, F P must be perpendicular 
 to A P B. 
 
 12. Field- strength.; and Rate of Change of Potential* 
 The diagram represents a portion of a field of force ; this por- 
 tion being so small that we may consider it to be uniform, 
 or the lines of force to 
 
 be parallel (see 13, 
 coroll.). Let A and B 
 be any two points in 
 this field, and let them 
 be separated by a dis- 
 tance of A B cms. 
 
 If A B be not perpendicular to the lines of force, i.e. if A and 
 B be not on the same equipotential surface, there will be a force 
 on a + unit of electricity acting along the line A B. We may 
 represent the force on a + unit that acts from B to A by the symbol 
 FA ; this will be measured in dynes, and may be + or as it 
 acts from B to A or from A to B. [See i (i.) ; and compare 
 Chapter II. 8.] 
 
 By the definition of * field-strength,' c., it is clear that F" measures 
 that component of the field-strength that has the direction B A. 
 
 Then, by the definition of work, we have 
 
 The work done in moving = F B x A B er(rs 
 our + unit from A to B 
 
 But, by the definition of potential, this work measures the 
 difference of potential between A and B ; or we have (V B V A ) 
 = F* x A B. 
 
 Whence F = 
 
 V,- V 
 
 AB 
 
 Now since A B is measured in centimetres, this last expression 
 represents the change of potential per unit length, or the rate of 
 change of potential, along A B. 
 
 So we may translate this formula (F* = B ~" A ) into words 
 
 A Jj / 
 
 somewhat as follows. 
 
136 ELECTRICITY en. x. 
 
 In a field of force, the component in any direction of the force 
 acting upon a + unit, i.e. the component in any direction of the total 
 field- strength, is measured by the space-rate of change of potential in 
 that direction. 
 
 When the potential is constant there is zero field-strength, and 
 conversely. 
 
 It is very important to remember that a field of force implies 
 varying potential, and conversely. 
 
 Examples. (i.) In the inside of a charged vessel we may havea 
 potential of very great magnitude ; but it is constant throughout, and 
 we have zero field of force. 
 
 (ii.) Half-way between two equal charges of the same sign we have 
 a certain potential ; but there is a point of zero force. 
 
 (iii.) Half-way between two equal charges of opposite signs we 
 have zero potential, but not zero field of force. 
 
 13. The Mapping Out of Lines of Force; Simple Case. 
 
 Since we can draw a line of force through every point in the field 
 such a line being defined as the direction in which a + unit of 
 electricity is urged it follows that the lines of force are infinite in 
 number. 
 
 But a little further consideration will show us that here, as in 
 the case of equipotential surfaces, we can mark a certain number 
 out of this infinite crowd of lines, in such a way as to represent 
 the various strengths of different parts of the field. 
 
 Let us as before consider the simple case of a single charge of 
 Q units collected at the point O (see fig. 7) ; and let us con- 
 sider first a spherical surface, described about O as centre, with 
 radius i centimetre. At any point on this sphere the force exerted 
 on a + unit will be Q dynes ; or the strength of the field will be 
 measured by Q. 
 
 The reader must remember that in our present case the lines 
 of force, which are infinite in number, radiate from O ; and that 
 they cut at right angles the surfaces of all spheres which have O 
 as centre. 
 
 Now let us mark out (we may suppose that we paint them red] 
 such a number of lines of force that exactly Q of them pierce each 
 i square centimetre of this spherical surface. Then the number 
 of lines piercing unit area placed perpendicularly to the lines of 
 
UNIVERSITY OF CALIFORNIA 
 
 CH x. 
 
 force does, over this spherical surface, represent numerically the 
 strength of the field. 
 
 At 2 centimetres distant from O these marked lines will have 
 spread out over a sphere of four times the area ; and therefore 
 there will be \ Q lines piercing each i square centimetre of the 
 sphere whose radius is 2 centimetres. And, in general, at any 
 distance r from O the lines will have so thinned out that the 
 number piercing each i square centimetre of the sphere, whose 
 
 radius is r, will be ^. 
 
 But , -, ... -^, also represent the strengths of the 
 
 field at distances 2 centimetres, 3 centimetres, r centimetres, from 
 O respectively. Therefore these marked lines will represent 
 numerically the strength of the field at any point, by the number 
 of them that at that point cut i square centimetre held perpendicular 
 to them. 
 
 The reader should carefully study this method of marking out 
 the lines. He will see that it consists in marking out such a num- 
 ber that they do over one equipotential surface (viz. over the 
 sphere of unit radius) represent, by the number piercing each i 
 square centimetre, the field- strength at all points over this surface. 
 He will see also that the result found to hold is that these lines 
 so chosen will indicate the field-strength at any distance from O, 
 in just the same manner. 
 
 Of course where the force on + unit is less than i dyne, or 
 the field-strength is less than unity, there will be less than i 
 line piercing each i square centimetre. We must then take 
 several square centimetres and divide the number of lines piercing 
 them by the number of square centimetres taken. 
 
 Corollary. Uniform field. Hence it follows that in a uniform 
 field the marked lines of force are parallel and equidistant from 
 one another. 
 
 14. General Case. We cannot in an elementary Course 
 prove that the same result holds in the general case ; but at least 
 the above discussion will prepare the reader for the following 
 statement. It may be well to point out that ' equipotential sur- 
 face,' and ' surface lying perpendicularly to the lines of force ' are 
 synonymous expressions. 
 
138 ELECTRICITY CH. x. 
 
 * If we mark out such a number of the lines of force that ovet 
 any one equipotential surface . they measure, by the number piercing 
 each i square centimetre, the field-strengths all over that eqtti- 
 potential surface, then will these same marked lines in any part of 
 the field measure the field-strength by the number piercing i square 
 centimetre held perpendicularly to the lines of force at the place in 
 question." 1 
 
 It is to be noted that when we use the expression * number of 
 lines of force,' we refer to these marked'Xvnes. 
 
 15. Total Number of marked Lines of Force. In the 
 simple case given in 13, since we mark Q lines of force for 
 each i square centimetre of the sphere of i centimetre radius, and 
 since such a sphere has an area of 4 TT square centimetres, it follows 
 that we have a number 4 TT Q of such marked lines. 
 
 And if there be any system of quantities of electricity + or 
 , and if the whole system of quantities be surrounded by 
 an envelope, then the total number of 4- lines piercing this 
 envelope will be 4 TT 2 Q ; where S Q is the algebraic sum of the 
 quantities. 
 
 1 6. Tubes of Force. <F<r is Constant.' We may, out of 
 the infinite crowd of lines of force, construct hollow tubes 
 bounded by the lines. Such tubes are called tubes of force. 
 
 Now lines of force cannot 
 cross ; for, if they did, it 
 would mean that at the 
 point where they crossed 
 there were two resultant 
 forces, which would be absurd. Hence, if such a tube once in- 
 clude any number n of the marked lines, it will always include 
 these same lines, neither more nor fewer. 
 
 Let us consider the simple case of 13, and let us suppose 
 the area of the perpendicular cross section of such a tube to be 
 <TJ square centimetres. Then, if the tube enclose n lines of force, 
 there are n lines of force piercing <r } square centimetres ; or the 
 
 average force on our + unit will be measured by , according to 
 
 *\ 
 
 the results of 13. If we name this average force Fj, then the 
 product of (average force over c?-oss section] into (the cross section] 
 wjll be 
 
CH. x. ELECTROSTATIC POTENTIAL I 39 
 
 -r- H 
 
 FI X <r, = X rr\ = n. 
 
 "] 
 
 If we take another section o- 2 of the same tube we have for 
 
 average force the value ; and the product F 9 x <r 2 still equals n. 
 ff 2 
 
 Hence we have for the same tube of force the result that 
 F 1 rT 1 =F 2 <T 2 = . . . = n = constant ; which is more shortly 
 stated as 
 
 F <r= constant. 
 
 If we take an oblique section <r', and take the force F' resolved 
 perpendicular to this oblique section, then F' = F cos 6, and v' = <r sec 0, 
 where 6 is the angle that the oblique section makes with the perpen- 
 dicular section. Hence the product FV = Fa- = constant. 
 
 We may state this property of tubes offeree as follows. 
 
 If a- be the area of cross section of a tube of force, and F be the 
 average force on H- unit over this area a- resolved perpendicularly to 
 it, then the product Fa- is constant ; it equals n, where n is the number 
 of marked lines of force enclosed by the tube in qttesiion. 
 
 We have proved the above only for the simple case of 13 ; but 
 we here state that it is true for any field, however complex. 
 
 A tube of force in which one marked line is enclosed, or a 
 tube where FT = i, is called a unit tube of force. 
 
 17. Statement of some further Theorems on Lines of 
 Force. 
 
 (1) It can be shown that a tube of force terminates against 
 equal and opposite quantities of electricity, + q and q respec- 
 tively. This statement includes Faraday's law ot electrostatic 
 induction referred to and discussed in Chapter IV. 16. 
 
 (2) Lines of force and tubes of force do so terminate when 
 they meet a conducting surface. Any lines of force on the other 
 side of the conducting surface are either lines that have curved 
 round part the edge of the conductor, or are independent lines 
 due to other charged systems. 
 
 (3) Hollow conductor acting as a screen. The diagram repre- 
 sents in section a hcllow conductor X Y separating two systems of 
 electrical charges, a b c external, and/ q r internal. 
 
 Now theory and experiment show that no lines of force due to 
 abc exist inside X Y (see Chapter IV. 13 (ii.)). Hence, if X Y 
 is to earth the system pqr\s totally unaffected by abc \ and if 
 
140 
 
 ELECTRICITY 
 
 CH. X. 
 
 X Y be not to earth the only influence of a b c will be to raise or 
 lower the potential of X Y and of its interior as a whole ; they do 
 not in either case alter the nature of the field inside. 
 
 Again, both theory and experiment show that the external effect 
 of the presence of the system pqr inside X Y is to cause there 
 
 to be on the external surface 
 of X Y a charge of magnitude 
 equal to the algebraic sum of 
 / + q + r ; but that this ex- 
 ternal charge is distributed 
 simply according to the shape 
 of X Y and to the distribution 
 of external bodies, not accord- 
 ing to the position of/,^, and 
 r(see Chapter IV. 16 (c)). 
 
 Hence X Y acts as a screen 
 between the two systems. 
 So we may use large conducting plates put to earth as fairly 
 complete screens. A wire-gauze case will completely screen an 
 electroscope (see note). 
 
 (4) Potential at an external point due to uniformly charged concen- 
 tric spheres. It-can be shown by the integral calculus that if we have 
 
 a series of concentric 
 spheres of centre C, 
 having charges Q 1} 
 Q 2 , Q 3 respectively, 
 then the potential at 
 an external point P 
 due to these charges 
 will be the same as 
 if they were all col- 
 lected at the com- 
 mon centre C. Or 
 
 FIG. ii. 
 
 Vp H~c~ 
 
 Note on insulated spherical screens. We will consider here two simple 
 cases of uninsulated and uncharged screens, showing what effect their presence 
 has upon the potential of an external point. The reader can easily draw for 
 himself the simple diagrams that represent the two cases considered. 
 
CH. x. ELECTROSTATIC POTENTIAL 14! 
 
 (i. ) Let there be a charged sphere, or a charged particle, A ; and let there 
 be an external point P at a distance R from A. Then if the charge on A be 
 
 Q units, the potential V P of P is measured by -i If now an insulated and un- 
 
 R 
 
 charged spherical screen B be placed round A concentrically with it, between it 
 and the point P, we know both by theory and experiment that the field at P is 
 due only to the ' free ' charge that has been ' repelled ' to the surface of B ; the 
 charge + Q on A, and the induced charge Q on the inner surface of B, 
 together give a zero field at any point P outside. But, by (4) above, the 
 potential at P will be the same as if the charge Q on the sphere B were col- 
 
 lected at its centre. That is, we have still V P = >-. Hence, when A and B 
 
 K. 
 
 are concentric, the presence of the insulated spherical screen B makes no 
 difference in the potential of external points. 
 
 (ii.) Next, let B be no longer concentric with A ; but let the distance from P 
 to the centre of A be R, while to the centre of B the distance from P is r. Then 
 
 without the screen we have V P = ^ as before. But, with the screen, we have 
 
 R 
 
 now Vp = ^> ; since it is only the free charge on the screen that gives a field 
 
 at the external point P. 
 
 (iii. ) If any charge Q' be given to B, we have merely to add it to the 
 induced charge Q that has been inductively given to its surface ; and the 
 potential at P becomes 
 
 in the two cases respectively. 
 
 18. The Potential of an 'Isolated' Body. Potential is a 
 property of a point in space, as explained earlier. But any equi- 
 potential surface has one potential, the same for all points on it. 
 Now any continuous homogeneous conductor (as a brass sphere 
 or cylinder,- a tin vessel, &c.) forms an equipotential surface ; for 
 no differences of potential could exist on a surface where electrical 
 readjustment takes place instantly. Hence we can speak of ' the 
 potential of a conductor.' 
 
 The potential may be a hard matter to calculate, owing to the 
 continuous nature of the distribution of electricity over the body, 
 
 which makes the expression 2^- of 9 (iii.) in general a matter 
 
 for the integral calculus. But we can easily deal with the case of 
 a sphere. 
 
 19. Potential of an Isolated Sphere. The figure represents 
 in section a sphere of centre O and radius R. Let it have on 
 
I4 2 ELECTRICITY CH. x. 
 
 it a charge Q, which we will consider to be divided into a series 
 of small charges q lt q^ q& &c., distributed over the surface of the 
 sphere. What is the potential of the 
 sphere? Or, what will be the work re- 
 quired to bring our + unit from infinity 
 up to the sphere ? 
 
 Now the sphere has the same poten- 
 tial all over it, and inside it. Hence we 
 may find the potential of the most con- 
 venient point, viz. of the centre O. 
 
 But by 13 this is 
 
 V, 
 
 V ,Q 
 
 Hence, for the sphere we have V = . 
 
 R 
 
 Now let us consider what we mean by 'an isolated sphere.' 
 There is. on the walls, &c., at a distance (let us suppose) of r centi- 
 metres from the centre O, a charge equal to Q. Hence the true 
 potential of O and of the sphere will be 
 
 v=Q-2 
 
 R r' 
 
 But if r is so large as compared with R that the latter term 
 
 may be neglected, then we have V = 5^. 
 
 R 
 
 This shows clearly the meaning of isolated ; and also. shows how, 
 for the same charge, the potential will be smaller as r becomes 
 smaller, or as the walls close in (see Chapter V. 8). 
 
 20. Capacity of an Isolated Sphere. By the definition of 
 capacity given in Chapter V. 4, we have for an isolated sphere 
 
 But we have just found that V=. Hence it follows that 
 
 R 
 
 K = R. Or we arrive at the result that 
 
 The capacity of an isolated sphere is measured numerically by its 
 radius expressed in centimetres. 
 
CH. x. ELECTROSTATIC POTENTIAL 143 
 
 Note. The reasoning just employed is of very general use. Suppose that 
 we find by calculation that 
 
 V = / . Q for an isolated body ; 
 or V = h . Q for a condenser ; 
 where / and h involve only one or more of the following quantities, viz. dimen- 
 sions of the plates, distance between them, and nature of the dielectric. We 
 may then argue that, by the definition of capacity given in Chapter V. 4, and 
 Chapter VI. 4, it follows that K = /, or K = /i, in the two cases respectively. 
 
 21. Distribution; from the Potential Point of View. In 
 
 hydrostatics it is frequently very useful to consider the distribu- 
 tion of a liquid to be such as will cause every part of the free 
 surface to be at the same level. The more immediate cause of 
 the distribution of a liquid is the action of gravitation-forces ; and 
 a level free surface rather follows from this. But still the former 
 view is often convenient. 
 
 So, in electrostatics, the cause of distribution is the direct 
 force acting on each elementary portion of the charge. But here 
 also it is often convenient to regard the distribution on a con- 
 ductor as such as will cause each part of it to be at the same 
 electrostatic potential. This last must be the resulting condition, 
 since otherwise we could not have equilibrium. 
 
 We will take one case as an example. 
 
 ' What must be the general distribution on an elongated cylinder 
 that it may be all at one V ;' or, we might say, 'that it may take the 
 same work to bring up our + unit to any part of it ? ' 
 
 If the distribution were uniform, it would certainly require more 
 work to bring up our + unit to the central portion, into the presence 
 of the whole charge, than to bring it up to the end, out of the way 
 of most of the charge. But, if the charge be more crowded at the 
 ends, we see how it may now require the same work in the latter case 
 as in the former. Thus, the well-known distribution on an elongated 
 cylinder is such as to cause the whole to be at one potential, though 
 it more directly results from the equilibrium of the tangential com- 
 ponents of the electrostatic forces. 
 
 22. Two Spheres of Different Radii. Let us consider two 
 spheres of different radii, Rj and R 2 . 
 
 (i) If they have the same charge Q, then 
 
 v Q . v __ Q . 
 
 1 ~ P~> 2 ~~ P~~ > 
 K { K 2 
 
 or Xl = % 
 
144 ELECTRICITY CH. x. 
 
 (2) If they have the same potential V, then 
 
 v Qi - v QI 
 
 T?~~ > P ' 
 
 K! K<> 
 
 or 21 = % 
 Q 2 R,' 
 
 These results should be translated into words, and noted carefully 
 by the student. 
 
 23. Potential, and Density, distinguished. By density of 
 charge, we mean quantity of charge per unit area. We use the 
 symbol p to signify the numerical value of density of charge. We 
 have, by definition, 
 
 where Q is the quantity on the area S ; all of course in C.G.S. 
 units. 
 
 Now we have seen that, on any conductor but a sphere, the 
 density is not constant all over, while the potential is. At a point 
 the density may be very great, though the potential be small. 
 
 We will illustrate the difference and the connection between density 
 and potential by the case of spheres. 
 
 (i.) Consider two spheres of equal radius R. Then, for each, 
 
 V = ~~ ; p = ^ . Hence, at the same potentials the densities 
 4 TT R* 
 
 are equal, and both V and p are proportional to O. 
 
 (ii.) But now consider two spheres of different radii R t and R.,, 
 and let them have the same potential V. Then for the two respect- 
 ively we have Q x = V R x ; Q 2 = V R 2 ; 
 
 Q, VR, V 
 
 and .< -IT p -tr 
 
 St2 
 
 Whence p t and p 2 are not the same, and Pl = ^? . 
 
 24. Force on a + Unit, acting Perpendicularly to a Conducting 
 Surface. 
 
 It is for many purposes very important to know how great is the 
 force on + unit at a surface, urging it normally to the surface. The 
 resultant force must evidently be normal, since any tangential force 
 would imply that the conducting surface was not at one potential. 
 
 We will take the case of a sphere. Let the diagram represent a 
 
CH. X. 
 
 ELECTROSTATIC POTENTIAL 
 
 145 
 
 sphere of radius R, and total charge O ; and let P be a point on the 
 sphere. It is required to find the force at P acting on + unit. 
 
 Now by the result of calculation, stated 
 in 17 (4), we learned that the action of 
 the charged sphere on a point external 
 to it is the same as if the whole charge 
 O were collected at the centre E. This 
 is true quite up to the surface ; or is 
 true when we take our point P to be 
 just on the external surface. Hence on 
 a + unit at P there is acting a normal 
 
 force equal to ^- r dynes. 
 
 But, if p be the uniform density of charge over the sphere, then 
 
 Q 
 
 Hence, for the force F acting on + 
 
 unit at the surface, we get F = 4 IT p in dynes. 
 
 Now if the radius of the sphere increase without limit, so that the 
 surface at P becomes ultimately plane, no change will occur in the 
 expression just found, provided that p remains constant. 
 
 Hence the formula is true for a plane surface on which the density 
 is p. It is indeed true for any surface, p being the density at the point 
 considered. This general result is, in more advanced Courses, proved 
 independently, without reference to a sphere. 
 
 25. Important Case of a Spherical Condenser. We will 
 now show how to find the capacity of a spherical condenser. 
 
 As in the case of an isolated sphere we found two expressions 
 for the potential of the sphere, and by equating them found the 
 capacity of the sphere ; so here, in the 
 case of two spherical surfaces forming a 
 condenser, we shall find two expressions 
 for the difference of potentials of the two 
 surfaces, and then by equating them shall 
 find an expression for the capacity of the 
 condenser. The figure represents the ideal 
 spherical condenser, in which there are 
 two concentric spherical conductors, the 
 inner being totally enclosed by the outer. In the practical con- 
 denser we must have an opening through the outer conductor, 
 and an insulated wire and knob connected with the inner ; as in 
 Faraday's condenser, Chapter IX. 4. 
 
 L 
 
146 ELECTRICITY CH. x. 
 
 Let C be the centre. Let the radius of the inner conductor 
 be R A cms., and the charge on it be + Q ; then there will be on 
 the inside surface of the outer conductor a charge of Q, by 
 Chapter IV. 16. Let the inner radius of this outer conductor 
 be R B cms., and its outer radius be r cms.; and let there be on the 
 outer surface of B a charge Q'. 
 
 As we wish to find an expression for the difference of potential 
 (V A V B ) between the two surfaces, we will find V A and V B sepa- 
 rately first. The reader must note that Q is the bound charge of 
 Chapter VI. 
 
 Now V A is the potential of any point on, or in, A, since A is 
 a conductor containing no insulated charged bodies ; and, there- 
 fore, V A is the potential at C. 
 
 .Hence V A = Q + =2 + 2 
 _K A K B r 
 
 by the results of 19. 
 
 And V B will be the same as the potential of a point just on 
 the outer surface of B. But, by 17 (4), this will be the same 
 as that which would be due to all the charges collected at the 
 common centre C. 
 
 That is, 
 
 v _ 
 
 B ~ 
 
 Q + Q'_Q' 
 
 B ~ ~r~ "7 
 Hence we get 
 
 as one expression for the difference of potential between A and B. 
 But, by the definition of capacity, we have also that 
 
 (V A -V B ) = | 
 
 since Q is the 'bound ' charge Chapter VI. 4. 
 Hence, equating, we have 
 
 The quantity Q', and the outer radius r, have both disappeared 
 from the expression. And this is what we should expect, since O' 
 has opposite to it on the walls, &c., a charge equal to - Q', and these 
 two charges with the intervening air as a dielectric form another 
 
 
ELECTROSTATIC POTENTIAL 
 
 (relatively insignificant) condensing system with which we are not con- 
 cerned in our present discussion. 
 
 The above calculation is based upon the supposition that the 
 dielectric is air. For our quantity Q is measured in terms of a 
 unit which repels another unit at i cm, distance with i dyne force 
 in air ; and our expressions for measurement of potential depend 
 on this condition. 
 
 Now, if we have any other dielectric of specific inductive capacity 
 <T, then by the definition 01 o- given in Chapter IX. we must write 
 <T . K instead of K ; and, if K' is the new capacity, we have 
 
 K.J 
 
 and (V - V )- - l - 
 
 and V V-____ 
 
 (r) 
 
 The reader will notice that this is the expression that we should 
 arrive at if we assumed, all through our calculations of potential as 
 measured by work done, that in the new medium the quantity Q acted 
 
 on our -h unit with a -force of - . -4 dynes at a distance r, instead of 
 
 a- r* 
 
 the -4 dynes with which it acts on the + unit when air is the medium. 
 
 Hence we conclude that, in a medium of specific inductive capacity o-, 
 the force between two quantities Qand Q' (measured in the absolute unit 
 
 of Chapter V. i), separated by a distance r^ will be L . _ 
 
 _~ 
 
 dynes. 
 
 We may write the above relation in the form 
 
 Q^-i^rl;^-^) - 
 
 26. The Plate Condenser. In the diagram we 
 give a simple section of a plate condenser A B. Let A 
 V A and V B represent the potentials of A and B re- 
 spectively ; let Q be the ' bound ' charges on the two 
 plates respectively ; let / be the perpendicular distance 
 between them, measured of course in centimetres ; let 
 Sbe the area of the inner surface of either A or B ; let 
 p be the density of charge on these inside surfaces ; let 
 a be the specific inductive capacity of the dielectric. 
 
 Now we shall assume that / is so small as compared with the 
 size of the plates that the field of force between A and B is 
 
 L.2 
 
148 ELECTRICITY CH. x. 
 
 uniform ; i.e. that it consists of lines of force that are parallel and 
 equidistant, which implies that p is uniform. 
 
 Then by 1 2 we have for the field strength F, or force acting 
 on a + unit, between A and B, 
 
 F = X^Z ? . 
 
 But by 24 we have that at the surface of the plate 
 
 F = 4 *> 
 Now it is clear that p = ^ ; since p is assumed to be 
 
 o 
 
 uniform. 
 
 Hence we have V A V B 4 TT p t 
 
 or 
 
 But by definition we have (V A V B ) = -^=. 
 
 rL 
 
 Whence 
 
 In the above we have assumed air to be the dielectric. But if 
 we have another dielectric of specific inductive capacity = n, then the 
 formulae become 
 
 (y) 
 
 4 7T / 
 
 27. Formulae for Capacities, &c. For convenience we here 
 put together the chief formulae that have occurred up to the 
 present point in this chapter, adding also one or two more. 
 
 The following remarks will apply to some of them, if not 
 to all. 
 
 (a) All the measurements are in the C.G.S. system of units. 
 
 (b) Electric quantity is measured by reference to force exerted 
 when air is the dielectric. When there is another dielectric of specific 
 
 inductive capacity v, the force exerted is -th that exerted through air 
 
 \see 3 25, end). The capacity of a condenser with this dielectric 
 will by definition be a times that of a similar air condenser. 
 
CH. x. ELECTROSTATIC POTENTIAL 149 
 
 (<r) An ordinary condenser is really a double system, as pointed 
 out immediately after equation (/?) in 25. The ' isolated ' body 
 is the limiting case of such a condenser, when this has become a 
 single system. The formulae for isolated bodies can be derived 
 from these for condensers, though it is often more simple, to find 
 them independently. 
 
 I. Formula connecting V, Q, and K. 
 
 (i.) For 'isolated ' bodies, Q. = K.V. 
 (ii.) For condensers, Q = K . (V^ V 2 ). 
 
 In the former case Q is the sole charge ; in the latter Q is the 
 bound ' charge. 
 
 II. Capacities of some ' isolated " bodies. 
 
 (i.) For a sphere of radius R, in air, K = R. 
 (ii.) For a disc of radius r and of negligible thickness, in air, 
 
 (iii.) For a cylinder of length / and of relatively small radius r^ 
 in air, K= -. 
 
 2 lOg, 
 
 A wire comes under the head of (iii.). The two last formulae 
 have not been proved in the foregoing. 
 
 III. Capacities of some condensers. 
 
 (i.) For a spherical condenser where the radius of the inside 
 sphere is R b and. that of the inner surface of the outer sphere is R 2 , 
 
 (ii.) For two co-axial cylinders of common length /, the radius 
 of the inner being r^ and that of the inner surface of the outer 
 cylinder being r> 2 , 
 
 (iii.) For two discs, that surface of either on which is the bound 
 charge being S, and the distance between them being /, 
 
150 ELECTRICITY 
 
 CH. X. 
 
 This formula applies to any case of parallel surfaces where the 
 dimensions are very large as compared with /. In fact, the formula 
 (i.) above can be transformed into this when R, and R 2 are very 
 great as compared with (R 2 RI). 
 
 28. Energy of Charging and Discharging 1 . When electricity 
 falls from a higher to a lower level, work can be done ; we lose a 
 certain quantity of electrical potential energy, and we must have 
 an equivalent in heat or mechanical work, &c. If conversely we 
 raise electricity from a lower level to a higher, we expend this 
 work and gain the above electrical potential-energy which is an 
 equivalent (see 4). 
 
 There is no more important matter in our present science of 
 electricity than this question of the application of the law of 
 * Conservation of energy ' to electrical charge and discharge. We 
 will consider the matter under three heads. 
 
 Case I. The case where an electrical quantity Q passes between 
 two conductors of fixed potentials V l andV^. Now by definition 
 and measurement of potential, we do on each + unit an amount 
 of + or work that is measured by (V^ V 2 ) ergs. Hence, 
 on the quantity of Q units we do + or work measured by 
 
 If one potential be V and the other be zero, the work is Q . V 
 ergs. 
 
 Hence, in this case the energy E of charge or discharge is 
 measured by 
 
 or E = Q. V ergs) ' 
 
 Case II. The case, more usual in electrostatics, where the 
 potential alters during charge or discharge. We will consider the 
 simplest case of charging a conductor, that is originally at zero 
 potential, with a quantity Q, up to a potential V. 
 
 Here the potential of the conductor rises in arithmetical pro- 
 gression with the charge given, since 
 
 potential^ - chai * e , 
 capacity 
 
 and capacity is constant. The result, therefore, is that the same 
 work is done as if the whole charge were raised to half the final 
 potential V. Hence the work of charging, or energy expended, 
 
CH. x. ELECTROSTATIC POTENTIAL 151 
 
 will be E = 
 
 or = 
 
 since V = . 
 
 Jv 
 
 This same expression gives the energy of discharge. 
 
 Analogies. (a) So, it we reckon the number Q{ foot-pounds-weight 
 done in building with P Ibs. of bricks a tower of H feet high, it being 
 assumed that the tower is cylindrical or rises in arithmetical progres- 
 sion with the amount of bricks used, we get as a result the same work 
 as if all the bricks were raised to half the vertical height H ; i.e. 
 we get 
 
 work = \ P . H . f o ot -pounds -iv eig Jit. 
 
 (b] A similar expression is obtained for work done by water flowing 
 out of a cylindrical reservoir of an average height of i H above the 
 lower level. 
 
 Case III. Case of discharge of a condenser. In the ordinary 
 condenser we have two plates of potentials V t and V 2 respec- 
 tively. 
 
 In this case the discharge will cause the equal and opposite 
 charges + Q and Q to disappear, leaving the whole condenser 
 at some potential v due to the ' free ' charge. Taking the two 
 charges + Q and Q separately, we have for the former 
 
 energy = J Q (V t - v\ 
 and for the latter 
 
 energy = \ ( - Q) (V 2 - v). 
 
 Adding the two we get 
 
 E = iQ.(V l -V 2 ) ...... ( 7 ) 
 
 which is the same as result (/3), if the one coating of the condenser 
 be at zero potential. 
 
 The reader should notice that in the case of the Leyden jar charged 
 to potential V of inner coating, the outer coating being to earth, we 
 have for the discharge of the jar the expression | Q V. 
 
 29. Examples in Energy of Discharge. 
 
 (i.) * A conductor is charged with 8 units to potential 7. What is 
 the energy of discharge? ' 
 
 Here E = QV = |.7.8 = 28 ergs. 
 
I5 2 ELECTRICITY CH. x. 
 
 ' And what was its capacity ? ' 
 
 (ii.) 'A conductor of capacity 10 is charged with 4 units. What is 
 the energy of discharge ? ' 
 
 (iii.) 'A sphere of radius 4 cm. is charged with 12 units. Find the 
 energy of discharge.' 
 
 Here K = 4 (see 19) 
 
 Two harder examples are worked out in 35. 
 
 30. Energy of Discharge in the Cascade Arrangement of 
 Leyden Jars. We must now ask the reader to turn back to 
 Chapter VI. 9, and to notice the arrangement there described 
 and the notation employed. 
 
 We found that if there were n equal jars in cascade, the differ- 
 ence of potential between the inside and outside coatings of each 
 
 jar was _th that of a single jar charged from the same source and 
 having its outside coating to earth, the charge of each jar being 
 also -th that of the single jar fully charged. 
 
 It follows that, in the case of the single jar thus fully charged, 
 we have for the energy of discharge E the value 
 
 E = lQV^ .... ..,, . (i.) 
 
 while, for all the n jars of the cascade, we have for the total energy 
 of discharge E' the value 
 
 E'=a x i.9.-= * . JQV= --E . (ii.) 
 
 n n n 2 n 
 
 This result may be better understood if we give an analogy. 
 
 Analogy. If instead of one tower 100 feet in height we build 10 
 towers each having ^th of the height, and th of the number of bricks 
 of the large one, a little thought will show us that the total work done 
 in the second case will be j^th of that done in the first case ; for in the 
 second case the same total weight of bricks will have been raised 
 through T a 5 th of the former average height. 
 
CH. x. ELECTROSTATIC POTENTIAL 153 
 
 31. Electroscopes and Electrometers, In general these in- 
 struments give their indications by movements. These move- 
 ments are due to the action of a field of force on electrical quantity, 
 the extent of the movement being such that equilibrium is finally 
 arrived at between electrical forces on the one hand and gravita- 
 tion-, magnetic-, or torsional-forces on the other hand. 
 
 We can have movements when there is only one charged body, 
 e.g. when this body is situated nearer to one wall of a room than 
 to the other. Here there is a field of force between the body and 
 the walls, and this field is strongest in one direction. Now, be- 
 sides the effect of a field on a charged body situated in it, we 
 know that the two sides of a field tend to close in ; or we may 
 say * the lines of force tend to shorten.' Hence the charged body 
 will move so as to close up the strongest portion of the field due 
 to it. 
 
 (a) In a gold-lea] electroscope each leaf is screened from 
 one wall by the other leaf. Hence, there is no wavering between 
 two walls about equally distant, but each leaf moves off towards 
 the wall from which it is not screened. 
 
 This view, or one like it, is more in accordance with present 
 ideas than is the view of ' repulsion between the two leaves.' 
 
 (b) Bohnenberger's electroscope. Here a single charged leaf 
 moves along the field of force that lies between two brass knobs ; 
 these brass knobs being maintained at different potentials by 
 being connected with the two poles respectively of a dry pile. 
 
 In both cases here given we indicate differences of potential by 
 the extent of the movement ; since in case (a) the field is stronger 
 according as the leaves are at a larger or smaller difference of 
 potential from the zero potential of the walls, and in case (b) the 
 movement of the single leaf in the fixed field of force depends on 
 the potential up to which this single leaf is charged. 
 
 32. Electrometers. The ' Attracted-Disc ' Form. In electro- 
 meters we aim at measuring differences of potential, or absolute poten- 
 tials with respect to the earth as zero. 
 
 Consider the plate condenser of 26, and make the same 
 assumptions as to uniformity of field and of p as were there 
 made. 
 
154 ELECTRICITY CH. x. 
 
 It can be shown that the total stress F between the plates is 
 given by the formula 
 
 S (V. - V 2 ) 2 
 8 TT P 
 
 This expression therefore gives in dynes the total force with 
 which one plate is urged towards the other ; and, if we measure 
 F by counterbalancing 'weights/ each gramme weight being 
 about 981 dynes force^ we can express (Vj V 2 ) in absolute 
 units. 
 
 But in the above apparatus there is error ; for the assumptions 
 made do not hold near the edges of the plates, and this error is 
 not one to be readily allowed for. 
 
 Hence, in the actual attracted-disc electrometer, Sir W. Thomson 
 adopted a contrivance called a guard- ring. The moveable disc 
 was cut out in the centre of a larger disc. 
 
 Then, since the fixed and moveable portions were connected 
 and at one potential, and since there was no perceptible break 
 between them, it followed that with respect to the moveable por- 
 tion of the one disc, and the central portion of the other fixed 
 disc that was opposite and parallel to the former, the assumptions 
 as to uniformity did hold. In fact, only the central portions of 
 the discs were used. 
 
 In the figure we have the arrangement indicated. The two 
 plates are h h and g ; and the formula applies to the moveable 
 
 portion /"of the one plate and 
 the portion opposite to this 
 of the other fixed plate. The 
 -^ moveable portion / is at the 
 end of a light lever whose 
 fulcrum, or axis of suspension, 
 is a wire m n. At the free 
 end of this lever is a hair /; 
 this is viewed by a lens, and 
 serves to indicate when the 
 plate /is flush with the guard- 
 plate h h. 
 
 The fane required to keep /in its place gives us F. Whence, 
 by knowing the dimensions, &c., of the instrument, we measure 
 
CH. X. 
 
 ELECTROSTATIC POTENTIAL 
 
 155 
 
 33. Sir William Thomson's Quadrant Electrometer. The 
 
 principle of the quadrant electrometer can be illustrated by the 
 Bohnenberger's electroscope. 
 
 If the gold-leaf were kept at a constant high potential, while 
 the difference of potential between the two knobs was the variable 
 quantity, then the amount of deflexion of the gold-leaf would in- 
 dicate the difference of potential between the two knobs. 
 
 The next figure gives a sketch, taken looking down directly 
 from above, of the essential parts of the quadrant electrometer. 
 
 Let us take a cylindrical box of thin brass about one inch high 
 and about five inches in diameter. Let us cut this into four 
 sectors as indicated in the figure, 
 and let us connect opposite sec- 
 tors respectively by wire, and sup- 
 port all four sectors on insulating 
 glass legs. 
 
 We thus have four hollow 
 brass sectors ; in the interior of 
 each of which, as an approxi- 
 mately ' closed vessel,' there is, 
 excepting near the edges, a con- 
 stant potential. The opposite 
 pairs of these are connected. 
 
 By u is represented a light aluminium needle suspended by 
 two parallel silk fibres this being called bi-filar suspension. 
 
 Hence there will be called into play, when the needle is 
 deflected from its position of rest, a couple tending to restore it. 
 The value of this couple for any given angle of deflexion can be 
 calculated. This needle swings horizontally in the interior of the 
 sectors ; and matters must be so arranged that it may come to 
 rest exactly along the line of one of the slits. The diagram repre- 
 sents the needle thus at rest, unacted upon by any electrical 
 forces. 
 
 This needle is maintained at some constant potential 1 V, by- 
 being connected with the inside coating of a charged Leyden jar. 
 
 If the one pair of sectors c and b be at a potential V l5 and 
 the other pair a and d at some lower potential V 2 , there will be 
 a field of force running across the gap or slit from c to a ; and 
 from symmetry, an equal field will run across the gap from b to d. 
 
155 
 
 ELECTRICITY 
 
 CH. X. 
 
 Moreover, this field will be confined to a region only somewhat 
 wider than these gaps ; the region in the interior of each sector 
 will be practically of a uniform potential, or will not be a field of 
 force. 
 
 From the symmetry of the whole, the needle will be acted 
 upon by a pure couple due to the electrical field. And, from its 
 peculiar shape, that amount of the needle which is in the field of 
 force (i.e. the portion lying under the slit) will remain constant, 
 since only very small deflexions are employed. We thus contrive 
 that there shall act on the needle, for any given values of V,, V 2 , 
 and V, a constant electrostatic couple whatever be its deflexion ; 
 provided that this deflexion do not exceed a certain maximum 
 depending on the construction of the instrument. 
 
 This constant couple will deflect the needle until the restoring 
 couple, due to the twisting of the two parallel silk fibres, is suffi- 
 ciently great to give equilibrium. If the potential V of the needle 
 be known, and if the ' constants ' of the instrument be known, 
 
 then the difference of potential 
 (Vj V 2 ) can be calculated from 
 the observed deflexion. 
 
 In the next figure we have a 
 sketch of the ' Elliott-pattern ' quad- 
 rant electrometer ; a comparatively 
 simple form of instrument. One 
 of the quadrants is represented as 
 removed, so that the needle may be 
 seen. This needle is maintained at 
 a high potential in somewhat the 
 following manner. From it there 
 hangs a platinum wire, dipping into 
 a glass vessel that contains strong 
 sulphuric acid. Outside this vessel 
 is a coating of tin-foil, so that it is 
 in fact a Leyden jar. Its capacity 
 being very great as compared with the capacity of an isolated 
 body, it serves to maintain the potential of the needle approxi- 
 mately constant for a considerable time. 
 
 From the needle rises a light stem, bearing a small mirror. 
 
CH. x. ELECTROSTATIC POTENTIAL 157 
 
 The usual 'lamp and scale ' arrangement gives us, in the reflected 
 spot of light, a very sensitive means of noticing and measuring the 
 deflexions of the needle. 
 
 Complete form of the quadrant electrometer. In the complete 
 form of Sir W. Thomson's quadrant electrometer there are many 
 details of construction that we shall not discuss here. But we 
 must mention two of the most important of these. 
 
 (i.) The gauge. Connected with the inside of the Leyden jar 
 (and so with the needle) is an attracted-disc electrometer. If the 
 weights or spring be so arranged that the hair (see 32) is in the 
 proper line of sight when the jar, needle, and disc/ therewith con- 
 nected, are at some fixed potential V, then any alteration in this 
 potential will be at once detected by the movement of the hair. 
 This is one detail. 
 
 (ii.) The replenisher. The other detail is a replenisher (see 
 Chapter VII. 5), in which one armature is to earth while the 
 other is connected with the Leyden jar. 
 
 We can thus remedy any error in V by turning the replenisher 
 the one way or the other, so as to raise or lower the potential of 
 the jar. 
 
 Formula for quadrant electrometer. It can be shown that the 
 following formula holds for the absolute quadrant electrometer. 
 
 -I_V (Vi _ Va) 
 
 where L is the electrostatic couple acting on the needle, balanced by 
 and equal to the mechanical restoring couple ; V the potential of the 
 needle ; V x and V 2 the potentials of the two pairs of quadrants re- 
 spectively, and h a constant depending on the construction of the 
 instrument. 
 
 If the needle be charged to a relatively very high potential, then we 
 
 may neglect 1 + 2 in comparison with V ; and we have 
 
 ' 
 
 L-A.V.^ - V 2 ) 
 This relation may be written as follows. 
 
 We see then that we could measure the difference of potential 
 : - V.J by observation of the deflexion ; provided that we knew 
 
158 ELECTRICITY CH. x. 
 
 the values of h, V, and L. In practice, however, these quantities 
 are not calculated ; but the instrument is graduated by means of a 
 ' standard voltaic cell,' or by comparison with an absolute electrometer 
 of the more simple ' attracted-disc ' form. 
 
 In instruments devoid of $t& gauge and the replenisher, as well as 
 in the above form, we may write . 
 
 V l - V, = b . tan 8 
 
 so long as the potential V of the needle remains constant. And, for 
 small deflexions, we have 
 
 v i - V 2 = ?>.B \nearly\. 
 
 Here 8 is the deflexion ; and the constant b may be determined on 
 each occasion of use by means ot a ' standard voltaic cell.' 
 
 34. Uses of the Quadrant Electrometer. 
 
 (i.) General use. The quadrant electrometer is mainly an 
 instrument for the measurement, or comparison, of potentials and 
 potential-differences. We have indicated in the above how it is 
 possible to use simpler forms also, as well as the ' standard ' pat- 
 terns of the instrument, for such purposes of measurement. 
 
 It may be added that usually one pair of quadrants is put to 
 earth, so as to be at zero potential ; the potential of the other pair 
 of quadrants being then measured with respect to the earth 
 as zero. 
 
 (ii.) Use as an electroscope. The instrument can also be used 
 for all purposes in which we desire merely a delicate electro.$w/<?, 
 and not an electrometer. For such purposes any delicate form 
 will serve, even though it be so faulty in construction that the law 
 of its action (see 33, end) is unknown. An important example 
 of such a use will be given in Chapter XIV. 10. 
 
 (iii.) Use in investigating the distribution on a conductor (see 
 Chapter IV. 16). In the investigation of distribution upon con- 
 ductors, of which we said something in Chapter IV. 16, &c., it 
 would at first sight appear impossible to use the electrometer. 
 For, it would be argued, where we are dealing with a conductor 
 whose potential is constant in every part, how can we employ an 
 instrument that indicates only potential-differences ? 
 
 A few words of explanation will enable the student to perceive 
 how the instrument may be used in such investigation. 
 
CH. x. ELECTROSTATIC POTENTIAL 1 59 
 
 When we lay the proof-plane against any part of the conductor 
 whose potential is V, it will also be at the potential V ; but the 
 charge that it acquires will depend upon the density p that exists 
 at the part touched. The fact is that the capacity of the proof- 
 plane changes as we change its position on the conductor; 
 and so, though its charge is proportional to-p, its potential is 
 always V. 
 
 But when we remove the proof-plane and place it in an isolated 
 position, its capacity assumes some constant value ; and its poten- 
 tial will therefore be proportional to its charge, and so will be pro- 
 portional to p. 
 
 When the proof-plane is connected with the one pair of quad- 
 rants of the electrometer, the combined conductors will have a 
 potential that is directly proportional to the charge, and inversely 
 proportional to their combined capacity. Since the latter remains 
 constant, it follows that the quadrants will assume a potential that 
 is directly proportional to f>. 
 
 35. Examples in Energy of Discharge, &c. (a) In all cases where 
 it is required to find the energy of discharge between two electrical 
 systems A and B, the following method will be found to be a good one. 
 First, find the energies of discharge of A to earth, and of B to earth, 
 separately; and add these quantities Secondly, find what will be 
 the energy of discharge to earth of the combined system A and B 
 after these have been connected. It will then follow, by conservation 
 of energy, that the difference between these two results will give the 
 required energy of discharge between the two systems when they are 
 connected with one another. Let us take an example. 
 
 ' There are 3 Leyden jars A, B, and C, equal in capacity, having their 
 outer coatings connected with earth. A is first charged. Its knob is 
 then connected with the knob of B. It is then disconnected from B, 
 and connected with C. Finally, the knob of B is connected with 
 the knobs of A and C. Find the energies of the several dis- 
 charges.' 
 
 (i.) The initial discharge of A to earth would give energy to the 
 
 amount of * Q V ; and the final discharge of A and B together to 
 
 earth would give energy measured by - Q . , or - Q V. Since energy 
 
 224 
 
 cannot have been lost, it follows that the difference between these 
 two quantities must measure the energy w 1 of discharge between A 
 and B when these two were connected. Hence 
 
160 ELECTRICITY CH. x. 
 
 (ii.) The discharge of A to earth would now give - . -^ . , or 
 
 * Q V. And the subsequent discharge of A and C together to earth 
 8 
 
 would give - . _~ . , or -- Q V. Hence, energy / 2 of discharge 
 224 10 
 
 between A and C is given by the difference, or 
 
 (iii.) The discharge of B to earth, and of A and C to earth, would 
 give together (1 . 2- . Y + I . S . Y\ = .3 Q V. And the discharge 
 
 of all three together to earth would give - . Q . -, or - V. Hence 
 
 2 36 
 
 the energy / 3 of discharge when B is connected with A and C is given 
 by the difference, or 
 
 It is to be noticed that all the discharges added together give the 
 energy 1 Q V, which was that of the original discharge of A to earth. 
 
 If this result had not followed there must have been some error in the 
 above work ; since ' Conservation of energy ' demands that no energy 
 be lost. 
 
 (/3) 'A sphere A of 9 cms. diameter is connected by a thin wire 
 (i.e. one of negligible capacity) with another sphere B of the same 
 diameter. Round this latter, and concentric with it so as to form a 
 spherical condenser, is a larger sphere of 10 cms. internal diameter, 
 connected with earth. A charge of 33 units is given to A. Find (i.) 
 in what proportion this charge is distributed between the two systems ; 
 and (ii.) the energy of discharge of the two systems separately.' 
 
 iLet V A be the potential of A, and therefore also of B. 
 C A be the capacity of A, and C B that of B. 
 Q A be the charge on A, and Q B that on B. 
 W A be the energy of discharge of A, and W B that of B. 
 
 9 
 
 2 
 
 condenser, C B = i -T- ( ~ -i ) = 45- 
 
 (i.) Then C A = radius = ? . And, by the formula of a spherical 
 
CH. x. ELECTROSTATIC POTENTIAL l6l 
 
 9 
 Hence Q A =* -^ of 33 = of 33 = 3 units, 
 
 2 + 45 
 and Q B = the remainder = 30 units. 
 
 (ii.) Again V A = V B = 9.A-2?-*. 
 '-A ^B 3 
 
 Hence W A = IQ A V A = Lx 3 *?- i, 
 
 
 1 
 
 lQ B V B = - x 30 x -= 10. 
 
 36. General Consideration of Electrostatic Fields of Force, 
 
 It is probable that among all the important conceptions that are 
 due in the first place to the genius and insight of Faraday, none 
 has done more to place the physical theory of electrical and of 
 magnetic phenomena upon a sound basis than his recognition of 
 the part played by the medium across which the forces act. The 
 former view of ' action at a distance,' and the purely geometric 
 conception of lines of force, were essentially mathematical ideas ; 
 to the physicist they were both unreal and unsuggestive. 
 
 Taking our present case of electrostatic fields, we may explain 
 the modern view, based upon Faraday's conception, somewhat as 
 follows. 
 
 There is no such thing as ' a + charge ' or ' a charge ' by 
 itself; but on the contrary, wherever electrostatic phenomena 
 occur there is an electrostatic field, on the two sides of which 
 occur equal + and charges respectively. The whole system 
 consists of these equal and opposite charges separated by a dielec- 
 tric which is the seat of the field of force. The lines of force have 
 a real physical meaning. Whether they are lines along which the 
 dielectric undergoes a kind of tension, or whether they are lines 
 along which the molecules of the dielectrics are ' polarised ' by 
 a separation of + and charges in them, is not yet known. 
 But it seems certain that the electrostatic potential energy, implied 
 by the existence of an electrostatic field, resides in the dielectric 
 that separates the two sides of the field ; somewhat as mechanical 
 potential energy resides in a bent spring, a stretched piece of 
 elastic, or a strained elastic solid. 
 
 Thus, when we speak of ' an isolated body charged with + Q. 
 
 M 
 
1 62 ELECTRICITY CH. x. 
 
 units to a potential V, whose energy of discharge is measured by 
 \ Q V,' we really mean that there is on the body a charge + Q, on 
 the walls or other surrounding surfaces an equal and opposite 
 charge Q, and that in the strained dielectric resides energy to 
 the amount of \ Q V. 
 
 Dielectrics are bodies in which this electrostatic strain can be 
 maintained ; while conductors are those in which no such strain 
 can "be kept up, and in which therefore lines of force and fields of 
 force cannot exist. In reality, bodies are not capable of sharp 
 division into these two classes, since the yielding to electrostatic 
 stress is only a matter of time. But in general we make use of 
 bodies at the two extremes of the series ; so that the foregoing 
 definitions will serve very well for usual cases. 
 
 It is part of the above view to say that tubes of force are 
 always terminated against surfaces so charged that at the two ends 
 of each tube there are equal and opposite charges + q and q 
 respectively (see 17). 
 
 It is also part of it to say that conductors act as screens for 
 lines of force cannot penetrate a conductor, and hence if an elec- 
 trostatic field exist on each side of the screen, these two fields are 
 independent of one another. 
 
 If lines of force were material, the phenomena of the electro- 
 static field would suggest that they act as stretched elastic threads, 
 which moreover repel one another. Thus the two sides of a field 
 are urged towards one another as if by tension of the lines of 
 force connecting them. 
 
CH. XI. 
 
 CHAPTER XI. 
 
 THE PHENOMENA OF ELECTRIC CURRENTS. BATTERY CELLS 
 AND BATTERIES. 
 
 i. Introductory. In Chapters IV. -X. we have dealt 
 mainly with the phenomena connected with the actions between 
 charged conductors, and with the strains of the intervening 
 medium. We have, in fact, considered mainly electrostatic fields 
 of force, or fields in which our test-charge of electricity is urged 
 along lines of force. 
 
 We did, indeed, examine to some extent the results of dis- 
 charge ; but in general the readjustment of electrical equilibrium 
 was so rapid that it was not easy to investigate the phenomena 
 accompanying discharge. With such machines as the Holtz we 
 might, however, have done so ; but not so conveniently as with 
 the help of other apparatus to be described in the present 
 chapter. 
 
 We shall find that, during discharge, new classes of phenomena 
 arise. In particular there are observed chemical, and heat, 
 phenomena ; and a new field of force, viz. a magnetic field (or 
 a field in which a magnetic pole is urged), springs into existence. 
 
 In our next division of the subject we enter into the consider- 
 ation of the phenomena accompanying electric discharge. As a 
 rule we shall have very small AV..J as compared with those 
 hitherto employed, but very large quantities of electricity, and 
 an even and continuous flow. W 7 here this is not .the case, atten- 
 tion will be drawn to the fact. 
 
 The above remarks will indicate that the popular terms 
 * statical electricity ' and ' current electricity ' must not be under- 
 stood in their literal sense of two kinds of electricity ; but must 
 be considered to refer to two classes of phenomena that require 
 different conditions of A V and of quantity for their investigation. 
 
 M 2 
 
1 64 ELECTRICITY CH. xi. 
 
 2. Galvani's Experiment. As stated in the preface, we 
 shall not enter into any historical account of this or of any portion 
 of our subject. But we must mention that experiment of Galvani's 
 which, perhaps more than any other, started Volta on his very 
 fruitful line of inquiry. 
 
 In the figure, Z C is a compound bar of zinc and copper, as 
 indicated by the two letters employed. The zinc end is put into 
 contact with the lumbar nerves of a frog's hinder quarters. It is 
 found that whenever the copper end touches the muscles of the 
 legs tnere is a sudden convulsion of the limb. This experiment 
 
 was first performed in 1786. Galvani considered that the metals 
 acted merely as conductors to complete the circuit of 'animal 
 electricity,' and followed put this idea in further investigations. 
 As we do not intend to discuss in the present Course the relations 
 of electricity to physiology, we shall not say more as to Galvani's 
 views. 
 
 Volta, on the other hand, fixed his attention on the metals, 
 and considered that they by their contact caused the current 
 which the nerves and muscles of the frog's leg conducted. We 
 shall have more to say as to his views, and as to modern modifica- 
 tions of them. For the present we will merely follow his experi- 
 
CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 165 
 
 ments and the explanations that he gave ; using, however, some 
 terms, such as * potential,' which he did not use. 
 
 3. Volta's Experiments and Views. For clearness we will 
 first state Volta's views, and will then give some of the experiments 
 by which he supported them. The student can see how far 
 each statement is supported or unsupported by the experiments 
 given. 
 
 Volta's views. (i) When two heterogeneous substances are 
 in contact they are found to be at different potentials. 
 
 (2) As a rule any A V. s between liquid and liquid, or metal 
 and liquid, are negligible as compared with the A V . s between 
 metals and metals. 
 
 (3) Metals can be arranged in a certain series (such as zinc, 
 tin, lead, iron, copper, silver, platinum, graphite}, with respect to 
 which the following facts hold : that any metal in contact with 
 another occurring later in the list will have a higher potential than 
 this latter ; and that in a series of several metals in contact the A V 
 between the first and last is the same as it would be were these 
 metals directly in contact. Thus, if we make the potential of 
 graphite zero, and find the potentials of all the other metals with 
 respect to this body when in contact with it, then we can calculate 
 the A V between any pair in contact by simple subtraction of their 
 potentials with respect to graphite. 
 
 (4) This law, however, does not hold good with respect to a 
 series in contact when composed partly of metals and partly of 
 liquids. 
 
 Thus, whereas in the series copper\zinc\gold\copper there is no 
 A V between the terminal metals, because it is the same as if 
 the initial and final metals (viz. copper\copper) were directly in 
 contact, yet in the series copper\zinc\dilute acid\copper there may 
 be, and as a matter of fact there is, a A V between the terminal 
 metals. 
 
 Experiments illustrating Volte? s views. In these experiments Volta em- 
 ployed the condensing gold-leaf electroscope, whose principle is explained in 
 Chapter VI. 14. 
 
 But far more satisfactory results can be obtained by use of any form of 
 quadrant electrometer (say the ' Elliott ' form of Chapter X. 33), in which 
 one pair of quadrants are to earth ; such an instrument indicates quantitative 
 results with an accuracy sufficient for lecture experiments. 
 
 (i.) The figure represents a simple experiment performed with the con- 
 
1 66 
 
 ELECTRICITY 
 
 CH. XI. 
 
 densing gold-leaf electroscope and a compound bar composed of a piece of zinc 
 and a piece of copper soldered end to end. 
 
 We may use the quadrant electrometer, putting one pair of quadrants to 
 earth, and connecting the other pair with an insulated terminal. 
 
 If the zinc be held in the hand, while we touch the lower condensing plate 
 of the gold-leaf electroscope (or terminal of the electrometer) with the copper, 
 it will be found that the condensing plate (or insulated pair of quadrants) is 
 now at a potential. 
 
 (For use of .electroscope and electrometer see Chapter X.) 
 
 FIG. i. 
 
 In this experiment it is assumed as sufficiently proved by various con- 
 vergent pieces of evidence that the zinc, held in the moistened hand, is 
 practically at the zero-V of the earth ; and that the contact of the copper with 
 the brass of the electroscope gives no A V. 
 
 Hence, it is argued, the electroscope or electrometer indicates the A V due 
 to the contact of the zinc and copper alone, and shows that zinc is + to 
 copper. 
 
 (ii. ) An electroscope is provided with an upper condensing plate made of 
 zinc. This is, as usual, provided on its under surface with an insulating layer 
 of lac varnish. Both plates are as usual carefully discharged by means of a 
 Bunsen's flame, until no movement of the leaves is observed on raising or 
 lowering the upper plate. 
 
CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 167 
 
 FIG. ii. 
 
 . Then a copper wire is made to connect the upper (unvarnished) surface of 
 the zinc plate with the under surface of the copper plate. 
 
 It is found that the zinc and copper have now assumed a A V in which the 
 zinc is + to the copper. This A V is of course (see Chapter X. ) magnified, 
 and so rendered evident, when the wire is 
 removed and the upper condensing plate is 
 lifted away. Here there is no doubtful con- 
 tact of metal and wet hand. 
 
 (iii.) If in experiment (i. ) we hold the 
 copper end of the compound bar, and touch 
 the plate of the gold-leaf electroscope with 
 the zinc, we get no result. This supports 
 Volta's conclusion (3). 
 
 So if we make a bar of zinc \ gold \ iron \ 
 copper ; and use it as in (i.), we get the same 
 result as in (i. ). 
 
 (iv. ) If we hold the copper end of the bar 
 and insert a piece of wet paper between the 
 zinc and the gold-leaf electroscope, we shall 
 find that, on removing the bar and raising 
 the upper condensing plate, the lower plate 
 is at a -- V. 
 
 Volta's view would be that the copper held in the moistened hand is 
 practically at zero-V ; the zinc at a + V ; the wet paper at practically the 
 same V as the zinc ; and the brass of the electroscope at practically the same 
 V as the paper, and therefore at the + V of 
 the zinc. 
 
 The reader should examine these 
 experimental results in connection 
 with the statements made above as to 
 Volta's views. 
 
 4. Volta's Pile, from Volta's 
 point of view. By the application of 
 the method of experiment (iv.) above, 
 it is easy to obtain A V . s high enough 
 to give shocks and to produce small 
 sparks. 
 
 We lay down a copper disc and 
 put it to earth. On this we lay a zinc 
 disc, which will be at some higher 
 potential that we will call v. On this we place a somewhat larger 
 disc of flannel, moistened with salt and water ; this will also be 
 
1 68 ELECTRICITY CH. xi. 
 
 practically at v. On this another copper disc, which will be also 
 at v. 
 
 Then a zinc disc on this will be at 2 v ; and so on. 
 
 We may conveniently solder together each pair of copper and 
 zinc discs. If there be n such pairs of plates, then we have the 
 bottom copper at zero-V, and the top zinc at n x v. If we solder 
 a copper terminal to this top zinc, it is to be noticed that this wire 
 will be at n i . v, or at the same potential as the n ih copper. 
 
 This arrangement is called ' Volta's pile.' 
 
 About fifteen or twenty such pairs will, when newly set up, 
 give a perceptible shock. After a time the surfaces of the metals 
 are altered from slow chemical action, and the pile must be taken 
 down and cleaned in order to give the initial results. 
 
 Volta's view was that the action is due to the series of con- 
 tacts ; but that when, owing to oxidisation of the surfaces, the sub- 
 stances in contact are altered, we do not have, nor can we expect, 
 the same results as initially. 
 
 Experiments. (i.) We put the bottom copper, and one pair of quadrants 
 of the electrometer, to earth. Then with an insulated platinum wire connected 
 with the other pair of quadrants we touch the 2nd, 3rd, 4th, &c., copper 
 plates in succession, putting the platinum wire to earth between each observa- 
 tion. 
 
 The deflexion of the needle will indicate the rise in potential up the pile. 
 
 (ii. ) We can make a similar observation on the momentary currents sent 
 through a delicate galvanometer of high resistance. But this assumes some 
 previous knowledge of the galvanometer. 
 
 Other forms of Vollds pile. There are other forms of Volta's 
 pile used ; of these we will mention one, viz. Zambonfs, 
 
 As a rule the drier the pile, the more lasting is its activity, but 
 the more easily is .it exhausted for the time. 
 
 In Zamboni's pile the arrangement answers to fixing together 
 a zinc, wet paper, and copper disc, as one element For each 
 element of his pile consists of a disc of paper that is ' silvered ' on 
 one side (i.e. coated with tin or lead}, and on the other side coated 
 with manganese dioxide. Such a pile gives a very high A V for 
 very little space occupied. It can be advantageously employed 
 to give the requisite A V in the Bohnenberger's electroscope, 
 where there is no completion of the circuit and therefore no running 
 down of the pile. 
 
CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 169 
 
 The Gymnotus. In certain animals, notably in the Gymnotus 
 or ' electric eel,' there have been found organs that appear to be 
 analogous to a voltaic pile. Severe shocks can be received, cur- 
 rents obtained, and other electric phenomena manifested. These 
 manifestations seem to depend partly upon the actual condition 
 of the organ, partly upon the general health and vigour of the 
 animal, and partly upon its volition. 
 
 The study of the relation of these electrical phenomena to 
 what we may call ' vital ' phenomena is exceedingly interesting ; 
 but we do not intend to include in the present Course anything 
 that belongs of right to the domain of physiology. 
 
 5. Volta's Cell, and the Couronne des Tasses, from Volta's 
 point of view, In order to obviate the 'running down' of the 
 power of his pile, resulting from the alteration of the surfaces of 
 the metals, Volta substituted dilute acid for wet paper. 
 
 His * pile ' then took the form of a number of cells or elements 
 joined end-on, forming what was called the Couronne des tasses. 
 We here give a sketch in section of this arrangement. 
 
 The zincs and coppers are plates connected by copper wires ; 
 they are immersed in vessels of dilute sulphuric acid. In the 
 figure we have supposed the copper wire that is soldered to the 
 zinc plate to be to earth and so at zero-V. 
 
 The rise of V through the arrangement is indicated. The 
 reader will see that, according to this view, the A V of a single ele- 
 ment is practically the same as that due to the first copper \ zinc 
 junction. When the terminals are connected a current flows, and 
 energy is given out in the circuit. 
 
 When pure water is used in the vessels we find the A V between 
 the terminals to be approximately the same as if dilute acid had been 
 used. But if we complete the circuit for a few moments, and again 
 break it, we find that this A V has almost or quite disappeared. The 
 
170 ELECTRICITY CH. XT. 
 
 battery has 'run down,' as did the pile. Examination will then show 
 that the surface of the zinc has been oxidised, while the copper is 
 covered with a film of hydrogen. 
 
 From VoltcHs point of mew we should say that we now have a new 
 series of bodies in contact ; viz. copper \ zinc \ zincic oxide \ water \ 
 hydrogen \ copper. The cell is now very different from its original 
 condition ; we could not predict whether there would be a A V between 
 the terminals, and in fact we find that there is none. This view of the 
 matter is doubtless unsatisfactory ; but it is of interest to notice that at 
 least the ' running down of the battery ' does not confute Volta's view. 
 
 But if we use dilute acid, the surfaces are continually renewed by 
 chemical action ; this action taking place far more freely in the present 
 case than in the case of the pile. 
 
 6. The ' Contact ' and < Chemical ' Theories, Volta fixed his 
 attention mainly on the A V that, as it seemed, accompanied the 
 contact of the dissimilar metals zinc and copper. His followers 
 exaggerated a certain one-sidedness that existed in his views ; and 
 the Contact school, as they were called, considered that the 
 chemical solution of the zinc played a subordinate part in the 
 action of the cell, serving mainly to keep the surfaces clean and 
 so to keep the same series of bodies in contact. In fact the word 
 contact was the key-note to their theory of the Voltaic cell. 
 They considered the A V between the terminals of the ' open ' 
 cell (i.e. of a cell in which the terminals were insulated) as the 
 algebraic sum of the different A V . s due to the different contacts ; 
 of which, in the ordinary Volta's cell, the only one of importance 
 was that where the copper wire was soldered to the zinc. 
 
 The * Chemical school ' of physicists considered the cell when 
 the circuit was closed and a current was running. They pointed 
 out how the strength of the current that flowed was proportional 
 to the vigour with which the chemical action proceeded ; and how 
 the power of the cell depended on having one plate as much 
 acted upon, and the other plate as little acted upon, as possible. 
 
 Faraday was the great exponent of this view. In modern 
 phraseology, the ' Chemical school ' insisted on the chemical action 
 as the source of the energy of the cell. 
 
 They, in their turn, were for the most part too one-sided ; and 
 many denied that dissimilar metals in contact did exhibit a dif- 
 ference of potential at all without chemical action. 
 
 In the next section we shall attempt to show the position of 
 
CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 171 
 
 modern theory and of modern knowledge in this matter ; and 
 shall conclude by giving a view of the Volta's cell, taken as a 
 whole, which can hardly involve any serious error. 
 
 But we should add that the whole question is still to a con- 
 siderable extent unsettled. 
 
 7. Theory of the Simple Volta's Cell, As is usual in such 
 cases there is truth in both the above extreme views ; though the 
 error of the extreme followers of the Volta school was perhaps the 
 greater, inasmuch as they neglected the source of energy in the 
 current. 
 
 There seems no doubt but that a difference of potential does 
 in general accompany the contact of dissimilar bodies ; though it 
 is quite beyond our present knowledge to say that the contact 
 causes the difference of potential. . . 
 
 But the phenomenon is not an easy one to observe ; nor is 
 the amount of A V as easy to measure as would at first appear. 
 
 When zinc and copper are in contact with each other, they are 
 also in general in contact with the air or other medium surround- 
 ing them. It has been asserted that while the zinc and copper 
 are nearly at the same V, the layers of (badly conducting) air or 
 other gas in contact with the two metals respectively will be found 
 to be at a considerable A V from each other ; and that this A V 
 alters both in magnitude and in sign with the nature of the gas. 
 
 It is said that in many electrostatic methods we really measure 
 the A V of these layers of gas, and not of the metals themselves. 
 That while zinc really has under all circumstances at ordinary 
 temperatures a very small V with respect to copper, yet when 
 the usual experiments are performed in air we find zinc to be 
 apparently very strongly + to copper, and when performed in an 
 atmosphere of hydrogen sulphide we find zinc to be apparently 
 very strongly to copper. 
 
 The A V . s between the metals and the gases are considered to 
 result fro n surface chemical action ; and the minute A V . s between 
 the metals themselves to result from some molecular action prob- 
 ably unaccompanied by any chemical change. These latter true 
 A V s between the metals have been measured by methods that 
 will be discussed in the chapter on thermo-electricity. 
 
 We have given here a view the difference of which from Volta's 
 view shows how necessary it is at present to be very cautious in 
 
172 ELECTRICITY CH. X[ . 
 
 making any dogmatic assertions as to the respective A V. s at the 
 different places of contact ; though at the same time we see that 
 even this view allows the Voltaists to have been right in asserting 
 that some A V does accompany mere contact. 
 
 The Chemical school, though they did not use modem expressions, 
 and had not fully grasped the laws of energy, at least understood 
 that we cannot get work done without an equivalent disappearance 
 of energy. They regarded the energy of the current, that ensues 
 when the circuit is closed, as derived from the chemical consump- 
 tion of zinc in the cell ; and they saw that the energy of the cur- 
 rent depended upon the intensity of the chemical affinities and 
 on the amount of the chemical action. 
 
 Perhaps at present it is safest to regard the cell as a whole* 
 Wherever it be that the A V . s occur, at any rate we may safely 
 say that each cell tends to keep its two terminals or poles at a 
 certain A V ; and, if the circuit be completed and a current be 
 allowed to pass, it is beyond dispute that the energy of the current 
 is supplied by chemical action in the cell. We may regard the 
 cell as a contrivance in which electricity is pumped up from a 
 lower level to a higher by the expenditure of chemical energy ; 
 the electricity running down through the external circuit to the 
 lower level again, and so on continuously. 
 
 In the simple diagram here given the cell is represented by 
 the vertical line CZ. This indicates that in the cell the elec- 
 tricity is raised from a lower to 
 a higher level by expenditure of 
 chemical energy ; and that then 
 the electricity flows down from 
 the higher level C to the lower 
 level Z through the external 
 circuit X. 
 
 We may here add that it is 
 usual -to call the zinc the + 
 plate, and the copper the plate ; while the copper wire soldered to 
 the zinc plate is the pole, and that soldered to the copper plate 
 is the + pole. In fact, that plate or pole is called ' -f ' from 
 which the current flows to the other plate or pole respectively. 
 The current flows from zinc to copper through the liquid, and 
 from copper to zinc through the external circuit. 
 
cn. xi. THE PHENOMENA OF ELECTRIC CURRENTS 173 
 
 The phenomena of the ' open ' cell are, we see, electrostatic ; 
 the A V of the terminals may be observed with a quadrant elec- 
 trometer. When the circuit is completed we get a flow of elec- 
 tricity ; the phenomena accompanying this flow will be surveyed 
 in Chapter XII. 
 
 8. Digression on the Galvanometer. In Chapter XVII. 
 the student will find discussed the construction and theory of the 
 galvanometer. We shall, therefore, deal with it very briefly here. 
 
 The figures of Chapter XVII. indicate that when a current 
 passes along a wire held parallel to a magnetic needle, above or 
 below it, the current produces a magnetic field that tends to set 
 the needle in a position at right angles to the wire. As the needle 
 is deflected, the earth's couple, tending to restore it to its 
 original position of rest in the plane of the magnetic meridian, 
 becomes greater ; while the disturbing couple due to the current 
 becomes less. Hence it will rest at some angle of deflexion from 
 the magnetic meridian ; and the magnitude of this angle can be 
 employed to calculate the strength of the current. 
 
 By ' strength of current ' we mean quantity of electricity passing 
 across any cross-section of the wire in one second of time ; as will be 
 further explained in Chapter XIII. 
 
 It is found that the galvanometer indicates the strength of 
 current as just defined. It makes no difference in the indications 
 whether this quantity per second flow quickly through a thin wire, 
 or slowly through a thick one. 
 
 9, The Solution of Zinc in the Voltaic Cell. We will now 
 examine the nature of the chemical action that goes on in the 
 Voltaic cell, considering first the open cell, and then the closed 
 circuit. 
 
 / In the open circuit. If we immerse a plate of ordinary 
 commercial zinc in the dilute acid of the battery cell (we usually 
 mix strong sulphuric acid with water in the proportion by weight 
 of one of the former to ten of the latter), we notice the usual dis- 
 engagement of hydrogen from the surface of the zinc. 
 
 But if we immerse either very pure zinc, or ordinary zinc 
 whose surface has been amalgamated with mercury, no such action 
 takes place. Here the zinc and copper plates, unconnected with 
 each other, stand opposite to each other in the dilute acid, equally 
 passive. 
 
174 
 
 ELECTRICITY 
 
 CH. XI. 
 
 Probably the action of the mercury with which we have amal- 
 gamated the surface of the commercial zinc is to soften that 
 surface, and to render the zinc superficially free from all strain ; 
 to give the zinc a mechanical homogeneity that apparently makes 
 it behave like chemically pure zinc in being unacted upon by dilute 
 sulphuric acid. 
 
 // In the dosed circuit. If we repeat the above simple ex- 
 periments, but now connect the copper and zinc plates, we get a 
 current. 
 
 In the figures, A is the zinc plate, B is the copper plate, C and 
 C' are copper wires. The first figure represents the fact that the 
 terminals C and C' in the open cell are at different V . s ; the 
 
 FIG. i. 
 
 FIG ii. 
 
 second figure represents the direction of the current that ensues 
 when the terminals are connected. 
 
 Now we shall find that, in the case of common zinc, when the 
 circuit is completed immediately there is an increase in the evolu- 
 tion of hydrogen, the additional gas being given off the copper 
 plate from the side that is turned towards the zinc plate. 
 
 In the case of pure, or amalgamated, zinc, we shall find that 
 on completing the circuit chemical action at once commences ; 
 hydrogen is given off from the copper plate, while the zinc is dis- 
 solved. In both cases we find, on testing the liquid, that it is the 
 zinc that is dissolved, the copper remaining unacted upon by the 
 acid. 
 
 In the case of pure, or amalgamated, zinc, it would seem from 
 this experiment that chemical action only occurs when a current 
 passes, and that then, while it is the zinc that is acted upon, yet 
 
CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS I?5 
 
 the hydrogen is set free from the surface of the copper plate. 
 Thus the hydrogen seems to travel in some invisible manner with 
 the current, and to be set free off that surface by which the 
 current leaves the liquid. 
 
 In the case of impure zinc it seems likely that the solution, 
 which proceeded while the cell was open, still goes on when the 
 circuit is closed, but that now there is additional solution of zinc, 
 the hydrogen corresponding to this portion being set free off 
 the copper plate. 
 
 Further investigations have made it almost certain that in no 
 case will the zinc dissolve, saving step by step with the passage 
 of a current ; but that in the case of impure zinc there are in- 
 numerable small currents circulating between portions of the 
 impure zinc, that differ from one another in chemical or in 
 mechanical properties. These currents are called local currents, 
 and the chemical action that occurs in the open cell is called 
 local action. Amalgamation then is to prevent the occurrence 
 of local action, which latter contributes no energy to the main 
 current. 
 
 10. Polarisation. In the simple Volta's cell in which we 
 use a copper and a zinc plate immersed in dilute sulphuric acid, it 
 is found that the current falls off rapidly from its initial strength. 
 We may show this by introducing a galvanometer into the circuit, 
 when a gradual decrease in the deflexion of the needle will be 
 observed (see 8). 
 
 Note. Mercury cups. In all experiments where it is necessary to make 
 and to unmake electrical connection between different conducting wires, it is 
 very convenient to use merctcry cups. These may be simply small round 
 wooden boxes filled with mercury, in the lids of which are bored holes. The 
 wire terminals of the battery, or of any piece of apparatus such as the galvano- 
 meter, may be rapidly dipped into or removed from these cups, and, if the 
 wires be well amalgamated, the connections thus made are good. 
 
 On examining into the cause of the falling off in current, we 
 find it to be due to the film of hydrogen that adheres to the 
 copper plate at which, as we have seen, it is set free. 
 
 Experiment shows that this hydrogen acts to lessen the current 
 in two very different ways. 
 
 Firstly, it covers the copper plate with a layer that conducts 
 the current very badly as compared with the dilute acid that 
 
ELECTRICITY 
 
 initially was in complete contact with the copper, thus lessening 
 the current by passive resistance. 
 
 Secondly, owing to the presence of the hydrogen on. the copper 
 plate there is a tendency to a back current opposing the main 
 current. This back tendency is called polarisation^ and is of a 
 very different nature from resistance. Indeed, if we now replace 
 the zinc plate of the cell by a clean copper plate, and complete 
 the circuit, we find that we actually get a reverse current ; this 
 continues until the hydrogen has combined with all the oxygen 
 that is mechanically dissolved in the liquid, as we shall see later 
 on (see Chapter XII. n). 
 
 It is pretty clear even at this stage that * polarisation ' can 
 never actually reverse the original current, for this latter causes 
 the polarisation. But the current can be nearly or quite brought 
 to a standstill. 
 
 Experiment. The figure indicates in a simple diagrammatic manner one 
 mode of showing this 'back tendency' or 'polarisation.' 
 
 G is a somewhat delicate galvanometer ; m m' are mercury cups ; Cu and 
 Zn are the plates of the cell, in which we here use common water instead of 
 
 dilute acid. C'u is a spare copper 
 
 ^ * - that may be readily substituted for 
 
 the plate Zn. 
 
 (If we find the current too 
 strong for the galvanometer, we 
 may protect this latter, to a greater 
 or less extent, by bridging over 
 the cups m m 1 with a wire of less 
 or greater resistance, as will be 
 explained in Chapter XIII. 10 ) 
 
 We first complete the circuit, as shown in the diagram, and notice that the 
 initial current soon almost, or entirely, ceases. We next replace the plate Zn 
 by the 'passive plate' C'u, and a reverse current, almost as strong as the 
 original current, will be observed. 
 
 If we use dilute acid in the cell, the current only falls off to some extent, 
 and the reverse current is proportionally weaker. 
 
 ii. Constant Batteries. The efforts of scientific men were 
 soon directed towards the contrivance of battery-cells in which 
 polarisation did not occur. 
 
 The hydrogen, which in the simple Volta's cell adheres to the 
 copper plate, was to be got rid of. One method of doing this was 
 
CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS J// 
 
 a mechanical method ; viz., to use as a negative plate some plate 
 to which the hydrogen could not stick in the form of a film. The 
 other method was to give the hydrogen, as it appeared in its 
 nascent condition on the surface of the negative plate (see 7), 
 some chemical reducing work to do. We will now describe 
 several forms of so-called ' constant ' battery-cells under the heads 
 of mechanical and chemical respectively. 
 
 Note. Electromotive force. The poles of the open cell exhibit a certain 
 A\ r ; the greater this is, the greater (cceteris paribus) will be the current when 
 the circuit is closed. When the current is running, that which urges it is 
 called electromotive force ; it can be measured in terms of the A V that would 
 appear were the circuit cut. We shall hear more of electromotive force 
 (usually called E.M.F.) when we come to Ohm's Law in Chapter XIII. 
 
 In Chapter XV. 13 we shall see that when the hydrogen is 
 got rid of by chemical means we not only obviate polarisation, but 
 we gain a positive advantage in a greater E.M.F. (see preceding 
 note). 
 
 I. Mechanical. The earliest form of cell in ordinary use that 
 comes under this head is the Smee's. In this the copper plate is 
 replaced by a sheet of thick silver foil that is covered with very 
 finely divided platinum (or ' platinum-black '). Off these points 
 the hydrogen rises to the top, and does not remain as a film 
 covering the negative plate. The substitution of silver for copper 
 gives, moreover, a slightly greater E.M.F. 
 
 II. Chemical. Single-fluid cells. The most important single- 
 fluid cell in which the hydrogen is used up in doing chemical 
 reduction is the 
 
 (i.) Bichromate cell. Here the negative plate is of carbon (a 
 special kind) instead of copper ; and the liquid is composed of a 
 solution of potassium-bichromate in water, mixed with sulphuric 
 acid. The nascent hydrogen is employed in reducing the chromic 
 acid ; so that chromic sulphate is formed. This battery has a high 
 E.M.F., yields no fumes, is simple in its arrangements, and there- 
 fore is adapted for general laboratory use. The zinc plates must 
 be removed from the liquid when not in use. 
 
 (ii.) Latimer Claris standard cell. For purposes of com- 
 parison rather than for practical use, Latimer Clark proposed the 
 following cell. 
 
 As a negative piate, pure mercury. Over this a paste obtained 
 
 N 
 
iyS ELECTRICITY CH. xi. 
 
 by boiling sulphate of mercury with a saturated solution of zinc 
 sulphate. 
 
 As a positive plate, zinc ; this resting on the sulphate paste. 
 Insulated wires are connected with both plates. 
 
 This cell can be easily prepared. Its value consists in the 
 fact that, when left with the circuit open, its E.M.F. is very con- 
 stant, and can be used as a standard of comparison. When, how- 
 ever, the circuit is closed, and a current flows, the E.M.F. does not 
 remain constant. 
 
 III. Chemical. Two-fluid cells. There are many elements 
 in which the negative plate is separated from the positive plate by 
 a porous pot or' partition. The positive plate is surrounded by a 
 liquid that acts upon it when the circuit is closed, and the negative 
 plate by a liquid containing something that the nascent hydrogen 
 can reduce. 
 
 (i.) Bunseris cell. In this battery the + plate is zinc, sur- 
 rounded by dilute sulphuric acid. The plate is gas-carbon ; it 
 
 FIG. i. 
 
 stands in a porous pot, and is surrounded by nitric acid, the 
 porous vessel thus preventing the two acids from mixing, while 
 yet allowing the chain of chemical changes to pass with the 
 current through its pores. 
 
 In the figure, P represents the entire cell ; while F Z V C 
 represent the outer vessel, zinc, porous pot, and carbon re- 
 spectively. 
 
 The zinc dissolves in the dilute acid, forming zinc sulphate ; 
 while the corresponding hydrogen, set free against the carbon, 
 reduces the nitric acid to lower oxides of nitrogen. 
 
CH. xi. THE PHENOMENA OF ELECTRIC CURRENTS 1/9 
 
 (ii.) Grove's cell. This is an earlier form than the preceding. 
 It differs from it only in having platinum instead of carbon. The 
 shape of the porous pot is flat, to suit the flat plates of platinum- 
 foil. 
 
 (iii.) Daniell's cell. In this the only essential difference from 
 the above is that we have copper in a saturated solution of copper 
 sulphate, instead of platinum (or carbon) in nitric acid. But it is 
 usually constructed having the zinc and acid in the porous pot, 
 while very often the copper itself forms the outside vessel. The 
 zinc may be surrounded by a semi-saturated solution of zinc 
 sulphate, or of common salt, instead of by dilute sulphuric acid. 
 
 Here the hydrogen reduces the copper sulphate ; sulphuric 
 acid is formed, while copper is deposited on the copper plate. It 
 is therefore necessary to keep up the strength of the copper 
 sulphate solution by a supply of crystals of that salt. 
 
 In one very portable form we have but a single vessel ; at the 
 bottom is a plate of copper on which is a layer of crystals of 
 sulphate of copper ; over this is 
 a layer of sawdust ; and, resting 
 
 on this, a plate of zinc immersed 
 
 in dilute sulphuric acid or in a 
 
 solution of sulphate of zinc. In- 
 sulated wires form connections 
 
 with the plates. 
 
 In ' gravity batteries ' the 
 
 liquids are kept apart simply by 
 
 the fact that the less dense liquid 
 
 forms a layer above the more 
 
 dense. 
 
 (iv.) LedanchPs cell. In this 
 
 the + plate is a zinc rod immersed 
 
 in a strong solution of ammo- 
 nium chloride. In the porous 
 
 pot is a carbon rod, round which 
 
 is tightly packed a mixture of 
 
 manganese dioxide and of pow- FIG .. 
 
 dered carbon. The porous pot 
 
 is closed at the top with pitch, a hole being left for the escape of 
 
 gases. 
 
 N 2 
 
180 ELECTRICITY CH. xi. 
 
 The ammonium chloride solution soon soaks through and 
 moistens the powder in the porous pot, and then the cell is ready 
 for use. 
 
 Zinc chloride is formed in the outer vessel, the zinc displacing 
 ammonium. In the inner vessel is set free ammonia, while the re- 
 maining hydrogen of the ammonium reduces the manganese dioxide. 
 
 12. Remarks on Cells and on Batteries. We have now men- 
 tioned the chief cells that are of interest to the general student. 
 A few remarks will be made in conclusion. 
 
 Chemical action in the cell, For a discussion of the manner in 
 which the chemical action takes place, in what way the hydrogen 
 displaced by the zinc * travels with the current ' and appears at the 
 negative plate, and what relation the amount of chemical action 
 bears to the strength of the current, the student is referred to 
 Chapter XII. , 
 
 Efficiency of different cells. We have stated that different cells 
 give a different A V at their terminals when the circuit is open, and 
 a different E.M.F. (which, when the other conditions are the same, 
 will cause different currents) when the circuit is completed. For 
 a further discussion of this matter we refer the reader to the sections 
 on ' Electromotive force ' in Chapters XIII. and XV. 
 
 Again, cells differ from one another in offering more or less re- 
 sistance to the passage of the current through them. This matter 
 will be discussed in Chapter XIII. 
 
 Coupling cells together. If we connect n cells as we connected 
 the elements in the Couronne des tasses we get a A V in the open 
 circuit, or an E.M.F. in the closed circuit, which is n times that of 
 a single cell. 
 
 Again, if we connect all the positive plates together and all the 
 negative plates together respectively, we have what amounts to one 
 large cell whose plates are n times the size of those of a single 
 cell. Such an arrangement has the E.M.F. of only one cell. 
 
 The advantages of the above two methods of coupling in dif- 
 ferent circumstances respectively will be discussed in Chapter XIII., 
 and the proper terms will be there given. 
 
 Amalgamating zinc. If the zinc be wetted with dilute acid it 
 is readily amalgamated with mercury. 
 
 Or, again, if a little sodium be added to the mercury it enables 
 one to amalgamate zinc and other metals with greater ease. 
 
CH xi. THE PHENOMENA OF ELECTRIC CURRENTS l8[ 
 
 Use of carbons. In order to obviate the tendency of the car- 
 bons to 'soak up' the liquids in which they are immersed a result 
 very unpleasant and very injurious to the binding screws attached 
 io the carbons it is usual to soak the upper part of the carbons 
 in melted paraffin wax. The surface must then be scraped at the 
 places where the binding screws make contact. 
 
1 82 ELECTRICITY 
 
 CH. XII. 
 
 CHAPTER XII. 
 
 THE CHEMICAL PHENOMENA ACCOMPANYING THE PASSAGE OF 
 THE CURRENT. 
 
 i. Introductory. In this chapter we shall consider the 
 chemical phenomena that accompany the passage of an electric 
 current. We shall assume that the student is acquainted with the 
 elements of theoretical chemistry. 
 
 Of the other classes of phenomena, the heat effects will be only 
 briefly mentioned here, a fuller treatment being found in Chapters 
 XV. and XVI. ; the magnetic effects will be discussed in Chapters 
 XVII. and XVIII., &c.; and the induction phenomena form a large 
 portion of uur subject and will occupy our attention in Chapters 
 XXI.-XXIV. 
 
 2. Heating Effects; a Brief Account. When a current 
 passes through a conductor it is found that the said conductor is 
 heated. The stronger the current, and the greater the resistance 
 of the wire, the greater is the quantity of heat evolved. The exact 
 law relating to this matter will be given in Chapter XV. A 
 battery of from three to six large bichromate cells will render 
 white-hot, or even melt, fine platinum wire, and will illumine 
 small ' glow ' lamps of ten candle-power. 
 
 Now, whence does this heat-energy come? Our source of 
 energy is the battery-cell. If in this a certain amount of zinc be 
 dissolved, there will be a certain amount of heat evolved. If the 
 circuit of action be contained within the cell, then all the heat 
 due to the quantity of chemical action will appear in the cell. But 
 if the cell drive a current and this current heat a wire external 
 to the cell, so much the less heat will appear in the cell. The 
 total amount of heat evolved during the solution of a given mass 
 of zinc will be constant ; but, by making the external circuit of 
 
CH. xii. CHEMICAL PHENOMENA 183 
 
 some substance that offers a very high resistance to the current, 
 we may cause nearly all the heat, due to the amount of chemical 
 action in the cell, to be evolved outside the cell. We lose 
 chemical-potential-energy, and we gain equivalent heat-energy 
 (see Chapter X. 4). 
 
 3. Chemical Effects ; General View. Bodies may be roughly- 
 divided into three classes with regard to their behaviour as to 
 allowing a current to pass through them. 
 
 (i.) Conductors. All those solids and liquids that allow a cur- 
 rent to pass, while themselves undergoing no change saving a rise 
 in temperature, are called conductors. Such are metals whether 
 solid or molten, carbon, the bodies of animals, &c. 
 
 (ii.) Insulators. Bodies that do not allow any appreciable 
 current to pass are termed insulators. A few examples of insu- 
 lators are glass, ebonite, paraffin oil, dry vapours, &c. 
 
 These two classes merge the one into the other. The reader 
 should refer to Chapter XIV. 14, for tables of resistances. 
 
 (iii.) Electrolytes. A great many bodies allow a current to 
 pass, but themselves suffer a chemical decomposition that pro- 
 ceeds step by step with the current ; not allowing any appreciable 
 current to pass without this chemical decomposition occurring. 
 
 Such bodies are termed electrolytes. This class consists almost 
 entirely of compound liquids or solutions of salts, or of molten 
 salts; the word 'salt 'being understood in its widest sense. A 
 few examples are aqueous solutions of adds, of metallic salts, or of 
 alkalis ; also such liquids as melted potash or soda, c. The 
 action, as we shall see hereafter, requires that the molecules of 
 the bodies shall move with freedom ; hence all electrolytes are 
 liquids or pastes. 
 
 Explanation of terms used. Referring to fig. (ii.), 4, we see 
 that usually the electrolyte is introduced into a glass vessel, while 
 the current enters by a metal plate A and leaves by another 
 plate B. This cell is called an electrolytic cell ; A is called the 
 anode, B is called the kathode. These plates are usually of 
 platinum, this being unacted upon by most liquids. In what 
 follows it is understood that they are of this metal, unless the 
 contrary is stated. 
 
 It is found that the electrolyte is invariably split up into two 
 molecular (or atomic) groups, the metallic radicle and non-metallic 
 
1 84 
 
 ELECTRICITY 
 
 ,CH. XII. 
 
 radicle respectively. Thus H. 2 SO 4 splits into H 2 and SO 4 , not 
 into H 2 O and SO 3 ; NH,C1 into NH 4 and Ci, not into NH 3 
 and HC1 ; CuSO 4 into Cu and SO 4 ; and so on. 
 
 These groups are set free at the two electrodes ; the metallic 
 groups (as H 2 , Cu, (NH 4 ) 2 , &c.) at the kathode ; the non-metallic 
 groups (as O, SO 4 , C1 2 , &c.) at the anode. 
 
 These groups are, from their 'travelling,' called ions ; the 
 metallic groups are called kations or electropositive, from being set 
 free at the kathode or negative plate ; the non-metallic groups are 
 called anions or electronegative, from a similar reason. 
 
 The process of chemical decomposition through the agency of 
 a current is called electrolysis. 
 
 Note. A slight acquaintance with Greek will enable the reader to discover 
 for himself the derivation of the terms anode, kathode, electrode, ions, &c. 
 
 Experiments in electrolysis. Experiments to illustrate electrolysis may be 
 devised almost ad infinitum. We here describe a few typical cases. 
 
 (i.) Decomposition of -water. It is not certain that pure water can be 
 electrolysed. Certainly as we approach purity the water becomes almost an 
 ' insulator ' ; it being remarkable that mere traces of 
 acids or salts in solution have a very great influence 
 in destroying the insulating power and in rendering 
 the water an electrolyte. 
 
 When we wish to decompose water we usually 
 mix it with about one-tenth its volume of strong 
 ILSO 4 . What part the FLSO 4 plays is not known 
 for certain. Here we will assume that it only renders 
 the water capable of electrolysis. 
 
 The figure represents the usual arrangement, or 
 one form of it. The vessel contains the acidulated 
 water ; A and B are the anode and kathode respec- 
 tively ; C and D are glass vessels at first filled with 
 the liquid and inverted over the electrodes in order to 
 collect the gases. 
 
 When the current passes, the H. 2 and the O of 
 the FLO are set free in chemical equivalents at the 
 kathode and anode respectively. Errors in the 
 volumes collected occur from the greater solubility of 
 oxygen, from part of the oxygen being set free in the form of ozone, and from 
 hydrogen being * occluded ' by the platinum electrode to a greater extent than 
 is oxygen. The first and third errors can be nearly eliminated by allowing 
 the action to proceed for some time before collecting the gases ; the second 
 error by heating the tube, or by having electrodes of such area that the current 
 is not too dense. 
 
CH. xii. CHEMICAL PHENOMENA 185 
 
 In several of the following experiments it is very convenient to 
 have a V-shaped tube ; the electrodes occupying the two arms of 
 the tube respectively. Or we may have an ordinary cell in which 
 the two electrodes are separated by a porous earthen diaphragm, 
 or two vessels connected by wet cotton wick. Each of these 
 methods enables us to examine at leisure the condition of the 
 liquid about the two electrodes after the action has proceeded for 
 some time ; the mixing of these two portions of the liquid being 
 to a greater or less extent prevented. 
 
 (ii. ) Electrolysis of CziSO r If we employ a solution of CuSO 4 , we find 
 Cu set free at the platinum kathode ; while from the platinum anode is set free 
 O, the liquid about this electrode at the same time losing colour and showing 
 the presence of free H 2 SO 4 . 
 
 If we employ copper electrodes we find fresh Cu coating the kathode, 
 while the anode is dissolved with the formation of CuSO 4 . 
 
 We shall in 5 argue that in both cases there is primarily set free Cu at 
 the kathode and SO 4 at the anode. 
 
 (iii. ) Electrolysis of Na. 2 SO v In this case (using platinum electrodes) we 
 find H 2 and 2NaHO appearing at the kathode ; from the anode is set freeO, 
 while the liquid about it shows signs of free H 2 SO 4 . We shall in 5 show 
 that this is equivalent to a setting free of Na 2 at the kathode, and of SO 4 at 
 the anode. 
 
 If some extract of red cabbage be mixed with the solution, and a drop or 
 so of dilute acid be added (if necessary) until the whole is of a dull purple 
 colour, the alkali and acid will be indicated by green and red colourations 
 about the kathode and anode respectively. 
 
 (iv. ) Electrolysis of JVff 4 . Cl. In this case we get Cl at the anode, and 
 NH 3 together with H at the kathode (NH 4 C1 = NH 4 + C1). If, however, 
 the kathode be of mercury, this latter swells up and forms what is by some 
 considered to be an amalgam of mercury and the metal NH 4 . 
 
 The solution of NH 4 .C1 must be weak and cold ; otherwise we may get the 
 very unstable and dangerous ' chloride of nitrogen ' formed. 
 
 (v.) Electrolysis of KHO. Some potassium hydrate is fused and is placed 
 on a piece of platinum foil, which forms the anode. In a cavity on the upper 
 surface of the salt is placed a drop of mercury which is made to form the 
 kathode. 
 
 The salt will have absorbed from the air, when it cooled after fusion, 
 enough water to render it an electrolyte ; it will, in fact, be a very stiff ' paste.' 
 After passing a current from a battery (say of four bichromate cells) for a few 
 moments, we drop the globule of mercury into water. It is seen to give off 
 hydrogen, while KHO is found in solution. ,, , 
 
 This indicates that the KHO (2KHO = K,O + HX>) was decomposed, 
 K._, being set free at the kathode and there forming an amalgam with the mer- 
 cury, while O was set free at the anode. 
 
1 86 ELECTRICITY 
 
 CII. XII. 
 
 (vi. ) Electrolysis of Pb. A.,. We here use a fairly strong solution of plumbic 
 acetate, and a pair of lead electrodes. 
 
 The Pb is set free at the kathode, there forming a beautiful ' lead tree.' 
 With a small cell this can be projected on the screen by means of a lantern. 
 A battery of four bichromates causes a very rapid growth of the tree ; any 
 battery-cell will give the result in time. 
 
 The anode will at the same time be dissolved, giving Pb.A 2 . 
 
 4. Grothiiss's Hypothesis. Nature of Electrolysis. In a 
 later section we shall see that, as we should have expected, the 
 ions set free at the two electrodes are always chemical equivalents 
 of one another. Thus (we assume the reader to be acquainted 
 with the exact meaning of chemical symbols) for H 2 and 
 2NaHO at the kathode we get O and H 2 SO 4 at the anode ; 
 we could not get O 2 , since that is equivalent to 2H 2 or to 
 4NaHO. 
 
 It may be noticed further that no signs of decomposition can 
 be detected saving at the surfaces of the electrodes. 
 
 We give here a view of electrolysis that' is consistent with the 
 above facts ; it is the view of Grothiiss, or Grothiiss's hypothesis, 
 slightly modified to suit the modem theory of compound liquids 
 or of salts in solution. 
 
 It is believed that, in a compound liquid, not only are the 
 molecules in continual 'slipping' motion, but the atoms them- 
 selves are being continually dissociated and recombined. Thus, 
 in water, neighbouring molecules of H 2 O are continually ex- 
 changing partners, the H 2 of one molecule taking the O of the 
 other, and reciprocally. This cannot be observed, because the 
 interchange is molecular only, and the average constitution of the 
 liquid remains the same ; for much the same reason, indeed, we 
 cannot observe the molecular motion called ' Heat.' 
 
 It is considered that the presence of the electrodes, connected 
 with the + and poles of the battery respectively, has the effect 
 of directing this interchange ; or, when one electrode is made + 
 and the other , then the interchange of partners is such that on 
 the whole the H .,..$ in their changes move towards the electrode, 
 and the O.s towards the + electrode. 
 
 Probably the ' current ' is thus, and thus only, conveyed ; 
 passing by convection in an electrolyte. In a metallic conductor 
 it passes by conduction. 
 
CHEMICAL PHENOMENA 
 
 I8 7 
 
 We thus have the H 2 .j- arriving at the kathode and the O.s 
 at the anode ; while the liquid between, being in the same condi- 
 tion as to average constitution as it was before we tried to pass a 
 current, appears to be unaffected. 
 
 This directed travelling of the groups has received the name of 
 the migration of the ions. 
 
 Experiment illustrating Grothiiss's hypothesis. The following experiment 
 illustrates how the electro + and electro groups (or kathions and anions) 
 travel from one electrode to 
 the other, while the inter- 
 vening liquid shows no 
 signs of change. 
 
 A, B, and C are three ves- 
 sels containing a solution of /^ 
 Na_,SO 4 coloured to a dull 
 purple with extract of red 
 
 cabbage. They are con- "^ 
 
 nected by moistened lamp- 
 wick, and in A is the kathode, and in C is the anode. If two Leclanche cells 
 be employed, and the whole be left for a day or so, it will be found that A has 
 
 become alkaline, and C is acid ; while B through which have passed both 
 
 the alkaline and acid groups is still unaltered. 
 
 We will now represent in chemical symbols the interchanges 
 that are continually taking place in an electrolyte when the current 
 is passing, choosing a few typical cases. The upper brackets 
 represent the condition of the chain of molecules before inter- 
 change ; the lower brackets give the new grouping after inter- 
 change, with the ions set free at ^ ^_^ 
 the two electrodes respectively. 
 
 I. The case of water. Here 
 we see that initially we have 
 
 while 
 
 Pt 
 
 FiG. ii. 
 
 groups of OH 2 on^y 
 
 finally we have an odd H 2 at 
 
 the kathode, an odd O at the 
 
 anode, with groups of OH 2 between, as before. 
 
 II. The DanieWs cell. Here we will consider the case of a 
 battery-cell itself, since all we have said of electrolysis applies 
 to the battery-cell equally with the electrolytic-cell, to chemical 
 combination equally with chemical decomposition. In the battery- 
 
i88 
 
 ELECTRICITY 
 
 FIG. iii. 
 
 cell the copper plate is the kathode, and the zinc plate is the 
 anode. 
 
 The 'Zn' against the zinc plate represents an atom of the 
 originally undissolved zinc. The arrangement of brackets repre- 
 sents that when the 
 circuit is closed this 
 Zn takes SO 4 from 
 the nearest H 2 SO 4 ; 
 this leaves H 2 to take 
 SO 4 from the next 
 H 2 SO 4 , and so 
 on ; Ho then passing 
 through the porous 
 cell takes SO 4 from 
 the next CuSO 4 , and 
 so on, until finally Cu is deposited on the copper plate (the 
 kathode). 
 
 As explained earlier, no change due to this interchange will 
 be observable saving at the surfaces of the battery plates. 
 
 5. Primary and Secondary Decompositions. We will now 
 complete the theory of electrolysis, explaining the results of de- 
 composition noticed in 3, (ii.), (iii.), (v.), &c. 
 
 There is every reason for, and no reason against, the view that 
 in electrolysis we have primarily the metallic, and the non- 
 metallic, groups set free against the kathode and anode respec- 
 tively. 
 
 But for reasons of ' chemical affinity ' such a condition of things 
 may not be stable. Thus if Na. 2 SO 4 be decomposed into Na 2 
 and SO 4 , the Na 2 will decompose water and give 2NaHO-f H 2 , 
 while the SO 4 with water will give HoSO 4 + O. 
 Thus we have by primary decomposition, 
 
 at anode SO 4 Na 2 at kathode ; 
 
 and then by secondary decomposition, 
 
 / tf;z<Mk O + H,SO 4 .... 2NaHO + H 2 at kathode. 
 
 The reader will notice that, in both cases, what we have 
 stated as to metallic groups moving with, and non-metallic groups 
 moving against, the current, still holds good. 
 
 In the case 'of the decomposition of CuSO 4 , the Cu is de- 
 
CH. xii. CHEMICAL PHENOMENA 189 
 
 posited at the kathode whether that be of copper or of platinum, 
 Cu being stable. But if the anode be platinum, the SO 4 gives 
 H 2 SO 4 + O ; while if it be of copper it will act on this and give 
 simply CuSO 4 . 
 
 So in the case of other salts ; the general principle of electro- 
 lysis determines the primary decomposition, and then a knowledge 
 of chemistry enables us to predict the nature of any secondary 
 action that may occur. 
 
 6. Simultaneous Decompositions, When the strength of 
 current per unit area of the electrode is great, we find that we get 
 in many cases a simultaneous decomposition of the salt and of 
 the water in which it is dissolved. 
 
 Thus, if we use very small electrodes of Cu in a solution of 
 CuSO 4 , and drive a large current through, we get H 2 set free 
 with Cu at the kathode, and O set free as well as Cu dissolved at 
 the anode. 
 
 Such is a case of what is called simultaneous decomposition. 
 
 7. Faraday's Laws of Electrolysis. The laws of electrolysis 
 were experimentally investigated, and enunciated, by Faraday. 
 We have somewhat forestalled them in what has preceded ; as it 
 was impossible to discuss the action in cells and the nature of 
 electrolysis in a purely historical order. 
 
 We have already (Chapter XL 8) stated that in the galvano- 
 meter we have a means of measuring current-strength, or quantity 
 of electricity passing per second. Hence we have a means of 
 investigating the relation between the amount of chemical action 
 per second, and the strength of the current. 
 
 Faraday found the following two laws to hold. 
 
 I. ' The amount of chemical action per second is directly pro- 
 portional to the current- strength. ' 
 
 The physical meaning of this law would seem to be twofold ; 
 ist, that the current does not pass through an electrolyte partly by 
 electrolysis and partly by ordinary conduction, but only by electro- 
 lysis ; znd, that an ion can only carry with it a fixed quantity of 
 -f , or of , electricity, so that additional current requires a pro- 
 portionally additional number of ions to carry it. The above law 
 involves also the statement ^ that a fixed current liberates per second 
 a fixed mass of each ion respectively ' (see 9). 
 
I9O ELECTRICITY CH xn, 
 
 II. ' If "the same or equal currents pasf through several electro- 
 lytic cells ', the weights of ions (which may be either atoms or mole- 
 cular groups) set free at the several electrodes will be in the propor- 
 tion of the chemical equivalents of these ions? 
 
 An example will make this clear, our modern system of 
 chemical notation rendering explanation an easy matter. Let the 
 same current pass in succession through four cells containing copper 
 sulphate, ammonium chloride, sodium sulphate solutions, and fused 
 silver chloride, respectively. Then, taking secondary actions into 
 account and considering the kathodes only, we get set free at the 
 successive kathodes (i.) copper, (ii.) ammonia and hydrogen (which 
 can be regarded as ammonium), (iii.) hydrogen, and (iv.) silver 
 respectively. Now these are, when in chemical equivalents, repre- 
 sented by Cu, 2NH 4 , H 2 , and Ag. 2 , respectively : giving us as the 
 numbers, to which the masses set free are proportional, (i.) 63 of 
 copper, (ii.) 36 of ammonium, (iii.) 2 of hydrogen, and (iv.) 216 of 
 silver. 
 
 The physical meaning of this law seems to be somewhat as 
 follows. Each equivalent atom or group such as Cu, H 2 , Ag 2 , 
 SO 4 , 2(NO 3 ), &c., carries with it the same unalterable quantity of 
 electricity as it migrates to that electrode at which it is ' set free.' 
 We have, in fact, a new significance attached to the expression 
 'chemical equivalent,' viz., that those groups or atoms which 
 are chemical equivalents have also the same power of carrying 
 electricity. 
 
 We may throw Law II. into the form . . . 
 When the same or equal currents pass through several cells, there 
 are the same number per second of equivalent atoms or groups set 
 jree against each electrode. 
 
 8. Further on Faraday's Laws of Electrolysis. 
 The case of simultaneous decompositions, In this case (see 6) 
 the form of the law last given is the more useful. 
 
 Now if we have P grammes of copper and P grammes of silver, 
 
 p 
 then the numbers of atoms in each are proportional to _ and 
 
 p 
 
 respectively. But, since copper is dyad and silver is monad, 
 106 
 
 the numbers of equivalent atoms or molecules are proportional to 
 
CH. xii. CHEMICAL PHENOMENA 19 1 
 
 P P ^ 
 
 . and _ respectively. If in successive cells we have set free at 
 63 216 
 
 the kathodes P grammes of copper, Q grammes of silver, and R 
 grammes of hydrogen, respectively, then the second form of the 
 
 P Q T? 
 
 law gives us the result that = -^- = - . 
 
 63 216 2 
 
 Now if in one cell we get simultaneous decompositions, it is 
 not hard to see what relation must hold. If, e.g., in the first cell 
 we get both copper deposited, and water also decomposed setting 
 free S grammes of hydrogen at the kathode, then we must have 
 
 S + P = Q = *. 
 
 2 63 2l6 2 
 
 But the proportion of S to P cannot be predicted ; it depends 
 upon the density of current and upon other conditions. 
 
 9. Electro chemical Equivalents. We shall in Chapter XIV. 
 2 explain in what units we measure current-strength. We shall 
 there find that the generally-employed unit of current is the 
 ampere ; this being a current in which a definite quantity of elec- 
 tricity called a coulomb passes across any section of the conducting 
 circuit in one second of time. 
 
 The quantity, one coulomb, of electricity performs a definite 
 amount of electrolysis in its passage through an electrolyte ; it 
 liberates a definite mass of each ion. If we express this mass Z 
 in grammes, the number is called the electro-chemical equivalent of 
 that ion. We usually, however, give Z in milligrammes. 
 
 From Faraday's laws it is clear that if we know this number 
 for H, we know it for all ions. Now, the chemical equivalent of 
 silver has been of late experimentally determined with great care. 
 And it seems certain that if we take i-u8 milligrammes of silver 
 to be set free by i ampere in i second, we shall be right to 
 within T V part in 100 parts. In these experiments the strength 
 of the current was of course measured independently of chemical 
 action. 
 
 10. Electro-plating. One case of electrolysis forms the 
 basis of an important industry. It is the case where the electro- 
 lyte is a solution of some metallic salt, the anode is a plate of the 
 same metal, and the kathode is a conducting surface that we 
 desire to coat with this metal. 
 
ELECTRICITY 
 
 Name of the body 
 
 Atomic 
 weight 
 
 Chemical 
 equivalent 
 
 Electro-chemical 
 equivalent Z, in 
 milligrammes per 
 ampere per second 
 
 Hydrogen -, . . . 
 
 I'O 
 
 I'O 
 
 0-0104 
 
 Potassium . . . . . 
 
 39'i 
 
 39'i 
 
 0-405 
 
 Sodium . ' . , . . 
 
 23-0 
 
 2 3 -0 
 
 0-238 
 
 Gold . ..-' ' . 
 
 196-6 
 
 6 5 -5 
 
 0-678 
 
 Silver , . ... 
 
 1 08-0 
 
 loS'O 
 
 1-118 
 
 Copper, ic-salts ,. 
 
 63-0 
 
 31'5 
 
 0-326 - 
 
 ,, ous-salts . . 
 
 63-0 
 
 63-0 
 
 0-652 
 
 Mercury, ic-salts . -. . 
 
 200'0 
 
 loo-o 
 
 1-035 
 
 ,, ous-salts . 
 
 2OO-O 
 
 2OO'O 
 
 2-070 
 
 Tin, ic-salts . ; ... 
 
 118-0 
 
 29^ 
 
 0-305 
 
 ,, ous-salts . " 
 
 118-0 
 
 59'0 
 
 0-611 
 
 Iron, ic-salts . . 
 
 56-0 
 
 18-3 
 
 0:189 
 
 ,, ous-salts . ' 
 Nickel . " . ' ' . - . ". 
 
 56-0 
 
 CQ-O 
 
 28-0 
 
 2Q-C 
 
 0-290 
 
 O'^O? 
 
 Zinc . . . . 
 
 jy w 
 6^"O 
 
 +>y 3 
 ?2*< 
 
 w J^J 
 O'^T.6 
 
 Lead '. " ' .' V 
 
 ^ j w 
 207-0 
 
 O J 
 
 103-5 
 
 oo 
 
 1-072 
 
 This table belongs to 9. 
 
 Sometimes it is desired to coat the body (as a baser metal) 
 with a permanent adherent layer (as of gold or silver) ; sometimes 
 with a non-adherent layer that, when detached, gives us a 're- 
 verse ' impression of the surface on which it was deposited. The 
 processes of electro -gilding and electro- silvering come under the 
 first head ; those of reproduction of engravings, casts of coins, c., 
 come under the second head. Sometimes also we employ the 
 process to etch designs on the anode. 
 
 In ii we shall see that under the above conditions, i.e. when 
 the solution remains unaltered in strength while there is a transfer- 
 ence of metal from the anode to the kathode, any single cell 
 however weak can effect the electrolysis (see also Chapter XV.). 
 
 As a general rule a single battery-cell with large electrodes 
 gives a compact film of metallic appearance, while a powerful 
 battery, with electrodes too small for the current, gives a spongy 
 non-metallic-looking coating. In a word, for good results the 
 current must not be too dense. 
 
CH. xn. CHEMICAL PHENOMENA 193 
 
 I. Copper-plating. In this case we wish to coat metals with 
 copper. In more technical books (such as Hospitaller's ' Formu- 
 laire Pratique de PElectricien ' and Sprague's * Electricity ') the 
 student will find recipes for copper solutions suitable for different 
 purposes, and directions as to current-density, &c., such as will 
 insure the best results. In the present Course we shall give little 
 but the principles of the different methods, and shall therefore 
 give but one recipe for a copper solution. ' A quantity of water is 
 mixed slowly with from one-tenth to one-twelfth its volume of strong 
 H 2 SO 4 , and the mixture when cold is saturated with CuSO 4 .' 
 
 As a battery we may use one Dani ell's cell of sufficiently large 
 size. This is connected up with two metal rods B and D as 
 
 shown, so that conducting bodies hung from B form the kathode, 
 while a copper plate hung from D forms the anode. All wires, &c., 
 which it is desired to protect from action must be coated with 
 wax, or. with caoutchouc, or other varnish. If the body is a non- 
 conductor it should be covered with some conducting substance, 
 and care must be taken, by encircling it with a copper wire or by 
 some other means, to make the action uniform over the surface. 
 The bodies to be coated may, more simply, be made to form the 
 negative plate in a Daniell's cell. In this case the substitution 
 of salt and water for battery acid, as the solution acting on the 
 zinc, will make the action slower and the copper-film more com- 
 pact. If the action is sufficiently slow, the deposit of a film of 
 f\ mm. should take from twelve to twenty-four hours (Hospitalier), 
 though a quicker rate of deposit may give good results. 
 
 II. Reproduction of engravings^ &c. If the object to be re- 
 produced be a delicate one, we may proceed as follows. We 
 
 o 
 
194 ELECTRICITY CH. xn. 
 
 will suppose, e.g., that it is required to get a facsimile of one side 
 of a rare coin. First, we take a wax cast (in reverse) of the coin. 
 This wax reverse is then coated with plumbago, surrounded with 
 a thin copper wire so as to distribute the current more evenly, 
 and made to form the kathode in an electrolytic cell. The copper 
 coating will, when removed, reproduce the surface of the coin in 
 question. Or we may make complete casts of bodies and obtain 
 electro- copper copies or reverses, according as the cast is a reverse 
 or not. 
 
 III. Electro- etching. By covering a copper plate with wax,, 
 tracing characters through the wax, and using the plate as an 
 anode, we may etch out designs to any required depth. It is, 
 however, not very easy to obtain any delicate results in this way. 
 
 IV. Obtaining designs in relief. If we use the plate, prepared 
 as in III., as the kathode in the cell, we shall obtain the same 
 designs in relief. 
 
 V. Silver-plating. For this purpose we must employ a silver 
 solution, and a silver plate as anode. One recipe for the silver 
 solution is as follows (Hospitalier). * Dissolve 150 grammes of 
 AgNO 3 in ten litres of distilled water ; add 250 grammes of pure 
 KCN ; stir until there is complete solution ; then filter.' 
 
 In silver-plating good results are more difficult to obtain than 
 in copper- plating, and more technical books than the present 
 Course should be consulted if the student intends more than mere 
 illustrative experiment. 
 
 ii. Polarisation of the Electrodes. In simple electrolysis,' 
 i.e. where no simultaneous decompositions (see 6) occur, there 
 are two cases to be considered. 
 
 I. Where there is no polarisation. In the cases of electro- 
 plating considered above, the condition of the plates and of the 
 solution remains constant. The fact that the kathode increases, and 
 the anode decreases, in thickness during the action obviously does 
 not affect the question of whether there will be a ' back tendency.' 
 Hence, there is no more tendency for the chemical changes to 
 take place in a reverse order after electrolysis has proceeded than 
 there was at first. There is, in fact, no chemical-potential-energy 
 stored up by the electrolytic action ; and no tendency, therefore, 
 for the electrolytic cell to drive a reverse current (see Chapter 
 XL 10 and Chapter XV. 8 and 9). 
 
CH. xii, CHEMICAL PHENOMENA 195 
 
 Note. There may be a very slight amount of polarisation owing to the fact 
 that the deposited metal is not of exactly the same quality as the anode 
 plate. 
 
 Experiment. We may pass a current through a Cu | CuSO 4 | Cu cell 
 for some time, and then prove the absence of polarisation by connecting the 
 terminals of the cell with a galvanometer, when no current will be observed. 
 Care must be taken that the original current be not so dense as to give simul- 
 taneous decomposition of water. 
 
 There being no back E.M.F,, we can perform such electrolyses 
 with any cell, however weak. (For further on this matter, see 
 Chapter XV. n.) 
 
 II. Where there is polarisation. But in all such cases as that 
 of the decomposition of water, i.e. cases where the condition of the 
 cell changes after electrolysis, we have a decomposition effected by 
 the current against chemical affinities. We have, set free at the 
 electrodes, ions that have a chemical tendency to reunite ; and this 
 recomposition can take place most readily by means of a reverse 
 current and a reverse chain of molecular interchanges. We have, 
 in such cases, done chemical work ; we have gained chemical- 
 potential-energy. The cell is now like a battery-cell ; and, like a 
 battery-cell, it will tend to drive a current. The cell will have 
 a certain ' electromotive force ' depending on the energy of the 
 chemical affinities of the ions ; and this E.M.F. must obviously be 
 opposed to that of the original current by which the electrolysis 
 was effected. (For E.M.F. see Chapter XI. u, note.) 
 
 Such cells are in fact battery-cells ; but, as a rule, they will 
 not act as such for a long time, as the supply of ions ready to re- 
 unite is limited. 
 
 Experiments. (i. ) Grovels gas-battery. The figure represents several 
 ' ceils ' for the decomposition of water. It will be noticed that the electrolysed 
 gases are collected separately, and that the platinum electrodes are so long 
 that their upper extremities are always immersed in the gas, however little of 
 this latter there may be. 
 
 First, a current is passed through each cell separately until a considerable 
 amount of gas has been set free in each ; this amount must be the same in all. 
 
 Then these cells are in reality battery-cells. They can be coupled up ' for 
 small internal resistance,' all the kathodes together and all the anodes together; 
 or ' for E.M.F'.,' anode to kathode and so on. 
 
 The hydrogen in the one tube and the oxygen in the other tube of each 
 tend to reunite ; and it is found that when the external circuit is completed 
 this reunion will take place step by step with the passage of a current, the 
 
 02 
 
196 ELECTRICITY CH. xn. 
 
 action being merely the molecular interchange, explained in 4, reversed in 
 direction. The current will be in the reverse direction to that by which the 
 electrolysis was in the first instance effected. 
 
 Here each cell answers to a Volta's cell in which the zinc 
 plate has been replaced by hydrogen in contact with platinum ; 
 the copper plate by oxygen in contact with platinum. 
 
 Notes. (i.) The recombination can occur only when the circuit is closed ; 
 and, further, only when there is a platinum plate that is in contact both with 
 the gas and the liquid. This action of the platinum will be familiar to those 
 who have studied elementary chemistry. 
 
 (ii.) There will be a current if we replace the platinum -oxygen plate by 
 plain platinum. This is due to the fact that there is always some oxygen dis- 
 solved in the dilute acid. 
 
 12. Secondary, or 'Storage,' Cells. Any arrangement by 
 means of which chemical-potential-energy may be used up in 
 such a manner as to give us the equivalent energy of an electric 
 current is called a ' battery-cell] or often more shortly 'a cell? 
 Where the cell is put together out of certain materials and then 
 does not require the passage of an electric current before it is fit 
 for use, it is called a ' primary -cell? All the cells described in 
 Chapter XL n, were of this class. 
 
 But where the cell requires the passage of a current before it 
 is ready for use, i.e. where the chemical-potential-energy is the 
 result of electrolysis, in such cases the term ^ secondary-ceir or 
 ' storage-cell] is employed. Such a cell, e.g., is the Grove's gas-cell. 
 The latter term was intended to imply that we * stored up ' elec- 
 
CH. xii. CHEMICAL PHENOMENA 197 
 
 tricity in the cell, since we could use a primary current to effect 
 the original electrolysis, and then could recover a current after a 
 greater or less interval of time. So understood, the term is a mis- 
 nomer. We use up some of the electrical energy of the primary 
 current, and get equivalent chemical- potential-energy stored up in 
 the secondary-cell ; later, when we complete the circuit of this 
 cell, we lose this chemical-potential-energy and get again the elec- 
 trical energy of a current. But we did not store up electricity ; a 
 condenser does this, not a secondary-cell. 
 
 Analogy. We might use up some wind-energy to turn a fan, and 
 this might wind up a weight from a lower to a higher level. We might, 
 later on, let the weight again descend, and so obtain a wind driven in 
 the reverse direction by the fan. W T e should have stored up mechanical- 
 potential-energy, not 'wind. 
 
 The problem of inventing a good secondary-cell is of a two- 
 fold nature. 
 
 (i.) How can we retain a large amount of the ions in contact 
 with the electrodes, ready to drive a reverse current ? 
 
 (ii.) How can we prevent action occurring while the cell is 
 lying idle with open circuit ? 
 
 In our next two sections we shall describe one form of cell 
 devised to satisfy these two conditions. 
 
 13. Plante's Secondary-Cell. In the original form of Planters 
 cell there are initially two lead plates immersed in dilute H 2 SO 4 . 
 
 I. Formation of the cell. A somewhat lengthy process is 
 needed in the first instance in order to get the cell ready to act 
 as a battery. 
 
 (i.) A current is passed through it ; this resulting in hydrogen 
 being given off at the kathode, while the surface of the anode is 
 oxidised into the condition of PbO 2 . 
 
 (ii.) The current is reversed, and is continued until the PbO 2 
 is all reduced to spongy lead, while the other plate is in its turn 
 per-oxidised. 
 
 (iii.) This process of sending currents in alternate directions is 
 repeated until the lead has been acted upon to some depth. 
 Thus, the plate that served last as anode is left coated deeply with 
 PbO 2 , that which served last as kathode is deeply coated with 
 spongy lead. This process is called ''Formation of the cell j and it 
 
198' 
 
 ELECTRICITY 
 
 CH. XII. 
 
 is said to be left charged' ; this is represented in fig, i. Subse- 
 quent chargings will need only a single passage of the current. 
 II. Discharging the cell. 
 
 In our discussion of the cell we will at present consider only the 
 degree of oxidisation of the lead, and will only touch on the part 
 played by the dilute acid. It is clear that if we have PbO, the form- 
 
 ation of PbSO 4 from this (PbO + H 2 SO 4 = PbSO 4 + H,O) can be re- 
 garded as a secondary action, and has no direct bearing on the 
 electrolytical theory of the cell. 
 
 If we connect the terminals of the cell, a current is set up in a 
 reverse direction to the current that was last passed through the 
 cell in its formation. One atom of oxygen from each molecule of 
 PbO 2 passes back to the other plate, giving PbO on each plate, 
 instead of PbO 2 on the one and Pb on the other. This action 
 from the one plate across the liquid to the other plate takes place 
 only step by step with the current ; and the current will continue 
 as long as there is PbO 2 left. When the cell has 'run down,' 
 each plate will be coated with PbO (or, by action with the dilute 
 acid, with Pb.SO 4 , which is equivalent to PbO as regards degree 
 of oxidisation). 
 
 III. For the cell to be recharged, a current must be again 
 passed through the cell. If we consider that there is PbO on both 
 sides, the action is simple ; by an electrolytic passage of H 2 to 
 
CH. XII. 
 
 CHEMICAL PHENOMENA 
 
 199 
 
 the kathode, and of O to the anode, we get again Pb on the 
 former and PbO 2 on the latter. 
 
 If there be Pb.SO 4 on both plates we may represent the 
 action as follows. [See fig. (iii.)] 
 
 For convenience we have written Pb.SO 4 +H 2 O (there is 
 plenty of available water in the dilute acid) in the two equivalent 
 forms H 2 SO 4 .PbO and OPb.H 2 SO 4 . 
 
 Lead Sulphate 
 Pb H^SO^ 
 Lead Free Acid 
 
 FIG. iii. 
 
 This indicates two main facts. 
 
 (i.) That the electrolytic action of ' charging ' the cells is the 
 same, whether there be PbO or Pb.SO 4 on the two plates of the 
 exhausted cell. 
 
 (ii.) That the recharging of the cell is practically a transfer- 
 ence of O from the kathode to the anode. 
 
 IV. Local action, and waste of energy, &c. It may naturally 
 occur to the reader that there seems no reason why the whole 
 action should not take place between the PbO 2 and the lead 
 plate upon which it rests, instead of between the PbO 2 and the 
 other lead plate that is separated from it by the dilute acid ; so 
 that after a short period of lying idle with open circuit, the cell 
 would have ' run down ' and be useless. 
 
 No doubt there is some such action at first. But it seems 
 that the insoluble layer of PbSO 4 , which is thus formed between 
 the PbO 2 and the lead plate on which it rests, hinders further 
 local action ; while the surface turned towards the other lead 
 plate, being a free surface, remains always more open and porous. 
 This appears to 'be a very important part played by the insoluble 
 PbSO,. 
 
 As regards the energy wasted in the charging of the cell, this 
 will be more fully discussed in Chapter XV. We will here only 
 
200 ELECTRICITY CH. xn. 
 
 remark that we waste more energy in heat when the back E.M.F. 
 of the secondary-cell is small as compared with that of the 
 primary battery, or when the charging current is large. 
 
 " It is of interest to remark that by charging cells arranged * in 
 parallel,' and then coupling them ' end-on,' we can obtain a second- 
 ary battery of as great an E.M.F. as we please ; one that could 
 drive a current back through the primary battery. 
 
 14. Fame's Accumulator. In order to obviate the necessity 
 for the lengthy and energy-wasting process of 'formation] Faure 
 devised the following important modifications in Plante's cell. 
 
 The two lead plates were coated with minium, this being 
 Pb 2 O 3 , or PbO.PbO 2 . One passage of the current then sufficed, 
 by the electrolytic setting free of H 2 at the one plate and of O at 
 the other, to convert the one layer into spongy lead, and the other 
 into lead peroxide. 
 
 Since, however, Pb 2 O 3 requires but O in order to per-oxidise it, 
 while it requires 3H 2 to reduce it, it is clear that we must have 
 three times the amount of minium on the anode (where the elec- 
 trolytic oxygen is set free) as on the kathode (where the hydrogen 
 is set free). Thus the electrolysis of 3H 2 O will reduce Pb 2 O 3 on 
 the kathode, and will per-oxidise 3(Pb 2 O 3 ) on the anode. 
 
 In all other respects the theory of the * FaurJs accumulator] 
 as it is called, is the same as that of the Plante's cell. The ques- 
 tion of energy will be discussed more fully later on. 
 
CH. XIII. 2O I 
 
 CHAPTER XIII. 
 OHM'S LAW. 
 
 i. General Ideas as to the Scope of Ohm's Law, Up to 
 the present point we have, in our treatment of the battery-cell, of 
 the current driven by it, and of the circuit through which the 
 current flows, used terms in a vague and qualitative, rather than 
 in an exact and quantitative sense. 
 
 But in the present chapter we propose to discuss at some length 
 the conditions which determine the magnitude of an electric cur- 
 rent in any particular case ; and to state and explain the law which 
 makes the calculation of the current a matter of simple arithmetic. 
 The law referred to is that known as Ohm's law ; it was enunciated 
 by Dr. G. S. Ohm in the year 1827. This law which must be 
 accepted as confirmed by countless direct and indirect experiments 
 and refuted by none states in the first place that when the circuit 
 of a cell is completed we are concerned with three quantities only; 
 and in the second place that these three quantities are connected 
 by a very simple relation, viz., that given in 2. 
 
 These three quantities are 
 
 I. Electromotive force. Each battery-cell possesses a certain 
 power of driving a current, which is directly proportional to, and 
 can be measured by, the A V that is found to exist between the 
 poles when the circuit is broken. Since that which moves matter 
 is called force, so by analogy that which moves ' electricity ' was 
 called l electromotive force? 
 
 The reader must remember that this term is an inexact one, as the 
 * electromoti 've force ' is not a force at all in the scientific sense defined 
 in Chapter IL 4. To prevent confusion we shall henceforth use the 
 letters E.M.F. instead; so that the actual word force will never be used 
 except in the strict sense of Chapter II. 4. 
 
2O2 ELECTRICITY 
 
 CH. XIII. 
 
 We may give an analogy from hydrostatics to make the relation 
 between E.M.F. and A V somewhat clearer. 
 
 Let us imagine two reservoirs, the levels of water in the two 
 being Li and L 2 respectively, connected by a pipe filled with coarse 
 sand. 
 
 A current of water will be urged through the pipe, and its 
 magnitude will depend, cczteris paribus, on the pressure due to the 
 difference of level. The pressure can be measured by the difference 
 of level (L, L 2 ) ; we might, indeed, if our system of units is suit- 
 ably fixed, say that the pressure ' is ' the difference of level ; this 
 would be scientifically inaccurate, but with our fixed system of 
 units would lead to no error in calculations. 
 
 So we can measure the E.M.F. of a cell by the A V that appears 
 at the poles when the circuit is broken ; or by the greatest A V 
 that we discover between two points in the circuit when the current 
 is flowing. And we may sometimes, somewhat inaccurately per- 
 haps, use A V as synonymous with E.M.F. 
 
 This E.M.F. of a cell depends solely upon the nature of the cell. 
 As we include the coatings of gases on the plates, &c., in the term 
 ' nature of the cell] the reader will see that we have taken into 
 account the phenomenon of ' Polarisation ' (see Chapter XL 10), 
 which diminishes the E.M.F. of the cell from its initial value. 
 
 However large a current be flowing, the E.M.F. remains 
 unaltered save by polarisation. 
 
 II. Resistance. Somewhat as, in the hydrostatic analogy given 
 above, the pipe will offer more or less resistance to the current of 
 water according to its length, its section, and the closeness with 
 which it is packed with gravel or sand, so a conductor in the elec- 
 tric circuit will offer more or less resistance according to its length, 
 section, material, and temperature. 
 
 Resistance is that which hinders the passage of a current and 
 limits its magnitude without producing any tendency to a back- 
 current ; it is purely passive. 
 
 III. Current. This term has been explained earlier. It 
 simply means quantity of electricity passing across any section of the 
 circuit in one second of time. We may measure this quantity as in 
 Chapter V. i ; but in dealing with electric currents another 
 system of units (explained in Chapters XIV. i, *and XVIII. 
 3 and 4) will be employed. 
 
CH. XIII. 
 
 OHM'S LAW 203 
 
 2. Exact Statement of Ohm's Law. Let AB be a con- 
 ductor including no source of difference of potential. And let there 
 
 be means of maintaining the points >^. >. 
 
 A and B at various A V . s t and of 
 measuring these A V.s by electro- 
 meter or other methods ; and also 
 of measuring by electrolysis or 
 galvanometer methods the current 
 that then flows in the conductor 
 AB. 
 
 Then the experimental law 
 known as ' Ohm's Law ' is that 
 
 * So long as the conductor A B remains unaltered in temperature 
 and in all other physical respects, the current flowing in it is 
 directly proportional to the difference of potential maintained be- 
 tween its extremities A and B? 
 
 We may in symbols express this by saying that 
 
 C ~ (VA-V B ). 
 
 On trying how the current varies when the conductor is 
 varied, the AV between the extremities being maintained con- 
 stant, our observations lead us to follow the hydrostatic analogy 
 and to say that the conductor * offers a greater or less resistance ' to 
 the current according to the dimensions, material, temperature, &c. 
 Symbolising this resistance by Rg, we define R A by the relation 
 
 p A 
 R B 
 
 It occurs to us to inquire next whether ' Ohm's Law ' applies 
 to the entire circuit, including both connecting wires and battery. 
 The difficulty now is to know what replaces the ' V A V B ' of the 
 experimental law given 1 above. Let E represent the electrostatic 
 A V between the terminals of the battery when the circuit is 
 broken. Let B and r represent respectively the resistances of 
 battery cell and of connecting wires (see 3 and 4, and Chapter 
 XIV.) Then a great mass of evidence has convinced investigators 
 that when the current is not dense enough to cause polarisation or 
 other alteration in the battery, we may state that 
 
 E 
 
204 ELECTRICITY CH. xm. 
 
 And if there are many cells in the circuit we have 
 c _ 
 
 where the E.M.F.s are to be reckoned as negative if they oppose 
 the resulting current. 
 
 Note. It is in fact believed that Ohm's law always holds, but that if the 
 current is too dense then E and B have some new values not known from our 
 previous measurements. 
 
 3. Resistance further Discussed, We will now discuss 
 further the nature of resistance, and the physical meaning of 
 Ohm's law. 
 
 Ohm's law connects the three quantities C, E, and R ; of which, 
 up to the present point, only the two quantities C and E have 
 received an exact meaning. 
 
 We might therefore be inclined to think that the law simply 
 
 p 
 
 defines R as such that 'C=~ ;' and therefore is no discovery at 
 
 s\. 
 
 all, but a mere truism. 
 
 If this were the case the law would be of little use to us, 
 since R might not be a fixed property of the conductor, one to be 
 found once for all, but might depend upon E, so that the law 
 would not then lend itself to the most important problem, viz., 
 that of determining the current from a knowledge of the E.M.F.s 
 and of the nature and dimensions of the conductor. 
 
 Thus it might have been the case that when once a current 
 was started with a certain E.M.F. E, the resistance of the conductor 
 was once for all broken down ; so that an E.M.F. of 2E with the 
 same circuit might give more than twice the current. 
 
 But direct experiment gives us two important results. 
 
 (i.) That R is a definite quantity for each conductor, and cap- 
 able of measurement as such. Hence we can predict the current 
 which any given E.M.F. will drive through any given conductor. 
 
 (ii.) That R depends in a very simple way on the length, cross 
 section, temperature, and material of the conductor. 
 
 4. The Exact Conditions on which Resistance Depends, If 
 the resistance of such bodies as wires be examined experimentally, 
 it is found that R is directly proportional to the length / of the 
 
CH. xui. OHM'S LAW 205 
 
 wire, inversely proportional to its sectional area A, and depends 
 also on its ' specific resistance ' c, and on its temperature. 
 
 We will define later the exact meaning of * specific resistance,' 
 and we will for the present omit the factor expressing the depend- 
 ence of the resistance on temperature. Thus we may state that 
 
 R-*.i/. 
 
 A 
 
 Here k is a constant depending on the unit of resistance employed. 
 The nature of the experimental methods of proving the above 
 is indicated in what follows. 
 
 Experiments. (i. ) External resistance. If we \vish to examine most 
 simply the law of resistance r of such external conductors as wires, we must 
 get rid of the internal resistance B of the battery (see 2, equation (ii. )). 
 
 We can practically make B = o by employing, instead of an ordinary 
 battery, a thermo-celL This can be constructed of metallic bars of such thick- 
 ness as to make their resistance relatively negligible ; and a constant E. M.F. 
 can be insured by keeping the two sets of junctions at two fixed temperatures 
 (^Chapter XVI. ). 
 
 With such a battery the formula of Ohm's law becomes C = 
 
 We can measure C by means of a galvanometer, as will be explained fully in 
 Chapter XVII. Then, by varying the length, material, and cross section of 
 
 the wire, we are led to the result that r = k 
 
 2\. 
 
 It is to be noticed that this result proves that the current flows 
 through the whole body of a wire equally, and not along its sur- 
 face. Provided that the temperature of the wire is constant, 
 it makes no difference whether a cross section of (e.g.) i sq. 
 cm. be circular, square, or of the form of a flat rectangle. 
 Here is a notable difference between conductors as used for 
 electrostatic purposes, and as used for conveying a current, re- 
 spectively. 
 
 (ii. ) Internal resistance of a battery. The internal resistance of a battery 
 is a far more difficult matter to investigate. Indeed the conditions are so 
 complex, that it is invariably the rule \ofind experimentally the resistance of 
 a cell or battery, instead of to calculate it as we do in the case of a wire 
 from a knowledge of the materials and their dimensions. 
 
 But still it is possible to show that in all probability the same general laws 
 
 hold ; or that B = k . . 
 
 A. 
 
 To investigate internal resistance we get rid of the external resistance r by 
 employing very thick copper conductors, of no appreciable resistance, outside 
 
206 ELECTRICITY CH. xnf. 
 
 the cell. The cell itself is constructed with plates of various sizes (so as to vary 
 A), whose distance apart can be altered (so as to vary /). The same general 
 results are found to hold ; though, since we cannot confine the current to that 
 portion of the liquid which lies directly between the opposed plates, we cannot 
 arrive at very exact results. 
 
 It is important to note that 
 
 (i.) n cells in series offer to the current n times the resistance 
 of one cell, since, in the former, the length / of liquid traversed is 
 n times that in the latter. 
 
 (ii.) If n cells be coupled zincs to zincs and coppers to coppers, 
 or ' in parallel,' we practically make one large cell of plates n times 
 the area of those of one cell. Hence the area A of the column of 
 
 liquid traversed is increased n-fo\d ; and the resistance is -th of 
 that of one cell, and of that offered by n cells in series. 
 
 5. Conductivity. Since C = E._*, we may call -- by the 
 
 _K Ix 
 
 name 'conductivity,' and say that the current is directly propor- 
 tional to the conductivity, instead of inversely proportional to the 
 resistance, of the circuit. 
 
 In 9, fig. iii., we represent two points A and B in a circuit 
 joined by several wires of resistances r lt r 2 , r 3 , r 4 . . . respectively. 
 
 We may call their conductivities c^ c^ c^ and <r 4 . . . re- 
 
 spectively ; where c^= -, &c. 
 
 Now it is possible to replace these wires by one wire of resist- 
 ance R', such that the current is unaltered. When this is the 
 
 case we call R' the equivalent resistance ; and if c = , then c' 
 
 will be the equivalent conductivity. 
 
 Now it hardly seems to require proof that 
 
 If we admit this, then it follows that 
 
 which enables us to determine that resistance which is equivalent 
 to, and might replace, the many branches. 
 
CH. XIII. 
 
 OHM'S LAW 207 
 
 Note. In case the above statement seem rather ' likely ' than convincing, 
 we here give a proof. 
 
 Let V A be the potential of A, V B that of B ; so that V A V B , which we 
 may write as E^ is the E.M.F. between A and B. 
 
 Let C be the total current passing, and let C,, C 2 , C 3 , &c., be the portions 
 of C that traverse r } , r v r 3 , &c., respectively. 
 
 Then by Ohm's law we have 
 
 E A E A 
 
 C* ^*i r ^_ B . o * 
 
 Whence C, r, - C 2 r^ = C 3 r 3 = &c. *= E, 
 C C C A 
 
 or r=r= ^=&c. = E B . 
 
 r, r^ r a 
 By ordinary algebra, each of these terms 
 
 r + Co + C, + &c. 
 
 But Cj + C 2 + C 3 + &c. = C, since the latter is the total current. 
 Hence E B = . 
 
 But if R' be the equivalent resistance, then C = 5. ; or E B = -. 
 
 R i 
 
 R> 
 
 Whence - = _L + + _i + & c . ; 
 R' r, r^ r 3 
 
 or the (equivalent conductivity^ (sum of the several conductivities]. 
 
 6. Application of Ohm's Law in a Simple Case. Suppose 
 that we have a battery of n cells, and wish to know the best way 
 of coupling them up in the two extreme cases of 
 
 (i.) Practically zero internal resistance. 
 
 (ii.) Practically zero external resistance. 
 
 Let B be the resistance of each cell, r the external resistance, 
 E the E.M.F. of each cell, and C the current. (Read again 4, 
 end.) 
 
 (I.) Let B, and even n B> be negligible as compared with r. 
 
 (i.) The current from a single cell will be 
 
 C = B ~- =? (approx.). 
 
2O8 ELECTRICITY 
 
 CH. XIII. 
 
 (ii.) The current from the n cells in parallel (see 4) will be . 
 
 p -p 
 
 Ci = -g -- = - = C (approx.). 
 
 + r 
 n 
 
 (iii.) The current from the n cells in series (see 4) will be 
 ^ n E n E ^ 
 
 2 ^ ^B~+> = ~V '' = H 
 
 Hence we see that with arrangement (ii.) we have no advantage 
 over a single cell ; while with arrangement (iii.) we get n times the 
 current that we get with a single cell. 
 
 (II.) Let r be negligible with respect to J3, and even with respect 
 
 (i.) With a single cell ...,,, 
 Co=. = (approx.). 
 
 (ii.) With n cells in parallel 
 E 
 
 (iii.) With n cells in series ... 
 
 17 *? "C* "FT 
 
 The reader should compare this result with that in (I.), and 
 should try to understand the physical meaning of these results ; it 
 is not enough to follow merely the algebraic reasoning. 
 
 Experiments. The figure represents diagrammatically three Leclanche cells 
 whose + poles are connected with three mercury cups, +(i), +(2), and +(3), 
 respectively, the poles being connected with the mercury cups (i), (2), 
 and (3), respectively. By means of thick copper pieces we can connect these 
 cells either in parallel or in series. 
 
 By means of other mercury cups, and of other thick copper pieces, we can 
 throw into the circuit of one or more cells the resistance-box r (see Chapter 
 XIV.), and the galvanometer G. This latter may either be a common tangent 
 galvanometer of no appreciable resistance, suitable for measurements of large 
 currents ; or may be a more delicate instrument by means of which much 
 smaller currents may be measured (see Chapter XVII.). 
 
 The mercury cups and thick copper pieces give us connections of practically 
 zero resistance. 
 
CH. XIIT. 
 
 OHM'S LAW 
 
 2O9 
 
 (i. ) Let us put into the circuit a very large external resistance r of (say) 
 l,ooo ohms ; and let us also put in our more sensitive galvanometer, as the 
 current will be small. 
 
 We can then, by coupling up our three cells in the different ways indicated 
 in (I.) above, verify the results there deduced from Ohm's law. 
 
 (ii. ) Next we may remove r, and have in the circuit only a tangent gal- 
 vanometer of practically no resistance. 
 
 The results of (II.) above may thus be verified. 
 
 7. Graphic Representation of Ohm's Law. By a graphic 
 method, familiar to those who are acquainted with the elements 
 of co-ordinate geometry, and capable of being understood also by 
 those of less mathematical knowledge, we may represent to the 
 eye in one view (i.) the difference of potential between two points 
 A and B in a circuit ; (ii.) the 
 resistance between A and B ; 
 (iii.) the current which flows 
 through this resistance on 
 account of this difference of 
 potential. 
 
 Let distances measured 
 along the line O r represent 
 by their length the magni- 
 tude of the resistances. If 
 the wire be uniform, then 
 
 equal lengths of wire will be represented by equal lengths along 
 O r. If the conductor be not uniform, this will not be the case. 
 A metre of a very thick good conductor may be practically a 
 
 p 
 
210 ELECTRICITY CH. xni. 
 
 point on our resistance-line O r, while a millimetre of bad con- 
 ductor may be represented by a considerable length on O r. 
 
 Let distances along O v, drawn at right angles to O r, repre- 
 sent potentials. If the reader will bear in mind that we are con- 
 cerned only with differences of potential, he will see that it does not 
 matter what potential we choose as our zero-potential, i.e. as the 
 potential of our starting-point or ' origin ' O. We can, in fact, take 
 the point O to represent any fixed point in the circuit that we 
 choose, and can measure off the potentials of other points as so 
 much above or below the potential of this point taken as our 
 arbitrary zero ; this will make no difference in the magnitude of 
 the line representing the A V of two points in the circuit. 
 
 Thus if A and B represent two points in the circuit, and if 
 lines A P and B Q be drawn perpendicular to O r, and if B M be 
 drawn parallel to O r, then the line A P represents by its magni- 
 tude the potential of the point A above our arbitrary zero, the line 
 B Q that of the point B, and AM the A V between A and B ; 
 while O P and O Q represent the resistances between our starting- 
 point and the points A and B respectively, and P Q represents the 
 resistance between A and B. 
 
 T? A 
 
 Now Ohm's law gives us that C = %-* ( see 2 ("*) ) '> wnence 
 
 J^B 
 
 C = ~T, if we have drawn our figure to proper scale ; or 
 
 C = k . A ^, where k is some constant, if we have taken our scale 
 
 AM 
 
 of drawing at random. Let us suppose that C = 
 
 But if < be the angle that the line A B makes with the axis 
 O r^ we have = tan < ; whence we have 
 
 C = tan <. 
 
 Thus the ' slope ' of the line A B, as measured by tan <, in- 
 dicates to us the current strength. Of course, since the current C 
 is the same throughout the circuit, the line A B has the same slope 
 whatever points in this circuit A and B may be. If we take in the 
 whole circuit we have 
 
 ordinate representing total E.M.F. in circuit 
 tan <j> = : -- = r . .- 
 
 abscissa representing total resistance in circuit 
 
CH. XIII. 
 
 OHM'S LAW 211 
 
 This last result, if known, obviously enables us to find the position 
 of the point A in the diagram when we know that of B, and when 
 we know also the resistance Q P between B and A ; just as we can 
 algebraically find the potential of A when we know the potential 
 of B, the resistance between A and B, and the current flowing in 
 the circuit (see n). 
 
 8. Applications of the Graphic Method. 
 
 We can thus represent to the eye each result arrived at algebraic- 
 ally from Ohm's law. We will indicate in a few cases how this may 
 be done. 
 
 (1) Referring to 6 (I.). To represent case (i.) the reader should 
 mark off from O v a distance O A representing the E.M.F. of one cell \ 
 and from O r a distance O a representing the external resistance r, 
 B being negligible. The current is then represented by tan ty^ where 
 < is the angle which A a makes with O r. 
 
 To represent case (ii.) we do not (appreciably) alter the resistance 
 O #, and the E.M.F. O A is unaltered. We have, therefore, the same 
 line A a making an angle c^ = < ; so that the current is unaltered. 
 
 In case (iii.) the resistance O a is practically unaltered, but we mark 
 off on O v a distance O N = n x O A, and we join N a. This line N a 
 makes with O r an angle $ 2 such that tan < 2 = ;z . tan </> , or the current 
 is n times greater than in case (i.). 
 
 (2) The reader can easily represent the three cases of (II.). 
 
 (3) It will be a useful exercise for the reader to draw the rises and 
 falls of potential through a circuit in which there are abrupt changes 
 at different points (due, let us suppose, to contacts of different metals 
 at those points), as well as a general fall following Ohm's law. The 
 method will be as follows. From a knowledge of the total E.M.F. and 
 the total resistance in the circuit we can find C, or can find tan < ; 
 thus we know the slope of the line between points separated by 
 homogeneous conductors. The abrupt changes of potential must 
 be given. It is easy then to trace the slope of, and abrupt rises 
 and falls of, potential from any point in the circuit round to that point 
 again. 
 
 (4) Note. For ordinary purposes it is sufficient to mark off from O v a 
 length representing the total algebraic sum of the E.M.F.s in the circuit, as 
 if this E.M.F. were an abrupt rise of potential occurring at one point in the 
 circuit and nowhere else. The slope (i.e. tan <>) ofHhe line is not affected 
 by this convenient assumption. 
 
 9. Divided Circuits, We will now discuss more fully than in 
 5 the case of a divided circuit. The notation employed is 
 
 UNJVER 
 
 p 2 
 
 
212 
 
 ELECTRICITY 
 
 CH. XIIT. 
 
 sufficiently explained by the diagrams and by what has preceded 
 in 5. (Read 5 again.) We will take several cases. 
 
 I. Two equal branches. Here we have for the equivalent 
 resistance .... ............. . . 
 
 Hence, if R be the resistance in the rest of the circuit and in 
 the battery together, and if E be the E.M.F. 
 of the battery, we have. ..... . .. 
 
 E 
 
 FIG i. 
 
 Total current C = 
 
 R' + R 
 
 And since E B = C } r C 2 r, it follows 
 that C l = C 2 = ; as is otherwise evi- 
 dent. 
 
 II. N equal branches. Here in a 
 similar manner we can show that . . .. 
 
 r 
 
 = 
 
 C, = Co = C, = &C. = 
 
 III. Two unequal branches. Here we have 
 
 or 
 
 Also, C = 
 
 And since E B = C^ = C 2 r 2 = CR', we 
 get the results that .......... 
 
OHMS LAW 
 
 213 
 
 r - ' 2 c 
 
 i - --- . V^ = -- . V^ 
 
 = . C = L_. . C 
 
 . . . (iv.) 
 
 IV. N unequal branches. In this most general case we have 
 ^-L + ^-K .&c. + l .) 
 
 or, 
 
 R' = 
 
 ^ + fj ^3 . . . r n + &C. 
 
 And by a similar reasoning to above we 
 find that . . . 
 
 R' + R 
 
 R' , 
 
 R/ 
 r 
 
 \^xO \^ 
 
 r> 2 
 
 R/ 
 
 Example in divided circuits. 1 Two wires, A D B and A E C B, of uniform 
 section, &c. , and of equal resistance, connect the points A and B. A third wire, 
 AFC, of equal resistance, connects the point A with the middle point C of 
 one wire. When a current flows 
 
 from A to B, find what fraction of D 
 
 it passes through the wire AFC.' 
 
 In the figure we represent the l""^- ^_ 1 
 
 arrangement intended, and we 
 have called the resistances of 
 ADB and AFC each I, while 
 those of AEC and of C B are 
 
 each _. We will first find the equivalent resistance of the paths from A 
 
 through C to B, and so find the fraction of the current that passes through 
 ADB; and next we will find what fraction of the remainder takes the route 
 
 AFC. Now the resistance of A D B = I, or -. And the equivalent resistance 
 
 FIG. iv. 
 
214 ELECTRICITY CH. xm. 
 
 the 
 
 R' between A and C is given by -L = - + I = 3 ; whence R' = . Hence 
 
 total resistance of the route by C is - + = >. Therefore the resistances by the 
 
 326 
 
 routes D and C are in the ratio of 6 : 5 respectively. Therefore Ji of the 
 current passes by D, and passes from A through C. 
 
 Again, this will be distributed between the branches A E C and A C 
 in the inverse ratio of their resistances ; that is, in the direct ratio of 2 : i. 
 Hence I of will pass by the route AFC; or the answer required is - 2 . 
 
 10. 'Shunts.' The case of two unequal branches has a very 
 important application. It is often necessary (i.) to protect a sen- 
 sitive galvanometer against a strong current by allowing only a part 
 of the current to pass through the galvanometer, and yet (ii.) to 
 measure the current. This is done by connecting the terminals 
 of the galvanometer by a wire of greater or less resistance, and 
 thus leaving a greater or smaller fraction of the current to pass 
 through the galvanometer. If this fraction be known, and its mag- 
 nitude be measured, we can easily calculate the total current. We 
 shall see, however, that the introduction of this 'short cut,' or 
 'shunt] has the effect of increasing the total current by decreasing 
 the total resistance ; and hence the fraction measured is a certain 
 fraction of the new total current, not of the original current. Let 
 R be the resistance of the battery and rest of the circuit combined; 
 let G be the resistance of the galvanometer which here takes the 
 place of r\ in 9, case (III.) ; let S be the resistance of the shunt 
 which here takes the place of r. 2 ; and let R' be. the equivalent re- 
 sistance of G and S. This R' is of course less than either G or S. 
 Let C be the total current ; C G the current passing through the 
 galvanometer ; and C s the current through the shunt. Let C be 
 the original current passing before the shunt was used. Then we 
 have ................. . . . . 
 
 ' _E . r E i i i 
 
 GTS J R' + R ' R' G "*" S " ' 
 
 We will now proceed to consider several points. 
 
CH. xni. OHM'S LAW 215 
 
 (i.) If we wish to allow -1th, th, ? th, &c., of the total 
 10 100 1000 
 
 current to pass through the galvanometer, then - must be - 
 
 G + b 10 
 
 (~* C* (~* 
 
 - 1 , or - 1 respectively ; or S must be , , , &c., re- 
 100 1000 9 99 999 
 
 spectively. 
 
 In general, if S = . G, then will -thof C pass through 
 
 n i n 
 
 the galvanometer. 
 
 (ii.) If G be negligible as compared with R, then will C = C , 
 whatever shunt we use. This follows from the above formulae, 
 since we there may neglect G, and therefore a fortiori may neglect 
 R 7 , which is less than G. 
 
 (iii.) If G, and -- G, be so great that R may be neglected, then we 
 findC = /zC ; and as a consequence we should have passing 
 
 through the galvanometer a current = -lc = . TZ C = C : thus 
 
 n n 
 
 exposing the galvanometer to a current as strong as the whole 
 original current. 
 
 By having in the circuit an adjustable resistance, we can intro- 
 duce a compensating resistance r, and so maintain C equal to C . 
 
 Note. To keep C = C , we must have the total resistance of circuit the same ; 
 or the compensating resistance r must be such that G + R = -^ 5- + R + r ; 
 
 or r = G ~ - = -. Thus if S = G, so as togive C G = - . C, we have 
 
 G + SG + S 11 i n 
 
 n\ 
 
 G. In this case C = C ; and C G = -- C , as was desired. 
 n n 
 
 ii. Fall of Potential through the Circuit. The following 
 view of a circuit, a view indicated by the figure to Chapter XI. 7, 
 is convenient, and, though not accurate in detail, it can for general 
 purposes be substituted for a more accurate view, and can lead 
 to no error in such considerations as follow. 
 
 In the figure referred to, it is supposed that in the cell the 
 E.M.F. occurs as an abrupt rise in potential ; in fact, that the 
 energy of the cell is spent in, as it were, pumping the electricity 
 straight up from the lower to the higher level. Thus, the line C Z 
 
2l6 ELECTRICITY CH. xm. 
 
 represents, in the graphic method, the total E.M.F. E of the 
 circuit. The potential falls round the circuit from the higher to the 
 lower level again ; the total fall being E. 
 
 Ohm's law, as expressed by the formulae given in 2, means 
 that this fall proceeds proportionally to the resistance. Hence, if R' 
 be the resistance of any portion of the circuit, and if R be the total 
 resistance of the whole circuit, then the fall of potential between 
 
 the beginning and end of R' will be Tr-E. We deduce the 
 
 following. 
 
 (a) In the case of negligible external resistance the whole fall 
 of potential will (approx.) take place in the cell itself. We find 
 (approx.) no AV between the beginning and the end of the external 
 circuit ; or the poles of the battery will have (approx.) the same 
 potential. 
 
 (b) In the case of negligible internal resistance the whole fall 
 of potential will (approx.) take place externally to the cell. We 
 shall find (approx.) between the beginning and the end of the 
 external circuit a AV equal to the whole E.M.F. of the battery ; 
 or the poles, of the battery will have (approx.) the same AV as 
 if the circuit were broken. 
 
 (c) Where internal resistance equals external, we shall have the 
 fall of potential equally divided. The poles of the battery will 
 
 exhibit a AV equal to - 1 - E. 
 
 (d) Where there is a multiple arc between two points A and 
 B, and we wish to find the AV Eg between A and B, we have 
 merely to calculate the equivalent resistance R'. We then have 
 
 E A = R; K 
 
 Total resistance ' 
 
 Note. All the above cases are readily exhibited graphically (see 7). 
 
 12. Kirchhoff's Two Laws. By a careful application of 
 Ohm's law, and of the principle that when the V of a point is 
 constant there must be as much electricity flowing away from it as 
 flows to it in each second, it is possible to investigate the distribu- 
 tion of current and potential in very complicated cases; cases 
 where we have any number of cells connected by a net-work of 
 conductors in any way whatever. 
 
CH. xin. OHM'S LAW 217 
 
 Kirchhoff has enunciated in the form of two ' Laws ' the 
 principles that must guide us in such an investigation. 
 
 Law I. If any number of conductors meet at a point, and if 
 all currents flowing to the point be considered + , and ail currents 
 flowing from the point be considered , and if the condition of 
 things be steady, or the potential at the point be not altering, then 
 the algebraic sum of the currents meeting at the point must be zero. 
 
 Or 
 
 2 . C = o. 
 
 Law IL Let us suppose there to be such a net- work of con- 
 ductors as that imagined above ; there being cells of various 
 E.M.F.S, and turned various ways, in this net- work. 
 
 If we imagine ourselves to start from any point in this net- work 
 and to make a circuit through the conductors back to our starting- 
 point, we shall have passed through conductors of various resist- 
 ances, shall have passed through various cells whose E.M.F.s are 
 directed so as to drive a current against or with us. and shall have 
 found various currents, some with us and some against us. 
 
 As long as we are passing along a single conductor r }t to 
 which there are no outlets, the current has some fixed value Cj ; 
 but on passing a point where two or more conductors meet we 
 may find a different current C 2 , which will remain constant over 
 the next piece of resistance r< up to the next place where two or 
 more branches meet. We can thus divide our circuit into por- 
 tions of resistances r lt r 2 , r 3 , &c., respectively ; along each of 
 which will be a current C )5 C 2 , C 3 , &c., respectively. If we call 
 these currents + or , according as they flow with us or against us 
 respectively ; and if we call the E.M.F.s that we encounter + or 
 according as they tend to drive a current with us or against us 
 respectively ; then it can be shown, by successive applications of 
 Ohm's law to these different portions of our circuit, that in the 
 complete circuit we must have 
 
 Ci r \ + C 2 r 2 + C 3 r 3 + . . . = e + e' + e" + . . . 
 where *, e', &c., are the various E.M.F.s that we pass ; attention 
 being paid to signs both of the C . s and of the e . s. 
 
 These E.M.F.s e, e', e", may occur in any way in the circuit, 
 and do not necessarily (or indeed usually) occur in the pieces of 
 circuit r^ r. ta &c., respectively. 
 
218 
 
 ELECTRICITY 
 
 CH. XIII. 
 
 This law is usually expressed thus. 
 
 ' In any complete circuit, . C R = 2 . e. 1 
 
 Proof of Kirchhoffs second law. We will here indicate the manner 
 in which Kirchhoff's second law may be proved. The figure repre- 
 sents part of a net-work of conductors, in which are introduced any 
 
 number of battery-cells, e^ e.,, e y &c. 
 Between any two consecutive junctions, 
 as A and B, the current has some con- 
 stant value CAB '> and between B and 
 the next junction C there will be a con- 
 stant current C BC , which will in general 
 be different from C AB . 
 
 The total resistance of conductor and 
 cells between A and B is represented 
 by r AB . 
 
 The symbols <?,, e.^ c., represent the 
 numerical values of the various E.M.F.s, 
 and we will suppose the signs of these E.M.F.s to be also included 
 in these symbols. 
 
 The potentials of the points A, B, C, c., will be represented by 
 V A , V B , &c. Then, by successive applications of Ohm's law, we have 
 
 CAB 
 C BC 
 CCD 
 C DE 
 CEF 
 CFG 
 CGH 
 CHA 
 
 r FG 
 
 = V A - V B + e r 
 
 = V B - V c . 
 
 = V c - V D + e. 
 
 = V D - V E . 
 
 = V E - V F + e z 
 
 = V F - V G . 
 
 = VG - V H + e 5 
 
 = V H - VA + e s 
 
 Whence, by addition, we have 
 
 S.CR-2.*. 
 
 These two laws may be applied in any complicated system of 
 E.M.F.s and of conductors, in such a way as to determine the currents 
 in all the branches, giving us always n independent equations to deter- 
 mine n unknowns. 
 
 As a simple illustration we may take the case of fig. iii. 9, where 
 there is but one E.M.F. E. Let us suppose that E, R, r^ r 2 , c., are 
 known, and that it is required to find C, C,, C 2 , c. 
 
 Applying Law I. to the point A (or B) we have 
 
CH. xin. OHM'S LAW 219 
 
 Applying Law II. successively to the circuit^. A r l B R, A r : B r. z A, 
 A r l B r 3 A, A r t B r 4 A, we get the equations ......... 
 
 C R + C l r 1 ="E ....... (ii.) 
 
 = O ....... (iv.) 
 
 C r r,-C 4 r 4 .0 ....... (v.) 
 
 These are five independent equations, and will enable us to deter- 
 mine the five unknown currents. 
 
 However, the method employed in 9 was in this case simpler to 
 pursue. 
 
 13. Maximum Current with a given Battery. If we have 
 a battery of n cells, each of E.M.F. E, and of internal resistance 
 B, it is of great importance to arrange them so as to drive the 
 largest possible current through a given external resistance r. 
 
 If n has factors m and /, we may couple the cells so as to 
 have ;;/ in series and / parallel. We thus get a battery of 
 
 E.M.F. = m E, and of internal resistance = - . The current 
 
 obtained will be 
 
 m E 
 
 If we investigate either algebraically, or by means of the 
 calculus, the arrangement that will give a maximum value of C, 
 we find that this occurs when ............. 
 
 ,f B-r, 
 
 Or, we obtain the maximum current tenth a given battery when it is 
 so arranged as to make the internal resistance as nearly as possible 
 equal to the external resistance. (For proof, see note.) 
 
 It is therefore convenient to have a battery in which the num- 
 ber of cells is a number having several pairs of factors. 
 
 Example. We will here assume current, E.M.F., and resistance to be 
 expressed in the units, amperes, volts, and ohms respectively ; these are units 
 which will be spoken of later. 
 
 Let us have n = 24, E = 2 volts, B = 2 ohms, and r 3 ohms. 
 
 Then we have the following arrangements possible. 
 
 (i. ) 24 cells end-on. 
 
 TJ r- 2A v 2 48 l6 .. 
 
 Here C, = " _B ? = _ amperes. 
 24x2 + 3 51 17 
 
220 ELECTRICITY CH. xm. 
 
 (ii. ) 12 ceils end-on, 2 parallel. 
 
 2 _ 24 _ 8 
 
 - _ _ -^amperes. 
 
 Here C 2 = _. I2x2 
 
 2 
 
 (iii.) 8 cells end-on, 3 parallel. 
 
 HereC 3 = A*. 2 -=' 
 
 3 X2 + 3 i 
 
 (iv.) 6 //y end-on, 4 parallel. 
 
 Here C, = 7 ^- 2 = I? - 2 amperes. 
 
 p* 1 
 
 (v.) 4 *//.$ end-on, 6 parallel. 
 
 TT /- 4^2 8 24 ,v 
 
 Here C 5 = - - - - amperes. 
 ~ x 2 + 1 T 3 J 3 
 6 ' '3 
 
 It is needless to continue, for it is clear that (iv. ) gives us the greatest 
 current. And case (iv. ) is that in which the internal resistance of the battery 
 is 3 ohms, or equals the external resistance. Of course we cannot in general 
 so subdivide our battery as to make the equality exact. 
 
 In such extreme cases as those of 6 we approximate most 
 nearly when the internal resistance is made as great as possible in 
 case (I.), and as small as possible in case (II.). 
 
 Note. Proof of the above rule. Let the number of cells be n, and let 
 n ml. We have C = . This may be written also in the form. 
 
 m- 1- E- n*-E- n- E 3 
 
 + Iry (;//B- /?)+ 4ml rB ~ (mW^Tr}- + 4 n rlt 
 
 Hence C is greatest when the denominator is least, since we cannot altej: 
 the value of the numerator by varying the arrangement. 
 
 This will be the case when the term (m B lr}-, which must be positive, 
 
 disappears ; and this again will be the case when ;// B = Ir, or when r = ~?. 
 
 But r is the external, and - is the internal, resistance ; hence our theorem 
 is proved. 
 
CH. xiv. 221 
 
 CHAPTER XIV. 
 
 MEASUREMENT OF RESISTANCES AND OF E.M.F.3. 
 
 i. Preliminary, on the Units Employed. In Chapter V. we 
 explained the unit of quantity of electricity that is employed in 
 electrostatics ; and in Chapter X. we explained the unit employed 
 in measuring differences of potential. Both these were based 
 upon the absolute C.G.S. system of units, and started from the 
 electrostatic repulsion between two very small charged spheres. 
 We might build upon these two units a further system of derived 
 units for current or quantity per second, E.M.F., resistance, &c. But 
 sucii a system would only have a theoretical interest, and would 
 not be the most convenient system to employ in dealing with 
 currents and with the phenomena accompanying currents ; for in 
 considering these phenomena we are rarely concerned with electro- 
 static actions. 
 
 The most important class of phenomena, accompanying elec- 
 tric currents, are the magnetic actions of a current. Hence, in 
 ' current electricity ' we employ a system of units, based on the 
 C.G.S. system, and starting from the action of a current on a 
 magnetic pole. 
 
 This system will be fully explained in Chapter XVII. Here we 
 will merely remark that the absolute units of E.M.F. and of R in 
 this system are so small that they are inconvenient for practical 
 purposes. We therefore use generally 
 
 (i.) As .unit of E.M.F., the volt. This is almost exactly the 
 E.M.F. of a cell consisting of a Cu and a Zn plate immersed in a 
 solution of ZnSO 4 . This unit is 100,000,000 (or lo 8 ) times the 
 absolute unit of E.M.F. In somewhat the same way we use a 
 kilometre, instead of the centimetre, in large measurements of 
 length. . 
 
 (ii.) As unit of resistance, the" ohm. This i is the- resistance of 
 
222 ELECTRICITY CH. xiv. 
 
 a column of mercury at o C. of i sq. mm. section and nearly 
 105 cm. in length. It is 1,000,000,000 (or io 9 ) times the absolute 
 unit. The ohm is usually designated by the symbol w. 
 
 Note. The ohm here given (called the 'B.A' ohm) is now found to be a 
 little too small ; i.e. it is not quite io 9 times the absolute unit. 
 
 (iii.) As unit of current, the ampere. This is the current given 
 by i volt E.M.F. when the total resistance is i ohm. From Ohm's 
 
 law it is clear that the ampere must be I0 9 , or y Lth, the absolute 
 
 unit of current. We have then, generally, 
 
 volts 
 
 Amperes = 
 
 ohms 
 
 2. Resistance Coils, and Resistance Boxes. In electrical 
 measurements we need, as standards of reference, resistances of 
 known magnitude ; just as in chemical investigations we need to 
 use ' weights ' of known magnitude. We must therefore have at 
 our disposal convenient multiples and sub-multiples of the ohm. 
 Now a column of mercury is obviously a very unmanageable unit ; 
 whereas coils of wire are very convenient, and should be durable 
 also, if carefully made. While, therefore, we can keep our mercury 
 column as the unit to which to refer, we can for purposes of use 
 make copies of it in wire. Coils are made some having i ohm 
 resistance, others having 2, 3, 4 ... io . . . 50 ... 100 ,< . . 
 1,000, &c., ohms resistance, and such standards of resistance are 
 very durable and occupy but little space. 
 
 For the making of coils we choose some wire which fulfils as 
 far as possible the following conditions : (i.) it should not corrode; 
 (ii.) its resistance should not alter with time or with a moderate 
 amount of bending ; (iii.) it should not alter much with tempera- 
 ture; (iv.) it should not be too expensive. The wire which is best 
 for the purpose is an alloy of copper, zinc, and nickel ; or of silver 
 and platinum. 
 
 The alteration with time, &c., can only be checked by periodic 
 testing. As to temperature, either the coils are arranged so that 
 they can be immersed in water at some standard temperature, or 
 there may be used a ' correction formula ' which will approximately 
 allow for the rise in resistance with rise in temperature. 
 
 The wire is covered with an insulating material, and is wound 
 
CH. XIV. 
 
 MEASUREMENT OF RESISTANCES 
 
 in a particular way, so as to obviate as far as possible induction 
 effects ; the wire being doubled on itself before winding, so that 
 all through the coil there are equal currents in opposite directions 
 lying side by side. When 
 wound it is soaked in melted 
 paraffin wax. Resistance coils 
 are usually arranged in resist- 
 ance boxes, in such a way that 
 we can readily throw into the 
 circuit any number of ohms 
 desired. 
 
 The first figure indicates 
 diagrammatically the arrange- 
 ment in a resistance box. Into the circuit we introduce the thick 
 brass piece T T'. This is not continuous, but is divided into 
 portions connected by the resistance coils. The gaps, however, 
 can be filled up by well-fitting brass plugs, and when all these 
 plugs are in their places, there is between T and T a continuous 
 thick brass conductor of practically no resistance. The removal 
 
 FIG. 
 
 of any plug throws into the circuit that resistance coil which in 
 the figure lies below that plug. Hence, by a suitable removal of 
 plugs we can throw into the circuit any resistance that is con- 
 tained in the box. 
 
 The figure indicates that the coils are so wound as to obviate 
 errors due to induction (see Chapter XXI.). 
 
22 4 
 
 ELECTRICITY 
 
 CH. XIV. 
 
 3. Wheatstone's Rheostat. We will now mention an instru- 
 ment intended for the measurement of resistances continuously, 
 the resistance box above described being evidently discontinuous. 
 
 Note. We may here remark that on account of certain defects of contact, 
 &c., that appear to be inherent in the very nature of a Wheatstone's rheostat, 
 this instrument cannot be used in any but very rough measurements. 
 
 Its chief use is as a continuously adjustable resistance, in cases where it is 
 not desired to measiire the resistance thus introduced. 
 
 A and B are two parallel cylinders, A being of brass and B of 
 wood. A fine uniform wire can be wound from A on to B, or 
 
 vice versa. The wire that is 
 on A is in good contact with 
 the brass of A, and thus forms 
 one conductor with the cylin- 
 der. The wire that is on B 
 is wound in a spiral groove 
 cut in the wood, and is thus 
 insulated. In the figure the 
 wire is represented as partly 
 on that portion of B which is 
 nearest to the spectator, and 
 partly on the further portion 
 of A. One end of the wire 
 is attached to a brass ring on 
 B, this brass ring being in connection with the terminal o by 
 means of a brass spring that presses against it. The other end of 
 the wire is attached to the far end of the brass cylinder A, and 
 then is in connection, by means of the cylinder itself and a spring 
 that presses against it, with the other terminal n. By turning the 
 handle d we can wind more or less of the wire on to the cylinder 
 B where it is insulated, and so throw more or less of it into the 
 circuit. That portion of the wire which is on A gives no appre- 
 ciable resistance, being then practically part of a very thick brass 
 conductor. 
 
 By noticing against what mark on the graduated scale (seen in 
 the figure between the two cylinders) the wire stands, we can read 
 off the integral number of turns of the handle ^, the fractions of 
 one turn .being read from a graduated circle round which the 
 handle d turns. 
 
 FIG. i. 
 
CH. xiv. MEASUREMENT OF RESISTANCES 225 
 
 We can use this instrument in two main ways, 
 
 I. Method of substitution. In the figure, R is the rheostat ; 
 m m' are two mercury cups between which either the unknown 
 resistance x, or a thick 
 
 copper piece of no resist- CHHa fL J 
 
 ance, can be inserted ; 
 and G is a galvanometer 
 
 We first measure the 
 deflexion a when x is in 
 the circuit. We then re- 
 place x by a thick copper 
 piece, and wind in a 
 measured length of rheostat wire of resistance / until we have the 
 same deflexion a as before. 
 
 Then, if the E.M.F. E has remained constant, we can easily 
 show by Ohm's law that 
 
 x = L 
 
 II. Method of comparison of deflexions. Here we use a 
 measuring galvanometer ; e.g. a tangent galvanometer in which, 
 if ri be the angle of deflexion, the current-strength C is given by 
 the relation C = k . tan a, where k is some constant. Let E be 
 the E.M.F. of the battery, x the unknown resistance, R the resist- 
 ance of rest of the circuit. 
 
 Let <*! be the deflexion when x is not in the circuit. .Then . 
 
 p 
 
 Ci=.tana 1 . AlsoC 1 = . 
 
 K. 
 p 
 Whence R = . 
 
 Now introduce the resistance x, and let the new deflexion 
 be a 2 . 
 
 Tj> 
 
 Then R -f x = -v- coto 2 <'... . (ii.) 
 
 K 
 
 Next remove x, and introduce some measured resistance /, 
 and let the new deflexion be a 3 . Then we have 
 
 R + / = -?. cot a 8 . . . . . . (iii.) 
 
 By subtracting (i.) from (ii.) and from (iii.) respectively, and by 
 
 Q 
 
226 
 
 ELECTRICITY 
 
 CH. XIV. 
 
 dividing the one result by the other, we finally eliminate both R 
 -p> 
 
 and ; the final result being 
 k 
 
 X COt a 2 COt n l 
 
 / COt U 3 COt a/ 
 
 which gives us x in terms of known quantities. 
 
 Both the above methods assume that E is constant. Now E 
 varies with time, rendering method (I.) inaccurate ; and in method 
 (II.) E varies not only with time, but with the strength of the 
 current flowing. 
 
 The Wheatstone's bridge method, next to be described, is free 
 from the above and other objections. 
 
 4. Wheatstone's Bridge, General Principle. Fig. i. repre- 
 sents diagrammatically the principle of Wheatstone's bridge. 
 
 FIG. i 
 
 FIG. u. 
 
 Let the current from A to C divide into two branches ABC 
 and A B' C ; and let the resistance A B be measured by r, B C by 
 r 1 , A B' by s, and B' C by s 1 . 
 
 Then A is at one potential V A , and C at a lower potential V c ; 
 and along both branches the potential falls from V A to V c propor- 
 tionally with the resistance. If (r + r 1 ) be great as compared 
 with (s + s'\ then the fall through A B C is gradual as compared 
 With the 'steeper' fall through AB' C. But whatever proportion 
 
CH. XIY. MEASUREMENT OF RESISTANCES 22^ 
 
 (r + /) bear to (s + s'), there will be on the branch A B' C points 
 of the same potentials as those of any points on ABC re- 
 spectively. 
 
 Now if we connect two points B and B' by a * bridge ' B G B' 
 containing a galvanometer G, then according as we see indicated 
 a current from B to B', no current, or a current from B' to B, we 
 shall conclude that B is above B' in potential, B and B' are at the 
 same potential, or B' is above B in potential, respectively. 
 
 The case of no current through G is the most important ; in 
 this case B and B' must be at the same potential. 
 
 Now in fig. ii. M P represents the potential V A ; and we 
 have taken V c as our zero potential, so that the point C is repre- 
 sented by O or by O'. The fall of potential down the one circuit 
 is given by the line P Q O, the fall down the other circuit by 
 P Q' O. The points Q and Q' represent B and B' respectively ; 
 when these latter are at the same potential, as proved by the 
 absence of current in G. The rest of the figure is clear after 
 what we have said in Chapter XIII. 7. 
 
 Then when B and B' are at the same potential, or when 
 QN = Q' N', we have, by Euclid, Book VI 
 
 r+7' = MP j and T+T? = M F ' 
 
 r 1 s' 
 
 and therefore 
 
 r' s + s' 
 
 r s 
 
 or = 
 
 Hence one of the resistances is determined by the other three. 
 
 Algebraic proof 'of the same. We will now use a notation that will be 
 readily understood with reference to fig. i. of 4, the notation being similar 
 to that employed in Chapter XIII. 9 and 10. 
 
 Since there is no current through G, we have C r C,/ ; and C s = C s >. 
 
 Applying Kirchhoff 's second law to the circuit A B G B' A, we have . . 
 C r r - C s . s = o ; or C r . r = C s . s. 
 
 Applying the same law to the circuit B C B' G B, and remembering that 
 C r = C r ', and C s = C s >, we have also in like manner ........ 
 
 Cr . r 1 - C s . s' = o ; or C r . r 1 = C, . /. 
 
 Whence we readily obtain the result that 
 
 Q2 
 
228 
 
 ELECTRICITY 
 
 CH. XIV. 
 
 It is clear that this formula remains true, however the E.M.F. 
 of the battery may alter. In this fact lies the main superiority 
 of this method, over the previous methods, of measuring re- 
 sistance. 
 
 5. Slide-form of Wheatstone's Bridge. The figure here 
 given represents roughly the simplest and least expensive form of 
 Wheatstone's bridge. 
 
 On a board are fixed three bands, A, B, and C, of stout copper ; 
 the pieces A and C being elbow-shaped. 
 
 Between A and B, and B and C, are gaps ; and here can be 
 inserted, most conveniently by means of mercury cups, any resist- 
 ances we please. 
 
 The other ends of the pieces A and C are joined by a thin 
 uniform wire of high specific resistance; and parallel to this runs 
 
 a scale that should 
 have its initial and 
 final points exactly op- 
 posite the points where 
 the wire is soldered 
 to the thick copper 
 pieces. This scale is 
 graduated to read both 
 ways ; and, on the sup- 
 
 / position that the wire is 
 of uniform resistance, 
 and that the scale corresponds properly to the beginning and end 
 of the wire, we can read off the ratio of the two resistances s 
 and s' into which any point such as B' divides the wire. 
 
 Over the wire slides a spring key B'; by pressing the button of 
 this key we make contact with the wire by means of a metallic 
 edge or fine wire. The point where contact is thus made should 
 be exactly opposite to the point on the scale at which the little 
 index carried by the key is then pointing. The ' bridge ' B G B' 
 connects the piece B and the key B'; the battery terminals are 
 connected with A and C. 
 
 Referring to fig. i. of 4, we see that we have complete cor- 
 respondence; only the points A, B, and C of that figure have been 
 replaced by the metal pieces A, B, and C, which are, from a resist- 
 ance point of view, equivalent to points. Further, we see that 
 
CH. xiv. MEASUREMENT OF RESISTANCES 22Q 
 
 the point B' is moveable, so that we can make the ratio s : s 1 
 anything that we please. 
 
 In using the instrument we insert our unknown resistance at 
 r, and a resistance box at r'. We then adjust the position of the 
 key B' until the making of contact with the wire at B' gives no 
 current through the galvanometer in the bridge. 
 
 When this is the case, B and B 7 must be at the 
 potential; and we then have 
 
 which determines r. 
 
 Note. There is often some want of exact correspondence between the ends 
 of the scale and the ends of the wire. Such a defect produces less error in the 
 result if we have s and s' not very unequal. We can contrive this by having 
 a resistance box at /, and by removing the plugs (i.e. by throwing in resist- 
 ance) until we get our zero deflexion of G when B' is not far from the centre 
 of the wire ; in other words we make r 1 not very unequal to r. 
 
 6. Wheatstone's Bridge; Resistance Box Form. The 
 
 above described instrument is liable to error. The wire, by much 
 use, loses its uniformity; so that equal lengfhs no longer correspond 
 to equal resistances. Moreover, there is, even in the best instru- 
 ments, some uncertainty as to the exact correspondence between 
 the point of contact B' and the point on the scale against which 
 the index rests. (This second defect can, however, be elimi- 
 nated.) In the instrument used for technical purposes B' is fixed, 
 and s and s' are resistance boxes containing coils of (e.g.) i, 10, 
 100, and 1,000 ohms. For the known resistance / we have another 
 resistance box. We can thus, by means of the two boxes s and /, 
 
 measure from I th'to 1,000 times any resistance in the box r' ; 
 TOOO 
 
 this is simply a matter of removing the suitable plugs. We can 
 
 thus with certainty measure to th of i ohm, which is better. 
 
 1000 
 
 than measuring to smaller fractions with uncertainty. 
 
 7. Resistance of a Galvanometer. This can be measured 
 as any other resistance ; but a very simple method is as follows. 
 
 Sir Wm. Thomson's method. Here we put the galvanometer 
 G, whose resistance is required, in the place of one of the resist- 
 ances r ; and, in the place of the bridge with its galvanometer, we 
 
233 
 
 ELECTRICITY 
 
 CH. XIV. 
 
 have a key K, by means of which we can make or break con- 
 nection between B and B' at will. 
 
 The galvanometer G will have a deflexion due to the current 
 through A B ; the battery power and resistance in the external 
 
 circuit A P C must be so 
 adjusted as to give a reason- 
 able deflexion of G. 
 
 We adjust the resist- 
 ances s and s' until the 
 making of contact with the 
 key K produces no alter- 
 ation in the deflexion of 
 G. When this is the case, 
 it can be proved that no 
 current then flows through 
 the bridge ; and that as 
 before, ....... 
 
 G = ^ . r\ where G is the galvanometer resistance. 
 
 8. Resistance of a Battery-Cell. The measurement of the 
 resistance of a battery-cell is not at all a simple matter. 
 
 For whereas in the above measurements we had only one con- 
 stant unknown, viz. the constant resistance x that we desired to 
 measure, in the case of the battery-cell we have two unknowns, 
 one of which is variable from moment to moment ; we have, in 
 fact, the resistance x, and also the E.M.F. y, which, owing to vary- 
 ing degrees of polarisation, is not constant. 
 
 We here give two methods for measuring x. 
 
 I. The E.M.F. reduced to zero. If we have any even number 
 2 m of similar cells, and if we set m against m, the total E.M.F. 
 will be zero. When the current, employed in measuring the. 
 resistance, passes through these 2 m cells, the polarisation pro- 
 duced should still be zero in total amount. With cells thus 
 matched m against m, we can measure the resistance of the whole 
 2 m cells as a * dead ' resistance ; that of one cell is found by 
 dividing by 2 m. 
 
 Against this method it may be stated that, as a rule, we require 
 the resistance of the cell or battery as it is at any given time ; and 
 
CH. XIT. 
 
 MEASUREMENT OF RESISTANCES 
 
 231 
 
 we cannot generally provide another cell or battery, equal to this 
 in E.M.F.. to oppose to it. 
 
 We may call this method the ' Method of opposition? 
 
 II. Mance's method. Let us arrange a Wheatstone's bridge as 
 in 5, but with the cell or battery, whose resistance is to be de- 
 termined, in the place of 
 the resistance r, and with a 
 simple contact key in the 
 place of the usual battery. 
 The cell at r will drive a 
 current ; and no arrange- 
 ment of resistances can pre- 
 vent there being a current 
 through the bridge and the 
 galvanometer G. Hence, 
 G will be deflected. 
 
 It can be shown that if the resistances be so adjusted that G 
 maintains the same deflexion whether the key K be opened or 
 closed, then we have the old relation V 
 
 The proof is not very simple. By application of KirchhofTs 
 laws we obtain equations by means of which the unknowns can be 
 determined. The current through G will be found to involve the 
 external resistance of A K C in such a way that it is only independent 
 
 of it when = . Whence it follows that, if this current is the 
 r s' 
 
 same whether the key is open (giving an infinite resistance) or closed 
 (giving a finite resistance), this relation must hold. 
 
 (For further discussion of this method see * Phil. Mag.,' Series V., 
 Vol. 3, 1877, Professor O. J. Lodge ; and also in Glazebrook and Shaw's 
 'Practical Physics.') 
 
 9. Measurement of E.M.F. Let us take our standard cell, 
 whose E.M.F. we suppose (for simplicity) to be just i volt, and 
 let us measure the AV between its terminals when the circuit is 
 
232 ELECTRICITY , en. xiv. 
 
 open (see Chapter X. 34). Let us measure in the same absolute 
 potential measure the AV between the terminals of any other cell 
 or battery, the circuit being open. Using the AV of the standard 
 cell as unit, we thiis can express the AV of the latter in volts. We 
 thus get a clear idea of what is the initial E.M.F. of any battery, 
 expressed in volts. But when the circuit is closed, polarisation 
 sets in ; and we find that the same cell has a different E.M. F. for 
 each strength of current passing through it. 
 
 'The E.M.F. of a cell' is, therefore, a term whose meaning is 
 undefined unless it be stated what current is running. As a rule, 
 therefore, the initial E.M.F., or the AV between the poles of 
 the open cell, is given ; and it is then stated with more or less 
 exactness to what extent this falls off with currents of various 
 magnitude. 
 
 The E.M.F. when the current is zero, or the statical AV (ex- 
 pressed in volts) between the poles of the open circuit, is therefore 
 what we record as ' the E.M.F. of a cell.' 
 
 To find the E.M.F. with any given current is a very difficult 
 matter. For, if we try to determine the E.M.F. and the internal 
 resistance of a battery by means of two equations derived from 
 Ohm's law, we must use two different currents ; and in this case 
 neither the E.M.F. nor the resistance of the battery will be the 
 same in the two equations. 
 
 10. Electrometer Methods; Open Circuit. (i.) If we have 
 an absolute quadrant electrometer we can measure the E.M.F. of 
 the open cell by measuring the AV between its poles. Dividing 
 this absolute value by that obtained for i volt, we express the 
 E.M.F. of the open circuit in volts. 
 
 (li.) If we have an ungraduated electrometer we may proceed 
 as follows. We take m of our cells, and n standard cells, arranged 
 end-on. We put one terminal (say the terminal) of each 
 battery to earth ; and connect the -f terminals of each battery 
 with one pair of quadrants respectively. 
 
 We arrange m and n until either we have no deflexion, or 
 until with n standard cells we get the deflexion of the electrometer 
 needle one way, and with n + i cells we get the deflexion the 
 other way. 
 
 Then, if e be the E.M.F. of the cell required, and if e be 
 that of the standard cell, we have ".*.. 
 
CH. xiv. MEASUREMENT OF RESISTANCES 233 
 
 (either e = e , 
 
 m 
 
 or e lies between e and - e ; thus obtaining a measure- 
 ment to within th of e . 
 m 
 
 u. Volt-meter Galvanometers. Let us suppose that we have 
 two points A and B in a circuit ; and that it is required to measure 
 
 the AV in volts between them, or the E.M.F. E B . We have above 
 indicated the electrostatic method of doing this. A current 
 method is also possible. For if we have a galvanometer G of a 
 very high resistance as compared with the resistance between A 
 and B, the current that passes through G when its terminals are 
 connected with A and B will be, by Ohm's law, directly propor- 
 tional to E B ; and at the same time, owing to the relatively very 
 high resistance of G, the A V E^ will not be appreciably altered 
 by connecting the points through G. Thus 
 
 (i.) If we connect the terminals of an open cell through G, we 
 do not appreciably alter the E.M.F. of the open cell, the resistance 
 of G being relatively so great (see Chapter XIII. n ()). 
 
 (ii.) If we connect G with two points in the circuit, we do not, 
 by introducing a branch circuit of so high a resistance, practically 
 alter the A V between A and B. 
 
 Hence, if we have a galvanometer of a high known resist- 
 ance G, and which measures currents accurately, we can from it 
 deduce the AV between the points A and B with which its 
 
 TT A 
 
 terminals are in contact, by means of the equation C = -* ; and 
 
 G 
 
 this will be practically the same AV as existed before we made 
 connection with the galvanometer. 
 
 Sir W. Thomson's graded volt-meter galvanometer is such an 
 instrument. 
 
 In Chapter XVII. we shall give some detailed description of 
 it, and shall indicate how it can be used to measure both large 
 and small E.M.F.s. Here we shall say little more about it 
 
 From the formula C = L the instrument can be experimen 
 G 
 
 \ 
 
234 
 
 ELECTRTCITY 
 
 CH. XIV. 
 
 ally or theoretically graduated, so that we can read off at once in 
 volts the A V E between the terminals A and B of the instru- 
 
 B 
 
 ment, answering to any particular current C. 
 
 12. Method of Opposition. Here, as in 10, we have zero 
 current ; but we use a galvanometer instead of an electrometer. 
 
 If m of the cells, whose E.M.F. we require, be opposed to n 
 standard cells, and if /;/ and n be so adjusted as to give if possible 
 zero-deflexion of a galvanometer in the circuit, then, as in 10, 
 
 we can measure to within th of the E.M.F. of a single standard 
 
 m 
 cell 
 
 13. Latimer-Clark's Potentiometer. In this instrument the 
 method of opposition is employed. 
 
 P is a battery ; R is a Wheatstone's rheostat ; A B is a uniform 
 graduated wire of high resistance, fixed between two terminals A 
 and B ; e is a standard cell ; e is a cell of a smaller E.M.F. that 
 we desire to measure ; G and G' are delicate galvanometers to 
 indicate when there are zero currents. Contact keys are not repre- 
 sented. 
 
 Note. As an example we may state that in a certain instrument the wire 
 A B has a resistance of about 53 ohms. It is about 53 metres long, and is 
 coiled in a screw-groove cut round an ebonite cylinder. 
 
 Now the total fall of V due to the battery P will be distributed 
 between the portions A R B and A Q B of the circuit, according to 
 
 their respective resistances. 
 Hence, by manipulating R, 
 we can arrange the AV be- 
 tween A and B until it equals 
 the E.M.F. e of the standard 
 cell ; this will be indicated 
 by the fact that when E* = ? , 
 then no current will pass 
 through G, since the two 
 equal and opposite E.M.F.s 
 are opposed. Since there is 
 
 no current through this cell, and so no polarisation, therefore e 
 remains constant, and therefore E is also constant. As the 
 
CH. XIT. MEASUREMENT OF RESISTANCES 235 
 
 E.M.F. of P gradually alters by polarisation, we can so diminish 
 the resistance of R as to maintain G undeflected, or E B constant 
 
 All the above may be regarded as a contrivance for keeping 
 the two points A and B at a constant AV equal to the E.M.F. e 
 of the standard cell. And the uniform wire A B gives us by its 
 graduations any fractional parts of e that we desire ; since, by 
 Ohm's law, the AV, e m falls uniformly from A to B. 
 
 Now, if we arrange the cell e as shown, and if we get zero- 
 deflexion of G' when we make contact at Q, where A Q = a and 
 
 B Q = b, it follows that then e = E Q , or that 
 
 
 since evidently E A = . E A . 
 * Q AB B 
 
 Thus we can measure e accurately when less than e . 
 
 Note. It may make matters clearer if we point but that the positive poles 
 of the battery P, and of the cells e and e, are all connected with the same end 
 A of the wire A B ; and their negative poles, are all connected with the other 
 end B. 
 
 If e be greater than e we have merely to interchange the posi- 
 tions of e and e, and the same experimental method gives us that 
 
 a 4- b 
 
 e = e . 
 
 a 
 
 14. Table of Resistances in Ohms (or w), We shall now 
 give some information as to the actual resistances of various sub- 
 stances ; and, in order that the statements may be scientifically 
 exact, we shall explain certain terms that will be used. 
 
 Microhms. A microhm is the T FOWTTo tn of an ohm ; or i 
 microhm = wX io~ 6 . 
 
 Megohms. A megohm = 1,000,000 times an ohm ; or i 
 megohm <u x io 6 . 
 
 Specific resistance p. The resistance offered by a conductor 
 of i centimetre length and i square centimetre section to a current 
 passing through it from one face to the opposite face is called the 
 ' specific resistance p ' of the substance of which the conductor is 
 made. We here use the word conductor to include all substances, 
 that do to any measurable degree conduct. 
 
236 ELECTRICITY CH. xiv. 
 
 The results given are derived from the results collected in 
 Hospitaller's small book, and from Lupton's tables of constants 
 (see below). We do not, in this Course, consider changes due to 
 temperature ; nor do we consider the slight difference between the 
 B A unit and the ohm (see Chapter XIV. i (ii.) note). 
 
 I. Specific resistances of metals in microhms. 
 
 Silver, annealed ..... 1*521 (co x io~ 6 ). 
 
 Silver, hard drawn ..... 1*609 > 
 
 Copper, annealed ..... 1*616 ,, 
 
 Copper, hard drawn . . . . 1*642 ,, 
 
 Gold, hard drawn .... . . 2*154 ,, 
 
 Aluminium, annealed . . . . 2*946 ,, 
 
 Zinc, pressed . . . . . 5*690 ,, 
 
 . Platinum, annealed . . . . .9*158 ,, 
 
 Iron, soft . . . . . . 9*827 ,, 
 
 Tin, pressed . . . . . 13*360 ,, 
 
 Lead, pressed ...... 19*847 ,, 
 
 Mercury, liquid . . . .- . 96*146) 
 
 to 99740) 
 
 German silver. * .21*170 ,, 
 
 Brass , \ - _--- . 5 '800 ,, 
 
 For further facts as to resistances, such as resistances of different 
 frnasses of wire of known gauge, we refer the reader to Hospitalier's 
 'Formulaire Pratique de l'Electricien'(Masson, Paris), and to Lupton's 
 ' Numerical Tables and Constants ' (Macmillan). 
 
 II. Specific resistances of liquids in ohms. 
 
 (Taken from Lupton's book.) 
 /Water at -75 C. . . '."'. . .. . ... i*i88x io s (a*) 
 
 4 Water at 4 C. . . ' * .'*<.'.- 9*100 x 10" (a) 
 
 (Water at 11 C. .'...:.. . . 3*400 x io 5 (w) 
 (Dilute hydrogen sulphate at 18 C. ; 5 per cent, acid . ,4*88 () 
 
 I Dilute hydrogen sulphate at 18 C. ; 30 per cent, acid . .1*38 (w) 
 
 Hydrogen nitrate at 1 8 C., density 1*32 1*61 () 
 
 Saturated solution of copper sulphate at i o C. .... 29*3 (w) 
 Saturated solution of zincic sulphate at 14 C. . . . . 21*5 (o>) 
 
 Hydrogen chloride, 20 per cent, acid, at 18 C 1*34 () 
 
 Ammonium chloride, 25 per cent, salt . . . . . . 2*53 (o>) 
 
 Sodium chloride saturated, at 13 C. . . . . . . 5*30 (w) 
 
 The above show the very great difference that exists between pure 
 water and solutions of salts, with respect to the conduction of a 
 current. 
 
CH. xiv. MEASUREMENT OF RESISTANCES 237 
 
 III. Specific resistances of bad conductors in megohms. 
 
 Ice at - 12-4 C 2240 ( x io fi ). 
 
 Glass; (soda-lime, density 2-54) at 20 C. . 9-1 x io 7 ,, 
 
 (Glass ; (crystal, density 2-94) below o C. . practically infinite. 
 
 (Glass; (crystal, density 2 -94) at 105 C. . I'i6 x io 7 (w x IO H ). 
 
 Shellac; at 20 C 9-00 x io 9 ,, 
 
 Paraffin; at 46 C. . . . . . 3-4 x io 10 ,, 
 
 Ebonite ; at 46 C 2-8 x ro 10 
 
 Air ; usual pressure . . . . practically infinite. 
 
 True vacuum ...... practically infinite. 
 
 Very dense gases, it would seem, do conduct (or ' convect ') to 
 some extent. 
 
 In very rare gases, or ' imperfect vacua,' there appears to be con- 
 duction ; but it seems certain that in this latter case, at all events, the 
 discharge is disruptive, and that there is no true conduction. 
 
 15. Table of E.M.F.s in Volts. We here give some results 
 as to the E.M.F.s of certain cells. The reader must, however, bear 
 in mind that the E.M.F. of a cell depends (i.) upon its general 
 make ; (ii.) upon details which are different in different cells pro- 
 fessedly of the same E.M.F. ; (iii.) upon the current. The 
 E.M.F.s here are therefore only approximately correct for cells in 
 general. They are the initial E.M.F.s ; and so we have some 
 anomalous results, as, e.g., that the E.M.F. of Volta's cell is greater 
 than that of a Smee's a result certainly not true when there is 
 much current running. 
 
 /Volta's cell ; Zn \ dilute If. 1 SO^ \ Cu ; about i-oo volt. 
 Smee's ; Zn \ diL H.,SO 4 \ platinised silver ; from 0-50 to I'OO volt. 
 Darnell's; Zn \ dil. H.SO^ \ CztSO 4 solution \ Cu\ about 1-12 volt. 
 Grove; Zn \ dil. H. i S^^ \ HNO 3 \ Pt ; nearly 2-00 volt. 
 Bunsen ; Zn \ dil. H.,SO^ \ HN0 3 \ C ; 175 to I -94 volt. 
 Clark's standard cell ; ~Hg \ Hg,.S0 4 \ Zn ; at I5C. 1-438 
 
 Leclanche ; Zn \ NH^Cl sohition 
 
 and C\ ' ' I<42 " 
 
 De la Rue ; Zn \ NH.Cl \ AgCl \ Ag . . . 1-04 ,, 
 
 Marie-Davy ; Zn \ dil. ff,SO 4 \ HgSO \ C . . 1-52 ,, 
 
 Bichromate ; initially ; solution new . . . 2-00 ,, 
 
 Plante secondary cell , , . about i'8o to 2-50 ,, 
 
 These results are for the most part those given in Lupton's book 
 above referred to. 
 
238 ELECTRICITY CH. xv. 
 
 CHAPTER XV. 
 
 JOULE'S LAW, AND CONSERVATION OF ENERGY. 
 
 i. General Survey. We have already pointed out how 
 (i.) all forms of energy may be measured in work-units (i.e. in 
 
 * r &Y> ; . 
 
 (ii.) energy is indestructible, or in any self-contained system 
 we have transformation of energy from one form to another, but 
 no loss or gain of energy. 
 
 In addition to these statements we may now add the law, which 
 seems to apply to all inanimate nature at any rate, 'that all energy 
 tends to run down to the form of .uniformly diffused heat.' This 
 law is called the law of ' Degradation of energy? 
 
 Now in the battery-cell our store of energy is in that form 
 which is called chemical potential -energy. In any particular cell 
 the total energy at our disposal is proportional to the total mass 
 of zinc (or other metal) to be dissolved. This energy also depends 
 upon the other chemical changes that take place ; and thus it is 
 that in different cells, in which the rest of the chemical action is 
 different, we have different amounts of energy answering to the 
 same mass of zinc dissolved. 
 
 This energy may pass (apparently) direct into the form of heat 
 energy, as when we simply dissolve unamalgamated zinc in acid. 
 
 Or we may obtain electrical energy as an intermediate form ; 
 and part of this electrical energy may be converted again into 
 chemical-potential-energy if it be caused to decompose an elec- 
 trolyte, or it may be transformed into mechanical energy if it be 
 caused to work an electro-motor (jw Chapter XXIV.), or it maybe 
 allowed to be entirely converted into heat-energy if it be .left simply 
 to heat the circuit. 
 
 But in every case we have simply the energy answering to the 
 
CH. xv. JOULE'S LAW 239 
 
 nature of, and amount of, the chemical changes that have taken 
 place in our sole source of energy, the cell or battery. 
 2. Units of Heat, Work, and Activity. 
 
 I. Unit of work. In the absolute system this is the erg, 
 already discussed in Chapter X. 
 
 II. Unit of heat. The calorimetric unit of heat is that 
 amount ef heat which will raise one gramme of water from o C. to 
 i C. ; or we may with sufficient accuracy measure the number of 
 calorimetric units of heat by the product of the (number of grammes 
 of water] into the (number of degrees Centigrade through which they 
 rise or fait in temperature]. This unit is called the calorie. 
 
 The calorie is not a C.G.S. unit, as the Centigrade scale of 
 temperature is independent of the C.G.S. fundamental units. We 
 have, therefore, to connect this unit with the C.G.S. system by 
 direct experiment, measuring the heat, as we can measure all forms 
 of energy, in ergs. 
 
 Now it has been shown experimentally that 
 
 i calorie 4*175 x /o 7 ergs = 42 x io 6 ergs [nearly], 
 
 or that one gramme-degree unit of heat would, if totally used up 
 in doing mechanical work, do 41,750.000 ergs of work ; or the 
 same would raise 424 grammes through one metre against gravity 
 (where g is taken as 981). 
 
 It was Joule who first established the definite measurement of 
 heat in units of work. The factor which reduces calories to ergs 
 will be called J, so that J = 4*175 x io 7 . We may, for ordinary 
 purposes, take J to be equal to 42 x IO G . 
 
 Note. We use the true calorie, the gramme-degree-Centigrade unit. 
 Another calorie is sometimes used, viz. the kilogramme-degree-Centigrade 
 calorie ; but its use will probably soon be entirely discontinued. 
 
 III. Unity of activity, or rate of work, or power. It is 
 often very important to consider the rate at which work is done, or 
 the work per second. It is convenient to give the name of 'activity ' 
 to rate of work. 
 
 The C.G.S. unit of activity is the activity of one erg per second. 
 (In English engineering practice the horse-power is much used 
 as a convenient unit of activity when work is being done on a 
 large scale.) All energy, then, can be measured in *rgs or units of 
 work, and all energy per second in units of activity. 
 
240 ELECTRICITY CH. xv. v 
 
 3. Energy of the Electric Current. Consider a current C 
 running steadily between the points A and B of a circuit, where 
 A is at the potential V A , and B is at some lower potential V B . 
 
 Each unit of electricity at A is at a higher potential than 
 when it comes to B ; in fact, there is a continual discharge from 
 
 the constant potential V A to the 
 
 / ^ n X- > *"*o! constant potential V B . Each 
 
 / \ unit, therefore, loses potential 
 
 / \ energy to the amount repre- 
 
 sented by V A V B (see Chap- 
 ter X. 28) in its passage from A to B ; and, if the current be C, 
 or if C units pass per second, there will be a loss of electrical- 
 potential-energy between A and B to the amount of C x (V A V B ) 
 per second. We may write this as C x E A . If C and V be 
 
 B 
 
 measured in suitably chosen units, the above product will give us 
 the electrical energy lost in ergs per second, or in absolute C.G.S. 
 units of activity. Now the absolute electro-magnetic system of 
 units, referred to in Chapter XIV. i, and explained in Chapter 
 XVIII. , is such that (current] x (E.M.F.) does give us this activity 
 in absolute C.G.S. units. If, however, we use amperes to measure 
 C, and volts to measure E M.F. or ^V, then the product C x E 
 gives us the activity in a new unit which we call a watt. Since 
 i ampere = T ^th absolute unit, and i volt io 8 absolute unit, it 
 follows that - r"-5 
 
 i watt = io 7 (ergs per second}. 
 
 To repeat ; when a current of C amperes runs between two points 
 whose difference of potential is measured by E volts, then electrical 
 energy disappears between these points at the rate of. .... 
 
 Cx^ watts (i.) 
 
 or C x E x i o 7 ergs -per- second. 
 
 We may add that 
 
 One English horse-power = 746 watts [nearly]. 
 If there be no source of E.M.F. between the points A and B, 
 
 A 
 
 then by Ohm's law we have C = . 1, where R is the resistance 
 between A and B. In this case, which is the case where A B is a 
 
CH. XV. 
 
 JOULE'S LAW 
 
 241 
 
 simple conductor, and includes no form of pile or electrolytic-cell 
 or thermo-cell, we have that 
 
 Activity = C 2 R watts (ii.) 
 
 But if we simply know E A as the AV between A and B, and if 
 
 B 
 
 we do not know whether or no there be any source of E.M.F. be- 
 tween A and B, then we cannot write C R instead of E B . For if 
 there be such an E.M.F., e, between A and B, then by Kirchhoffs 
 extension of Ohm's law, we have that C R does not equal E B) 
 
 but equals (E B db e). Hence the expression (i.) given above 
 is of the more general application. 
 
 In the case of a battery of E.M.F. E, through which is passing 
 a current C, the above considerations will show us that the rate at 
 which chemical-potential-energy is used up in giving electrical 
 energy must be represented by an activity of C E, since electricity 
 to the amount of C per second is being raised through a AV 
 represented by E. In fact, the rate at which chemical energy is 
 used up in ' pumping up ' electricity in the cell from the lower 
 level to the higher (the difference of level being E volts) is repre- 
 sented by the activity of CE watts. 
 
 Inside the cell, then, we are using up chemical or other energy 
 in raising electricity from a lower to a higher level, i.e. in gaining 
 electrical energy ; while partly inside and partly outside the cell 
 this efectrical energy is being again lost 
 giving us, as we shall see later on, 
 other forms of energy. 
 
 4. Joule's Law. The same Joule 
 who first clearly showed that heat was 
 a form of energy, and could therefore 
 be measured in work-units, was also 
 the first to point out the exact connec- 
 tion between the electrical energy lost 
 in any portion of a circuit and the 
 heat evolved in that portion of the 
 circuit. 
 
 He showed experimentally that if 
 a current C be flowing through a conductor of resistance R, then 
 the heat evolved per second is proportional to the product C 2 R ; 
 
 R 
 
242 ELECTRICITY CH. xv. 
 
 and further investigation has shown the exact relation that the 
 amount of heat evolved bears to the electrical energy lost in this 
 portion of the circuit. 
 
 The nature of his experiments is indicated by the accompany- 
 ing figure. 
 
 A wire of known resistance is immersed in a known mass of water, 
 a current measured by a galvanometer is passed through the wire for 
 a measured time, and the rise in temperature of the water is noted. 
 We thus measure C, R, and the heat evolved. From a comparison of 
 many experiments in which C and R were varied, the law above given 
 was deduced. 
 
 Further experiment showed that the same law held with respect 
 to the battery-cell itself. It is then true that in the whole circuit 
 or in any part of it, the ; . 
 
 (Heat evolved per second} is proportional to (the product C 2 R). 
 
 If we express our heat, current, and resistance, in suitable 
 units, then this proportionality becomes equality, and we have . . 
 
 (H per second) = C 2 R. 
 
 We shall now for convenience introduce a new symbol IS 
 having a very simple meaning. Let IS be the algebraic sum of the 
 E.M.F.s in any portion of the circuit considered ; so that, by Ohm's 
 
 law or by KirchhofFs extension of it, we have the relation C = ' 
 
 R 
 
 or C R = IE.- (Thus if we consider the portion of the circuit A B, 
 and if no source of E.M.F. lie between A and B, we have } the 
 same as Eg or (V A V B ) ; but if there be a direct E.M.F. e, or an 
 opposed E.M.F. e, between A and B, then IB = EB e). We 
 can then write C IS instead of C 2 R ; and we have, when suitable 
 units are employed .....' 
 
 H (per second) = C 2 R = C IE. 
 
 If C, R, and IS be measured in amperes, ohms, and volts, then 
 the product C 2 R or C 2 gives us the heat-activity in watts. 
 
 If watts and calories per second be compared, by referring both 
 to ergs per second, it can be shown by the data given earlier that . 
 
 i watt = '24 calories per second (approx.). 
 This relation may be verified directly by experiment, 
 
CH. xv. JOULE'S LAW 243 
 
 We may then express Joule's law as follows. If in any circuit 
 or portion of a circuit C be the current in amperes, R the resistance 
 in ohms, and 15 be the algebraic sum of the E.M.F.s (in such a sense 
 that by Ohm's law C R = !S) measured in volts, then there will 
 be evolved in that circuit or portion of circuit heat to the amount of 
 C-R x -24, or C3&X '24, calories per second. 
 
 We have thus two formulae for the heat evolved in calories per 
 second ; the formula C 2 R x '24 is always true ; and C IB X '24 is 
 
 3S 
 true if IS be such that C =^ . 
 
 5. The Heating of Uniform Wires. The most important 
 practical application of Joule's law is the application to the case 
 of conducting wires. 
 
 We will use formula (i.) of the last section. 
 
 (I.) If a wire have resistance R, and there be flowing through 
 it a current C, then there is evolved 
 
 C 2 R x '24 calories per second. 
 
 With a given current we can thus calculate the calories per second 
 evolved if we know the length, section, and specific resistance p, 
 of the wire. For, if p be given in ohms, or be reduced to ohms, 
 we have that " . 
 
 R = p : ohms. 
 
 cross section 
 
 If r be the radius of a wire of circular section (the most usual 
 make of wire) then / _.; 
 
 R = p . _. ohms. 
 
 TT r- 
 
 (II.) The temperature to which a wire is raised. If the heat 
 were not dissipated by convection, conduction, and radiation, no 
 limit (saving that imposed by the fusion of the wire) could be fixed 
 to the temperature to which a wire would rise. As a matter of fact, 
 however, there soon obtains a state of equilibrium between the heat 
 evolved per second and the heat dissipated per second. Hence 
 the temperature will be, with more or less exactness, proportional 
 to C 2 and to R ; or is higher as the current is larger, and as the 
 wire offers a greater resistance. 
 
 Experiment. Put in the same circuit lengths of Pt and of Ag wire of the 
 same radius. The platinum is the worse conductor of the two ; and therefore 
 
 R 2 
 
244 ELECTRICITY CH. xv. 
 
 .for it the resistance is greater, while the current is the same for both. It there- 
 fore is seen to attain a white-heat, while the silver remains much cooler ; in the 
 final condition, the greater dissipation due to the higher temperature com- 
 pensates for the greater amount of heat evolved per second. 
 
 Note. If in the same circuit we have (were it possible to find such) two 
 wires of same radius, same conductivity, same surface power of dissipating heat, 
 but of different masses and specific heats, then it is sometimes stated that the 
 temperatures will depend upon the masses and on the specific heats, varying 
 inversely as both. This, however, is not the case. That wire whose water- 
 equivalent (or mass x specific -heat] is the least, will first attain the final tempera- 
 ture. But, when this is once attained, only the resistance and the dissipating- 
 power come into the question ; and, sines by hypothesis this is the same for 
 both, it follows that the final temperatures will be the same. 
 
 III. Temperature as dependent on radius. Let us now consider 
 wires of different radii, but of the same material, included in the same 
 circuit, so that the same current necessarily passes through each. 
 And let represent the number of degrees by which the temperature 
 of the wire exceeds the so-called ' temperature of the air,' when a 
 steady condition of temperature has been arrived at. 
 
 We have then, for each unit length of the wire 
 
 t Heat produced by current per~\ _ VHeat lost by radiation, &>c., per~\ 
 second J L second J 
 
 If we assume [but it is not at all accurate to do so] that the 
 resistance of the wire does not depend upon its temperature, we may 
 under the conditions assumed say that . 
 
 {Heat produced per second} oc _L 
 
 where r is the radius of the wire. 
 
 Again, if we assume [but at such high temperatures this is not 
 accurate] that the rate of loss of heat is simply proportional to 6 and 
 to the area of surface exposed, we have 
 
 (Heat lost per second] oc 2 TT r . 6. 
 
 With these assumptions, then, we have that for wires of same 
 material in the same circuit 
 
 (The rise in temperature, &} oc - _ _ . 
 
 6. Distribution of Heat in the Circuit, Let the current in 
 a circuit be C ; the resistances of battery ^and of various parts of the 
 
CH. xv. JOULE'S LAW 245 
 
 circuit be B, R 1} R 2 , R 3 , &c., respectively ; the total resistance 
 be R ; the total heat evolved per second be H calories ; the several 
 portions evolved in the above different portions of the circuit be 
 H B , HJ, H 2 , &c., respectively ; then we evidently have the follow- 
 ing relations holding. 
 
 ( H = C 2 R x -24 = C 2 (B + Rj + R 2 + &c.) x -24. 
 
 J H B = C 2 B x -24. 
 
 [Hi = C 2 R! x -24. &c. 
 
 Whence also 
 
 HB ' Hj : H 2 : . . . = B : 
 
 HB = |.H. ; ! 
 
 H, =|l H. &c. 
 
 7. Heat Evolved with various Arrangements of n Cells, 
 
 Let there be n cells, each of resistance B and E.M.F. E. Let R be 
 the total resistance of the whole circuit including the battery. Let 
 H be the total heat per second evolved ; and let H } and H e be 
 the portions evolved internally to the battery, and in the external 
 circuit, respectively. 
 
 () Let the cells be arranged ;;/ end-on and / in parallel ; and 
 let the external resistance be r. Then we have H = C 2 R x "24 
 calories. 
 
 We have also C R = m E by Ohm's law ; and R = r + "~ 
 
 (see Chapter XIII. 13, &c.). We can thus substitute for C and 
 R in terms of known quantities ; and we have . 
 
 H = C 2 R x -24 = x '24 calories-per-second. 
 
 r+m i 
 
 m B 
 ^ H = &c., calories-per-second. 
 
 V 
 
 ^ H = &c., calories-per-second. 
 
246 ELECTRICITY 
 
 CH. XV. 
 
 (/?) If the cells be so arranged that the internal resistance 
 
 v ~ = r, then we have 
 
 H = C 2 R x -24 = n ~ x -24 calories. 
 
 2 r 
 
 [, = H e = H = I H. 
 2 r 2 
 
 8. Case of no Back-E.M.F. in the Circuit. - Where there 
 is in the circuit only the E.M.F. E of the battery, and no other 
 E.M.F., then by Joule's law : t \ ; 
 
 H = C 2 R = C E x "24 calories-per-second. 
 .or H = CE watts. 
 
 And by 3 we know that the total electric activity is 
 W = C E waits. 
 
 In this case, therefore, all the electric activity runs down into 
 the form of heat-per-second. 
 
 Experiment and theory, moreover, concur in establishing it to be 
 a fact that in a cell where there is no local action, i.e. in which no 
 chemical action occurs until the circuit is closed, all the chemical- 
 potential-energy lost per second appears as electrical activity. 
 Hence in this case the total chemical action gives first the equiva- 
 lent electrical energy, and then, finally, the same amount of heat 
 that would have been given had the chemical action taken place 
 without the intermediency of a current. 
 
 9. Case of a Back-E.M.F. e in the Circuit. Now let us 
 have in the circuit an electrolytic cell, or some other arrangement 
 giving a reverse E.M.F. -e the total resistance being still R. 
 
 As argued in 3, the battery is expending energy at the rate 
 of W = C x E watts. 
 
 By Joule's law we have heat evolved at the rate of H = C 2 R 
 watts. 
 
 And since by Ohm's law C = g-~j this heat activity H may 
 
 R 
 
 be written as C (E e) or C E - C e. 
 
 Hence of the activity C E expended by the battery, we have 
 accounted for part in heat ; but we have not yet accounted for the 
 remainder C e. 
 
CH. xv. JOULE'S LAW 247 
 
 ' Conservation of energy ' alone would, therefore, lead us to 
 conclude that activity measured by C e watts is being expended 
 on the electrolytic cell. And, in complete accordance with this 
 result, we have the argument that in order to drive a current C 
 against an E.M.F. e we must expend an activity of C e watts. 
 
 In an electrolytic cell it is fairly evident that this activity can 
 only be expended in the storing up of chemical-potentiai-energy, 
 since no other work is being done. 
 
 Then we have expended by the battery an activity of C 2 R or 
 C E watts ; this amount of chemical-potential-energy being lost 
 each second in the chemical changes there taking place. Of this 
 C (E e) watts reappear as heat evolved per second ; while C e 
 watts are stored up each second in the electrolytic cell in the 
 form of chemical-potential-energy, chemical decomposition being 
 there effected. 
 
 Case of charging a secondary battery. In the charging of a 
 secondary battery whose back-E.M.F. during charging is e, we 
 store up the activity C e, and waste in heat the activity C (E - e). 
 Hence it is more economical to charge the cell with a battery 
 whose E.M.F. E is only a little greater than e ; for then we store 
 up nearly the whole activity. But if E nearly equals e, the current 
 C will be small, and the process of charging will be a long one. 
 As a rule we compromise matters, and do waste a good deal of 
 activity in heat in order to get the cell charged within a reason- 
 able time. 
 
 10. Numerical Examples. 
 
 (i.) A battery has E.M.F. = 50 volts ; the total resistance is 20 
 ohms. Find the current in amperes, and the activity (or work per 
 second) in ergs per second, in watts, and in horse-power. 
 
 Here, 
 
 C = = ^ = 2 1. amperes. 
 R 20 2 
 
 Activity = CE = lx5o=i25 watts. 
 
 2 
 
 Activity = CExio 7 = i25xio 7 ergs per second. 
 
 Activity - C E -f- 746 = -?| horse-power. 
 746 
 
248 ELECTRICITY CH. xv. 
 
 (ii.) In the same case the external resistance R e is 15 ohms, and 
 the internal resistance R ; is 5 ohms. Find the external and internal 
 activities both in watts and calories per second. 
 
 Here, 
 
 'External activity = x 125 =-? x 125 watts. 
 K 4 
 
 External activity = 3 - x 125 x -24 calories-per-second. 
 4 
 
 Internal activity =- ixi25 = -xi25 watts. 
 K 4 
 
 \Internal activity = - 1 . x 125 x -24 calories-per-second. 
 4 
 
 (iii.) A battery has E.M.F. 20 z/0//.y and current 10 amperes. Find 
 the total heat per second in calories. 
 
 Here H = C x E x -24 -= 10 x 20 x -24 calories-per-second. 
 
 (iv.) A battery has E.M.F. = 10 volts ; current is observed to be 
 6 amperes ; and in the circuit are a set of electrolytic cells of total 
 back-E.M.F. equal to 4 volts. 
 
 What is the chemical activity (or ejiergy-per- second) stored up in 
 the electrolytic cells ? And how many calories per second are given 
 out? 
 Here, 
 
 - J Chemical activity stored up = ^=4x6 = 24 watts. 
 {Heat activity = C (E ^) =6x6 = 36 watts. 
 
 ii. Failure of a Smee's Cell to Decompose Water. If the 
 back-E.M.F. e of an electrolytic cell would be greater than the 
 E.M.F. E of the battery, then such a battery will fail to drive a 
 current through, and decompose, such a cell. 
 
 For if the action be supposed to take place, we get . * : . . 
 
 (i.) C = - = a negative quantity; which would mean that 
 _K 
 
 the electrolytic cell would be driving a reverse current through the 
 battery-cell ; and that the latter would not be decomposing the 
 former. 
 
 (ii.) Again, we should have the activity C E expended by the 
 battery less than the activity C e expended on the electrolytic 
 cell, which would be a breach of conservation of energy. 
 
 It is for this reason that a Smee's cell cannot decompose 
 water. 
 
CH. xv. JOULE'S LAW 249 
 
 If E = <?, then we have a stand-still ; and this will be the case 
 whatever the relative size of battery- cell and of electrolytic cell. 
 
 The current is C = "^ = o (by hypothesis). 
 _K 
 
 12. Partial Polarisation in the foregoing Case. In such a 
 case, however, we get a partial polarisation. Referring to Grothiiss's 
 hypothesis (Chapter XII. 4), we may add that we do not get the 
 full back-E.M.F. e until the chain of interchanges of ions is com- 
 pletely established. At first, no doubt, some + ions will during 
 their temporary disengagement from their nP partners attach them- 
 selves to the kathode and anode respectively ; without, however, a 
 complete interchange of partners occurring throughout the entire 
 chain between the electrodes. As this action occurs to a greater 
 and greater degree, so will the back-E.M.F. rise, until (in the case 
 of the last section) there is finally a balance and a stand-still. 
 
 13. Connection between E.M.F.s and 'Heats of Combina- 
 tion.' 
 
 (I.) The passage of a unit quantity of electricity through the 
 battery cell is accompanied by a definite amount of chemical 
 action. Thus, in the case of a voltaic cell in which platinum and 
 zinc are immersed in dilute sulphuric acid, the passage of a unit of 
 electricity is accompanied by the solution of an electro-chemical 
 equivalent of zinc, and the setting free of an electro-chemical 
 equivalent of hydrogen. In all cases, if the whole action be con- 
 fined to the cell and no current be sent through an external 
 circuit, each such chemical action fixed in nature and in amount 
 is accompanied by the evolution of a fixed quantity of heat. This 
 heat is the full measure of the chemical potential energy lost by 
 the cell, and its determination for each particular case is a matter 
 of experiment. 
 
 Let us denote it, in the general case, by H calories. Then we 
 
 have that ; 
 
 In the passage of one unit of electricity through the circuit the 
 cell loses chemical potential energy whose measure in heat is H 
 calories. 
 
 (II.) Now Ex i, or E, measures the electrical work done by 
 the battery upon unit quantity of electricity passing through it; 
 or we must write (E x -24) if we express this work in calories. 
 (See p. 24 r, second paragraph ; and p. 242, last two lines.) 
 
250 ELECTRICITY GH. XT. 
 
 Hence ............. . ". " '. , ' .. . 
 
 In the passage oj one unit of electricity through the circuit the 
 cell does electrical work measured by E x '24 calories. 
 
 (III.) Now, experiments tend to show that in cells devoid of 
 local action the former energy equals the latter ; or that . . . .L 
 
 24 
 
 This theory can be and has been tested by comparing the 
 values of E calculated in this way from an experimental know- 
 ledge of H, with values measured by electrical methods. So far, 
 then, as experiment has been able to go, there appears to be a 
 very intimate connection between intensity and quantity of chemi- 
 cal action on the one hand, and E.M.F. and quantity of electrical 
 movement on the other, respectively. 
 
 (IV.) Different E.M.F.s of different cells. The above reason- 
 ing and results throw much light on the differences in E.M.F. that 
 exist between various cells in all of which zinc is dissolved in 
 dilute sulphuric acid. 
 
 The difference lies in the different methods of getting rid 
 of the hydrogen. In every case this involves an absorption of 
 energy. 
 
 In Smee's cell the hydrogen is liberated in the free state from 
 the platinised silver plate. The separation of this hydrogen from 
 the sulphuric acid absorbs a considerable part of the energy sup- 
 plied by the solution of the zinc. The energy of the current and, 
 consequently, the E.M.F. are comparatively small. 
 
 In Daniell's cell the hydrogen, instead of being liberated in the 
 free state, is allowed to decompose sulphate of copper in solution. 
 This decomposition is accompanied by evolution of energy which 
 restores in part that absorbed by the liberation of hydrogen from 
 the acid. Here, therefore, the current energy and E.M.F. are 
 higher. 
 
 In Grove's cell the hydrogen decomposes nitric acid, yielding 
 thereby nearly as much energy as its original liberation had absorbed. 
 Here, therefore, the energy supplied is nearly that yielded by the 
 solution of the zinc, and the E.M.F. is very high. 
 
CH. xv. JOULE'S LAW 2$l 
 
 Note. It may naturally occur to the reader to ask why in the table of 
 Chapter XIV. 15 we are given different values for the E. M.F.s of a Volta's 
 cell and of a Smee's cell, although in both the hydrogen is simply set free. 
 
 We may say that the E. M.F.s as calculated chemically correspond to the 
 E. M.F.s of the batteries when a current is running and when there is neverthe- 
 less no unknown polarisation. 
 
 It is, therefore, almost hopeless to expect agreement between the values of 
 E.M.F. as calculated chemically, and as measured with open circuit, respec- 
 tively, when we are dealing with ' non-constant ' cells ; for we do not quite 
 know our data for calculation. With ' constant ' cells the agreement is 
 closer. 
 
 There is, however, little doubt but that there would be exact agreement 
 if we knew the exact chemical action in the cell at the time when its E.M.F. 
 was electrically measured. 
 
 (V.) The expression for the E.M.F. given in (III.) shows us from 
 another point of view why a Smee's cell cannot decompose water. 
 Since in this case the back-E.M F. e is greater than the E.M,F. E 
 of the Smee's cell, it follows that the heat per second used up in 
 decomposing the water would be greater than the heat per second 
 that could be supplied by the Smee's cell. 
 
252 ELECTRICITY CH. xvi. 
 
 CHAPTER XVI. 
 
 THERMO-ELECTRICITY. 
 
 i. Introductory. In the cells with which we have up to 
 the present point been concerned, we have had certain arrange- 
 ments by means of which chemical-potential-energy could be 
 converted into electrical energy; the materials of the cell under- 
 going permanent chemical change. Any such arrangement (gene- 
 rally of two metals and one or two liquids) was called a voltaic 
 cell. 
 
 If we passed a current through such a cell we either lost 
 chemical-potential-energy and gained heat energy, or we gained 
 chemical-potential-energy and used up electrical energy that would 
 otherwise have appeared in the form of heat energy \ according to 
 the direction in which we drove the current. (We refer to the 
 <CV of Chapter XV. 9.) 
 
 In the present Chapter we shall see that it is very easy to 
 make certain arrangements (consisting simply of metals, without 
 any liquid or other non-metallic body) by means of which heat 
 energy may be converted direct into electrical energy \ the metals 
 employed undergoing no chemical change. Such an arrangement, 
 which gives us an electric current by the application of heat only, 
 is called a thermo-celL 
 
 We shall see further that if we pass a current through a 
 thermo-cell we have certain effects (called Peltier and Thomson 
 effects) consisting in disengagement or absorption of heat. These 
 effects are entirely distinct from the Joule effect of Chapter XV. 
 4, and answer to the chemical work Ce that is done when we 
 pass a current through an electrolytic- cell (see Chapter XV. 9). 
 
 In what follows we shall discuss in an elementary manner the 
 subject of thermo-cells, taking matters in the following order. 
 
 (I.) We shall first give the main facts, established by experi- 
 
CH. XVI. 
 
 THERMO-ELECTRICITY 
 
 253 
 
 ment, as to the construction of a thermo-cell, and the dependence 
 of the E.M.F. on the nature of the metals employed and on the 
 temperatures at which the different parts of the cell are main- 
 tained ( 2 to 7). 
 
 (II.) We shall give some account of the facts with respect to 
 the Peltier and Thomson effects referred to above ( 8 and 9). 
 
 (III.) We shall say something with respect to the theory of 
 the subject ( 10 and n). But, as is usual in elementary books, 
 we shall not deal at all fully with the theoretical part of the subject. 
 
 2. The Simple Thermo- Cell. If a circuit be made of two 
 metals A and B, having soldered junctions [one of which we de- 
 note by A|B, the other by B|A], we have seen that the two metals 
 are as a rule at different potentials, but that there is no current (see 
 Chapter XL 3) ; it being assumed as a condition that the whole 
 circuit is at one temperature. 
 
 If, however, the two junctions are at different temperatures, we 
 find in general that a current flows through the circuit, or that 
 
 FIG. L 
 
 there is in the circuit under these conditions a resultant E.M.F. 
 (The reader is requested to notice the simple manner in which we 
 denote the junctions across which we pass from A to B, and from 
 B to A, respectively.) 
 
 The figures here given indicate two equivalent forms of the 
 simple thermo-cell. We may either have the two metals A and B 
 only, with A|B at tf C. and B|A at // C; or we may connect A 
 
254 
 
 ELECTRICITY 
 
 and B by a third metal D, having both the junctions A|D and 
 D|JB at /i C. It can be proved experimentally, and it follows 
 from Chapter XI. 3, that the two forms are exactly equivalent. 
 
 3. The Thermo-Pile. The E.M.F.s of thermo-cells are very 
 small, while their internal resistance, as they consist entirely of me- 
 tallic bars or thick wires, is very small. Hence it is usual to couple 
 up many simple cells in series. If n such cells be coupled up we 
 have an E.M.F. that is n times the E.M.F. of one cell, while the 
 internal resistance is still small. 
 
 The figure here given, fig. i., indicates how several cells may 
 be joined in series. It is evident that if we have an E.M.F. in 
 case (ii.)of the last section, then we shall in the present case have 
 the several E.M.F.s acting in the same direction. 
 
 In the next figures, figs. ii. and iii., it is shown how such 
 cells may be packed into a very small space. Nobili made his 
 
 cells of thick Bi and Sb wire; 
 and, by folding the cells into 
 layers with insulating material 
 between, he packed a large 
 number of cells into the form 
 
 FIG. ii. 
 
 FIG. 
 
 of a cube. The opposite faces 
 of this cube consisted of the 
 alternate sets of junctions; so 
 that if the two faces were ex- 
 posed to different tempera- 
 
 tures /! and / 2 , then the alternate junctions were heated to /j and 
 
 ^2 respectively, as in fig. i. above. 
 
 4. Thermo -Electric Series. Referring to the simple cell of 
 
 2, it may be stated that the E.M.F, depends on three conditions. 
 
 They are ................ , ; . 
 
CH. XVI. 
 
 THERMO-ELECTRICITY 
 
 o:> 
 
 (a) The nature of the metals. 
 
 () The difference of temperature of the two junctions. 
 
 (c) The absolute temperature of the junctions. 
 
 Thus we have (a) a different E.M.F. if we make our cell (cceteris 
 paribus] of Bi and Fe from what we have if we make it of Bi and 
 Sb. Again, we have in general (b) a different E.M.F. if the differ- 
 ence of temperatures of the junctions be 20 C., from what we 
 have if (cateris paribus) it be 50 C. And, finally, we have in general 
 (c) a different E.M.F. if the junctions be at, e.g., 20 C. and 22C. 
 from what we have if (cceteris paribus] they be at 50 C. and 52 C. 
 
 Thermo-electric series at 20 C. Hence we cannot give a list 
 of metals showing their thermo-electric relation to one another 
 for all temperatures, but only for some stated temperature. 
 
 In the following are given in micro-volts the E.M.F.s of 
 various cells, one metal in each case being lead, and the other the 
 metal opposite to which the number stands. The junctions in 
 each case are at 20^ C. and 19^ C. respectively ; or there is a 
 temperature-difference of i C., while the mean temperature is 20 C. 
 
 For the metals that occur higher in the list than lead, the 
 current passes from that metal through the hotter junction to 
 lead ; for those lower in the list than lead, the current passes from 
 lead to that metal through the hotter junction. This is indicated 
 by the sign of the E M.F. Experiment shows also that the E.M.F. 
 of a cell composed of any two metals is simply the algebraic 
 difference of their respective E.M.F.s with reference to lead. 
 
 (The list is taken from Lupton's Numerical Tables.) 
 
 E.M.F.s, in micro-volts, of cells whose junctions are at 20^ C. and K)\ C. 
 respectively ; lead being one of the metals in each case. 
 
 Bismuth, pressed ; comm. 
 
 + 97-0 
 
 Antimony, pressed . ' y 
 
 - 2-8 
 
 Bismuth, pressed ; pure . 
 
 89-0 
 
 Silver, pure; hard. . * 
 
 -3-0 
 
 Cobalt . . . . 
 
 22 -O 
 
 Zinc, pressed ; pure . 
 
 - 37 
 
 German silver " . 
 
 H75 
 
 Copper, electrolytic ; . 
 
 -3-8 
 
 Mercury . 
 
 418 
 
 Antimony, pressed ; 
 
 
 Lead . - , . - . 
 
 O'O 
 
 Commercial ... ^ 
 
 - 6-0 
 
 Tin 
 
 -I 
 
 Iron wire, soft . .. 
 
 - I7-5 
 
 Copper, comm. . 
 
 -I 
 
 Antimony crystal, axial . 
 
 -22-6 
 
 Platinum 
 
 - -9 
 
 Antimony crystal, equatorial 
 
 - 26-4 
 
 Gold . 
 
 - 1-2 
 
 Selenium . * . 
 
 - 807-0 
 
 UNIVERSITY OF CALIFORNIA 
 
256 ELECTRICITY CH. xvi. 
 
 Explanation of table, The following two or three cases may 
 explain further what the table means. 
 
 (i.) If we make a cell of cobalt and lead, the junctions being at 
 20^ C. and 19^ C.,the E.M.F. will be 22 x io- b volts ; and the cur- 
 rent will pass from cobalt to lead through the hotter junction. 
 
 (ii.) With lead and silver the E.M.F. is 3 x io~ 6 volts ; and the 
 current passes from lead to silver through the hotter junction. 
 
 (iii.) With pure bismuth and cobalt \ht E.M.F. is (89 22) x io~ 6 = 
 67 x 10 ~ 6 volts ; and the current passes from former to latter through 
 the hotter junction. 
 
 (iv.) With cobalt and pure silver the E.M.F. is [22 - ( - 3)] x io~ 6 = 
 25 x io- 6 volts ; and the current passes from former to latter through 
 the hotter junction. 
 
 Such a series is called a * thermo-electric series at the tempera- 
 ture (i.e. 20 C.) in question.' 
 
 5. Thermo -Electric Powers, The effectiveness of any par- 
 ticular cell (i.e. a cell composed of any two specified metals), at 
 any specified mean temperature f C., may be reasonably measured 
 by the E.M.F. of the cell when the junctions have a temperature 
 difference of i, the mean temperature being t C. 
 
 To express this ' effectiveness ' we use the term thermo-electric 
 power. The meaning of this term may be seen from the following 
 statement. 
 
 The thermo-electric power of any pair of metals at t C. is 
 measured by the E.M.F. of a cell composed of these metals when there 
 is a temperature difference of i C. between the junctions, the mean 
 temperature being t C. The E.M.F.s are usually expressed in 
 micro-volts. 
 
 The more exact meaning of thermo-electric power may perhaps 
 be further explained with some advantage. It is found experi- 
 mentally that if the junctions are at any two temperatures 
 
 / 2 and /! of which the mean is *, and if E^ be the E.M.F. of the 
 cell under these conditions, then .-, 
 
 E' J 
 The thermo-electric power at t C. = ^ 
 
 where t = 
 
 
CH. xvi. THERMO-ELECTRICITY 257 
 
 Example. Thus, if 30 C. were taken as / C. , we have the thermo-electric 
 
 < < E S : 
 
 power at 30 C. given by any such fraction as ^-, , , &c. 
 
 Or we may say that if we know the thermo-electric power at 
 t C. in micro-volts, then we get the E.M.F. in micro-volts when 
 the junctions are at temperatures (/ + 0) C. and (t 6) C. by 
 multiplying the thermo-electric power at / C. by the temperature 
 difference 2 0. 
 
 Note.\ide be the infinitesimal E.M.F. due to the infinitesimal difference 
 dQ in temperature of the junctions, we may in infinitesimal notation say that . 
 
 The thermo-electric power at t C. = f 
 
 If the thermo-electric power of a single metal be spoken of, it 
 is understood that lead is the other metal. 
 
 A metal A is said to have a greater thermo electric power than 
 a metal B at t C. when it stands higher in the list for /C.; in 
 this case the current passes from A to B through the hotter 
 junction. 
 
 6. The Neutral Point It is found experimentally that for 
 every pair of metals there is a particular temperature possessing 
 properties to be described in the present section. This tempera- 
 ture is, for reasons to be given later on, called ' the neutral point ' 
 for that pair of metals, and is usually designated by the letter T. 
 This temperature, T C., is different for each pair of metals. Sup- 
 posing that we are dealing with two metals A and B, whose neutral 
 point is T, the facts observed are as follows. 
 
 (I.) If one junction, say A | B, be at T while the other junc- 
 tion be at some higher temperature t^ there will be a certain 
 
 E.M.F. which we may call E^, and the current will pass from one 
 
 metal to the other (say from A to B) through the hotter junction, 
 i.e. through that at /,. We should for these temperatures say that 
 A is above B in the thermo-electric series. The higher /j be- 
 comes, the greater is the E.M.F. 
 
 (II.) If the temperature of the other junction be lowered from 
 T to some lower temperature / 2 > step by step, the E.M.F. (which 
 
 we may now call E ') decreases step by step ; and when t^ and t. 2 
 
 s 
 
ELECTRICITY 
 
 CH. XVI. 
 
 are equidistant from T, or when /, T = T / 2 , the E.M.R is 
 zero. 
 
 (III.) If the temperature of the junction that was at T be 
 lowered still further, so that T / 2 is greater than ^ T, an E.M.F. 
 acting in a reverse direction to the former is observed. This in- 
 creases step by step as the temperature / 2 is still further lowered 
 step by step. 
 
 (IV.) If now the upper temperature t be brought down to T, 
 this reverse E.M.F. is still greater. 
 
 We find, in fact, that the current flows from A to B, or from 
 B to A, across the hotter junction, according as /, T is greater 
 or less than T / 2 respectively ; or that A and B change their 
 
 thermo-electric order when the mean temperature of the 
 
 junctions crosses the neutral temperature T. 
 
 (V.) As we should perhaps have anticipated from the fact that 
 when /! T = T / 2 we get no E.M.F., we find that, e.g., if one 
 junction be 20 above the neutral point T while the other is 10 
 below it, we have the same E.M.F. as if the former were 20 above 
 T, and the latter were 10 above T. It is usually said that A and B 
 have zero thermo-electric power at T ; and that above T and below 
 T their thermo-electric powers are of opposite sign respectively. 
 
 Note. In more advanced treatises it will be found that certain pairs of 
 metals have more than one neutral point. But such cases will not be considered 
 here. 
 
 Experiment. The following experiment will illustrate the above statement. 
 Round the ends of a piece of iron wire are twisted two pieces of copper 
 
 wire. The ends of 
 these latter are con- 
 nected, by means of 
 mercury cups m m', with 
 a galvanometer G. This 
 latter must be delicate, 
 and, on account of the 
 higher resistance due to 
 the bad contact of the 
 Cu and Fe, it may be 
 of higher resistance than 
 the form commonly used with a simple thermo-cell. The end A is left at the 
 temperature of the air, say about 20 C, while the end B is heated in a 
 Bunsen's flame, For iron and copper the neutral point T is about 260 C. 
 
 Te wire 
 
 B 
 
CH. XVI. 
 
 THERMO-ELECTRICITY 
 
 259 
 
 As B rises in temperature an increasing current is observed ; its direction 
 can be shown to be from Cti to Fe across the hotter junction. This current 
 increases until B is at 260 \ It then decreases, and becomes zero when Bis at 
 500 C. , i.e. is zero when /, - T = T / 2 . On raising B above 500 C. a current 
 in the reverse direction is observed ; and this increases as the temperature rises 
 still further. 
 
 It is clear, from what has been said in this section and in 4, 
 that two metals A and B have zero thermo-electric power at .their 
 neutral point T. 
 
 7. Ther mo-Diagrams. From the experimental and theoreti- 
 cal investigations of Professor Tait, Sir W. Thomson, and others, 
 it is found that the thermo-relations of most metals for all temper- 
 atures (at least for a wide range of temperature) can be simply 
 exhibited in one diagram. We shall give and explain this graphic 
 method ; the reader understanding that the construction of the 
 diagram follows from 
 the experimental lesults 
 given in 5 and 6. 
 In fig. i. O v is the axis 
 of ordinates, along 
 which measurements 
 represent micro-volts ; 
 and O t is the axis of 
 abscissae along which 
 measurements repre- 
 sent degrees Centigrade. 
 The diagram is drawn 
 to scale suitably, so 
 that the lengths of or- 
 dinates and abscissae 
 give micro- volts and de- 
 grees respectively ; but 
 the ordinates and ab- 
 scissae need not be 
 measured in the same 
 units of length respec- 
 tively. The three straight lines given represent the metals lead, 
 metal A, and metal B, respectively ; in what sense they ' repre- 
 sent ' these metals will soon be seen. 
 
 Let us erect ordinates from the points / 2 > A and t^ on the line 
 
 \? 
 
 
 </ 
 
 ^ 
 
 Lead' 
 
 t, T 
 
 FIG. i. 
 
260 ELECTRICITY CH. xvi. 
 
 O /, answering to the temperatures /./, /, and tf C., respectively ; 
 and let A L and B L measure the thermo-electric powers of 
 metal A, and metal B, respectively, with respect to lead at / J C., 
 so that also A B represents the thermo-electric power of A with 
 respect to B at t C. Let t { and / 2 be equidistant from /, so that 
 /,_/=/_/,. 
 
 Then by simple geometry the area of the trapezium AJ A 2 L 2 L^ 
 is given by the .product AL x Lj L 2 ; that is, by the product 
 of the thermo-electric power at t C., measured by A L, into 
 the difference of temperature / 2 - t { of the junctions, measured 
 
 by L,L 2 ; where / = -^ -. 
 
 But by the result of experiment as given in 5, this product 
 gives us the E.M.F. E^ 2 of a cell composed of the metal A and 
 
 lead, in which the junctions are at / 2 an d /]. 
 
 Hence we have that if the line A L measure the thermo-electric 
 power of metal A with respect to lead at t C., then the area 
 
 A! A 2 L 2 L! gives the E.M.F. E^ 2 of the cell for temperatures t^ 
 
 and / 2 of the junctions ; where / is the mean of /j and t. 2 . 
 
 By experiment it is shown that this is true for all temperatures, 
 i.e. wherever / be taken. 
 
 From mathematical reasoning it follows that if the line repre- 
 senting lead be straight, then the line representing the metal A 
 must also be straight. This line can therefore be determined by 
 finding two points ; or by determining the thermo-electric power 
 of metal A at two temperatures. 
 
 The same reasoning applies to metal B ; so that the area 
 B! B 2 L 2 L! measures the E.M.F. of a cell of metal B and lead at 
 temperatures t\ and / 2 . 
 
 The area Aj A 2 B 2 B, is the difference of these areas ; and, as 
 experiment shows, measures the E.M.F. of a cell composed of 
 metal A and metal B at temperatures ^ and t 2 . If, therefore, we 
 draw a straight line for lead, and determine straight lines for other 
 metals (by determining their thermo-electric powers with respect to 
 lead for two temperatures), the diagram so formed gives us 
 measurements of the E.M.F. of any cell at any two temperatures 
 of junctions, by simple measurement of areas. Such a diagram, 
 then, expresses the results of experiment given in 5. 
 
CH. xvr. THERMO-ELECTRICITY 26 1 
 
 Now we will show more clearly how the properties of the 
 neutral point are exhibited in the diagram. 
 
 Suppose that the lines of the metals A and B cross at a point N 
 lying in the ordinate from the temperature T on the axis of abscissae. 
 
 We noticed that A B measures the thermo-electric power of 
 A with respect to B at t C. Then at T C. this thermo-electric 
 power becomes zero, since the intercept A B is here a point. Hence. 
 T C. must be the neutral point of the two metals. 
 
 On the other side of this neutral point the thermo-electric 
 powers and E.M.F.s are reversed in sign, but are otherwise 
 measured as shown above. 
 
 If, therefore, we reckon any area such as N B 3 A 3 as negative 
 in sign, it is clear that the algebraic sum of the areas A, N B, and 
 N B 3 A 3 , or the total area (having regard to sign) between the 
 lines of the metals and the two ordinates from /, and / 3 respec- 
 tively, will agree with the experimental results of 6 in measuring 
 the E.M.F. of the cell when the junctions are at t { and/ 3 respec- 
 tively. When /, T = T f 4 the areas are equal and their algebraic 
 sum is zero. If / 4 were still further on, so that T / 4 were greater 
 'than /, T, the sum of the areas would change sign ; indicating the 
 change in sign of the E.M.F. of the cell. 
 
 Thus the diagram conveniently embodies the experimental 
 results referred to in 5 and 6. 
 
 On the next page we give a diagram in which the lines of many 
 metals are given. After the above discussion it needs no comment. 
 
 Taifs formula. The area of any trapezium, such as A, A 2 B 2 B b 
 is given by the product of A B into Lj L 2 , or by the breadth of the 
 trapezium into the mean between its parallel sides. But, from the 
 triangle N A, B,, it is evident that A B is proportional to T t\ 
 
 i.e. to T - / ' + -. And the length Lj L 2 measures t^- t v 
 Hence, the area A 1 A 2 B 2 B 1 is proportional to the product 
 
 (/._, /j) (T - 1J ; or we may say that, for the two metals 
 
 in question, A 
 
 where k is a constant depending on the units of measurement 
 
262 
 
 ELECTRICITY 
 
 CH. XVI. 
 
CH. xvi. THERMO-ELECTRICITY 263 
 
 employed and upon the situation of the lines of the metals, that is 
 upon the metals. When k is once found for any pair of metals, 
 the formula can then be used for all changes made in t\ and / 2 for 
 those two metals. 
 
 The student can, though not quite so easily, show that the 
 above formula represents the result arrived at graphically when 
 the two temperatures are on the opposite sides of the neutral 
 point, attention being paid to the fact that in this case the areas 
 in the diagram are of contrary signs. 
 
 8. Peltier Effect ; Observ-d Facts, We have seen that, when 
 a current is passed through a conductor, heat is disengaged at a 
 rate proportional to the resistance of the conductor and to the square 
 of the current- strength. This is usually called the Joule effect. 
 
 In 1834 Peltier discovered that when an electric current 
 passes from a metal A to a metal B across their junction, heat is 
 absorbed or given out at this junction. The absorption takes 
 place if A have a higher thermo-electric power than B, and the 
 disengagement if A have a lower thermo-electric power than B. 
 Thus, if the direction of the current be reversed, we have heating 
 at a junction where before there was cooling, and vice versa. This 
 rate at which the heat is absorbed or evolved is found to depend 
 upon the nature of the metals ('not upon their resistance), upon 
 the temperature of the junction, and upon the current- strength (not 
 upon the square of this). It is thus quite distinct from the Joule 
 effect. This latter may mask the Peltier effect if the current be 
 too strong or the resistance at the junction be too great. How- 
 ever, if we drive the same current first in one direction and then 
 in the other, and measure the heat per second evolved, we obtain 
 (Joule effect) + (Peltier effect), and (Joule effect} (Peltier effect), in 
 the two cases respectively. 
 
 If the reader will refer to the meaning of higher ' thermo-electric 
 power,' given earlier in 4 and 5, he will see that the above- 
 stated discovery of Peltier's involves the following : that when a 
 current is passed across a junction this will be so heated or cooled 
 as to originate always a thermo-electric E.M.F. opposed to that of 
 the current in question. Thus, if A have a higher thermo-electric 
 power than B, and if the current pass from A to B, we have stated 
 that the junction is cooled : and this will create an E.M.F. tending 
 to drive a current from B to A across this cooler junction. The 
 
 Jk- 
 
. 
 
 264 ELECTRICITY CH. xvi. 
 
 Peltier effects are thus analogous (in a degree) to the creation of 
 counter E.M.F.s arising from the passage of a current through 
 electrolytic cells, or through electro-motors (see Chapter XXIV.). 
 
 Experiment. A cross may be made of fairly thick bars of bismuth (B B') 
 and antimony (A A'), having a soldered junction at m ; bars 3 inches long and 
 inch diameter will answer very well. Wires from the ends A B can be 
 connected with a galvanometer G ; and wires from A'B' with a battery not 
 drawn in the figure. By means of a suitable arrangement of mercury cups 
 these connections, of A B with the galvanometer and of 
 A' B' with the battery respectively, can be made alter- 
 nately with ease j it being necessary only to ' rock ' the 
 cross slightly. 
 
 If the current be first passed from Bi to Sb through 
 B' A' (the former metal being of higher thermo-electric 
 power than the latter at usual temperatures), and if 
 A and B be then connected with G, the deflexion will indicate a cooling of 
 the junction. [We can cool the junction with ether and observe a deflexion 
 in the same direction.] 
 
 If the current be passed through A' B' from Sb to Bi, and if then A B be 
 connected with G, we observe a deflexion in the opposite direction, indicating 
 that the junction has been heated. 
 
 Caution. If the current be too strong the general 'Joule effect' 
 heating may mask the Peltier effect. It is well to try two or one 
 Smee's cells ; and, if the current is too strong even with one cell, to 
 interpose resistance by means of a rheostat or resistance box until 
 the Peltier effect is sufficiently marked. The Joule effect falls off, as 
 the current weakens, as the square of the current ; the Peltier effect 
 only as the current. 
 
 9. The < Thomson Effect.' Sir W. Thomson pointed out 
 the existence of another very important heat phenomenon ; and 
 Professor Tait continued and added to his work. The combined 
 result may be given somewhat as follows. 
 
 1 In general, when a current passes through any portion of a 
 homogeneous conductor where the temperature is not uniform, it tends 
 to heat or to cool such a portion ; the amount of the action at each 
 point being proportional to the absolute temperature of that point' 
 
 Thus, in some metals, as (e.g.) in copper, the current passing 
 from a cold to a hot region absorbs heat throughout, but most in 
 the hotter region whose absolute temperature is the higher, thus 
 tending to reduce temperature-difference ; while when it passes 
 from hot to cold it disengages heat throughout, but most in the 
 hotter region, thus tending to exaggerate temperature-difference. 
 
CH. xvi. , THERMO-ELECTRICITy 265 
 
 In other metals, as (e.g.) in iron, the effect is just the reverse ; 
 in these the passage of a current is accompanied by an absorption 
 of heat when passing from a hot to a cold part, and by an evolution ot 
 heat when passing from a cold to a hot part. These phenomena 
 are known as the ' Thomson effects/ and they have been found to 
 accompany the passage of an electric current in all simple metals 
 with the exception of lead. 
 
 A little consideration will show that, so far as the Thomson effects 
 are concerned, an electric current in a copper conductor behaves in an 
 analogous manner to a material fluid having specific heat and flowing 
 sufficiently slowly to acquire at each point the temperature of the con- 
 ductor at that point. For this reason the effects are sometimes 
 ascribed to a specific heat possessed by electricity, the amount of heat 
 absorbed by the passage of a unit quantity of electricity in passing 
 from one part of a conductor to another one degree higher in tempera- 
 ture being called the specific heat of electricity in the given conductor. 
 The specific heat of electricity is not a constant quantity, but depends 
 upon the conductor in which it flows. To account for the Thomson 
 effect in iron, electricity must be supposed to possess a negative 
 specific heat in that metal, while in lead it is zero. In all cases the 
 specific heat of electricity is found to vary as the absolute temperature. 
 
 10. Theory of the Simple Thermo-Cell, We shall now give 
 some account of the theory of the simple thermo-cell. But, as 
 stated earlier, we cannot enter at all fully into the subject. 
 
 (I.) Two metals in contact, the junctions at the same temper- 
 ature. It seems beyond, dispute that at all temperatures, excepting 
 the 'neutral temperature,' two metals in contact are at a A V ; the 
 amount of this A V depending on the temperature. Could we 
 make a cell of two metals A and B having one junction A | B, and 
 then join the other ends by some conductor that was at the same 
 potential aseac/i metal, we should have a circuit with an E.M.F. in 
 it and should get a current. But, as seen in Volta's law of Chap- 
 ter XI. 3, whether we join the other extremities direct or through 
 another metal, we must always have an equal and opposite A V at 
 
 the other junction, and so have a resultant zero E.M.F. in the cir- 
 
 t 
 cuit. If A | B represent the E.M.F. from A to B at the one junc- 
 
 t 
 tion, and B | A the E.M.F. from B to A at the other junction, it 
 
 t t 
 
 is clear that A | B = B | A. Hence, as we go completely round 
 
 the circuit, we have, in the notation of 5 
 
266 ELECTRICITY CH. xvi. 
 
 EWA|fc AJB = O. 
 
 (II.) Two metals ; the junctions at different temperatures ; the 
 E.M.F.s at the junctions only being considered ; compare with 
 
 (IV.\ If now the junctions be at different temperatures t\ and A 2 > 
 
 /, * 3 
 
 we shall in general have A | B different from A | B. Hence, as 
 
 regards the E.M.F.s that occur at the junctions only, we have 
 a resultant E.M.F. in the circuit given by ; . 
 
 / , t l 
 
 E*~A|B-A|B, 
 
 The reader must observe that we do not state that the E.M.F.s 
 at the junctions are the only E.M.F.s in the circuit. We shall see 
 in (IV.) that in general such is not the case. These junction- 
 E.M.F.s occur abruptly, and may be measured by the abrupt 
 change of potential that occurs at the junctions, the circuit being 
 broken at some point where it is homogeneous (i.e. not at a 
 junction). 
 
 (III.) One metal ; its extremities at different temperatures. When 
 any metal (the one known exception being lead) has its extremities 
 at different temperatures / 2 and f l9 there is in general an E.M.F. 
 either from hot ter cold or from cold to hot, according to the 
 nature of the metal. Thus, in the two cases respectively, it will 
 be found that the hotter end has the lower or the higher potential ; 
 the end towards which the E.M.F. acts being raised to the higher 
 potential. This rise or fall of potential is 1 'gradual. If a complete 
 circuit of one unequally heated metal be made, there will be no 
 resultant E.M.F. if the temperatures change gradually from point 
 to point ; for in that case the E.M.F. due- to the rise of tempera- 
 ture through one part of the circuit will be balanced by an exactly 
 equal but opposite E.M.F. due to the fall of temperature through 
 the rest of the circuit. This will be true even if the rise and fall 
 be very different in * steepness ' ; provided that there is no abrupt 
 change of temperature at any section. If, however either owing 
 to two very unequal heated ends touching, or owing to a very 
 abrupt alteration in the diameter of the conductor at any point 
 there be some abrupt discontinuity in the variation of temperature 
 from point to point, then there will be in general a resultant E.M.F. 
 in the circuit. It is generally said that this is the case when there 
 
CH. xvt. THERMO-ELECTRICITY 26? 
 
 is a finite change of temperature within a distance so small that 
 it may be considered to be within the limits of molecular action. 
 In such a case there is a A V occurring abruptly, as in (II.). 
 
 (IV.) Two metals ; the junctions at different temperatures ; the 
 E.M.F.s all round the circuit being considered ; compare with 
 
 (//.). 
 
 If there be two metals A and B with junctions at different tem- 
 peratures, the total E.M.F. in the circuit will be the algebraic sum 
 of (the E.M.F.s at the junctions) + (the E.M.F.s in the metals 
 A and B). The former were considered in (II.). As regards 
 the latter, these w r ould have a zero sum were the metals the same, 
 as stated in (III.). But in general a metal A w r ith its ends at 
 to and /j does not give the same E.M.F. as another metal B with 
 its ends at / 2 and /, ; and so in general the sum of these ' gradual 
 E.M.F.s' (as we may call them to distinguish them from the 
 abrupt junctioh-E.M.F.s) is not zero. In the case of lead 
 there is no E.M.F. due to its unequal heating. Hence, in a cell 
 of which one metal is lead w r e have a somewhat simpler sum of 
 E.M.F.s in the circuit. 
 
 (V.) Thus when the current is running in a thermo-cell of 
 junctions at /._> and /j respectively, the distribution of potential 
 in the circuit is a resultant distribution comprising three com- 
 ponents. 
 
 (i.) The abrupt changes of potential at the junctions. These 
 measure w r hat we may call the juncticn- E.M.F.s or Peltier- 
 E.M.F.s. 
 
 (ii.) The rises and falls of potential occurring gradually along 
 the conductors, due to the unequal heating of these conductors. 
 These measure what we may call the Thomson-E.M.F.s. 
 
 (iii.) The regular fall of potential that follows Ohm's law, 
 discussed in Chapter XIII. 
 
 The total E.M.F. in the circuit could be measured by break- 
 ing the circuit at some place where it is homogeneous (i.e. not at 
 a junction) and measuring the A V between the broken ends. 
 
 (VI.) Experiment tends to show that at the neutral point T C. 
 of two metals A and B there is no junction E.M.F. between 
 them, though this is hardly certain ; and most certainly shows that 
 in a cell where the junctions have T C. as their mean tempera- 
 ture, the total E.M.F. in the circuit is zero. 
 
268 ELECTRICITY 
 
 CH. XVI. 
 
 The above statements represent, in a necessarily very imper- 
 fect form, the present views as to the sources of the E.M.F. in a 
 thermo-cell. 
 
 For practical purposes it is simpler to depend upon the diagram 
 and formula of 7 ; since these (when fully given) embody all the 
 known results of experiment. 
 
 ii. Theory of the Peltier and Thomson Effects. 
 
 (I.) Let there be a cell of two metals A and B whose junctions 
 are at / 2 C. and t\ C. respectively. We will suppose, for the sake 
 of more clearly defining the particular case that we at first con- 
 sider, that / 2 ^ s higher than /j, and that both temperatures are 
 above T ; 'further we will suppose that, for temperatures above T, 
 the metal B is of higher thermo-electric power than A. This is 
 the case in which the total E.M.F. in the circuit is represented by 
 the area B 3 B 4 A 4 A 3 in the diagram, fig. i., of 7. 
 
 (II.) When a current is running we have manifested electrical 
 energy. If this energy be not converted (in part) into chemical- 
 potential-energy or into mechanical or other energy, it will all be 
 converted finally into heat energy. 
 
 Now this electrical energy must have been derived from some 
 original form of energy of which an equivalent amount must have 
 disappeared. 
 
 In the voltaic-cell it was chemical-potential-energy that was so 
 used up. 
 
 But in the thermo-cell the only available source of energy is 
 the heat energy supplied to the cell by a flame or other source of 
 heat. 
 
 (III.) When, therefore, the current does no other work it must 
 be that (i.)heat energy is absorbed, and disappears, some where in 
 the circuit ; (ii.) this is transformed into an equivalent amount 
 of electrical energy ; (iii.) and this again is finally transformed 
 into an equivalent of heat energy, distributed over the circuit 
 according to the resistances of the several portions of the circuit. 
 
 The question arises, from what part of the circuit is the original 
 heat energy derived ? 
 
 (IV.) In attempting to answer this question theoretically, we 
 must be guided mainly by two considerations. Firstly, we may 
 feel sure that the well-established law of ' Degradation of energy ' 
 will hold here as in all the phenomena of inanimate nature. That 
 
CH. xvi. THERMO-ELECTRICITY 269 
 
 is, the source of heat must be on the whole the hotter part of the 
 cell j and the general result of the action must be that the tem- 
 peratures of the cell tend to become equalised. Secondly, we must 
 remember that when a current runs against an E.M.F. work is 
 done ; and, if the work takes no other form, it will appear as heat 
 evolved. Further, it seems almost certain that this heat will be 
 evolved in that portion of the conductor in which the work is done, 
 i.e. in which the E.M.F. lies. Conversely, when a current runs 
 with an E.M.F., this E.M.F. does work on the current ; and, when 
 no heat is supplied, this work will be done at the expense of heat 
 absorbed from the conductor ; further, it seems almost certain 
 that this heat will be absorbed from that portion of the conductor 
 in which the E.M.F. lies. 
 
 (V.) The consideration of (IV.) will lead us to predict that, on 
 the whole, the hotter portion of the cell will be cooled and the 
 cooler portion will be heated ; while some of the heat derived 
 from the hotter portion of the cell will, after passing through the 
 intermediate form of electrical energy, reappear as heat distributed 
 round the circuit in accordance with Joule's law. 
 
 (VI.) Where the current crosses a section (e.g. a junction of 
 two different metals) at which occurs an abrupt E.M.F., or [see 
 10 (V.)] a 'Peltier E.M.F.,' there we should expect to have 
 heat absorbed or given out according as the current flows with the 
 E.M.F. or against it respectively. 
 
 Such an absorption or disengagement of heat, occurring at a 
 mere section and not over any finite length of the conductor, is 
 the true Peltier effect referred to in 8. 
 
 There is probably no Peltier effect when the junction of the 
 two metals is at their neutral temperature T ; for it seems probable 
 that at that temperature the two metals are as one. 
 
 (VII.) Where the current flows through an unequally heated 
 metal, it flows (in all metals excepting lead) with or against the 
 Thomson E.M.F. spoken of in (III.) and (V.) of 10. 
 
 We should then expect heat to be absorbed or given out re- 
 spectively, over a finite length of the conductor in question. This 
 is, in fact, observed in the Thomson effect of 9. 
 
 Since the unequally heated ends of each metal are at the two 
 junctions, the result of the Thomson effect will be to alter the 
 temperatures of the two junctions. Unless, therefore, special ex- 
 
2/0 ELECTRICITY CH. xvi. 
 
 periments are performed from which we can calculate each effect 
 separately, we shall in general observe the sum of these two effects 
 (i.e. of the Peltier and Thomson effects) at the junctions, and shall 
 not be able to ascribe the heating or cooling of the junctions to 
 the Peltier effect only. 
 
 (VIII.) We may then say that the transformations of energy re- 
 ferred to in (IV.) and (V.) take place through the intermediency of 
 the Peltier and Thomson effects together. 
 
 When the hotter junction is at the neutral temperature T, 
 then if the Peltier effect be zero, heat energy must be supplied 
 through the intermediency of the Thomson effect only. 
 
 (IX.) Now let us suppose that we cease to supply heat from 
 external sources. The junctions will arrive at the same tempera- 
 ture, by the cooling of the hotter and the heating of the cooler 
 junction. 
 
 If now the current be continued from some external source in 
 the same direction as before, there is no reason for supposing that 
 the above Peltier effects would cease, though there would now be 
 no Thomson effects, since the metals are at one temperature. We 
 should predict then that the junction which was the hotter would 
 now become the cooler, and conversely. 
 
 This would raise up an E.M.F. opposed to the former E.M.F., 
 and therefore opposed to the current that is flowing. 
 
 Some such reasoning as the foregoing would therefore lead us 
 to predict the ordinary case of the Peltier effect ' ; the case, namely, 
 where a current is sent across a junction of two metals A and B, 
 and where it is found that the junction is so heated or cooled as 
 always to raise up an E.M.F. opposing the current, provided that 
 the junction is not at the neutral temperature. In other words, 
 there will be heating or cooling according as the current flows from 
 the metal of lower, to the metal of higher, thermo-electric power, 
 or vice versa. 
 
 The whole of the above reasoning is necessarily rather of the 
 nature of guessing at probable results than of strict argument 
 The fact is that the theory of thermo-cells is beyond the scope of 
 an elementary book. 
 
CH. XVII. 2/1 
 
 CHAPTER XVII. 
 
 GALVANOMETERS ; WITH A PRELIMINARY ACCOUNT OF THE 
 MAGNETIC ACTIONS OF CURRENTS. 
 
 i. Magnetic Field about a Simple Rectilinear Current. We 
 will now turn our attention to the very important class of pheno- 
 mena coming under the head of ' the magnetic actions of currents} 
 
 On these magnetic phenomena depend the construction and 
 use of that important class of instruments called galvanometers, 
 whose use, as current detectois and current measurers, is so essen- 
 tial in the modern science of electricity. 
 
 In the present Chapter we propose describing various forms of 
 galvanometers. But, in order the better to understand their theory, 
 we shall give some preliminary account of the magnetic fields due 
 to electric currents ; leaving, however, the main part of this impor- 
 tant subject to be pursued further in Chapters XVIII.-XX. 
 
 When a conductor is charged statically with electricity, we 
 have about it what we call an electrostatic field. This field acts 
 on a -f unit of electricity in lines of force that run to or from the 
 conductor, as explained in Chapter X. The case of a conductor 
 carrying a current is very different. It is true that still a + unit 
 of electricity would in general find a field of force about the con- 
 ductor, for this conductor will be in general at potentials different 
 from that of the earth, and different from point to point of the 
 conductor. 
 
 But this electrostatic field is quite unimportant and negligible 
 compared with the new field of force that springs into existence 
 directly the 'electricity' moves, or directly there is a current. This 
 new field is a magnetic field ; and we shall therefore consider its 
 action, not on a -f unit of electricity, but on a + unit magnetic 
 pole. 
 
272 
 
 ELECTRICITY 
 
 It is easily shown by direct experiment that the lines of force 
 in which this field urges our unit pole form circles round the wire 
 that carries the current, so that a pole is urged, not to or from the 
 wire, but continually round and round it (the wire being assumed 
 straight). Each line of force is a closed circle (not a spiral), lying 
 in a plane perpendicular to the rectilineal wire, and having its 
 centre in this wire. The field is weaker further from the wire, 
 and stronger nearer to the wire. 
 
 As with the electrostatic field in Chapter X. 13 and 14, so 
 with a magnetic field we can mark out a certain number of lines 
 of force and leave in the field such a selection that the number 
 piercing i sq. cm. held perpendicularly to the lines at any place in 
 the field measures the strength of the field at that place, i.e. gives 
 the number of dynes with which a unit magnetic pole would be 
 urged at that place. 
 
 Experiments. (i. ) A hole is bored (with a bow drill and turpentine) in a 
 sheet of glass, and a wire carrying a current is passed through this hole per- 
 pendicularly to the glass plate. 
 
 On passing a strong current, and scattering iron filings on the plate, these 
 latter will (on tapping the plate) be observed to arrange themselves in con- 
 
 FIG. i, 
 
 centric circles about the hole as centre. If the wire be not perpendicular to 
 the plate, the lines will be ellipse-shaped. This indicates that the lines of 
 force are circles lying in planes perpendicular to the wire. 
 
 (ii. ) If the wire lie on the plate, we find the filings arranged in straight 
 lines perpendicular to the wire. These straight lines are the sections of the 
 concentric circles, made by the plate. 
 
CH. xvn. GALVANOMETERS 273 
 
 Now let a magnetic needle, so balanced as to turn any way, be 
 placed near such a current ; and let us for the present consider 
 only the field due to the current, leaving ,the earth's field out of 
 the question. It is clear that the needle can be in equilibrium 
 only when it lies in the plane passing through its point of sus- 
 pension and perpendicular to the wire carrying the current ; and 
 when, further, its poles are equidistant from the wire. If it is very 
 small it may be said to come to rest when it lies in (or is tangent 
 to) one of the above circular lines of force. (Compare Chapter II. 
 ii.) In such 'a position its poles are urged by equal forces 
 tending in opposite directions round the wife ; these equal forces 
 being inclined at equal angles with the needle, since this forms a 
 chord of the circular line of force. 
 
 This pair of forces will direct the needle as stated, and will 
 also give a resultant urging the needle broadside-on towards the 
 wire. The reader should make the above clear to himself by 
 means of a figure. 
 
 (iii.) A magnetic needle is fixed on a cork, and so floats on the surface 
 of water. To one side of the needle is placed a wire perpendicularly 
 to the surface of the water, and a strong current is passed through the wire. 
 It is better so to place the wire, and to pass the current in such a direc- 
 tion that the action of the current is not opposed to that of the earth, but 
 acts with it. It will then be observed that the needle will be dragged into a 
 position in which its poles lie on the circumference of a circle which has for 
 its centre the point where the wire cuts the water, i.e. lie on the same circular 
 line of force ; and that then the needle will be urged broadside-on towards the 
 
 (iv.) In the same way are steel or iron filings urged broadside-on towards 
 a wire carrying a current, and caused to adhere to it, the filings having become 
 ' magnetic needles ' by the inductive action of the field. They are thus 
 arranged in very close circles round the wire. (In the case of a magnetised 
 steel wire the filings adhere end-on, pointing along the lines of force that in 
 this case radiate from the wire.) 
 
 2. The -f and Directions of the Lines of Force. It is 
 easily shown, by experiment with a magnetic needle, that the + 
 and directions of the lines of force are given by either of the 
 following rules. 
 
 (I.) Amperes rule. 'If one swims with the current (i.e. so 
 that it flows from feet to head) and looks at a n-seeking (or +) 
 pole, this will be urged to one's left hand. A s-seeking (or ) pole 
 will be urged to one's right hand.' 
 
274 ELECTRICITY CH. xvn. 
 
 Oersted's experiment. The his- 
 toric experiment of Oersted illus- 
 trates well this rule of Ampere. The 
 figure here given sufficiently explains 
 this simple experiment. 
 
 (II.) Another form of the 
 rule is often very useful. No 
 one who has driven in an ordi- 
 nary (or right-handed} screw 
 can forget how the hand turns 
 as tne screw advances away from the person driving it. As one 
 drives it from one, the hand turns as do the hands of a clock 
 that faces one. Now the above experimental rule of Ampere 
 means that if the current advances with the screw, a + pole is urged 
 round it in the direction in which the screw turns. And since 
 the direction in which a + pole is urged gives us the + direction 
 of the lines of force, we may give the following rule. 
 
 ' The -f- direction of the lines of force about a wire carrying a 
 current is associated with the direction of the current in just the same 
 way as the direction of rotation of an ordinary (or right-handed) 
 screw is associated with the direction of the onward movement of the 
 screw? 
 
 (III.) Field due to a circle of ivire carrying a current. If the 
 wire be bent into a circle it will easily be understood that at the 
 centre of the circle all the lines of force there combine to give a 
 line of force running perpendicularly to the plane of the circle. 
 To any side of this line the lines of force bend away ; they ulti- 
 mately curve completely round the wire and run into themselves 
 again. Thus there is, if we wish to be very exact, only one straight 
 line of force, viz. that which runs through the centre of the circle 
 perpendicularly to its plane. [Mathematicians would say that the 
 two ends of this line join at infinity, so that it also forms a -'closed 
 curve.'] But practically, if we take a portion of the field that is 
 near the centre of the circular current and is small as compared 
 with the diameter of the circle, we may consider this portion to be 
 uniform, and to have its lines all running perpendicular to the 
 plane of the circle. 
 
 A little consideration will show us that the same ' right-handed 
 screw relation ' holds here in this somewhat converse case. If we 
 
CH. xvii. GALVANOMETERS 2/5 
 
 follow this central line of force in its + direction (i.e. if we travel 
 as a + pole is urged), then the current round the circle is asso- 
 ciated with our direction of movement as the direction of rotation 
 of a screw is associated with its onward movement. .g., if we face 
 a 'circle in which the current goes dock-wise (i.e. as do the hands 
 of a clock when we face it) our + pole is urged towards this 
 circle. 
 
 3. Simple Form of Galvanometer. If we balance a mag- 
 netic needle so that it moves in a horizontal plane, it will come to 
 rest when it is in the plane of the magnetic meridian. If we de- 
 flect it from this position of rest, there will be a couple due to the 
 horizontal component of the earth's field tending to restore it to 
 its original position of rest. 
 
 Now let a loop of wire be passed round the needle, above and 
 below it, so that its plane coincides with the plane of the magnetic 
 meridian. The figure here given exhibits 
 this simple arrangement. 
 
 If a current be passed round this wire 
 loop, it is easy to see that the upper and 
 the lower wires give fields about the 
 needle in the same direction, tending to 
 set it at right angles to the plane of the 
 loop, i.e. at right angles to the plane of # 
 the magnetic meridian. 
 
 Now the earth's couple is greatest 
 
 when the needle stands E. and W., and is zero when the needle 
 stands N. and S. ; while the converse is the case with the current's 
 couple. 
 
 Hence the needle will settle at some angle of deflexion from 
 the magnetic meridian ; the magnitude of Q (cceteris paribus] de- 
 pending upon the current-strength. 
 
 Such is the principle of the galvanometer. In 5, 6, 12, 13, 
 14, 15, we shall describe instruments designed for measurement of 
 currents. In 7, 8, 9, 10, u, the instruments described are 
 mainly for detection of very small currents. (For further general 
 remarks on galvanometers, see 16.) 
 
 4. Relation of Strength of Field to Current-Strength, 
 It can be shown that strength of field is (cater is paribus] directly 
 
276 ELECTRICITY CH. xvn. 
 
 proportional to strength of current ; the strength of field being 
 measured, e.g., by the vibration method of Chapter III. 3 and 6. 
 
 We make a vertical frame, round which insulated wire may be 
 wound, and place it in the plane of the magnetic meridian. In 
 the centre of this frame the needle vibrates under the field due to 
 earth and current together. 
 
 We then try the effect of having one, two, three, &c., turns of 
 wire round the frame. By including in the circuit a rheostat, by 
 means of which more or less resistance may be introduced, and 
 an auxiliary galvanometer whose deflexion we thus keep constant, 
 we can insure the constancy of the current that we employ. 
 
 We then compare the fields, as in Chapter III. 3, and show 
 that the above law holds. It is easy to show that the field depends 
 strictly upon current-strength and not upon current- density ; that 
 is, upon the quantity per second passing over any section of the 
 conductor, not upon whether this current forms (so to speak) a 
 broad slow stream or a quick narrow one. 
 
 Experiment. One way of proving the above is to show that when a current 
 flows through a thick wire, and returns through a thin wire wound round the 
 former, the total resultant magnetic field is zero. This is another example of 
 proving equality by means of the 'zero method.' 
 
 The reader will observe that from this it follows, as assumed 
 above, that with the same current n turns of wire act as one turn 
 carrying n times this current. Hence, the law stated at the begin- 
 ning of this section is proved by the method given, in which we 
 employ one, two, three, &c., turns of wire. 
 
 5. The Tangent Galvanometer. The simplest form of true 
 galvanowgte/' (or current- measurer] is that called the tangent- 
 galvanometer. Referring to 2 (III.), we shall see that about 
 th centre of a circular wire carrying a current the field may be 
 considered uniform. Or, if a needle whose length is small as 
 compared with the diameter of the circle (see note (iii.) at end of 
 this section) be suspended at the centre, then, however much it 
 be deflected it will remain practically in the same strength of field 
 as long as the current remains constant, 
 
 If then we can measure the strength of this field by the amount 
 of deflexion, we measure at the same time the current that gives 
 rise to the field ; whereas, if the needle were too long or situated 
 unsymmetrically with respect to the circle, the needle would, for 
 
GALVANOMETERS 
 
 277 
 
 FIG. i. 
 
 the same current, pass into different strengths of field according 
 to the extent of its deflexion ; in which latter case we could 
 measure the fields but not 
 the current. 
 
 Fig. i. gives a general 
 view of the tangent galvano- 
 meter. There is usually a 
 short needle provided with 
 a long and light index that 
 moves round a graduated 
 circle. Very often arrange- 
 ments are made by means of 
 which we can either use one 
 thick circle of copper for very 
 powerful currents, or two or 
 more turns of wire for cur- 
 rents not so strong (see 7). 
 This circle or coil is placed 
 in the plane of the magnetic 
 meridian. In fig. ii. we are 
 supposed to be looking down 
 from above, and so to be 
 viewing the top of the circle 
 of copper and the needle, 
 projected together on to a 
 horizontal plane. 
 
 G G' is part of the pro- 
 jection of the copper circle ; 
 it lies, as mentioned in 3, 
 in the magnetic meridian. 
 n s represents the needle ; 
 this lies in reality below 
 GG', at the centre of the 
 circle of which GG' is the 
 top. The dotted lines run- 
 ning parallel to GG' repre- 
 sent the direction of the 
 
 earth's horizontal field, H ; those perpendicular to G G' represent 
 the field I due to the current. 
 
2/8 ELECTRICITY CH. xvn. 
 
 Using the notation of Chapter III. we have that 
 
 (i.) Earth's couple acting on needle is H /u I sin a. 
 (ii.) Current's couple acting on needle is I fj. I cos . 
 When there is equilibrium we must have these couples equal ; or 
 
 H n I sin a = I p. I cos a. 
 Whence I = H . tan o. 
 
 Now, in the present case we have said that whatever be the 
 deflexion of the needle we may consider I to be constant while 
 the current is so, and to be directly proportional to the current C. 
 Hence we may write C = k I, where k is some constant depending 
 on the dimensions of the coil, the number of turns of wire, and 
 the unit of current that we employ. 
 
 We have then C = k H . tan a. 
 
 Hence with the same instrument, and with H the same, the 
 current is proportional to the tangent of the angle of deflexion ; 
 whence the name of this instrument. 
 
 It is clear that it can be used for currents up to any strength, 
 since tan a can have any value. But it fails to give accurate 
 measurement if a exceed (say) about 70, since then large changes 
 in current give small changes in a. The best angle of deflexion 
 is about 45, as may be shown mathematically. 
 
 Notes. (i.) Other forms of the instrument. In Gaugain's form we have the 
 needle placed in the axis of the coil, at a distance from the centre of the coil 
 about one-fourth of its mean diameter. In Helmholtz's modification of 
 Gaugain's instrument there are two coils lying symmetrically on either side of 
 the needle, and the wires are wound on small portions of cones, of which, if 
 completed, the centre of the needle would be the common apex. Such an 
 arrangement gives a more uniform field, and permits one to use a somewhat 
 larger needle. 
 
 (ii. ) To set the coil in the magnetic meridian. When the coil is in the 
 magnetic meridian, the same current, passed in opposite directions, should give 
 opposite deflexions of equal magnitude. The coil should be adjusted until this 
 is the case. 
 
 (iii.) As regards the relative dimensions of needle and diameter of coil, 
 giving results accurate enough for practical purposes, we quote here the follow- 
 ing authorities. 
 
 Wiedemann ('Die Lehre von der Electricitat,' iii. 247) gives length of 
 needle at most one-eighth diameter of coil. 
 
 Kempe (' Handbook of Electrical Testing,' p. 18) gives needle about f -inch 
 for a 6-inch or 7-inch ring, as accurate enough for most purposes. 
 
 S. P. Thompson gives length of needle about one-tenth of diameter of coil. 
 
CH. XVII. 
 
 GALVANOMETERS 
 
 279 
 
 B rough gives needle one-sixth of diameter of coil. 
 
 Andrew Gray gives needle I centimetre for coil of diameter 15 centimetres , 
 in a standard instrument. 
 
 Hence, needle one-tenth of diameter of coil seems a good relative dimension. 
 
 6. The Sine Galvanometer. Fig. i. shows us a somewhat 
 different form of galvanometer. Here the coil turns round on a 
 vertical axis, the movement being measured over a horizontal 
 graduated circle. As before, the coil is initially in the plane of 
 the magnetic meridian. 
 
 We may here turn the coil until the current does not affect the 
 needle at all. In this position it must be that the lines of force due 
 to current coincide with v 
 
 those of the earth, or 
 the coil is due magnetic 
 east and west. We then 
 turn the coil back through 
 90 over its graduated 
 scale, and it will lie in 
 the plane of the magnetic 
 meridian. 
 
 When the needle is 
 deflected the coil is 
 turned in the same di- 
 rection ; and finally, if 
 the current be not too 
 strong, can be brought 
 again directly over the 
 needle. When this is 
 the case we note the 
 angle through which 
 the coils have been 
 turned. The current will 
 be proportional to sin a. 
 
 Fig. ii. shows us the theory of this instrument. In it is re- 
 presented the final position of equilibrium, in which the coil G G' 
 lies again over the needle at an angle a from the magnetic meri- 
 dian N S. Since now the lines of force due to the current are 
 perpendicular to the needle, we have 
 
 j (i.) Current's couple = I /u /. 
 
 1 (ii.) Earth's couple = H p I . sin a. 
 
 FIG. 
 
280 
 
 ELECTRICITY 
 
 CH. XV] T. 
 
 With similar reasoning to that given in the last section we have 
 finally . ." .-^ . . 
 
 C = k H . sin a 
 
 whence the name of the instrument. 
 
 It is evident that we cannot, without using shunts, measure 
 any current exceeding k H, since the greatest value of sin u is i, 
 
 which it has when a = 90, 
 or when the needle stands 
 E. and W. With a larger 
 current we should chase the 
 needle completely round. 
 
 Since the coil is always 
 over the needle we may 
 have this of any length we 
 please, for it will always be 
 in the same part of the cur- 
 rent's field. 
 
 7. The Multiplying 
 ,.f Galvanometer. The extent 
 of deflexion of the needle 
 depends upon the strength 
 of the magnetic field due to 
 the current. 
 
 With the same current, 
 
 this can be multiplied many-fold by passing the wire many times 
 round the needle. In fact, we wind the wire in a coil and suspend 
 the needle inside it. 
 
 The advantage of this can be demonstrated as in 4. 
 A figure of a multiplier is shown in the next section. 
 8. Astatic Galvanometer; Two Needles. In the last section 
 we showed how to increase the current's action on the needle by 
 increasing the field- strength due to the current. 
 
 There is another very effective manner in which the deflexion 
 for a given current may be almost indefinitely increased. This 
 other method is based upon the device of making the earths re- 
 storing couple as weak as we please, while we leave the current's 
 deflecting couple unaltered. 
 
 We cannot do this by weakening the magnetic moment of a 
 
CH. XVII. 
 
 GALVANOMETERS 
 
 28l 
 
 single needle ; since, as we see in the formulas of 5 and 6, we 
 should then weaken equally the current's action on the needle. 
 
 But we can so combine two needles that, they form with re- 
 spect to the earth a single needle of as small a magnetic moment 
 as we please ; while, by coiling the wire round one needle only, we 
 can cause the current to act on a single needle of considerable 
 magnetic moment. 
 
 A system of needles on which the earth has little (or, strictly, 
 no) action, is called an astatic system ; and a galvanometer in 
 which such a system is 
 employed is called an 
 astatic galvanometer. 
 
 There are two com- 
 binations that give us 
 convenient astatic sys- 
 tems. 
 
 (i.) Let us connect 
 rigidly two needles ns 
 and ri s' of nearly equal 
 strength, so that, when 
 they are suspended, 
 they are in the same 
 vertical plane ; the poles 
 being directed opposite 
 ways. If ns be some- 
 what the stronger, the 
 two needles will act with 
 respect to the earth as 
 one needle of the same 
 length /, but of pole- 
 strength measured by 
 the difference (u /*') ; FIG. i. 
 
 so that the magnetic 
 
 moment with respect to the earth is l(pp'\ and may be very 
 small indeed by having the pole- strengths p and // nearly equal. 
 
 The coils of the galvanometer, however, as seen in the figure, 
 are so arranged that one needle of moment /* / is in the strong 
 field within the coil ; while the other needle, //, inasmuch as it 
 is reversed in direction and is also in the reverse field above the 
 
282 ELECTRICITY CH. xvn. 
 
 coil, is acted upon by a weaker couple in the same direction as is 
 the lower needle. 
 
 Thus we can, by making the pole-strengths ^ and p' very 
 nearly equal, cause the earth's restoring couple to become very 
 small, without diminishing the deflecting couple ; and so can 
 cause the galvanometer to become very sensitive. 
 
 (ii.) Next let two needles be fixed and suspended as before, 
 but let the vertical planes through them make some small angle 
 with each other. In fig. ii. we give a projection of the two 
 
 needles as viewed from above. 
 
 Let them further be of equal 
 
 strength, so that p = //. 
 
 Then it is pretty clear that 
 
 with respect to the earth they will 
 act as a short needle n" s" whose axis is at O, standing at right 
 angles to the bisector of the angla ; its pole-strength being 2 p, 
 
 f\ 
 
 its length being n" s" or I sin -. It follows that its magnetic mo- 
 ment will be 2 p. /sin. This can be made as small as possible 
 
 by making as small as we please. It is to be observed that the 
 needles will stand so that. ".$" is in the magnetic meridian ; or 
 the general direction of the two needles themselves will be E 
 and W. The action of the current is as before. 
 
 9. The ' Controlling Magnet ' Method . A needle balanced 
 so as to move in a horizontal field is influenced by the horizontal 
 component of the earth's field only, as we have said before. 
 Now we may superimpose on this horizontal field another due to 
 a magnet ; and, if the lines of force of this field run parallel to 
 those of the earth's, but in an opposite direction, we may weaken, 
 to any desired extent, the resultant field in which the needle 
 moves, by suitably adjusting the strength of the magnet's field. The 
 most convenient arrangement is to have a magnet directly above 
 the needle, capable of being slided up or down a metal stem so 
 as to make the field about the needle weaker or stronger ; and 
 also capable of revolving in a horizontal plane about the stem,. so 
 that it may either stand in the magnetic meridian or may make 
 any angle with the meridian. 
 
en, xvii. GALVANOMETERS 283 
 
 For our present purpose our object is to leave the needle 
 directed in the magnetic meridian by a very weak resultant 
 field. 
 
 We thus should have the magnet's field a little weaker than the 
 earth's, and opposite in direction. 
 
 The needle is acted upon by the current as before ; and, as 
 the restoring couple is very small owing to the weakness of the 
 restoring field, the instrument may be very sensitive. 
 
 We may add that in the most sensitive instruments the methods 
 of 8 and 9 are combined. 
 
 10. Sir W. Thomson's Mirror Galvanometer. The trans- 
 mission of signals by means of interruptions and reversals of 
 currents will be described in a future Chapter. At present we 
 will merely observe that galvanometers will indicate such reversals 
 and interruptions, and so will serve as instruments to receive 
 signals of this nature. We shall see in a later Chapter that when 
 such signals are sent through long submarine cables, the currents 
 are very small on account of the great resistance ; and further, 
 that interruptions and reversals lose their abrupt character and 
 become mere fluctuations in the weak current transmitted. 
 
 To indicate such weak currents and such slight fluctuations, 
 Sir W. Thomson invented a form of galvanometer without which 
 signals by cable would hardly have been possible. 
 
 In his mirror galvanometer we may notice the following main 
 points of interest. 
 
 (i.) As the instrument is usually intended to be used in circuits 
 where the resistance is already very great, there are often many 
 thousand turns of (necessarily) fine wire ; this ' multiplies ' the 
 strength of the field due to the current without perceptibly 
 diminishing this latter. For the total resistance will not relatively 
 be much increased. 
 
 (ii ) The ' needle ' is .a very small bit of magnetised watch- 
 spring. Having little mass and inertia, its movements follow any 
 fluctuations in the current almost instantaneously. 
 
 (iii.) A controlling magnet enables one to vary the sensitive- 
 ness of the instrument at will, as explained in 9. 
 
 (iv.) The same controlling magnet enables one, if need be, to 
 bring the needle to rest in any position ; and. aided by the method 
 of suspension mentioned below, to work the instrument even on 
 
284 
 
 ELECTRICITY 
 
 CH. xvir. 
 
CH. XTII. GALVANOMETERS 285 
 
 board ship. In such cases we over-compensate the earth's field 
 and direct the needle by the magnet alone. The fibres supporting 
 the needle are fixed at top and bottom, and so prevent it from 
 swinging against the sides of the coil in which it works. 
 
 (v.) To the needle is attached a very light mirror. A beam 
 from a lamp falls upon this, and the reflected ray gives as index 
 a spot of light that moves over a graduated scale. 
 
 We thus (as in the case of the quadrant electrometer of 
 Chapter X. 33) can have an index as long as we please, possess- 
 ing no mass or inertia. 
 
 The figure on the opposite page indicates the coil, lamp, beam 
 and its reflection, scale, and controlling magnet placed above the 
 coil. The needle and mirror are inside the coil and are not seen. 
 An additional controlling magnet T is sometimes of service in 
 bringing back the spot of light to the zero mark. 
 
 Notes. (i.) The mirror maybe concave or plane. In the latter case an 
 
 auxiliary lens is used, in order to bring the reflected beam to a focus on the 
 scale. 
 
 (ii.) Either in the reading of, or in the construction of, the scale, we must 
 
 make allowance for the fact (proved by elementary optics) that if the needle 
 
 and mirror be displaced through an angle 0, the reflected beam is displaced 
 through an angle 20. 
 
 ii. The Differential Galvanometer. In the differential 
 galvanometer the needle is suspended symmetrically between two 
 coils. These coils must fulfil the following requirements. 
 
 (i.) Their resistance must be exactly equal, so that equal 
 E.M.F.S at the terminals give equal currents. 
 
 (ii.) Equal currents in the same direction must give equal but 
 opposite fields about the needle ; so that, when the same current 
 in the same direction is passing through the two, the needle is un- 
 affected. 
 
 An exact test that these conditions are fulfilled is to send the 
 same current through two coils coupled end-on, so that the current 
 passes in the same direction through each coil. The needle 
 should not be affected. Usually a small portion of one coil is left 
 moveable, allowing exact adjustment to be made when the in- 
 strument is used. 
 
 It is clear that by coupling the coils suitably this instrument 
 
286 ELECTRICITY 
 
 CH. XVII. 
 
 may be used as an ordinary galvanometer, the two coils giving 
 fields acting on the needle in the same direction. 
 
 The differential galvanometer has several uses, (i.) One use is to 
 compare resistances ; the method is somewhat as follows. A current 
 is divided into two branches, passing through the two coils respectively. 
 Since the resistances of these are equal, the currents will also be equal 
 and the needle will be unaffected. The unknown resistance is now 
 introduced into one branch, thus causing an unequal distribution of 
 the two currents. Then known resistances are introduced into the 
 other branch until the needle returns again to zero. When this is the 
 case the resistances must be equal. In this method we are indepen- 
 dent of changes in the E.M.F. of the battery. 
 
 (ii.) Another, less simple, use of this instrument is given in 
 Fleeming Jehkin's 'Electricity,' p. 242, to which book we refer the 
 student for details as to this method. 
 
 We will only say here that by shunting one branch of the galvano- 
 meter we may either measure a small resistance to a small fraction of 
 an ohm, or may measure a resistance which is a large multiple of the 
 greatest resistance contained in the resistance box. 
 
 12. The Ballistic Galvanometer. As we have seen, a 
 current can be measured by the couple that it exerts (if we may 
 use the expression) on a needle in an instrument of known con- 
 struction ; this couple depending also in a known way on the 
 angle that the needle makes with the plane of the coil. 
 
 Now quantity of electricity is given by (current} x (time}. 
 Hence we can, in the case of steady currents, measure the total 
 quantity of electricity that has passed by means of the, tangent 
 galvanometer, by observing both current and time of duration. 
 
 But supposing that we wish, eg., to measure the quantity of 
 electricity in a condenser by discharging it through a galvano- 
 meter. In the ordinary case we have a current whose strength 
 rises from zero to a maximum, and then falls again to zero. While 
 it passes the needle is deflected, and part acts when the arm of 
 the couple is the full length / of the needle, part when the arm is 
 I cos (see 5), where rises from zero to a maximum and then 
 decreases again. It is impossible, under these conditions, to calcu- 
 late in any simple way the total quantity that has passed. In order 
 to do this an instrument is devised analogous to the ballistic 
 pendulum used in mechanics. In this latter instrument we can 
 estimate what are usually called ' impulsive forces,' or ' impulses,' 
 
CH. XVII. 
 
 GALVANOMETERS 
 
 28 7 
 
 by allowing them to act on such a massive and slowly-moving 
 pendulum that there is no appreciable change of position until 
 the impulses have ceased to act. The impulses thus act on the 
 pendulum in one known position (viz. when it is hanging vertically); 
 and the sum total of these impulses can be deduced from the 
 extent to which the pendulum swings, its mass and dimensions 
 being known. 
 
 In the ballistic galvanometer we have a needle of such a mass 
 that it does not move appreciably from zero until the discharge has 
 entirely passed. Thus each portion of the discharge acts on the 
 needle with coilples whose arms are all the length / of the needle. 
 The extent of swing of the needle then depends solely upon the 
 sum of (each cur rent- strength) x (the infinitesimal time during 
 which it remains constant}. In spite of the fact that the current 
 changes continuously, thus making this 'sum ' a matter for the 
 ' infinitesimal calculus/ even a beginner can see that the sum of 
 all the products above given measures the total quantity of elec- 
 tricity that has passed. If Q be this quantity, and be the maximum 
 angle of deflexion of the needle, it can be shown that . . . 
 
 where k is a constant depending upon the construction of the in- 
 strument and upon the strength H of the earth's horizontal field. 
 This constant can be found or 
 calculated. 
 
 Note. The student can find the 
 proof of the above formula in more 
 advanced books. 
 
 Use of ballistic galvanometer. 
 We may use the foregoing in- 
 strument to measure, or compare, 
 the capacities of condensers. In 
 the accompanying figure xy a b is a 
 condenser. The plates ab are to 
 earth, while the opposite plates xy 
 can be connected, by means of a 
 key M, either with the pole P of a battery, wwith the one terminal of a 
 ballistic galvanometer G. The other pole C of the battery, and the 
 other terminal of the galvanometer, are to earth. When the plates 
 
288 ELECTRICITY CH. xvn. 
 
 xy are connected with the pole P of the battery, the condenser is 
 charged so that xy are at a potential V, while a b are at zero poten- 
 tial. (It is clear that V measures the unpolarised E.M.F. of the 
 battery, for the other pole C is to earth, and there is no permanent 
 current.) 
 
 Then if K be the capacity of the condenser, the quantity Q of the 
 charge will be 
 
 where Q x is in coulombs, if V is in volts and K t is in farads (see 
 Chapter XVI 1 1. 4). 
 
 Now let the condenser be discharged through the ballistic galvano- 
 meter. 
 
 The total impulse given to the needle is, we have seen, directly 
 
 proportional to Q ; and this impulse is also, we have stated, propor- 
 
 /\ 
 tional to sin - 1 , for mechanical reasons that we have not given ; 1 
 
 being, as above, the ' angle of throw ' of the needle. 
 
 J? A 
 
 Hence we have K x = ^ . sin -* 
 
 k 
 
 So, for another condenser, we have K 2 = . sin -% ; whence 
 
 sin ^ 
 K 
 
 K 2 sin - 2 
 
 2 
 
 We can thus compare any condensers with standards, or with each 
 other. 
 
 Note. Let / be the time in seconds of a complete to-and-fro oscillation of 
 the needle when no current passes. Let 6 measure the angle of throw in some 
 unit of angle such as degrees. Let r\ measure in ohms that resistance through 
 which the E.M.F. measured by V (see above) would drive a steady current, 
 giving a permanent deflexion of one unit of angle the same unit as is used in 
 measuring 9. And let K be the capacity of the condenser \n farads. Then it 
 can be shown that ...... ... ........ . . 
 
 . 
 
 t . sin - 
 
 K-- -i 
 
 7T. fj 
 
 This formula is given in Fleeming Jenkin's ' Electricity.' The more mas- 
 sive the needle, and the less friction there is in its movement, the truer is this 
 formula. 
 
r ui~ 
 
 DEPARTMENT OF P'- 
 CH. xvir. GALVANOMETERS 289 
 
 13. Sir W. Thomson's 'Graded' Potential Galvanometer 
 
 To meet the present requirements of exact electrical measure- 
 ment, Sir W. Thomson has invented two galvanometers of con- 
 venient form and of wide range of sensibility, the one designed es- 
 pecially for measurements of E.M.F.s or of potential differences, 
 the other for measurements of currents, In both there are arrange- 
 ments for altering to a known extent and through a wide range 
 the sensibility of the instrument ; and both are so marked by the 
 makers that, with simple corrections for the strength of the 
 earth's horizontal field H at each place, results can be read off in 
 volts and amperes respectively. For these reasons the instru- 
 ments are called graded galvanometers. A very full account of 
 them is given in Gray's ' Absolute Measurements in Electricity 
 and Magnetism.' They have not, however, for reasons alluded to in 
 17, satisfied practical requirements. Sir W. Thomson has since , 
 invented other instruments, alluded to at the end of Chapter XXV. 
 
 The volt-meter galvanometer. The general principle of a volt- 
 meter galvanometer has been already explained in Chapter XIV. n. 
 
 In the figure C is the coil. This has a great number of turns of 
 wire, and a large resistance, generally exceeding 5,000 ohms. 
 
 M is the magnetometer, or system of needles. This consists of a 
 system of short needles, parallel to one another, disposed symmetri- 
 cally about a common pivot. We can consider this system to be 
 equivalent to one short needle, and shall often speak of it as 'the 
 needle.' To this is rigidly fixed a long and light aluminium index, 
 standing at right angles to the direction of the needle. It is this index 
 that is seen in the figure. It moves over a scale properly graduated. 
 
 Over the magnetometer and resting on the magnetometer box, but 
 removable at pleasure, is seen a semicircular magnet. By the use of 
 
 U 
 
2QO ELECTRICITY CH. xvn. 
 
 this magnet the { restoring-field ' (which would otherwise be only the 
 earth's horizontal field H) may be much increased in strength. Further, 
 when the magnet is used, any error in the measurement of H at any 
 particular place will be a smaller percentage of the whole field than it 
 would be if H alone were the whole field, and so will cause a smaller 
 error in the measurements made with the instrument, providing that 
 the field due to the magnet is known. (This latter has, in practice, to 
 be re-determined from time to time.) But it must be remembered that, 
 according to the principle of 9, any strengthening of the restoring- 
 field diminishes the sensibility of the instrument. 
 
 The instrument is so constructed that the axis of the coil C, along 
 which runs the central line of force (see (III.) of 2), passes through 
 the point of suspension of the needle, and thus the current tends to 
 direct the needle along the line of the axis of the coil. 
 
 On the other hand, the magnet is so placed that its magnetic axis 
 passes through the point of suspension of the needle, in a direction 
 perpendicular to the last. 
 
 When used, the instrument is levelled, and is so placed that the 
 planes of the coil and of the magnet are in the magnetic meridian. 
 When this is the case, the index should stand at zero when no current 
 passes, and equal currents in opposite directions should give equal 
 deflexions from zero. 
 
 In this position we have as a restoring-field the earth's horizontal 
 field H, to which may be added at pleasure the stronger horizontal 
 field due to the magnet, coinciding in direction with H. The deflect- 
 ing-field &OK. to the coil is horizontal, and is perpendicular to the last 
 field. Thus the theory of the instrument^ since its needle is relatively 
 short and moves in a uniform field, is that of the tangent galvanometer. 
 
 The sensitiveness of the instrument can be varied, both abruptly, 
 by means of the magnet, which may be used or removed, and also, 
 step by step, by sliding the magnetometer box nearer to, or further 
 from, the coil. The feet of the box slide in a groove in such a way 
 that the whole moves parallel to itself, while the point of suspension 
 of the needle moves along the axis of the coil. Instructions as to the 
 conversion of the readings into volts under all conditions of sensitive- 
 ness are sent with each instrument. 
 
 The terminals are of an ingenious construction, designed partly to 
 obviate accidents in measuring very large AV.s, and the ' leads ' are 
 so arranged that the current passing through them has no action on 
 the needle. 
 
 14. Sir W, Thomson's Graded Current Galvanometer. In 
 
 the construction of the last instrument it was desired to measure 
 
CH. xvn. GALVANOMETERS 291 
 
 the AV between two points without appreciably diminishing this 
 -W by lessening the resistance between them. Hence we saw 
 that the coil was made of a very high resistance. 
 
 In the present case it is desired to measure currents without 
 appreciably diminishing them by the introduction of resistance. 
 The coil, therefore, must be of as low a resistance as is consistent 
 with the requisite sensitiveness. 
 
 The current galvanometer, or am-meter, is in general construction 
 almost identical with the last. But the coil is composed of a few turns 
 
 of thick copper wire, or copper strip, of a resistance equal to about 
 jJ^ ohm. There is also a special arrangement of the terminals, to 
 obviate inconveniences or accidents in dealing with large currents, and 
 the ' leads ' are so arranged that the current passing through them has 
 no action on the needle. 
 
 15, Weber's Electro-Dynamometer. 
 
 It is sometimes desired to measure currents by some instrument 
 that does not depend upon the magnitude of the earth's field of force 
 at any particular place in fact, to construct an instrument that can be 
 used without reference to the variable quantity ' H,' that has appeared 
 in the formulae of the preceding instruments. 
 
 Now we shall see in later Chapters that a coil carrying a current 
 acts as a magnet of the same shape whose poles answer to the two 
 faces of the coil. It is found that the magnetic moment of such a coil 
 depends solely upon its shape and the strength of the current flowing 
 through it ; so that, as long as these remain constant, we have an 
 absolutely invariable magnetic needle. 
 
 Let the large coil of 5 (or one with more turns of wire, if suitable 
 for the purpose) stand with its plane vertical and coinciding with the 
 
 U 2 
 
2Q2 ELECTRICITY CH. xvn. 
 
 plane of the magnetic meridian. And let there be suspended at its 
 centre by a * torsion wire ' a relatively small coil, whose faces are turned 
 due magnetic N and S, and whose plane therefore is perpendicular to 
 that of the larger coil. * 
 
 If a current be passed in the right direction through the small 
 moveable coil, it will be in equilibrium with respect to the earth's hori- 
 zontal field, since its faces (or poles) are turned N and S. If another 
 current be passed through the larger coil, the field due to this will act 
 on the smaller coil, tending to set it parallel to the larger. We might 
 have given the small coil free suspension, and have used the small coil 
 as we used the needle in 5. But since we wish to get rid of H from 
 the formula of calculation, we twist the torsion wire until we bring the 
 small coil back to its zero position, in which the earth's field did not 
 act on it. The angle of torsion then measures the couple with which 
 (to speak intelligibly enough, even if not very accurately) the large coil 
 acts on the smaller. 
 
 We have then a balance between (i.) the torsion of the wire 
 on the one hand, and (ii.) the electro-magnetic couple between 
 the coils on the other. The former can be determined by ordinary 
 mechanical methods, and the latter depends (in a manner known to 
 those who have studied the theory of electro-magnetic actions) on the 
 dimensions of the coils, and on the product of the strengths of the two 
 currents. Thus, for each instrument a formula can be once for all 
 constructed, giving the value of one current when the other is known. 
 
 If we pass the same current through both coils- then we can measure 
 this current by measuring only the above given angle of torsion. This 
 last form of the instrument is adapted for the measurement of currents 
 that pass in rapidly alternating directions, since in this case the reverse 
 current in the large coil finds a needle of reverse polarity upon which 
 to act. The angle of torsion will in this case be proportional to the 
 square of the current-strength. 
 
 1 6. Some General Observations on Galvanometers. We 
 
 will end this Chapter with a few general remarks on galvano- 
 meters. 
 
 (I.) Long-coil and short-foil galvanometers. Fleeming Jenkin 
 uses these two very clear terms to indicate instruments in which 
 the current passes many times, or few times, round the needle 
 respectively. 
 
 For obvious reasons we cannot increase the field acting an 
 the needle by the device of passing the current many times round 
 without at the same time unavoidably increasing the resistance of 
 
CH. xvn. GALVANOMETERS 293 
 
 the instrument, for we cannot make coils having many turns of 
 any but fine and therefore highly-resisting wire. 
 
 In using an instrument we must, then, consider whether the 
 resistance R of the rest of the circuit be large or small as com- 
 pared with the resistance G of the galvanometer. If G be 
 negligible as compared with R, the field is multiplied n-fold by 
 using n turns of wire, and the deflexion for a given current is 
 thus made much greater. But if R is negligible as compared 
 
 with the resistance of one turn of wire, then we get but th of the 
 
 n 
 
 current when we use n turns, and thus gain nothing but incon- 
 venience in using a long-coil instrument. 
 
 17. Galvanometers for Practical, or Commercial, Use. 
 
 Now that electrical measurements have to be made every day in 
 places where the presence of large magnets and powerful electrical 
 currents renders the magnetic field unknown and variable, it is neces- 
 sary to have instruments whose action is independent of the earth's 
 field of force, and whose accuracy is not perceptibly impaired by such 
 disturbances as are likely to occur. 
 
 At the end of Chapter XXV. will be found some explanation of 
 the principles of several of the more recent types of instruments. 
 
 1 8. Calibration of Current-meters and of Voltmeters 
 
 The system of units employed in modern electrical science is based 
 upon the electro-magnetic actions of currents, as will be explained in 
 Chapter XVI 1 1. And the practical unit of current employed, the 
 ampere, can be determined by observing the attraction or repulsion 
 between accurately constructed coils of wire carrying a steady cur- 
 rent. [It was (e.g.} in some such way that Lord Rayleigh measured 
 in amperes the currents he employed.] The ampere being thus 
 measured electro-magnetically, it can be determined how much silver 
 or copper is set free in i second by a current of i ampere. 
 
 From the experiments t>f Lord Rayleigh and others it appears that 
 about i 1 1 8 milligramme of silver is set free by I ampere in i second. 
 
 In practical text-books the student will find it explained how this 
 result renders it a comparatively easy matter to calibrate any given 
 ammeter in amperes. 
 
 As regards voltmeters, we may [see Chapter XIV. n] graduate 
 them also by means of currents measured electro-chemically. Or we 
 may make use of the known E.M.F.s of certain standard cells ; these 
 E.M.F.s having been previously determined once for all, in absolute 
 value, by long and careful laboratory experiment. 
 
 For further on ' Calibration ' we refer the student to practical 
 text books. 
 
294 ELECTRICITY OH. xvur. 
 
 CHAPTER XVIII. 
 
 ACTIONS BETWEEN CURRENTS AND MAGNETIC POLES. MAGNETIC 
 EQUIVALENT OF A CURRENT. - ACTION BETWEEN CURRENTS 
 AND CURRENTS. 
 
 i. Action of an Infinite Eectilinear Current on a Magnetic 
 
 Pole. If we pass a current through a vertical rectilineal wire A B, 
 and observe its action on a horizontally-balanced magnetic needle 
 
 ns placed near it, we arrive experimentally at several 
 
 important results. 
 
 (i.) We find (as also was indicated by the experiments 
 
 of Chapter XVII.) that the lines of force form circles 
 
 whose centres lie on the wire and whose planes are per- 
 
 pendicular to it. 
 
 (ii.) We find that when once the wire is so long that 
 
 it makes the angle A OB nearly 180, then, as far as 
 
 its action on the needle is concerned, it is practically 
 
 infinite. 
 
 This suggests the idea that the current gives a field at 
 O only so far as it gives a ' broadside-on ' projection when 
 viewed from O ; and that a current running directly towards 
 O would give no field 
 
 (lii.) We find that the field strength at O is directly propor- 
 tional to the current-strength, as has been otherwise shown in 
 Chapter XVIT. 4. 
 
 (iv.) We find that, for the same current, the field-strength is 
 inversely proportional to the perpendicular distance from O to 
 the wire. 
 
 2. Action of an Element of a Current on a Pole. If we 
 wish to be in a position to calculate the action of a current of any 
 shape on a pole having any position with respect to the current, 
 
CH. xvin. CURRENTS AND MAGNETIC POLES 295 
 
 we must know the action of each little bit (or element) of the cur- 
 rent on the pole, and must sum up all these actions in order to 
 get the total action of the entire current. 
 
 Such summing-up belongs to the integral calculus in general ; 
 but in one case it can be done very simply. 
 
 For the benefit of those readers whose mathematical knowledge is 
 elementary, we shall explain further the meaning of an element of a 
 current, and shall then discuss in a simple manner the law of action 
 that is found to hold, and the application of this law to determine the 
 field at the centre of a circular current (see 6). 
 
 Let ab represent a bit of a current so small that the following 
 results may be considered to be true. 
 
 (i.) a b may be considered to be straight, although it be part of a 
 curved circuit. Its length will be designated by ds^ where in general 
 ds is a very small fraction of a 
 centimetre. 
 
 (ii.) The length p. a (where p 
 represents the position of the 
 pole acted on, the pole- strength 
 being \L units) may be considered as equal to /* b, in whatever posi- 
 tion a b stand with respect to p.. Thus, in the figure, r centimetres is 
 the length of either p. a or pb. This is because we suppose ds to be 
 exceedingly small with respect to r. 
 
 (iii.) If we draw b a' so as to make an isosceles triangle bp.a, we 
 may consider b a to be at right angles to both /j.a' and ph. 
 
 (iv.) And, following from (iii.), we may speak of If a as making 'the 
 angle 6 ' with either /z a, or with /LI b produced ; we shall generally say 
 ' with r? 
 
 When the above conditions hold, ab is called an element. These 
 conditions do not hold in the figure, since that is exaggerated in order 
 to make details sufficiently plain. 
 
 From the experiment of i it was not difficult for mathe- 
 maticians to guess at the law of action of each element of current 
 on the pole or on the short needle. The law guessed at was 
 such as, applied to the case of i, would enable us to predict the 
 results there experimentally arrived at. It was then applied to 
 other cases. The total action of a circuit was predicted by means 
 of the integral calculus from the assumed law of action of an 
 element. In each case thus tested, experiment fulfilled the predic- 
 tion. When any assumed law stands thus the test of experiment, 
 
296 ELECTRICITY 
 
 CH. XVIII. 
 
 we must be satisfied to consider it to be true. We now give 
 the law, thus established, of the action of an element of current 
 on a pole. 
 
 Let n be the pole-strength, r the distance in centimetres of the 
 element d s from the pole, d s the length of the element, d s . sin 
 the projection of ds perpendicular to r (or the length b a' in the 
 figure), C the current-strength. Then it is found that the pole 
 
 is urged with a force proportional to ^ ' 2 ' ' in a direction 
 
 perpendicular to the plane containing the element ds and the pole ^, 
 the direction of this force being further determined by Ampere's rule. 
 If we analyse this law with the aid of the figure given, we see that 
 istly. The force acting on /j. is directly proportional to p and 
 to C. This fact is what we should expect. In the last Chapter it 
 was partly proved, and partly assumed as following from the defi- 
 nition of IJL and C. 
 
 2ndly, For the same length ds of the element, the force is pro- 
 portional to sin 6 ; that is, the element acts solely in so far as it 
 presents a ' broadside-on projection ' (which is a! b in the figure) to 
 the pole. 
 
 3rdly. The force varies as - 1 - ; thus following the same law 
 
 as we found to hold in gravitation, in electrostatics, and in 
 magnetism. 
 
 . 4thly. If the current have the direction of the arrow, and the 
 pole be + or north-seeking, this latter is urged straight up from 
 the plane of the diagram. 
 
 5thly. The action is not in the line joining the pole and the 
 element, but is perpendicular to the plane containing the two. 
 The action then is very different from any with which we had to 
 do in electrostatics or in magnetism. 
 
 Note on the experiment 'of I. In the case of the infinite rectilinear cur- 
 rent, we found the total field at ns to vary as -, whereas the elemental law 
 
 r 
 
 has -4 This may perhaps perplex the learner. 
 
 The reason is as follows. As we recede further from the current A B, we 
 certainly increase the distance from each element. But at the same time the 
 upper and lower parts of the wire begin to present an increasingly great 
 ' broadside-on projection ' to the pole or needle. 
 
CH. xviir. CURRENTS AND MAGNETIC POLES 297 
 
 3. The Absolute System of Electro-Magnetic Units. The 
 reader should at this point look again at Chapter XIV. T, and 
 Chapter XV. 2. We intend now to explain what are the funda- 
 mental units to which we there referred. 
 
 (I.) Unit of current. \i a wire be bent into the arc of a 
 circle, and a pole be placed at its centre, we see by the results of 
 2 .that the forces due to all the separate elements will act in the 
 same direction on the pole ; and, further, since each element is 
 the same distance from the pole, the total force on this will be 
 simply proportional to the current- strength, the pole-strength, the 
 length of the arc, and the inverse square of the radius. 
 
 Now we have already denned the units of force, pole-strength, 
 and length. It is, therefore, simplest to define the electro- 
 magnetic unit of current to be such as makes the force unity 
 when all the other above quantities are unity. Or ..... 
 
 The elect?-o -magnetic unit of current is such that flowing through 
 an arc of unit length (i.e. i centimetre), whose radius is unity (i.e. 
 i centimetre), it acts with unit force (i.e. i dyne) on a pole of unit 
 strength (see Chapter II. 7) placed at the centre. 
 
 As has been already stated, the practical unit, called the 
 ampere, is one-tenth of the above absolute unit. 
 
 (I I.) The unit of E.M.F.If the E.M.F. betiveen two points in 
 a circuit be such that unit current flowing for unit time does unit 
 work between these two points, then this E.M.F. is called unit 
 E.M.F. in the electro-magnetic system. 
 
 It will be evident to the student that this definition of unit 
 E.M.F. is, in theory at least, simple and clear; for the work can 
 be conceived of as measured by observation of the number of 
 calorimetric heat units given out between the points (as in 
 Chapter XV. 4) ; while the current can be measured in the 
 above given absolute units by means of a galvanometer of known 
 constants. 
 
 Such a definition of unit E.M.F. is far better adapted to the 
 requirements of electro-dynamical measurements than is the 
 electrostatical definition given earlier. 
 
 As has been already stated, the above given unit is incon- 
 veniently small. The practical unit, called the volt, is io 8 times 
 this absolute unit. 
 
29$ ELECTRICITY 
 
 CH. XVIII. 
 
 (III.) The unit of resistance. Ohm's law defines the unit of 
 resistance as that through which unit E.M.F gives unit current. 
 The practical unit of resistance, called the ohm, is io 9 times the 
 absolute unit. Thus, as stated, we have ..... 
 
 T O^ T 
 
 The ampere = = th absolute unit of current. 
 
 T/VJ T/-\ 
 
 4. Summary of Electro-Magnetic Units (see 3). 
 
 ( Absolute unit of current as defined in (I.). 
 
 i Practical unit of current, the ampere . . = JL absolute unit 
 
 io 
 
 (Absolute unit of E.M.F. or of A V v . . ' . . . . as defined in (II.). 
 
 (Practical unit of E.M.F., the volt = io' s absolute units. 
 
 f Absolute unit of resistance as defined in (III.). 
 
 ( Practical unit of resistance, the ohm = 10 absolute units. 
 
 /Absolute unit of activity is the rate of work when unit current runs 
 
 under unit E.M.F., and is i erg per second. 
 
 j Practical unit of activity, the watt, is the rate of work when one 
 ( ampere works under one volt E.M.F. ; it = io 7 ergs per second. 
 | We get watts by multiplying the number of volts by the number of 
 j amperes ; or, if we are considering only the heat given out in a 
 ( conductor, by (number of amperes} 2 x (number of ohms]. 
 
 (about -24 calorie per second, 
 
 I or about ^ English horse-power. 
 
 (Absolute unit of quantity is that which crosses any section of a con- 
 ductor in one second when absolute unit of current flows. 
 Practical unit of quantity, or the coulomb, is the same when one 
 ampere flows. 
 
 Absolute unit of capacity is that of a condenser in which the bound 
 charge is absolute unit of quantity when the AV of the plates is 
 absolute unit of E.M.F. or AV. 
 
 Practical unit of capacity, or the farad, is that of a condenser in 
 which the bound charge is one coulomb when the AV of the plates 
 is out volt. We may add that the more usual practical unit is the 
 micro-farad (see below). 
 
 The megohm, mega-volt, c., are respectively i,coo,ooo times the 
 ohm or volt, &c. 
 
 The microhm, micro-volt, &c., are respectively - of the ohrn 
 
 i oooooo 
 
 or volt, &c. 
 
CH xvm. CURRENTS AND MAGNETIC POLES 299 
 
 Note. Determination of the units. The three quantities, C, R, and E, 
 are related to each other by Ohm's law. It may be well to indicate how the 
 absolute units, or any convenient multiple of them, might be determined 
 independently of one another. 
 
 Current. When the dimensions and construction of a galvanometer are 
 known, and when the magnetic field H in which it is situated has been deter- 
 mined by the method indicated in Chapter III. 15 and 16, it is possible to 
 measure current in absolute electro-magnetic units by observations of its action 
 upon the galvanometer needle. For such a measurement no knowledge of E 
 or of R is either directly or indirectly implied. 
 
 Resistance. The absolute measurement of the resistance of a conductor is 
 one of the most important experiments in physical science, and one of extreme 
 difficulty. Of the several methods that have been employed to obtain a direct 
 measure of the resistance of a wire, the simplest consists in passing a current 
 through the wire which is contained in a calorimeter similar to that shown in 
 the figure on page 241. Theoretical considerations give for the heat developed 
 per second in the calorimeter the value JH = C-R. Here C is the current, the 
 value of which can be calculated from the indications of a tangent galvano- 
 meter placed in the circuit ; H is the heat developed per second, which can be 
 found from the rise of temperature of the liquid by the principles of calorimetry ; 
 and J is the mechanical equivalent of heat (p. 239), which is known from 
 Joule's experiments to be very near to 41,750,000. The only quantity remain- 
 ing is R, which can therefore be calculated from the other three. Other methods 
 depend upon the laws of electro-magnetic induction given in p. 357 ; for the 
 details of these the student must refer to' more advanced works upon the subject. 
 
 Electromotive force. Having methods of measuring current and resistance 
 absolutely, E.M.F. may be found by the arrangement shown on p. 234. e is 
 the cell whose E. M. F. is required ; a tangent galvanometer is placed between 
 R and B, and the current Ae (l GB is not required. The current given by the 
 battery P is first adjusted to a convenient strength by the rheostat R, then the 
 contact piece Q is moved along the wire A B until no current flows in G. 
 When this is the case the reasoning of 13, p. 234, shows that e = aC ; a being 
 the resistance of the wire A Q, and C the current in the circuit P R B Q A P, 
 which is given by the tangent galvanometer : thus e is determined. 
 
 5. The Dimensions of the Derived Units. 
 
 When any system of units has been constructed, each derived unit 
 can be expressed in the fundamental units. The manner in which 
 each derived unit involves the fundamental units can be exhibited in 
 a simple form, giving what is called the dimensions of that physical 
 quantity. Thus, in Chapter II., the dimensions of force were given by 
 the relation ..... -" .............. 
 
 In any one system of units physical quantities are of different 
 natures if their dimensions (i.e. the way in which they involve the 
 
3OO ELECTRICITY 
 
 CH. XVIU. 
 
 fundamental units) are different. This statement may be taken as 
 axiomatic. 
 
 But the same physical quantity may have different dimensions in 
 two different systems of units respectively. This is not so easy a 
 matter to understand, but the fact is clear enough, as will be seen 
 from the table given on the next page. Thus the dimensions of current 
 are different according as we regard it from the point of view of electro- 
 static quantity (see Chapter V. I passing across a section in unit 
 time, or from the point of view of magnetic actions. 
 
 If two physical quantities have different dimensions in the same 
 system, it follows that not only are they different in essential nature, 
 but that in general any alteration of the fundamental units will alter 
 in different ratios the numerical values of the two quantities. 
 
 On p. 301 we give a table exhibiting the dimensions of the various 
 physical quantities in the two systems, electro-magnetic and electrostatic, 
 respectively. It would be out of place in this Course to show how the 
 dimensions of each are found, but we give one example. 
 
 Example. The dimensions of magnetic pole-strength. By defini- 
 tion and experiment we have .:"; 
 
 force between two equal poles = no e-s reng ^ 
 
 (distance oetween tnemj- 
 
 Using symbols to represent the units of pole-strength, force, and 
 
 length, we may write -. " 
 
 ^ = F x L 2 = ML x L o = M L 3 T _ 2> 
 
 And therefore /u, = M . L . T" 1 gives the dimensions required. 
 
 With respect to the fact that in the electro -magnetic system the 
 dimensions of R are LT~', or are those of a velocity, the reader is 
 referred to Chapter XXI. 7 (II.), note. 
 
 The ratio-velocity v. We must now call attention to the very 
 remarkable fact that when a quantity is expressed both in electrostatic 
 and in electro-magnetic measure, the ratio between the two sets of 
 dimensions always involves simply the dimensions of a velocity ; this 
 we have designated by v in the last column of the table on the next 
 page. Direct experiment shows (within the limits of experimental 
 error) that this velocity is a constant. 
 
 If C and C' represent the numerical magnitude of the same 
 current measured electro-magnetically and electrostatically respec- 
 tively, and if the same convention be assumed with respect to the 
 other symbols, then by the table on the next page we have . . ., .^ 
 
 v = 2: = c : = E = /IT = /K 7 
 
 Q C E' VR> </ K 
 
CH. xvni. 
 
 CURRENTS AND MAGNETIC POLES 
 
 301 
 
 Table of Dimensions of Units. 
 
 
 
 In the (mag- 
 
 
 
 Symbol 
 
 The magnitude measured 
 
 netic or) elec- 
 tro-magnetic 
 
 tn the electro- 
 static system 
 
 Ratio of latter to 
 former 
 
 
 
 system 
 
 
 
 u 
 
 Strength of magnetic pole 
 
 LjM-T~ 
 
 
 
 
 
 V 
 
 Magnetic potential . . . 
 
 L-M^T" 1 
 
 
 
 
 
 I 
 
 Strength of magnetic field 
 
 L^M^T' 1 
 
 
 
 
 
 m 
 
 Magnetic moment . . . 
 
 T ^M^T 1 " 1 
 
 
 
 
 
 P 
 
 Intensity of magnetisation 
 or (magnetic moment)-:- I 
 
 L^-M-T" 1 
 
 
 
 
 
 
 (volume) 
 
 
 
 
 Q 
 
 Quantity of electricity . . 
 
 L*M 
 
 L'M'T"' 
 
 i i ^ ' 
 
 c 
 
 
 L^M-T" 1 
 
 L^M^T" 2 
 
 L T~ ( = /) 
 
 v ) 
 
 Potential, difference of x 
 
 
 
 
 AV 
 
 andEj 
 
 potential, and electro- I 
 motive force 
 
 LtM^T- 3 
 
 L/M'T- 1 
 
 ~* } 
 
 R 
 
 Resistance 
 
 LT- 1 
 
 L~'T 
 
 L~ 2 T 2 (- I ) 
 
 K 
 
 Capacity 
 
 
 L 1 
 
 L 2 T- 2 l\ 
 
 
 Specific inductive capacity 
 
 
 
 [a simple 
 
 
 
 
 
 
 ratio or 
 
 
 
 
 
 number ; 
 
 
 
 
 
 zero di- 
 
 
 
 
 
 mensions] 
 
 
 The value of the constant v has been determined from double 
 measurements of the same quantity, the same E.M.F., or the same 
 capacity, respectively ; and each investigation agreed in giving to v a 
 value approximately agreeing with the velocity of light, and of the 
 propagation of electro-magnetic induction (see Chapter XX. 18). It 
 also agrees with the velocity with which two bodies carrying electro- 
 static charges must move parallel to each other in order that the 
 electrostatic repulsion between them may just balance their electro- 
 magnetic attraction (see Chapter XXII. 15 (e) ). It would appear 
 that when rapid motion is given to electrostatic charges of electricity, 
 these give electro-magnetic fields as do currents ; but the velocity 
 given to them must be very great in order to render this action at all 
 sensible. 
 
 We may then regard the velocity v as one of the great constants 
 in nature, that which expresses the physical constitution of the ether 
 with respect to the propagation of waves in which the vibrations are 
 transverse. 
 
302 ELECTRICITY en. xvni. 
 
 6. Field due to a Circular Current, We will now complete 
 what was said about a circular current in Chapter XVII. 2. 
 Near the wire the lines of force will practically be small circles 
 about the wire, as in Chapter XVII. i, since the rest of the 
 circuit will be relatively unimportant. But in mean positions 
 the lines will be the resultants due to the whole circuit, and will 
 not be circles, but curves of some other form situated symmetri- 
 cally with respect to the central line and to the plane of the 
 circular current. 
 
 Field at the centre* of the circular current. Now consider the 
 field at the centre of the circle. Referring to 2 and 3, we see 
 that the following considerations will determine the field. 
 
 (i.) All the elements of this circular current are the same distance 
 r from the centre, and therefore the field-strength at the centre is 
 directly proportional to the length 2 TT r of the circumference. 
 
 (ii.) The field will also be proportional to the inverse square of the 
 
 common distance r, or to -,, where r is the radius of the circle. 
 
 (iii.) If the current be given in electro-magnetic measure, it will be 
 easy to find the magnetic field-strength at the centre, or the number 
 of dynes with which unit pole would there be urged in a direction 
 perpendicular to the plane of the circle. 
 
 From the above it follows that if c be the current in absolute 
 measure, and I the magnetic field-strength at the centre, then . . 
 
 ,- 2 Trr. c 2 TT. c 
 
 The force with which a pole of strength ^ is urged will be . . 
 F = 2 J^J dynes. , (ii.) 
 
 The couple acting on a small needle of magnetic moment ;, 
 placed at the centre and lying in the plane of the circuit, will be . 
 
 in /= ' 2rn L'' K units of couple (iii.) 
 
 If the current C be given in amperes, we must divide each of 
 the above results by 10. 
 
 7. Magnetic Shells ; and the Fields due to Them. If we 
 regard only the external magnetic field produced by an electric 
 circuit, we find that we can replace the circuit by a thin steel she^t 
 
CM. xvm. CURRENTS AND MAGNETIC POLES 303 
 
 whose faces are of opposite polarities ; such a sheet magnet, 
 or magnetic shell, giving exactly the same magnetic field as did the 
 current, provided that certain conditions as to its thickness and 
 its degree of magnetisation be observed. 
 
 But it will be necessary first to give some account of the 
 nature of a magnetic shell, or, as it is otherwise expressed, of 
 lamellar distributions of magnetism. We return to ' currents ' 
 in 9. 
 
 Consider an enormous number of small straight magnetic 
 needles placed side by side, until . they form a sheet whose area is 
 very large as compared with the length of each little needle ; that 
 is, a sheet very large as compared with its thickness. If the needles 
 be all turned one way, so that the one side of the sheet be 
 formed of their north poles and the other side of their south poles, 
 then we have what is called a lamellar distribution of magnetism ; 
 or we have a magnetic shell. 
 
 In the case of magnetic needles, whose length was very great as 
 compared with their thickness, it was convenient to speak of their 
 poles and of their length ; the product of the length and pole- 
 strength giving us that magnetic moment of the needle (called /j. I 
 or m) upon which so much was found to depend. The reader 
 will see from Chapter III. 16, and from Chapter XVIL, that 
 upon the magnetic moment of the needle depended the external 
 field that it gave, and also the couple with which a magnetic field 
 acted upon it in turn. 
 
 Now in the case of a magnetic shell we must express ourselves 
 somewhat differently. We must here consider the intensity p of the 
 surface magnetisation, and the thickness / of the shell. 
 
 To take the case with which we shall be concerned, viz. that 
 of a uniform shell, we may suppose all the little needles to 
 have been equal both in length /, and in pole-strength. Then, if 
 the sum of all the little poles that form i square centimetre of the 
 surface of the shell would, if concentrated at one point, form a 
 single pole of strength measured by p units, we may very naturally 
 say that the density of magnetism on the surface of the shell is 
 measured by p ; the word ' density ' being here used in the same 
 sense as in electrostatics ; viz. quantity [in this case magnetic 
 quantity] per unit area. 
 
 If the thickness of the shell be called /, the product t.p is 
 
304 ELECTRICITY CH. xvm. 
 
 called the strength of the shell. It is the same as the magnetic 
 moment of a needle of length / and of pole-strength p, repre- 
 senting the concentrated magnetism of i square centimetre of the 
 surface. We will use the single symbol / for the strength of a 
 shell. 
 
 Now let us consider any point P external to the shell. Let 
 the boundary of the shell subtend a solid angle l measured by il 
 at the point P ; and let the strength of the shell be /. Then it 
 can be shown that the field at the point P depends upon /, upon 
 O, and upon the way in which ft alters in value as we move from 
 point to point. As long as the thickness / is relatively very small, 
 it is found (by mathematical methods belonging properly to the 
 integral calculus] that we are concerned with the strength j ; not 
 with / and p separately. 
 
 The reader must notice that (according to the mathematical 
 investigation not given here) the field at P does not depend upon 
 the shape of the shell, so long as its edge remains the same. If 
 the shell be bent into a surface nearly closed, but having a hole of 
 a certain shape and size left unclosed, it will give the same external 
 field as would a shell that just fitted this hole ; since the two 
 would have the same edge. And a completely closed shell gives 
 no external field. 
 
 8. Magnetic Potentials due to Magnetic Shells. The 
 reader is now referred to what was said of potential, and of force 
 
 1 Note on solid angles, Let us consider a point P and a sphere of unit radius 
 described about P as centre. Any portion of the surface is said to subtend 
 a solid angle at the centre, this angle being measured by the fraction 
 
 portion of surface in square centimetres c . ., r ,. ., 
 
 \ ? L. : Since the radius is one centimetre, 
 
 (distance from centre in centimetres)' 
 
 we may say that the solid angle is measured by the area of the portion of the 
 surface considered, this area being expressed in square centimetres. Now let 
 lines be drawn from P to the edge of the magnetic shell. These will intercept 
 a portion of the surface of our unit sphere. The solid angle H subtended by 
 the boundary of the magnetic shell at the point P is measured by the portion 
 of the surface, expressed in sqtiare centimetres, which is thus intercepted on the 
 surface of the unit sphere described about P as centre. 
 
 Thus, since the total surface of unit sphere is 4 TT square centimetres, the 
 total solid angle about any point is 4 IT. 
 
 If a point is relatively very close to the surface, the surface subtends (very 
 nearly) a solid angle of 2 ?r at the point ; and subtends exactly 2 TT when the 
 point is actually on the surface. 
 
CH. xvin. CURRENTS AND MAGNETIC POLES 305 
 
 as measured by rate of change of potential, in Chapter X. 9-14 
 and elsewhere. If ' -t- unit pole ' and * unit pole ' be substi- 
 tuted for ' + unit electricity ' and ' unit electricity,' all that was 
 there said applies to magnetic fields. (Thus, for example, a pole 
 p gives out 4 TT p marked lines of magnetic force.) It is then 
 readily understood that if we know the magnetic potential at a 
 point P due to a magnetic shell or due to an electric circuit, and 
 if we know at what rate this potential changes in different direc- 
 tions, then we know all about the magnetic field at P. 
 
 In the case of the magnetic shell, it can be shown that the 
 magnetic potential on unit pole at P due to this shell is measured by 
 the product j'Q. The force at P in any direction will, for the general 
 reasons given in Chapter X., be measured by the rate at which this 
 potential changes its value in this direction ; and this, since j does not 
 alter for the same shell, depends upon the rate at which Q changes in 
 that direction. Thus a small shell close by and a large one far off may 
 give the same value of Q, at P, or the same potential at P ; but in 
 general the rate at which Q, changes, or the force at P, will be different 
 in the two cases. 
 
 Inside a completely closed surface formed of a magnetic shell, the 
 potential is everywhere 4?r.y, as explained; since now Q = 477. It 
 does not, therefore, change from point to point ; or there is a constant 
 potential and no force in the hollow of such a closed surface. Outside 
 it the potential and force are both zero t since, as explained, O now 
 equals zero. 
 
 As was seen in Chapter X., the direction in which the potential 
 changes most rapidly is the direction of the resultant force, or is the 
 direction of * the ' lines of force of the field. 
 
 Of course the potential on a pole of \i units will be measured by 
 
 It follows also that the work done between two positions of the 
 pole /z, at which the shell subtends solid angles Q.^ and Q 2 respectively, 
 will be measured by 
 
 The mathematical investigation of the potentials and fields 
 due to a magnetic shell demands a knowledge of the infinitesimal 
 calculus. But an investigation in which infinitesimal notation, at 
 any rate, is not used, can be found in Cumming's ' Theory of 
 Electricity ' (Macmillan). 
 
306 
 
 ELECTRICITY 
 
 CH. XVIII. 
 
 9. Magnetic Equivalent of an Electric Circuit. We now 
 
 return to that which was the main reason for the digression of 
 7 and 8. 
 
 If we have on the one hand a circuit carrying a current 
 measured by C in absolute electro-magnetic units, and on the 
 other hand a magnetic shell of the same boundary and of a 
 ' strength ' j numerically equal to C ; and if we examine the 
 potentials and fields at any external point, due to the two re- 
 spectively, then it is found that the two give identical results. 
 
 The proof of this equivalence demands a knowledge of the 
 infinitesimal calculus ; and in the present Course the reader must 
 be content to accept as a result of mathematical analysis confirmed 
 by expeiiment the statement that 
 
 1 As regards the external magnetic field produced, a circuit of 
 current-strength C (as measured in absolute electro-magnetic units) 
 is exactly equivalent to a magnetic shell having the same boundary 
 and having an uniform strength j numerically equal to C> 
 
 So the formulae of the last section become, for an electric circuit, 
 p C Q and ft C (Oj - O 2 ). 
 
 Experiment. De la Rive's floating battery. A small battery-cell is con- 
 structed, composed of two strips the one of zinc and the other copper immersed 
 
 in dilute acid in a short 
 wide test tube. The cir- 
 cuit is completed by an 
 insulated wire wound into 
 a small circular coil that 
 has its plane vertical when 
 the test tube is vertical. 
 This small cell and cir- 
 cular current is floated 
 on the surface of a large 
 vessel of water, by means 
 of a cork or other float 
 through which the test 
 tube passes. 
 
 We thus have a 
 vertical circular current 
 capable both of rotation 
 on its vertical axis and 
 of horizontal translation. 
 
 If we test it by means of a large bar magnet we shall find that it acts as 
 would a thin circular magnetic shell. As could be deduced from Chapter 
 
CH. xvin. CURRENTS AND MAGNETIC POLES 3O/ 
 
 XVII. 2, that face in which (as one stands opposite to it) the current 
 appears to circulate clockwise will act as a south-seeking magnetised surface, 
 while the face in which the current appears to run coimter-clockwise acts as a 
 north-seeking surface. The coil will thus turn until the face of opposite polarity 
 to that of the magnetic pole presented is turned towards this latter, and will 
 then be ' attracted ' and move towards it. 
 
 If the current be strong and the water be still enough, the coil will under 
 the earth's action only ' set ' with its faces turned north and south respectively. 
 
 The equivalence of a magnetic shell and an electric circuit was 
 discovered and stated by Ampere. 
 
 10. This Equivalence is for the External Field only. There 
 is one obvious difference between the circular current and its 
 ' equivalent' magnetic shell. In the circuit, the field is continuous; 
 and a pole is urged through the circuit as well as up to it. In the 
 magnetic shell the external field stops at the surface of the 
 shell. 
 
 ii. Principle of Sinuous Currents. Referring to 2, we 
 see it stated as the result of experiment that the element a b acts 
 on the pole as does the element a' b. 
 
 Extending this to the case of wires of finite length we should 
 predict that a circuit of any form in which the wire has small 
 sinuosities, loops, and other irregularities, will act on an external 
 pole just as does a circuit of the same general form in which the 
 wire is without such irregularities ; provided that the irregularities 
 are of very small size as compared with their distance from the 
 pole acted on. 
 
 This equivalence of simple and sinuous wires is called the 
 * Principle of sinuous currents. ' 
 
 Experiment. Employing the 'zero method,' as before, it is easy to show 
 that a circuit, in which the current goes one way round a simple wire, and 
 returns by another wire wound over the former in small sinuosities, has zero 
 action on a pole. This shows that simple and sinuous currents, which have 
 the same general form, are equivalent as regards action on an external pole. 
 
 Thus, a complete circuit in which the wire has sinuosities that 
 are very small as compared with the size of the circuit, is equi- 
 valent to a magnetic shell of the same average boundary as the 
 circuit and same * strength.' 
 
 Thus, we may regard the arrangement of the above experiment 
 to be equivalent to two equal shells superimposed on one another, 
 
 X 2 
 
308 
 
 ELECTRICITY 
 
 CH. XVIII. 
 
 the faces being turned contrary ways ; such a magnetic arrange- 
 ment giving neutrality. 
 
 12. Keaction of a Pole on an Element of Current. We 
 have spoken of the force exerted on a pole by an element of 
 current, assuming the pole to be moveable and the current to be 
 fixed. But by Newton's laws, which may be regarded as so con- 
 firmed by universal experience that they are axioms, there must 
 be between the pole and the element a reciprocal action of which 
 either half becomes manifest as we fix the pole or fix the current 
 respectively. The pole and the element are urged round each 
 other in a direction given by the 'screw' analogy of 2. If we 
 fix the pole, then the current will be urged round it, the rotation 
 following the same direction as before. We can thus give the 
 rule that 
 
 1 If we stand in a + pole, so that we are exactly parallel to, and 
 facing, a person swimming with the current in the element (see 
 Ampere's rule, 2), then the element is urged off to our left in a 
 direction perpendicular to the plane containing the element and the 
 
 .pole ; the force being measured by ' - as in 2.' Of 
 
 course a pole gives the force in exactly the opposite direction. 
 
 13. Action of a 
 Pole upon a Closed 
 
 Circuit. What fol- 
 lows applies to any 
 closed circuit ; but, 
 since the mechanical 
 reasoning is then 
 simpler, we will take 
 only the case of a 
 plane closed circuit. 
 
 Let N be the north 
 pole of a magnet, and 
 N.A' A"X a straight line through N ; we suppose the pole to be con- 
 centrated at the point N and to be of strength /*. Let the circuit C 
 be so fixed that it can only turn about the axis N X, swinging round 
 upon the points A' A" as a door swings upon its hinges. 
 
 We will consider whether or no the pole N has any tendency to 
 make the circuit C revolve as a whole round the axis N X. 
 
CH. xvin. CURRENTS AND MAGNETIC POLES 309 
 
 Take two lines N a' a" and N b' b", so close together that a' b' and 
 a" b" may be considered to be elements (see 2), and drop perpen- 
 diculars on N X from a" and a'. 
 
 Since the circuit is closed, each such pair of lines will intercept 
 two elements a b' and a" b", in which the current flows opposite ways. 
 
 As seen in 2, we may, as regards the action of N, replace a' b' 
 and a" b' f by the intercepts made on circles through b' and b" respec- 
 tively ; these intercepts are represented by dotted lines, and will be 
 called s' and s" respectively, their distances from N being called 
 r' and r". 
 
 Now the pole N urges a" b" perpendicularly down into the page 
 
 C s' f 
 with a force measured by "- f ] ; while a' b' is urged upwards from the 
 
 page with a force '-^- . 
 
 (r ) 
 
 But from plane geometry we have the proportion s" : s' = r" : r' ; 
 and hence the force on a" b" : force on a' b' = _L : L. 
 
 The effect these will have in turning the circuit as a whole about 
 N X as axis depends upon the product of these forces into the arms, 
 a" A" and a' A' respectively. 
 
 But, again by simple geometry, it is clear that 
 
 arm a" A" : arm a' A = r' f : r'. 
 
 Combining this result with that obtained for the two forces, we 
 have that the two moments are equal, being in the proportion of 
 
 (r" x JL\ : (r' x M or oif I : i. They are also opposite in direc- 
 tion. Hence they neutralise each other. In the same way the 
 whole closed circuit may be divided into pairs. of elements that, 
 together, give zero moment about N X. Hence the whole has zero 
 moment about the axis N X. 
 
 By similar, but more advanced, reasoning it can be shown that for 
 any closed circuit [whether plane or not] there is zero moment about' 
 any axis through a pole N. 
 
 By the above method of investigation it is shown that a com- 
 plete circuit has no tendency to revolve as a whole about any axis 
 through the pole of a magnet ; of which axes one particular case 
 is the magnetic axis that joins the N and S poles of the magnet. 
 This means, when interpreted mechanically, that the two poles N 
 and S act on the closed circuits in lines passing through the poles > 
 
ELECTRICITY 
 
 CH. XVIII. 
 
 since, if they did not act in such lines, we should have a 'moment' 
 about axes through the poles. In other words, the action between 
 the circuit considered as a whole and the poles is a direct action, 
 and not such as that considered in 2. This result, arrived at 
 from the consideration of the action of a pole on the elements 
 of current forming the complete circuit, agrees with the view 
 which regards the circuit as equivalent to a magnetic shell. For 
 the action between a pole and a magnetic shell is also a direct action. 
 
 The line of action between N and the circuit C will pass 
 through some point in C ; and, if the circuit be moveable about 
 some axis through which the line of action does not pass, the 
 circuit will revolve into a position of equilibrium, or will ' set 
 itself,' as would a magnetic shell. In this position it will then 
 rest, being prevented, owing to the fixed axis, from moving up to 
 the pole. 
 
 14. Action of a Pole on an Incomplete Circuit. If there be 
 such an arrangement that the circuit external to the pole be not 
 
 FIG. i. 
 
 complete, then this 'incomplete circuit' will be urged round the 
 pole. Every voltaic circuit must be complete ; but part of it 
 may be completed through the magnet itself, leaving the portion 
 external to the magnet incomplete. 
 
 In fig. i. we see that, as regards rotation round the pole N, the 
 portion P Q of the circuit C is uncompensated, and therefore 
 the whole will be urged round an axis through N. 
 
CH. xvrn. CURRENTS AND MAGNETIC POLES 311 
 
 In practice this can be effected by arranging a circuit A C B A 
 so that the part C is mobile about the axis A B ; the part A B of 
 the circuit being the axis of the magnet N S. 
 
 It can be shown that if the points A B lie both between N and S, 
 or both outside N S, the resultant action of the two poles N and S will 
 be such that C does not revolve about the axis. In the case drawn it 
 will so revolve, 
 
 15. Action of the Earth's Field on Currents Completely 
 or Partly Mobile. 
 
 When currents are ai ranged so as to be completely or partly mobile 
 about vertical or horizontal axes, there will be in general movements 
 due to the action of the earth's field. 
 
 If we resolve the earth's field into its vertical and horizontal com- 
 ponents, the nature of the movements referred to can be predicted 
 without much difficulty, by means of the principles explained in 
 Chapter XIX. 4, 5, and 7. 
 
 There is, however, no especial interest attaching to these actions, 
 and we shall accordingly omit further discussion of them. 
 
 1 6. Actions Between Currents; Ampere's Laws. The first 
 law arrived at by means of direct experiment is very simply ex- 
 pressed. It is as follows. 
 
 Law I. Parallel currents in the same direction attract, parallel 
 currents in opposite directions repel, one another. 
 
 There are many pieces of apparatus by means of which this 
 fact can be shown ; the form that is perhaps of the most general 
 use is the ' Ampere's stand. 5 It is not necessary to describe in 
 detail the arrangements in this or in similar pieces of apparatus. 
 It is sufficient to say that, by means of pivots and mercury con- 
 nections, wires bearing currents are given, to a greater or less 
 degree, freedom of movement ; and that thus the action of other 
 currents on them can be observed. 
 
 Experiment. The figure on the next page represents the portion M of a 
 rectangular coil M N acting upon the portion B of the moveable piece B C. As 
 drawn, the currents in M and in B are in opposite directions, and repulsion 
 between M and B will be observed. 
 
 When two wires bearing currents are not parallel, but are in- 
 clined to one another, the following law is found to hold (see also 
 Chapter XIX. 8). 
 
312 
 
 ELECTRICITY 
 
 CH. XVIII. 
 
 Law II. When two currents make an angle with one another 
 they attract one another if they run both towards or both from the 
 vertex of the angle, and repel one another if they run the one toivards 
 and the other from the vertex. 
 
 FIG. i. 
 
 Or, two currents crossing one another tend "to move into a posi- 
 tion in which they are parallel and in the same direction. 
 
 Experiment. This may be illustrated by 
 means of the arrangement here indicated. The 
 |jf[ p current PQ acts mainly on the portion CD of 
 
 >-!J _ A the mobile current, and movements ensue which 
 are in accordance with the above law. 
 
 It may be observed that P.Y acts on the 
 portions Y C and C B, and the portion Q Y acts 
 on D Y and on A D, so as to give movement in 
 one and the same direction. The action on the 
 upper part BA will be in a contrary direction, 
 but this action will be negligible if B is relatively 
 very far from P Q. 
 
 [Law ITT. In a rectilinear current there is repulsion between each 
 two consecutive elements of the current. Ampere devised experiments 
 
CH. xviii. CURRENTS AND MAGNETIC POLES 313 
 
 in which it appeared that collinear elements of a current repelled one 
 another, as if, indeed, this were an extreme case of Law II., in which 
 the angle between the two currents was 180. It is, however, doubtful 
 whether any experiment has shown directly this action between two 
 elements of currents that are truly collinear. It is indeed true that 
 calculations based upon the assumption of this action have been veri- 
 fied. But the simple statement of this law, as given above, does not 
 fit in well with the modern theory of electro-magnetic and electro- 
 dynamic actions, and, as no direct application of it will be made in 
 this Course, we think it best to pass it by without laying any stress 
 upon it.] 
 
 For the more modern view of Laws I. and II. we refer the 
 reader to Chapter XIX. 8. 
 
 17. Continuous Rotations of Currents. 
 
 If O A be a current mobile about O as centre, and if P Q be a 
 rectilinear current, the above first two laws show us that there will 
 
 be continuous rotation of _-- x 
 
 O A through the posi- 
 tions O A', O A", O A'", 
 &c., round and round. 
 
 So again, if A B C be 
 a circular current and 
 p m be a current mobile p 
 about p as centre, then 
 from the same laws we can again predict continuous rotation. 
 
 Experiments have been devised that illustrate these actions. For 
 the more modern view of the same we refer the reader to Chapter XIX. 
 9- 
 
 1 8. Ampere's Laws of the Actions Between Elements of 
 Currents. 
 
 By means of experiments on finite currents, and by deductions 
 from these aided by arguments based upon the principle of ' symmetry,' 
 Ampere investigated the laws governing the action of 'elements of 
 current ' (see 2) upon one another. 
 
 These laws once established, he employed them to calculate the 
 actions of circuits upon each other in various cases, the calculations 
 being as a rule long and laborious. In fact he worked out, in a manner 
 which commands our admiration, a very complete system and method 
 of attacking electro-dynamical problems a method based upon the 
 * action at a distance ' between the current elements. 
 
 Now, however, this method is not employed. Faraday introduced, 
 
3H ELECTRICITY CH. xvm. 
 
 and Clerk Maxwell further worked out and systematised, the electro- 
 magnetic method of treating all electro-dynamical matters. In this 
 method the currents are treated as magnetic systems giving magnetic 
 fields and acted upon by magnetic fields. This view is far more fruit- 
 ful than is Ampere's, and probably represents more accurately the 
 physical nature of the actions. The views and methods of Ampere 
 were rather those of a mathematician than of a physicist. 
 
 With respect to his views we must, however, allow that for 
 many purposes, e.g. for predicting the nature of the movements 
 that will ensue in certain cases, the Laws I. and II. given in 
 16 are very convenient ; and they should certainly be committed 
 to memory by the student, as expressing and summing up a group 
 of experimental, results. 
 
CH. XIX, 
 
 CHAPTER XIX. 
 
 LAWS OF THE MOVEMENTS OF CURRENTS AS DEDUCED FROM THE 
 CONSIDERATION OF MAGNETIC FIELDS AND POTENTIALS. 
 
 i. Magnetic Fields and Potentials. In Chapter X. we dis- 
 cussed, with reference to electrostatic phenomena, various matters 
 of importance connected with fields of force, lines of force, poten- 
 tials, equipotential surfaces, and the like. The reader must ob- 
 serve that all the general propositions there given hold good, 
 mutatis mutandis, for any field where the forces that give rise to 
 
 the field vary as -^. They hold good, e.g., for magnetic fields. Thus 
 
 we can test and investigate a magnetic field by the magnitude 
 and direction of the force acting on + unit-magnetic-pole in the 
 various parts of the field : and again, we can map out a magnetic 
 field in just the same way as, in Chapter X. 13, 14, we mapped 
 out the electrostatic field. 
 
 So also magnetic-potential-differences can be measured by the 
 work done on + unit-pole, the work being measured in ergs. 
 And the 'tubes of force' property given in Chapter X. 16 
 applies also to magnetic tubes of force. 
 
 2. Movements are from Higher to Lower Potentials, When 
 a pole, system of poles, or electric circuit equivalent to a magnetic 
 system, is placed in a magnetic field, it will as a rule be urged by 
 forces. By the very definition of ' potential ' // will be urged from 
 a place or position of higher, to one of 'lower, potential ; or it moves 
 from a place or position to which it could have been brought from 
 an infinite distance with the expenditure of more work, into a 
 place or position to which it could be brought from infinity with 
 less work. This, in the case of electrostatics, amounts only to 
 saying that a -f charge moves down, and a charge moves up, 
 the lines of force ; for, in electrostatics, we deal always with 
 
ELECTRICITY CH. XTX. 
 
 charges that are either + or . In the case of magnetic fields 
 we may be dealing with simple + or poles, in theory at least ; 
 and with regard to these the same may be said. But we may be 
 dealing with magnetic shells or with electric circuits that are equi- 
 valent to magnetic shells ; and in such cases the movements may 
 be of both rotation and translation, or of either alone. The italic- 
 ised statement given above is therefore the more general form of 
 the law. 
 
 To make matters clearer we will consider the case of a field 
 due to a + magnetic pole N. A simple + pole placed in this 
 field will move off to infinity from N, and a simple pole will 
 move from infinity up to N ; in both cases the final position is 
 that to which the + pole could be brought from infinity with least 
 positive (or with most negative) work. A magnetic shell placed in 
 the field will turn until its face is turned towards N, and will then 
 move up to N ; the final position being that into which it could 
 have been brought from infinity with the most negative work pos- 
 sible. If the shell were compelled to slide on rails so as to keep its 
 + face towards N, it would then move off to infinity ; this being, 
 under the assumed conditions of constraint, the position of lowest 
 potential possible. 
 
 Whether the final position of the pole or magnetic system be 
 one of low positive-, of zero-, or of negative-potential, depends 
 upon the conditions of each particular case. But this final position 
 will always be that from which the system could be restored to its 
 initial position only with the expenditure of the greatest amount 
 of work that was, under the existing conditions as to freedom of 
 movement, &c., possible. 
 
 3. Potentials on Poles and on Circuits. Where we have 
 magnetic /<?/<*$ in fields due to magnetic shells, or to electric circuits 
 equivalent to magnetic shells, it is convenient to use imiO potential 
 on these poles the expression given in Chapter XVIII. 8 and 9. 
 This expression, mutatis mutandis, is derived mathematically from 
 the fundamental form of Chapter X. 7 and 9. 
 
 If we have a magnetic shell, or an electric circuit equivalent 
 to this, in a field, it is convenient to derive another form for the 
 expression giving the potential on the shell. Referring to the 
 method of mapping out fields, given in Chapter X. 14, we may 
 consider the number of marked lines of force that pierce the shell or 
 
CH. XTX. LAWS OF THE MOVEMENTS OF CURRENTS 317 
 
 circuit. These are said to pierce the circuit in a + direction if 
 they run into its + face and out at its face ; that is, if they run 
 in the opposite direction to the circuit's own lines of force. 
 
 By considering various cases the reader will perceive . . . . 
 
 (i.) That when a shell or circuit is moved from infinity into a 
 position in which lines of force pierce it in a + direction, then 
 positive work is done on the shell. 
 
 (ii.) And when moved into a position in which on the whole 
 no marked lines of force pierce it, or in which [if plane] it lies 
 ' edgewise ' to the field, then no work is done. 
 
 (iii.) And when moved into a position in which the lines 
 pierce it in a direction, then negative work is done. If then 
 we so move the shell that more lines pierce it in a + direction, 
 or fewer lines pierce it in a direction, we do positive work on 
 the shell ; and conversely. The former position is the one of 
 higher potential. 
 
 The foregoing is reasoning of a general nature : we will now 
 examine this matter in a more exact manner. It was stated in 
 Chapter XVIII. 8 and 9 that the potential of a shell or circuit 
 on a pole, or of a pole on a circuit (this potential must by 'Con- 
 servation of energy 3 be mutual] is given by the expression j*/Q, or 
 H C fl, respectively. Here C is in absolute electromagnetic units. 
 
 Now a pole p. gives out 477 /M lines of force all round it, or over the 
 whole solid angle 4?r ; this following, mutatis mutandis, from Chapter 
 X. 14 and 15. Hence unit solid angle will enclose /z of the lines, 
 and a solid angle 1 will enclose p 2 lines. 
 
 Hence we may write the expressions pjQ. and p C Q. in the equi- 
 valent forms 7 N and C N ; where N is the number of marked lines, 
 due to the pole /*, that are included within the boundary of the shell 
 or circuit. 
 
 Since this expression for the potential on the shell or circuit holds 
 for each such pole as /*, and since we may consider that any magnetic 
 field is due to a system of poles, it follows that when a circuit of cur- 
 rent-strength C is in a magnetic field, the potential on the circuit is + 
 CN. 
 
 So, again, the work done in any change of position will be measured 
 (in ergs, of course) by the value of 
 
 C (N x - N 2 ) 
 where N x and N 2 are the number of marked lines embraced by the 
 
318 ELECTRICITY 
 
 CH. XIX. 
 
 circuit in the first and second positions respectively ; C being expressed 
 in absolute electro-magnetic units. The sign of the work done is 
 easily settled in any particular case by the principle explained earlier 
 in this present section. 
 
 4. General Law of Movement of (Magnetic Shells or of) 
 Electric Circuits. From 2 and 3 considered together it fol- 
 lows that , , j, 
 
 * A circuit (bearing a current) when placed in a magnetic field 
 tends to place itself in such a position that as many lines of force 
 as possible pierce it in the negative direction.' 
 
 In other words, it tends to move into the position of greatest 
 negative potential. If so constrained that it cannot turn itself so 
 as to embrace the lines of force negatively, then it will move so as 
 to embrace as few positive lines of force as is possible ; or will 
 move into the position of lowest positive potential. 
 
 And a circuit, completely free to move, tends to place itself, 
 firstly, so that the lines of the field run with its own lines, or run 
 into its south-polarity face and out of its north-polarity face ; 
 secondly, so that as many lines as possible are enclosed by the 
 boundary wire. 
 
 5. The Case of a Uniform Field, In a uniform field the 
 lines offeree are parallel and equidistant. 
 
 Hence a circuit cannot here gain more lines by moving from 
 one part of the field to another ; only by turning into a suitable 
 position. 
 
 Thus in the earth's uniform field the floating circuit, described in 
 Chapter XVIII. 9, will only ' set' itself so that its plane, which is 
 vertical, may be perpendicular to the horizontal component of the 
 field ; it will have no movement of translation. In this it behaves as 
 would its equivalent magnetic shell. 
 
 It is possible to place the circuit so that the field lines run 
 exactly against the circuit's lines ; in this case it is in unstable 
 equilibrium, and cannot so turn as to obey the above ' law ' until 
 slightly displaced. 
 
 6. The Case of a Field not Uniform. In this case the cir- 
 cuit can, by moving from one part of the field to another part, 
 embrace more or fewer lines of force. 
 
 If quite free to move it will turn itself so that the field lines 
 run with its own lines, and will also move into the strongest part 
 
CH. xix. LAWS OF THE MOVEMENTS OF CURRENTS 319 
 
 of the field where the lines are closest together. It will then 
 be embracing as many lines as possible. If constrained by any 
 means to keep so turned that its lines run against those of the 
 field, it will then move into the weakest part of the field ; for 
 there it will embrace as few + lines as possible, which is all that 
 it can do under the assumed conditions. 
 
 7. The Case of Incomplete Circuits. What we have said 
 above gives very simply the law by means of which we can predict 
 the general nature of the movements of magnetic shells, or of 
 complete electric circuits equivalent to magnetic shells. 
 
 But we have, in Chapter XVIII. 15, 16, and 17, given 
 cases of movements on which it is not at all easy to see at first 
 glance how the general law of 4 can have any bearing at all. 
 In the cases referred to we seem to be dealing with portions of 
 currents, often rectilinear ; not with circuits. 
 
 It is more direct, and sometimes simpler, to treat the move- 
 ments of rectilinear currents (and of other incomplete circuits) 
 from the point of view of the fundamental law of action given in 
 Chapter XVIII. 12. We may throw the law, there given, into 
 the following somewhat more general form . 
 
 ' A rectilinear current placed in a magnetic field is urged to more 
 at right angles to its own length and perpendicularly to the lines of 
 force ; the direction of movement being to the left for a person ivho is 
 swimming with the current and looking down the lines of force? 
 
 The word ' down ' means along the direction in which a + pole is 
 urged ; or along the + direction of the lines. 
 
 It is, however, important to observe that we can also treat 
 these movements, and the above law, from the point of view 
 of 4- 
 
 If we consider an indefinite rectilinear current, we have as we 
 know a field of force composed of lines that form circles about 
 the wire, the planes of the circles being perpendicular to the 
 wire (see Chapter XVII. i). 
 
 Now if we imagine a plane magnetic shell whose edge coincides 
 with the wire, and which has no other boundary excepting this 
 edge but extends to infinity, it can be shown that the field of force 
 due to this magnetic shell would also consist of lines forming 
 circles about the edge ; in fact, it can be shown that the fields due 
 
32O ELECTRICITY CH. xix. 
 
 to the wire, and due to an infinite magnetic shell whose one 
 bounding edge coincides with the wire, would be exactly the 
 same; provided, of course, that the shell has a suitable 'strength.' 
 So long as the edge of the shell coincides with the wire, it 
 makes no difference in what direction from the wire the shell is 
 supposed to extend. Thus, if the wire were vertical the shell might 
 extend to infinity in any azimuth, to N., to S., to E., or to W. ; the 
 field given by it remains constant. But it is an accepted principle 
 that if the fields due to two magnetic systems are the same, we 
 may for all electro-magnetic purposes consider these systems to be 
 equivalent. 
 
 We may thus consider any indefinite rectilinear current to be 
 replaceable by a magnetic shell as described above ; or, as we may 
 fairly deduce from the results summed up in Chapter XVIII. 9, 
 we may consider it to be one side of an electric circuit the rest of 
 which is at an infinite distance. We will now show that the law 
 
 of movement given (in italics) 
 ._i _c aDC) ve could have been pre- 
 
 * dieted from the principle 
 
 enunciated in 4. 
 
 In the accompanying dia- 
 
 gram the dots represent in 
 section a field of force, the 
 lines running from above 
 down into the plane of the 
 ^ diagram. 
 
 A B represents a mobile 
 
 rectilinear current, the direction of the current being from B to A. 
 Let AC and B D extend 'to infinity' in a direction e.g. per- 
 pendicular to the line A B ; and let C D, which completes the 
 circuit, be at infinity, or at least so far away as to lie outside the 
 magnetic field considered. 
 
 Then, as far as movements in this field are concerned, we 
 may consider the wire A B to be one side of the ' infinite ' circuit 
 AC DBA. 
 
 In the case drawn, the circuit ' embraces lines of force ' (see 
 3, end) for .the lines of force are running down into a face in 
 which the current runs clockwise, i.e. into a S face. Hence, there 
 should be a movement of A B to the left, so that the circuit may 
 
CH. xix. LAWS OF THE MOVEMENTS OF CURRENTS 3'2I 
 
 embrace more lines of force, according to the principle of 4. 
 And this movement, thus predicted, is what would follow from 
 the principle of Chapter XVIII. 12, as expressed in the italicised 
 law given above. 
 
 If the circuit A C D B lay away to the left of A B then the 
 lines would be piercing the circuit in a + direction ; and A B 
 would still move to the left, in order that A B C D might 
 embrace fewer + lines of force. And so for all positions of our 
 infinite circuit. 
 
 We have thus shown that the simple law of Chapter XVIII. 
 12 does fit in with the more general principle of 4 of this 
 Chapter, though perhaps the agreement may seem a little artificial. 
 
 8. Reconsideration of Ampere's Laws. We will now re- 
 consider, from the point of view that we have explained in the 
 present Chapter, the two main laws of currents enunciated by 
 Ampere ; viz. the laws of parallel, and of angular, currents. The 
 reader is referred back to Chapter XVIII. 16, Laws I. and II. 
 
 For simplicity we will suppose the currents to be rectilinear 
 and indefinite in length, and we will consider them to be replaced 
 by infinite plane magnetic shells as explained in the last section, 
 or by the equivalent infinite circuit. Each will give a field of the 
 nature explained in Chapter XVII. i, and elsewhere. And each 
 circuit will embrace more or fewer -f or lines of force due to 
 the other. If the wires are parallel and the currents run in the 
 same direction, then the lines of each current cut the other circuit 
 negatively ; while if the currents run in opposite directions, the 
 lines of each will cut the other circuit positively. This will be 
 true in whatever ' azimuths ' the two infinite circuits lie. Hence 
 in the former case we should, from the principles of 4, predict 
 that the wires would move towards each other ; and in the latter 
 case, from each other. In the former case the movement causes 
 each circuit to embrace more lines of force ; and in the latter 
 case the movement causes each to embrace fewer + lines of 
 force. The movements actually observed would thus have been 
 predicted. 
 
 In the case of c angular currents ' the same general reasoning 
 would be followed. If the currents are at some small angle, and 
 are in opposite directions, then each circuit is embracing + lines 
 
 Y 
 
322 ELECTRICITY CH. xix. 
 
 of force due to the other current. If this angle opens out, the 
 number of + lines embraced is decreased ; and it becomes zero 
 when the currents are at right angles to each other, since now 
 each circuit is edgeways to the lines of force due to the other. 
 Further rotation will cause each circuit to embrace lines due 
 to the other ; and the number embraced will be greatest when the 
 currents are parallel and in the same direction. 
 
 9. Cases of Continuous Rotation. Referring to the cases of 
 continuous rotation given in Chapter XVIII. 14 and 17, we 
 may say that it is not easy to apply to them in a direct manner the 
 principle of 4 above. Such direct application will always appear 
 to be forced and artificial. 
 
 It is better to apply the italicised law of 7 of the present 
 Chapter, being content with having shown that this law is in 
 accordance with the more general law of 4. 
 
 Applying this law we see that . . . . , . . . . ;; ', 
 
 (i.) In Chapter XVI.II. 14, fig. ii., the mobile portion of the 
 circuit is urged continually across the lines of force that radiate 
 from the magnet pole. 
 
 (ii.) In Chapter XVIII. 17 the mobile radial wire is con- 
 tinually urged across the lines of force due to the circular current ; 
 these lines, as seen earlier, cutting the plane of the circuit at right 
 angles. 
 
 10. Potentials Due to Circuits (continued}. 
 
 We have said that, as regards the external field, circuits are equi- 
 valent to magnetic shells of the same boundary and ' strength. 5 
 
 But there is an important difference between the two ; inasmuch 
 as, in the case of circuits, a pole can thread its way through the cir- 
 cuit and can come back to its original position. It is clear then that 
 we can do any amount of + or work on a pole by causing it to 
 perform complete tours through a circuit, without any change in the 
 initial and final solid angle Q subtended by the circuit. The question 
 arises, What work is done in one such complete tour of a pole p. ? 
 
 Without going fully into the matter we may explain to some extent 
 as follows. In a plane, a line may perform a complete revolution 
 round a point and may come back to its initial position ; it will be in 
 the same place as before, but will have described a plane angle 
 measured by one complete revolution, i.e. by 2 jr. Similarly, when 
 the pole has performed a complete tour through the circuit, the solid 
 angle subtended by the circuit has really changed by what we may 
 
CH. xix. LAWS OF THE MOVEMENTS OF CURRENTS 323 
 
 from analogy call ' a complete solid revolution ' ; and this is a solid 
 angle of 4 IT. 
 
 Hence, if in one position of the pole p the solid angle subtended is 
 Q 15 and in another position it is & , and if further the pole have also 
 performed a complete tours in one and the same direction through the 
 circuit, then the solid angle has changed by an amount measured by 
 ( + 4 a TT + &! - Q 2 ) ; the double sign being due to the fact that the 
 tours may have taken place in the one or the other direction respec- 
 tively. 
 
 Hence, from 8 and 9 of Chapter XVI 1 1., we have for the work 
 done on. or. by the pole in the above movement the expression ;>?. 
 
 JU C ( + 4 <2 7T + Oj - Q 2 ). 
 
 If the pole comes back to the same position, so that Qj = Q g , we 
 have for the work the expression 
 
 + 4 an p C. 
 
 In the case of an indefinite rectilinear current, the same formula 
 holds with respect to complete tours round the wire ; the case of in- 
 complete tours will not be noticed here. We regard the indefinite 
 rectilinear current as the edge of an indefinitely large circuit; and the 
 work done in a complete tours of the pole /* will be given by the 1 
 formula + 4<37TfiC. [Here C is in absolute electro-magnetic units.] 
 
 n. Potentials on Circuits (continued). 
 
 So, again, the expression 
 
 ; C<N iT .NJ 
 
 of 3 does not give the total change of potential on a circuit due to a 
 magnetic field, if this field be due to a pole which performs tours 
 through and round the circuit. 
 
 As in statical electricity (see Chapter X. 14) we saw that + O 
 gave 4 TT Q marked lines of force, so a pole of strength /u, gives 4 n p 
 lines of force. Each time that the pole performs the tour through and 
 round the circuit, all these lines of force are caused to pierce the circuit. 
 Thus each tour may be regarded as adding 4 TT ju marked lines to those 
 piercing the circuit. Hence, to the above expression we must add 
 4 a TT p. C, if the pole p have performed a complete tours between 
 the two positions, all in one or the other direction. As before, the 
 infinite rectilinear current may be regarded as a particular case. 
 Hence, a complete revolutions of the pole round the current give 
 work measured by + 4 a TT p. C. 
 
 Y 2 
 
324 ELECTRICITY CH. xx. 
 
 CHAPTER XX. 
 
 SOLENOIDS, ELECTRO-MAGNETS, DIAMAGNETISM, AND ELECTRO- 
 OPTICS. 
 
 i. Cylindrical Magnet built up of Circular Laminae. Just 
 
 as in Chapter I. 5 we considered the ideally perfect linear 
 magnet as built up of small linear elements each of which was a 
 perfect magnetic needle, so in the case of solid cylindrical magnets 
 perfectly magnetised we may consider the whole to be built up of 
 a series of thin magnetic laminae in each of which the magnetic 
 distribution is 'lamellar' (see Chapter XVIII. 7). 
 
 Each element will be a magnetic disc in which the one face 
 is uniformly + , and the other uniformly . The total external 
 action of the whole uniformly magnetised cylinder will be that of 
 the two end surfaces, the one -f and the other . 
 
 2. The Ideal Solenoid. Now by what we have stated in 
 Chapter XVIII., a circular current is exactly equivalent to a 
 plane circular magnetic shell. We should suppose, then, that if 
 we built up a cylinder composed of such circular currents, placed 
 with their + faces all turned in the same direction, we should 
 have the equivalent to a cylindrical magnet. 
 
 In such a system of plane circular currents the opposed + 
 and faces of the successive elements would neutralise each 
 other with respect to external action, leaving only the two end 
 faces, -f and respectively, to act. 
 
 An arrangement of this sort is called the ideal solenoid. 
 
 3. The Practical Solenoid. It is not, however, practicable 
 to form such an arrangement of plane circular currents, each 
 complete in itself. 
 
 The nearest approach to it that we can make is a current of a 
 spiral form, the turns of the spiral being flat and close together. 
 
CH. xx. SOLENOIDS AND- ELECTRO-MAGNETS 325 
 
 By the principle of Chapter XVIII. n, each turn AB<r of 
 this flat spiral is approximately equivalent to two components : 
 
 (1) a plane circular current B<<r, the projection of the spiral; 
 
 (2) a linear current of length A<r, lying perpendicular to the plane 
 of the circle, and passing through its centre (see fig. i.). 
 
 Thus, the whole spiral will be equivalent to (i.) an ideal sole- 
 noid composed of plane circular currents, its length and diameter 
 
 FIG. i. FIG. ii. 
 
 being those of the spiral ; and (ii.) a linear current, coinciding 
 with the axis of the spiral. 
 
 If, then, we cause the current to return along the interior of 
 the spiral, we can, for the external field, neutralise the linear com- 
 ponent of the spiral. Such an arrangement, represented in the 
 accompanying fig. ii., is the practical solenoid. 
 
 A spiral in which the linear component is unneutralised will 
 give us both a field similar to that of a cylindrical magnet, one in 
 which the lines of force lie in planes passing through the axis of 
 the spiral, and also a field similar to that of Chapter XVII. i, 
 composed of circles about the axis of the spiral and lying in planes 
 perpendicular to it. These two components give, of course, a 
 single field which is nearly that of a cylindrical bar-magnet, but in 
 which the lines of force do not lie exactly in planes containing the 
 axis of the spiral.. 
 
 Experiments. (i.) Experiment with a spiral current, in which the linear 
 component is uncompensated, on a balanced magnetic needle. It can be shown 
 that as a whole the action of a spiral, if its coils be flat and close together, is 
 mainly that of a weak bar-magnet of the same shape. 
 
 Now hold the spiral vertically, so that its axis is situated with respect t,o 
 
326 
 
 ELECTRICITY 
 
 tne needle as was the linear current in Chapter XVII. I. It will be found 
 that there is an action on the needle showing the existence of the circular lines 
 of force about the axis. 
 
 (ii.) Experimenting with 'practical solenoid,' or spiral in which the linear 
 component is neutralised by the return current led through the interior, 
 we find that the second action disappears ; the field is that of a weak bar- 
 magnet. 
 
 (iii.) Experimenting with one solenoid on another balanced upon an 
 Ampere's stand, we can show the resemblance to the action of two bar-magnets 
 on each other. 
 
 (iv. ) A magnet acts upon a balanced solenoid as on a balanced magnet 
 
 (v.) A solenoid suitably balanced will show the declination and inclina- 
 tion. 
 
 That end of the solenoid in which, as one faces it, the current 
 runs clockwise will be or S-seeking ; that end in which the 
 current is counter-clockwise will be + or N-seeking. 
 
 (vi. ) If we hold a solenoid vertically and place a sheet of glass horizon- 
 tally above it, iron filings will show us lines of force radiating from its poles, 
 as in the case of a bar-magnet. If we treat a common spiral current, with 
 the linear component uncompensated, in a similar manner, the lines of force 
 radiating from the poles will have a slight spiral twist. 
 
 4. Ampere's Theory of Magnetism, Struck with the close 
 resemblance between a solenoid and a bar-magnet of the same 
 
 shape, Ampere constructed 
 a theory of magnetism 
 which should account 
 for this resemblance. He 
 considered that in each 
 molecule of the magnetic 
 substance there circulates 
 continually an electric cur- 
 rent ; this current being 
 self-sustained and con- 
 stant, and pursuing an invariable path round the molecule. 
 
 As in all molecular hypotheses of magnetism (for which see 
 Chapter I. 5), neutrality or evident magnetisjn are merely a matter 
 of the arrangement of the molecules. 
 
 In Ampere's theory, a bar would be magnetised to a maximum 
 when the molecules had so turned that all their currents were 
 parallel. 
 
CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 327 
 
 This condition of things is represented in the figure, where the 
 two ends of the bar are shown. It is reasonable to suppose that, 
 except on the surface, the currents of neighbouring molecules 
 would neutralise each other with respect to external effect ; since, 
 as shown, the neighbouring pairs of contiguous currents would be 
 equal and opposite. There would, therefore, be left only the out- 
 side ring of molecular currents ; and these may be supposed to 
 be equivalent to one continuous circular current. Thus, if we 
 consider the bar to be cut into flat discs as in i above, each 
 disc would be equivalent to a plane circular current ; and the 
 whole bar would be equivalent to a solenoid, as regards the external 
 field. 
 
 5. Solenoid, and Hollow Cylindrical Magnet, Contrasted. 
 In fig. i. we represent in section a solenoid, and a few of its lines 
 
 .- w^-r - \ 
 
 x s - % / N / f s 
 
 ^~~- "*. s " *' ____ _ *_ **_**.) .' ,' ,**','" .** 
 
 '" ,''',.'"' '** /i"i " "/ " x ( *~ s v t "^-s^""** S> 
 
 ' /' \ *v *""*-* -*--*'"" / ' ^ 
 
 FIG. i. 
 
 . g ; 
 
 of force are also given, the arrows representing their + direction. 
 It will be seen that inside the solenoid the lines run in a contrary 
 direction to that which they have outside. 
 
 In fig. ii. is given in section a hollow cylindrical magnet, 
 in like manner. Here, the lines along the inside have the same 
 direction as outside. The outside field has the same general 
 character. But while all the lines of the solenoid run continuously 
 through and round the hollow tube, in the case of the hollow 
 magnet, on the other hand, all the lines run into, and end in-, 
 the solid steel that forms the side of the hollow cylinder. In fact, 
 we must remember that the complete solenoid is equivalent to a 
 magnet, as regards external field, but any longitudinal slip of 
 it taken alone is not equivalent to a magnet ; while each such 
 
328 
 
 ELECTRICITY 
 
 CH. XX. 
 
 longitudinal slip of the hollow magnet is itself a complete magnet, 
 the hollow cylinder being not a simple whole, but being a com- 
 pound arrangement formed of a system of magnets arranged as 
 are the staves in a barrel. 
 
 6. Matter Placed in a Uniform Magnetic Field of Force. 
 
 Let us consider a uniform field of magnetic force, and a cylinder 
 of any material, whose two ends are plane faces standing perpen- 
 dicular to its axis, so placed as to lie with its axis along the lines 
 of force of the field. 
 
 In the case of soft iron, or of steel previously unmagnetised, 
 we find that the magnetisation is such that we have evident mag- 
 netism at the two end-surfaces only ; if we neglect, as relatively 
 unimportant, the irregularities that occur at the edges where the 
 molecules are free towards the outside and in contact with other 
 molecules towards the inside. Over these end-surfaces the density 
 p of magnetisation (see Chapter XVIII. 7) will be approximately 
 uniform. There is no experimental reason for supposing but that 
 cylinders of any material are, if sensibly magnetised at all, mag- 
 netised in a similar. manner to the above. 
 
 The value of this density p depends (i.) upon the field-strength 
 I, and (ii.) upon the nature of the material of which the cylinder 
 is composed. , Now experiment indicates that, so long as the bar 
 is far from saturation (see Chapter I.), then p is directly proportional 
 to the field-strength I. Thus we may write , ...... 
 
 where k is a quantity depending upon the nature of the material. 
 When the field-strength I = unity, then k is numerically equal to p. 
 
CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 329 
 
 This quantity k is called the coefficient of magnetisation of that ma- 
 terial, and is measured by the value which t> has when I = unity. 
 
 Without at present discussing whether the following assump- 
 tions are physically possible (we shall see later that they are), let 
 us assume that k may be a + quantity, zero, or a quantity ; 
 and let us consider what would be the observed condition of the 
 cylinder in the three cases respectively. 
 
 (I.) Let k = a + quantity. This would make p positive. 
 That is, remembering what is the + direction of the lines of 
 force, we should observe a north-seeking polarity at the end lying 
 furthest down the lines of force, and a south-seeking polarity at 
 the other end. Or this case is the usual one of soft iron or other 
 magnetic matter placed in a field of force. 
 
 (II.) Let k zero. Here we observe no polarity, since f>= o. 
 That is, the material is one whose presence in the field makes no 
 difference to it 
 
 (III.) Let k = a quantity. In this case p will have a con- 
 trary sign to that which it had in case (I.)., Or the material would 
 be one in which induction takes place in a contrary direction to 
 that observed in iron ; thus a north-seeking pole of a magnet 
 would, at any rate apparently, induce in a bar of such material a 
 north-seeking pole at the end nearest to the former. 
 
 In order that we may now use convenient names for different 
 classes of bodies, we shall to some extent forestall what will be 
 discussed more fully later on in this Chapter. 
 
 We may, therefore, state that there is in the first place a class 
 of bodies for which k is + ; or for which induction takes place 
 down the lines of force, as in the case of iron. Such bodies, of 
 which iron (including steel) is by far the most important, are called 
 magnetic, or more properly paramagnetic. In the case of very 
 pure soft iron, k has a large value ; thus the presence of a bar of 
 such iron in a magnetic field may increase the field-strength near 
 the poles even fifty-fold. 
 
 There is, secondly, another large class of bodies for which k 
 is ; or for which induction takes place up the lines of force, or 
 in the contrary direction to the above. For such bodies k is very 
 small. For example, if a bar of bismuth be placed in a magnetic 
 field, this field will appear to be slightly weakened near the poles 
 of the bar, in virtue of the opposed induced polarity of the bar j 
 
33O ELECTRICITY CH. xx. 
 
 but, from the smallness of k, the field as a whole is but very 
 slightly affected. Such bodies are called diamagnetic. (For 
 further discussion see 14 and 15.) 
 
 7. Movements of Small Bodies in a Non-Uniform Magnetic 
 Field. Let us now consider a field that is not uniform, and a 
 small body placed in the field. By small body we here mean one 
 so small with respect to the whole field that it can be considered 
 to be all of it in a stronger or weaker part of the field at the same 
 time ; and yet not so small but that one side of it is, in our non- 
 uniform field, in a part of the field of somewhat different strength 
 to that in which the other side finds itself. We will consider 
 what will be its behaviour. 
 
 (I.) Small magnetic bodies. It can be shown that a small 
 magnetic body, such as an iron pellet, for example, is urged from 
 weaker to stronger parts of the field. 
 
 Thus, if a small iron pellet be presented to a pole of a magnet 
 (the usual case of a non-uniform field) there will be induced an 
 opposed polarity on the side next to the pole, and a similar 
 polarity on the side more remote. The former will be attracted, 
 and the latter will be repelled, by the pole ; these two polarities 
 are equal in magnitude but opposite in sign. Now the former 
 polarity is in a stronger field than is the latter, and hence attrac 
 tion will predominate, and the pellet will move towards the 
 pole. 
 
 (II.) Small diamagnetic bodies. For similar reasons a small 
 diamagnetic body is urged from a stronger into a weaker part of 
 the field. 
 
 8. The ' Setting' of a Long Body in a Uniform Magnetic 
 Field. Next let us consider the case of a long cylinder placed in 
 a uniform magnetic field, at an angle with the lines of force of 
 the field. We will represent our cylinder as composed of a series 
 of small spheres placed near to one another. This is a convenient 
 representation, and though not an accurate one, will not invali- 
 date the very general results at which we shall arrive. 
 
 (I.) A magnetic cylinder. Fig. i. represents a cylinder of iron. 
 Each of the little spheres A B C D would, if it stood alone, be mag- 
 netised in the direction of the lines of force of the field. Tnis 
 is represented by the lettering n s in each. There is, however, 
 inductive action between the spheres, each n or s inducing an s or n 
 
CH. 
 
 SOLENOIDS AND ELECTRO-MAGNETS 
 
 331 
 
 respectively in the nearest portion of the neighbouring sphere. 
 The total result .will be that each little sphere is magnetised, not 
 along the lines of the field, but in a direction represented by n' s 
 of fig. ii., lying between the direction of the field and the direc- 
 tion of the cylinder. 
 
 Thus each little sphere is acted upon by a couple tending to 
 drag n' s' into the direction of the lines of the field (see Chapter II. 
 12). Hence there will be a couple acting upon the whole 
 
 FIG. ii. 
 
 cylinder, and there will be stable equilibrium' only when this 
 lies along the lines of force. It is to be noticed that in this 
 position the external and internal actions concur to give the 
 maximum magnetisation. 
 
 When the cylinder is perpendicular to the field there is also 
 equilibrium, but unstable. In this position the internal induction 
 acts against the external, and the magnetisation is at a minimum. 
 
 We may therefore state that 
 
 A magnetic cylinder tends to set along the lines of force of a uni- 
 form field ; that is, to assume the position in which its magnetisation 
 is at a maximum. 
 
 (II.) A diamagnetic cylinder. In this case the external and 
 internal induction will both be the 
 reverse of what it was in case (I.). 
 Thus we must interchange the letters 
 n and s in the above given figures ; 
 and must further remember that each 
 n or s in one sphere induces an n or s 
 respectively in the nearest portion of the neighbouring sphere. 
 The total result will be that each little sphere will be magnetised 
 
 FIG. iii. 
 
332 ELECTRICITY CH. xx. 
 
 somewhat as in fig. iii. Hence there will be a couple acting on 
 each sphere : and this will be in such a direction that the whole 
 cylinder will be urged to lie along the lines of force. In this 
 position the internal and external inductions act against one an- 
 other, and the magnetisation is at a minimum. 
 
 When the cylinder is perpendicular to the field there is un- 
 stable equilibrium, and the magnetisation is at a maximum. 
 
 Hence a diamagnetic (see 6, end) cylinder tends to set along 
 the lines of a uniform field ; that is, to assume the position in which 
 its magnetisation is minimum. 
 
 We may, however, add that with diamagnetic bodies the above 
 action is very feeble ; and no observation has as yet detected any 
 ' setting ' of such bodies in a uniform field. 
 
 9, A Long Body in a Non-Uniform Field, When a magnetic 
 cylinder is suspended in such a non-uniform field as that between 
 the two poles of a powerful magnet, the results of 7 and 8 
 concur to show that it will set along the lines running between the 
 two poles. 
 
 When a diamagnetic cylinder is so suspended, the action de- 
 scribed in 7 would urge it to stand in the weakest part of the 
 field ; i.e. at right angles to the lines of force between the poles. 
 But the action given in 8 would tend to make it set along 
 these lines. In all cases that occur in practice, the former action 
 gives rise to the greater couple ; and so the diamagnetic cylinder 
 stands at right angles to the lines running between the poles. 
 
 To exhibit these phenomena, very powerful magnets are 
 needed. We therefore proceed to consider how powerful tem- 
 porary magnets may be obtained. 
 
 10. Solenoid With, and Without, an Iron Core. According 
 to the view that appears to be most in accord with experiment, 
 magnetic matter possesses innate magnetism. This magnetism is, 
 however, 'molecular ; and we have evident magnetism, or an ex- 
 ternal field, only when the molecules are suitably arranged. 
 
 We see that, on this view, there is nothing very surprising in a 
 moderate field * producing ' a strong magnet ; for the magnetism 
 is in the soft iron already, and the field may be able to arrange the 
 molecules suitably. 
 
 We may contrast the magnetic action of an ordinary solenoid 
 
CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 333 
 
 with that of one in which there is a soft iron core. It is often 
 said 'the core strengthens the solenoid.' But it is perhaps more 
 in accordance with facts to say ' the solenoid renders evident the 
 innate magnetism of the iron.' 
 
 Experiments. (i. ) We can compare the action of the two on a balanced 
 needle. 
 
 (ii. ) We can examine the fields of the two respectively by placing them 
 under glass and sprinkling steel filings above. It will be seen that in the case 
 of the solenoid the field is weak, and that some of the lines of force ' leak out ' 
 between the turns of the wire. In the case of the solenoid with iron core, 
 the field is far stronger, and the strength is more concentrated at the poles. 
 
 The iron is, from its symmetrical position, magnetised in the 
 direction of the lines of force due to the solenoid ; i.e. in the 
 direction of the solenoid's axis. The north pole of the core will 
 be that at the north end of the solenoid, where the current, to 
 one facing the end, appears to run counter-clockwise. 
 
 1 1. Electro-Magnets. Thus, when a soft iron core is wrapped 
 round with many turns of wire, and a current is passed through 
 the wire, the core becomes temporarily a magnet. Such magnets 
 are called electro -magnets, and can be made far more powerful, 
 mass for mass, than can any permanent steel magnets. 
 
 We may regard the external field to be made up of two com- 
 ponents ; the one due to 'evident magnetism ' now evoked in the 
 iron, the other due to the spiral or solenoidal current. These two 
 fields are superimposed upon one another. As long as the iron 
 is far from saturation, the field due to it is approximately propor- 
 tional to the field-strength (see 6) and therefore to the current- 
 strength. But when saturation is reached, any further increase 
 in current will only increase the comparatively insignificant com- 
 ponent field due to the spiral alone. In winding the wire about 
 an iron core, and in passing a current through the wire, it is 
 necessary to have regard to the following considerations. 
 
 (i.) The wire must not be too thick, or it will not be possible 
 to give a number of turns sufficient for the production of a strong 
 field with a current of reasonable magnitude. 
 
 (ii.) The wire must not be so thin as to give great resistance 
 and consequent loss of energy in heat. 
 
 (iii.) It is of little use producing a field-strength greater than 
 that necessary to magnetise the iron nearly to saturation. 
 
334 
 
 ELECTRICITY 
 
 CH. XX. 
 
 (iv.) The distribution of the wire about the core must be 
 adapted to the shape and dimensions of this latter. 
 
 The accompanying figure 
 shows one form of electro- 
 magnet. 
 
 Experiment. It is possible, in 
 the case of a powerful electro- 
 magnet, to trace the lines of force 
 in a very striking manner. Instead 
 of filings we may use small pieces 
 of steel knitting-needles. If the 
 field be powerful, the effect of 
 gravity on these pieces of steel will 
 be relatively insignificant ; and we 
 may cause them to attach them- 
 selves end to end, to one another, 
 and so trace out the lines of force 
 in any direction in space, while 
 with permanent magnets we were 
 able to trace the lines only over a 
 horizontal plane. 
 
 Note. Field due to an electro- 
 magnet. In calculating the field 
 due to an electro-magnet we can 
 consider the solenoid, and the core 
 which has become a cylindrical 
 
 magnet, separately. But in general the field due to the former is relatively 
 
 insignificant, and we need only regard the core. 
 
 12. Paramagnetic and Diamagnetic Phenomena. When 
 powerful electro-magnets are employed it is found that all bodies 
 are influenced by the magnetic field (see 6). 
 
 If pellets of various materials are suspended, by means of a 
 light and long thread, near one of the poles of such a magnet, it 
 is found that certain substances are attracted by the pole, while 
 others are repelled. 
 
 Those bodies which are attracted are called paramagnetic, or 
 magnetic; such are iron, nickel, cobalt, manganese, platinum, 
 carbon, many salts of magnetic metals, solutions of such salts, and 
 oxygen gas. 
 
 Those bodies which are repelled are called diamagnetic ; such 
 are bismuth, antimony, zinc, tin, mercury, lead, silver, copper, 
 
CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 335 
 
 gold, phosphorus, glass, quartz, alum, sulphur, sugar, hydrogen, 
 nitrogen, water, alcohol, and most other liquids and gases not 
 here named. 
 
 Iron is the most strongly magnetic, and bismuth the most 
 strongly diamagnetic, body known. 
 
 If we make bars of various substances, those which are mag- 
 netic will set axially, or in a line with the poles ; while those 
 which are diamagnetic will set equatorially, or at right angles to 
 the line joining the poles (see 9). 
 
 The accompanying figure represents experiments with magnetic 
 and diamagnetic liquids respectively. The liquid is placed in a 
 watch-glass and rests on the poles. When 
 the current is passed, the magnetic liquid 
 B rises up in a heap over each of the 
 two poles ; while the diamagnetic liquid 
 A is repelled into a heap between the 
 two poles. Such effects are very small, 
 and must be magnified by means of re- 
 flected light if they are to be made clear. 
 The difference between the two classes of 
 liquids is clearer if we employ thin glass 
 
 tubes filled with the one or the other respectively, and observe 
 whether these set axially or equatorially. It must, however, be 
 remembered in this case that the glass itself is diamagnetic ; its 
 action can be allowed for. 
 
 13. Pseudo-Diamag-netic Phenomena. In certain cases a 
 bar may set equatorially when its material is magnetic, or axially 
 when its material is diamagnetic, owing to peculiarity of structure. 
 Thus, a bar may be made composed of short steel needles 
 separated from each other, lying side by side, running transverse 
 to the length of the bar. Such an arrangement will, as a whole, 
 set equatorially ; each little needle lying axially. So again, if in a 
 bar of bismuth the crystallisation have a certain direction with 
 respect to the length of the bar, this may set axially. The repul- 
 sion or attraction of pellets from or to a pole is the best way of 
 dividing bodies into the two classes. 
 
 14. Relative Magnetism or Diamagnetism, By Archi- 
 medes' principle we know that bodies immersed in any fluid 
 medium appear to have a +, zero, or weight according as they 
 
336 ELECTRICITY 
 
 CH. XX. 
 
 displace less than their own, their own, or more than; their own 
 weight of that medium respectively. By this principle we can 
 predict, e.g., whether a body immersed in water will sink, remain 
 where it is, or be forced upwards. 
 
 Similar reasoning applied to the case of a magnetic field leads 
 us to predict what can be verified by experiment that when a 
 body, whose coefficient of magnetisation with respect to vacuum 
 is k, is immersed in a medium whose coefficient is h, the body will 
 behave as though it were in vacuo and had a coefficient k' equal 
 to k h. If this be true reasoning, then if k> h the body will 
 appear to be magnetic ; if k = h it will be neutral, and if k < h it 
 will appear to be diamagnetic. 
 
 These predictions have been experimentally tested and verified. 
 Thus, a weaker solution of ferric chloride appears diamagnetic when 
 in the midst of a stronger solution, though in vacuo it is distinctly 
 paramagnetic. 
 
 15. Is there Absolute Diamagnetism ? The question 
 naturally arises: 'Is there then such a thing as true diamagnetism, 
 or is it merely that some bodies are less magnetic than that which 
 we call " vacuum " ? ' 
 
 Some bodies which appear diamagnetic in the magnetic 
 medium oxygen, may very well be found to be magnetic when 
 tested in vacuo. But most diamagnetic bodies (e.g. bismuth) are 
 still diamagnetic in vacuo. 
 
 It can be shown that all phenomena of repulsion and of equa- 
 torial-setting with which we are acquainted could be accounted for 
 by supposing ' vacuum ' to be a medium slightly magnetic. 
 
 But the phenomenon referred to in 16, viz. the contrary 
 directions in which a ray of plane polarised light is rotated in 
 different media, seems to imply an essential difference in the sign 
 of k for the two classes of media respectively. Before, therefore, 
 we can accept unreservedly the view that all phenomena come 
 under the head of paramagnetism or relative paramagnetism, it will 
 be necessary to show that such a view can be reconciled with the 
 fact quoted. 
 
 The whole question is at present unsettled. Quite recently 
 (1886) experiments have been tried tending to show that bodies 
 may change from diamagnetic to paramagnetic behaviour, or vice 
 versa, according to the strength of the field in which they are- 
 
CH. XX. 
 
 SOLENOIDS AND ELECTRO-MAGNETS 
 
 337 
 
 placed. It has even been suggested that perhaps these qualities 
 are not permanent, but change with time under the action of a 
 field. Curiously enough, Ampere's theory of ' molecular currents ' 
 has been revived as a somewhat fruitful view. 
 
 1 6. Rotation of the Plane of Polarisation in a Magnetic 
 Field. In giving some account of certain phenomena that show 
 a remarkable connection between magnetic and electric stresses on 
 the one hand, and radiant energy on the other, we must assume 
 that the reader has some acquaintance with the elements of 
 physical optics. If this is not the case he is advised to read 
 enough of the subject to understand (i.) what is meant by a ray of 
 light or of other radiant energy ; (ii.) what is meant by a plane 
 polarised ray, and by the plane of polarisation \ (iii.) what a Nicols 
 prism is, and how it is used to obtain a plane polarised ray; (iv.) 
 how a second NicOl's prism can be used as an analyser to detect 
 whether a ray is plane polarised, and whether the plane of polarisa- 
 tion has been rotated or has changed its 'azimuth.' 
 
 Now it is found that if a plane polarised ray be passed through 
 .a transparent medium that ordinarily has no power to rotate the 
 
 FIG. i. 
 
 plane, and if this medium be placed in a powerful magnetic field, 
 then the plane of polarisation of the ray is in general slightly 
 rotated in its passage through the medium. It was Faraday who 
 discovered this. The accompanying figure indicates how the 
 
 z 
 
338 ELECTRICITY CH. xx. 
 
 ' electro-magnetic rotation of a plane polarised ray ' may be de- 
 monstrated. If more exact results are desired the necessary 
 experimental arrangements are less simple. M and N are powerful 
 electro-magnets, provided with hollow iron cores, their unlike poles 
 being opposed so as to give a powerful field at c. Here is placed 
 the piece of glass or other transparent body, whose rotating powers, 
 when subject to a powerful magnetic field, are to be observed. At 
 b is \\\Q polarising Nicol, and at a the analyser. 
 
 If (as we here suppose) the body at c has no innate rotating 
 power as has (e.g.) quartz, then before the current passes the plane 
 of polarisation of the ray is not rotated ; and no light can be seen 
 through a when the principal planes of the polariser b and of the 
 analyser a are at right angles to one another. But when the 
 current passes so that a powerful field is produced at c, then some 
 light is received through a, and we must turn the analyser a about 
 the line a b as axis, in order to recover the initial darkness. When 
 monochromatic light is used we can then readily measure the 
 amount and direction of the rotation by the amount and direction 
 of the rotation that has been given to a. If the light be not mono- 
 chromatic, e.g. if it be white light, we can never recover the white 
 light again ; this showing that the rays of different wave-lengths 
 have been rotated to a different amount. 
 
 By experiments that we have not space enough to give here, 
 Verdet and others established several results. These results will be 
 
 FIG. H. 
 
 made easier to understand by the simple diagram here given. The 
 dotted lines represent a magnetic field as usual ; A and B are two 
 points in it at which the magnetic potentials are V A and V B respec- 
 tively, the former being [in the case represented] at the lower potential ; 
 A B or B A is the direction of the ray ; /3 is the angle that this direction 
 makes with the lines of force of the field ; I is the field-strength ; / is the 
 
CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 339 
 
 distance in the medium traversed by the ray ; and 6 is the angle 
 through which the plane of polarisation is rotated. 
 
 It was found that 
 
 (i.) 6 depended upon the field-strength, being directly proportional 
 to it. 
 
 (ii.) 6 is directly proportional to cos ft. 
 
 (iii.) 6 is directly proportional to /. 
 
 (iv.) 6 depends upon the wave-length of the radiation in question. 
 
 (v.) The sign of 0, or the direction of rotation, is not the same in 
 all media. If we call the rotation positive when it is as a right-handed 
 screw would rotate as it advanced down the lines of force, then we 
 may say that in vacuo there is zero rotation (as far as experiment 
 indicates), in diamagnetic media the rotation is + , and in magnetic 
 media (such as a solution of ferric chloride) it is . 
 
 (vi.) This rotating power seems to be innate in the molecules pro- 
 vided that they are in a magnetic field. A substance preserves its 
 power when in solution or when mixed with other bodies, and so we 
 can with certainty prepare mixtures of the two classes of solutions 
 which shall give zero rotation. 
 
 (vii.) The direction and magnitude of rotation are the same 
 whether the ray pass from A to B, or in the reverse direction. Hence 
 the rotation can be multiplied by causing the ray to be reflected to 
 and fro many times up and down the field, this being equivalent to 
 multiplying /. It would not be practicable to make / great in any 
 direct manner, if we wished still to preserve a strong field. 
 
 (viii.) With reference to (v.) we may add that all wave-lengths 
 have the same direction of rotation in the same medium. 
 
 Theory suggested by the above. The above phenomena suggest 
 very forcibly that in a magnetic field the ether may be eddying about 
 the lines of force. For, were this so, the wave would naturally be 
 rotated in azimuth during its passage. 
 
 Remembering how the lines of force pass down the axis of a 
 solenoid or spiral, it would seem as though a spiral current caused the 
 ether to eddy about the axis of the spiral. It has even been suggested 
 that perhaps electricity is itself ether, so that the eddy is produced by 
 a spiral current of ether through stagnant ether. 
 
 At present, however, there is in this direction little but conjecture. 
 Some day, no doubt, electro-optical phenomena such as the above will 
 lead to the establishment of very important results as to the real 
 nature of magnetic, electrical, and radiant phenomena. 
 
 We may observe that the opposite directions of rotation in different 
 media indicate a real difference between the two classes of bodies, or 
 tend to show that diamagnetism is not merely a relative phenomenon. 
 
 Z 2 
 
340 ELECTRICITY CH. xx. 
 
 17. Other Electro-Optical Phenomena. 
 
 There are other phenomena that show how the propagation of a 
 wave of radiant energy is affected by the magnetic or electric condi- 
 tion of the medium through which the ray passes. 
 
 Kerr discovered that when a plane polarised ray is reflected from 
 the polished surface of the pole of a magnet, the plane of polarisation 
 is rotated. He believes that this is due, at any rate in part, to the 
 action at the reflecting surface, and is not simply due to the air or 
 other gas that fills the magnetic field. 
 
 He also discovered that when a simply refracting transparent 
 dielectric is subjected to electric stress, as e.g. is the glass of a charged 
 Leyden jar, it becomes doubly refracting. The effect was on the 
 whole similar to that which was produced if it was subjected to com- 
 pression along the direction of the lines of force. It seemed, in fact, 
 as if there was tension along the lines of electric force, tending to 
 compress the glass. 
 
 18, The Electro-Magnetic Theory of light, 
 
 The genius of Clerk Maxwell founded a theory which should con- 
 nect all these phenomena that are usually classed under the heads of 
 magnetism, electrostatics, and electrodynamics, on the one hand, with 
 the phenomena of light, heat, and of radiant energy in general on the 
 other. But before this theory could grow into a complete structure 
 the author of it died, and the work has never been completed. 
 
 In what follows we have tried to indicate in an elementary manner 
 the nature of the theory, and also to show how much uncertainty still 
 exists with respect to it. We have made use of the suggestions put 
 forward by Dr. Lodge, Professor S. P. Thompson, and others, as such 
 suggestions show the directions in which Clerk Maxwell's funda- 
 mental views tend'. 
 
 (a) The ether. By the use of the expression the ether is implied a 
 belief, now general among men of science, in a universal medium per- 
 vading all space and penetrating solids and liquids. 
 
 The ether is not matter, but possibly all atoms of matter are merely 
 indestructible vortex-rings of ether, so that this latter is the parent of 
 matter. Again, the ether is not energy ; but yet it is only through the 
 ether that energy can be transmitted from one group of matter to 
 another. 
 
 The ether penetrates all bodies, but it may be that in some cases 
 the matter has, to a greater or lesser degree, a ' hold ' upon the ether 
 that pervades its intermolecular spaces, and gives to it something of 
 the properties of a rigid solid, while in other cases the ether may be 
 lett quite free. 
 
CH. xx. SOLENOIDS AND ELECTRO-MAGNETS 34! 
 
 (3) Radiant energy ; heat and light. We may here remind the 
 reader that the transmission of heat and light is believed to be 
 effected by means of waves propagated in the ether ; the vibrations 
 being at right angles to the direction of propagation, somewhat as in 
 the case of a shaken rope. Further, it depends upon the wave-length 
 of the radiation whether it give the impression of light when received 
 upon the retina, or in what respect its action is most remarkable. 
 Heat and light are partial expressions, radiant energy is the more 
 general term for transmissions of all wave-lengths. 
 
 (y) Nature of electricity. It has been suggested that if anything 
 can rightly be called 'electricity,' this must be the ether itself; and 
 that all electrical and magnetic phenomena are simply due to changes, 
 strains, and motions in the ether. Perhaps negative electrification (as 
 we believe has been suggested) means an excess of ether, and positive 
 electrification a defect of ether, as compared with the normal density. 
 
 (8) Electrostatic phenomena. It is possible that all electrostatic 
 phenomena follow from strains in the ether, These strains (or defor- 
 mations) can only occur where the ether possesses some degree of 
 rigidity. On this view then it would appear that dielectrics are bodies 
 which have what we may style so much 'hold' upon the ether that 
 pervades their intermolecular interstices, that this is given, to a certain 
 degree, the properties of an elastic solid. Thus the phenomena of the 
 stored-up energy of a charged Leyden jar may imply that the ether 
 pervading the glass, and probably the glass itself, is in a state of 
 elastic strain analogous to that of a bent spring. Co?iductors, on the 
 other hand, would be bodies in which the ether is free and, to speak 
 from analogy, fluid, or in which it is incapable of retaining a strain. 
 
 (e) Dielectrics are transparent, conductors are opaque. Now from 
 mechanical principles we should say that through bodies in which the 
 ether has some rigidity, waves of transverse vibration can be propa- 
 gated ; while through those in which the ether is free and fluid, only 
 waves of longitudinal vibrations can be propagated. From (#) and 
 (8) we should therefore predict that dielectrics would be transparent 
 to the transverse waves of radiant energy (i.e. to rays such as those of 
 heat and light), while conductors would be opaque to the same. If 
 we take into account such disturbing influences as those of internal 
 and irregular reflexion, it can be fairly said that dielectrics are, as a 
 rule, transparent ; while the conducting metals are certainly the most 
 opaque of bodies. Here, then, is one fact that tends to confirm directly 
 the theory. 
 
 (f) Electric currents. There are reasons that would lead us to 
 assume that when ' an electric current flows ' through a conductor, 
 there is either a direct translation of the ether along the conductor, or 
 
342 ELECTRICITY CH. xx, 
 
 a propagation of longitudinal vibrations similar to those of sound in 
 air. This view is consistent with the hypothesis that in conductors 
 the ether is free and not rigid, and therefore not capable of trans- 
 mitting waves of transverse vibrations. 
 
 (77) Electro-motive force. We might then suppose that electro- 
 motive force may be a kind of ' ether pressure ' due to un-uniformity 
 of distribution, or to be, at least, of a nature analogous to what is sug- 
 gested by such a term. 
 
 (&) Electro-magnetic induction. Let us suppose a ' current ' to be 
 sent along a wire, this wire being surrounded by a dielectric such as 
 air. Now we shall see in Chapter XXI. that when any change in 
 current-strength occurs, all neighbouring conductors are affected ; 
 temporary currents being ' induced ' in them. Since the possibility of 
 action at a distance is now denied, we must suppose that this action 
 is due to some kind of transmission of energy through the dielectric ; 
 and it is most consistent with the views expressed above, and with the 
 observed facts of induction, to imagine that something of the nature of 
 a wave in which the vibrations are transverse is propagated at right- 
 angles to the wire carrying the current. If this is the case we should 
 expect to find the velocity of propagation of electro-magnetic inductive 
 action to be the same as the (average) velocity of radiant energy, since 
 both are supposed to be waves of transverse vibrations, propagated in 
 the ether. Now this is actually the case ; many measurements having 
 shown that the rate at which electro-magnetic inductive action is pro- 
 pagated is nearly the same as the mean velocity of light. In this 
 agreement we have what is probably the strongest corroboration of 
 the theory that we are discussing. 
 
 (i) There is another piece of confirmatory evidence in the relation 
 that certainly exists between the specific inductive capacity of a 
 dielectric and its refractive index. 
 
 (K) Lines of magnetic force. In 16 we pointed out how the 
 phenomena there described suggested to us that lines of magnetic 
 force are in some way axes of eddies formed in the ether. Here we 
 may add that the eddy may be one in which the ether moves continu- 
 ously round, or may be formed of circular vibrations in which there is 
 no final displacement of the ether. 
 
 Considering the wire bearing a current, the circular lines of mag- 
 netic force about it, and the radiating lines in which electro-magnetic 
 inductive action is propagated, we may say that probably the follow- 
 ing relation holds. ' When there is a discharge of electricity in one 
 direction the lines of electro-magnetic induction are at right-angles 
 to this direction and radiate from it, while the lines of magnetic force 
 are at right-angles to both these directions.' 
 
343 
 
 CHAPTER XXI. 
 
 ELECTRO-MAGNETIC INDUCTION. 
 
 i. General Account of Induction Phenomena. Faraday, 
 discovered that when a conductor moves in a magnetic field, or 
 when the field in which a conductor is situated is caused to vary 
 in strength, then there is in general a current induced in the con- 
 ductor, this current lasting only as long as the movement or 
 variation lasts, and ceasing when conditions are again stationary. 
 
 Since a current implies an electro-motive force, we may say 
 that such movements or changes give rise to induced E.M.F.s in 
 the conductors. The ejection of the E.M.F., and of the conse- 
 quent current, depends upon the nature of the movement or 
 change. If there be an E.M.F. (and current) in the conductor 
 already, the induced E.M.F. if in one direction will be added to 
 the initial E.M.F., and in the other direction will decrease, reduce 
 to zero, or even overpower and 'reverse' the initial E.M.F., 
 according to the relative strengths of the two. 
 
 It is soon noticed that the cases in which an E.M.F. is in- 
 duced in a conductor are mainly as follows. 
 
 (a) When a conductor cuts across the lines of force of a 
 magnetic field. If a complete circuit so cut across the lines, it 
 may happen that the E.M.F.s induced in the opposite sides of the 
 circuit are equal and opposed, giving a zero resultant E.M.F. in 
 the complete circuit. 
 
 (I)) When a magnetic field is created in the midst of a cir- 
 cuit, or is/ caused to cease. This case is shown in Experiment (i.). 
 
 (c} When the field enclosed by a circuit is varied in strength. 
 This case is shown in Experiment (iii.). 
 
 The changes in question can be brought about in various 
 ways : by the continuous movement of a simple rectilinear wire 
 across a magnetic field ; by a movement of translation of a circuit 
 
344 
 
 ELECTRICITY 
 
 CH. XXI. 
 
 across a non-uniform field ; by rotation of a circuit in any field ; 
 by making and unmaking an electro-magnet (or solenoid) in the 
 midst of a circuit ; By the approach of a current, solenoid, or 
 magnet towards a circuit, or its withdrawal from the same. And 
 in many other ways. In the following experiments, which are 
 here given without further comment, we use coils instead of 
 simple circuits in order to obtain more remarkable results. 
 
 Experiment. (i.) A coil is made of a quantity of fairly stout insulated 
 wire, and the ends of this wire are connected with the terminals c and d, so 
 that a current can be sent through it when desired. This coil A is concealed 
 in fig. i., but is seen in fig. ii. A second coil B is made ; it is hollow and 
 encloses the coil A, but is entirely insulated from it. The ends of this latter coil 
 are connected with the terminals a and b. A galvanometer is also connected 
 
 FIG 
 
 wit'h these terminals, so that any current in B will be detected. The effects 
 are more marked if the wire of which B is made be very long. It, therefore, 
 must be of fine wire, to admit of great length and many turns within a reason- 
 able compass. The coil A is called the primary, and the coil B the secondary. 
 
 When a current is sent through the primary, the galvanometer indicates a 
 sudden current in the secondary ; this current lasting only until the primary 
 current is established and steady. When the current is broken again we 
 observe another current induced in B, this time in the other direction. 
 
 Thus the creation or destruction of a field within B produces currents in B, 
 the currents being in opposite directions respectively. There is no current in 
 B, while the current through A is steady ; that is, while the field within B is 
 constant. 
 
 (ii. ) In the next figure the arrangements and connections are the same, 
 but now A can be thrust into B or withdrawn from B. It is found that by 
 thrusting A, while a current is flowing in it, into B, we induce in this latter a 
 current in the same direction as would have been induced had we left A within 
 B and had then sent the current through A. So also the withdrawal of A 
 
CH. XXI 
 
 ELECTRO-MAGNETIC INDUCTION 
 
 345 
 
 answers to stopping the current in it. We find, also, that the currents induced are 
 more or less violent according as we move A more or less rapidly respectively. 
 
 FIG. ii. 
 
 (iii.) Similar effects are produced if we thrust a magnet into B, or with- 
 draw it again. We should expect this, having seen that a coil bearing a 
 current acts as a weak magnet 
 
 As regards the relation of the direction of the induced cur- 
 rents to the direction of the primary currents, or to the sign of the 
 
346 ELECTRICITY 
 
 CH. XXI. 
 
 magnetic pole which is presented to the coil B, enough will be 
 said later on. 
 
 2. General Keason for ' Induced Currents.' The actual 
 reason why such movements or changes in field-strength as those 
 described should have the effect of inducing E.M.F.s in the con- 
 ductors concerned, cannot be said to be truly known. For it 
 must depend upon the real nature of electric currents and of 
 magnetic fields, and upon their relation to 'the medium that 
 pervades all space.* 
 
 But we have certain great laws established by experiment, 
 which have included and bound together in one theory all the 
 phenbmena that have been noticed up to the present point. If 
 we can show that these new phenomena of ' induction ' come also 
 under these laws and could be predicted from them, we shall in a 
 sense have shown the * reason for ' induction. The laws to which 
 we refer are Conservation of energy, Faraday's laws (Chapter XII. 
 7), andfoule's law (Chapter XV. 4). 
 
 Let us consider the case of a circuit A which includes a 
 battery-cell, opposite to which is a magnetic pole N. (Instead of 
 a single pole we might consider the one end of a very long 
 magnetic bar whose other end is far away and need not be 
 considered.) 
 
 This circuit will attract or repel the pole N, according to their 
 relative position ; since the circuit acts as a magnetic shell, and 
 produces a magnetic field. If it attracts N, then work can be done 
 by N as it moves up towards A ; and this work comes out of the 
 system of the two bodies A and N, so that there must be an equiva- 
 lent of energy lost from this system by ' Conservation of energy' 
 
 How is this energy supplied ? 
 
 If the two bodies A and N were a gravitation system, i.e. 
 merely two masses attracting one another by the action of gravi- 
 tation alone, we know that we should have lost an equivalent of 
 potential energy ; that is, we could not restore N to its initial 
 position save by doing upon it just as much work as we previously 
 got out of it. In moving towards A it moves with the lines of 
 force, in moving from A it moves against them. 
 
 But in our electro -magnetic system of the circuit A and the pole 
 N matters are very different. We can, without any work, make 
 or break the current, i.e. create or destroy the magnetic field at 
 
CH. xxi. ELECTRO-MAGNETIC INDUCTION 347 
 
 will. Hence we could turn on the current when we are pulling N 
 towards A, and thus get work done ; and could turn it off while 
 restoring N to its initial position, and thus require to do no work 
 on N. Thus, in a gravitation system the law of conservation of 
 energy demands only that we consider the change in relative 
 positions of the bodies, or ' the configuration of the system ' ; 
 while in an electro-magnetic system we must, if the law is to hold, 
 seek for some source of energy other than that depending on 
 relative positions ; for we have shown how we could get un- 
 limited work done and yet end up in the initial positions. 
 
 We are driven to conclude then that the energy must come 
 out of the circuit A, and shall be able (see 3) to see how the re- 
 quisite energy can be supplied if we suppose E.M.F.s to be in- 
 duced in the circuit A whenever work is being done on or by the 
 pole N ; the E.M.F.s being in opposite directions in the two 
 cases respectively. 
 
 3. More Exact Reasoning, in a Simple Case. Let us con- 
 sider a vertical rectilinear current A, and a single pole N pivoted 
 so as to be capable of movement about A. (We cannot in practice 
 have a single pole, but we can contrive an arrangement in which 
 a current acts on one pole only of a magnet, so as to produce 
 continuous rotation.) When the current passes, the pole will be 
 urged round and round the wire A continuously. Hence we can 
 have unlimited work done, while the pole continually returns to 
 its initial position. As argued in' the last section, there must 
 be an equivalent of energy lost from the circuit of which A is part. 
 
 We will employ a notation similar to that used in Chapter XV. 
 Let E be the E.M.F. of the battery in the circuit ; R the total 
 resistance of the circuit ; C the current that flows when the pole 
 N is stationary, i.e. when no work is being done external to the 
 circuit ; C the current when the pole N is moving, i.e., when 
 external work is being done. [We here use absolute units.] 
 
 (In order to understand better what follows, the student is 
 recommended to read Chapter XV. 3, 4, 8, and 9 again.) 
 
 Initially, i.e. before the pole N begins to move and to do work, 
 
 Tf 
 
 we have a current C = , energy lost in the cell at the rate of 
 
 ix 
 
 E C per second, and an exact equivalent of heat energy appear- 
 ing in the circuit at the rate of C 2 R or E C per second 
 
348 ELECTRICITY CH. xxi. 
 
 Now let the pole move and external work be done. How can 
 the circuit supply the necessary energy ? At first sight it seems 
 simplest to suppose that C remains unaltered, while less heat is 
 evolved ; the diminution of heat evolved per second being the 
 exact equivalent of the mechanical work per second done by the 
 pole. In this way the current C would be unaffected, only 
 it would give us less heat with an equivalent of mechanical 
 work. Such an hypothesis would save the law of ' Conservation of 
 energy ' from being broken ; but it is neither in accordance with 
 experiment, nor with Joule's law which is supported by experi- 
 ment. For, by Joule's law, as long as the current is C so long 
 
 E 
 
 is there heat C 2 R given out per second. And since C = -n> 
 
 this heat is equivalent to the whole energy E C given out by the 
 battery per second ; so that there is none left for the external 
 work. 
 
 Hence, when the pole moves we conclude that the current 
 cannot remain the same. Let us suppose, then, that it decreases 
 and becomes C instead of C ; and let us see if such an alteration 
 will give us the energy needed for the external work done in 
 moving -the pole. The total activity of the cell is now E C, and 
 the heat activity is C 2 R. Since E = C R, and since C is less 
 than C , it follows that C 2 R is less than EC; or we have less 
 heat activity evolved than is the equivalent of the cell's activity. 
 Thus, if the current decrease from C to C while E remains 
 the same, we have a balance of activity left to account for the 
 mechanical work done on the pole. 
 
 Now, since the battery and its E.M.F. E remain unaltered, 
 and since R is itself unaltered also, it follows from Ohm's law 
 that the current cannot fall from C to C unless an E.M.F. e arise 
 contrary to E. 
 
 If such an E.M.F. e be supposed to arise, we have (see 
 
 Chapter XV. 9). 
 
 E C = energy per second expended by battery. 
 
 T7* T7* 
 
 C = ^ , and is less than C which = -^. 
 J R R 
 
 | C 2 R, or (E e] C, = energy per second appearing as heat. 
 E C C 2 R, or e C, = energy per second available to do the 
 external work on the pole. 
 
CH. xxi. ELECTRO-MAGNETIC INDUCTION 349 
 
 The reader will notice at once the close resemblance of the 
 above distribution of energy to that which occurs when a battery 
 is employed to electrolyse any body, i.e. to do chemical work. In 
 both cases there is a reverse E.M.F. e ; and we have total energy 
 is E C, heat energy is (E e) C, and energy expended upon the 
 chemical or mechanical work is e C. (All these reckoned per 
 second.) 
 
 If the pole N be caused to revolve in the other direction, we 
 must do external work upon it, since we move it against the lines 
 of force. Thus we keep doing work on the system without gain- 
 ing energy of position. So we conclude that we must in some 
 way be giving energy to the circuit. But we cannot, by Joule's 
 law, give heat to the circuit so long as C remains constant. As 
 before, then, we conclude that C must alter ; and, this time, must 
 increase to some value C'. This implies an E.M.F. d in the same 
 direction as E. And we have energy of battery E C', heat energy 
 (E + e') C f , energy given to the circuit from outside e' C (all 
 reckoned per second). In fact, we merely change the sign of the 
 induced E.M.F. 
 
 Thus we have shown how, by reasoning founded upon the 
 laws quoted above, we should predict that E.M.F.s would be 
 induced in conductors whenever movements or changes are 
 made in which + or work is done by the system. The case 
 above was chosen for its simplicity ; but the same reasoning 
 applies to all cases. The main facts of 'induction' were dis- 
 covered and investigated by experimental methods. But it is 
 satisfactory to see how readily they fall into place under the same 
 few great laws as are followed by all the electrical phenomena 
 with which we were previously acquainted. 
 
 4. General Expression for Induced E.M.F. More generally, 
 let there be any system of currents and poles, or of currents 
 alone. Let + or work be done by movements occurring in 
 this 'electro-magnetic system' ; the work being due to the existence 
 of the currents. Then the same arguments as those given in 
 2 and 3 show us .:... .... . . .,-> . *. . . 
 
 (i.) That the equivalent of this + or work is not to be 
 sought iri, the change in potential energy due to the changed 
 relative positions of the bodies forming the system, as would be 
 
350 ELECTRICITY CH. xxi. 
 
 the case with a gravitation system, but is found as a diminution 
 of or addition to the electric energy of the system. 
 
 (ii.) That, in consequence of Faraday's and Joule's law, this 
 loss or gain of the electrical energy is possible only if an E.M.F. 
 be induced in one or more of the circuits. 
 
 (iii.) That if this induced E.M.F. be , and there be a current 
 C in the circuit, then the loss or gain of electrical energy per 
 second, answering to the work done per second in consequence of 
 the movements referred to, will be measured by e C. 
 
 In what has preceded we have for simplicity spoken only of 
 cases where the work done per second is constant. 
 
 When this is the case the induced E.M.F. e will be constant, 
 
 as will also the current C which = - -_ (see 3) ; and we have a 
 
 ix 
 
 constant gain or loss of electrical energy C e per second, the 
 equivalent of the work done per second. 
 
 In the case of a circuit moved in a magnetic field, the work done 
 between any two positions is measured in ergs by C (N, N 2 ), 
 as explained in Chapter XIX. 3 ; the symbols having the 
 meanings there given to them. 
 
 Now let (n\ 2 ) be the change per second in the number of 
 marked lines of force piercing the circuit. Then the work per 
 second due to the movement is measured by C (n { n 2 ). But 
 we have seen that the equivalent of this is e C. 
 
 Whence it follows that .............. 
 
 Or, the E.M.F. induced in a simple circuit is measured by 
 the change per second in the number of marked lines of force 
 piercing the circuit. 
 
 Here e is in absolute units of E.M.F., not in volts. (In fact, 
 whenever we do not state to the contrary the reader may take it for 
 granted that in this Chapter absolute C.G.S. units are intended.) 
 
 Note. We must remark that where a pole of strength ju. threads through a 
 circuit and round again any number of times, returning to its initial position, 
 the above formula is not sufficient by itself. This was shown in Chapter XIX. 
 
 |. 
 
 Thus for each complete circle made about the circuit by the pole /* we 
 must consider 4 IT /j. marked lines of force to have been added to those piercing 
 the circuit (see Chapter XIX. Ii). As a particular case, when the pole n 
 
CH. xxi. ELECTRO-MAGNETIC INDUCTION 351 
 
 revolves a times per second about a rectilinear wire, it is as though there had 
 been added 4 a IT /i marked lines per second to those piercing the infinite plane 
 circuit of which we may consider the wire to form the edge. In this case, 
 therefore, we have e = 4 a IT ^, this being the form of the ' general expression ' 
 that applies to this particular case. 
 
 If the circuit be made of m turns of wire, all of them pierced by 
 the lines of force, then the E.M.F.s induced in them all severally 
 are added, and give a resultant E.M.F. in the circuit that is m 
 times as great as that in one turn. Hence if a change (N\ N 2 ) 
 in the number of marked lines piercing the coil occur in / seconds 
 of time (where / may be a small fraction), we have . . . , , 
 
 Resultant E.M.F. e= <Ni^_N,_)jo 
 
 5. Induction where there is no Initial Current 
 
 (I.) If we consider the case of a conducting circuit, in which 
 there is no current flowing, moving across a magnetic field so as to 
 change the number of lines of force piercing the circuit, there is 
 no reason from the point of view of ' conservation of energy ' why 
 there should be an induced current at all ; for the circuit when 
 stationary in any part of the field has no potential at all, not being 
 subject to any force. 
 
 (II.) Or again, let us consider a circuit in which there is an initial 
 E.M.F. E ; and let us suppose that, by making the rate of change 
 in the number of lines of force piercing the circuit great enough, 
 we induce an opposed E.M.F. e exactly equal to E. In such a 
 case the current will first cease, since the total E.M.F., E *, is 
 zero. There is no reason, from the conservation of energy point of 
 view, why there should be any further induction if we still further 
 inaease the rate of change of the lines. For we might now have 
 any velocity of change without work being done, since the 
 current C is zero, and therefore C (n l 2 ) is a ls zero - 
 
 But we may fairly conjecture that, as no doubt the true cause 
 of induction lies in the reaction between the conductor that forms 
 the circuit, and the magnetic field existing in the surrounding 
 medium, therefore there will in case (I.) be induction, and in case 
 (II.) further induction. 
 
 If there is such induction, so that there is a current C, then 
 since the work per second due to the induced E.M.F. e is 
 
352 ELECTRICITY CH. xxi. 
 
 measured' by e C and is also measured by C . (n l n 2 ), it follows 
 that as before ................ 4 1 
 
 where (n l n. 2 ) is the change per second in number of lines of 
 force piercing the circuit. 
 
 Thus in case (I.) we should have an induced E.M.F. 
 
 e (n l n. 2 ), and a current C = . We should do work per 
 
 R 
 
 second on the circuit measured by C (n l n 2 ), and should gain 
 equivalent electrical energy measured by e C. In case (II.) we 
 should do negative work e C as long as the current flowed against 
 the E.M.F. e, i.e. as long as E was > e ; zero work as long as E = e 
 and C was zero ; and positive work e C when C had changed sign in 
 consequence of e being now > E. [All reckoned per second.] 
 
 Experiment fully confirms this conjecture, and proves the 
 formula e = (n\ -~ n^ to be universally true for conducting cir- 
 cuits ; e being added to or subtracted from the initial E.M.F. E if 
 there be such, or appearing alone if there be no initial E. 
 
 This occurrence of induced E.M.F.s in all cases, whatever be 
 the initial E.M.F. E, must be regarded as a phenomenon due to 
 the ' nature of things ' ; it could not have been predicted from the 
 laws of ' Conservation of energy,' &c., alone. 
 
 Note. In the simple case of 3, where a pole revolves about a wire that 
 now we will suppose to bear no current, there will be an induced E.M.F. 
 measured by 4 air ^ ; the symbols having the same meaning as in 4, note. 
 
 6. Direction of the Induced Currents ; Lenz's Law. We 
 
 have stated that the direction of the induced E.M.F.s depends 
 upon the nature of the movement. It is not hard to see how we 
 can, from the law of 'Conservation of energy,' predict this direc- 
 tion. . 
 
 There are three main cases to be considered, 
 
 (I.) Where the -electro-magnetic system does positive mechanical 
 work. Under this head come all cases in which magnets, or wires 
 bearing currents, are caused to move in consequence of electro- 
 magnetic attractions or repulsions. We will take as a typical 
 case that of a pole urged (by the electro-magnetic field) round a 
 wire carrying a current ; the Case, in fact, of 3. Here there is 
 mechanical work done upon the pole, since this latter is either 
 
CH. xxi. ELECTRO-MAGNETIC INDUCTION 353 
 
 gaining kinetic energy or perhaps is set to work some machine at 
 a constant speed. According to the argument of 3, there is an 
 induced E.M.F. e opposed to E. That is, there is an E.M.F. 
 induced in such a direction as would alone give a current in the 
 opposite direction to that which is actually flowing, and which is 
 causing the pole to move. We may say then that the induced 
 E.M.F. opposes the movement that induces it. 
 
 In all similar cases where there is work done by the system and 
 electrical energy expended, the induced E.M.F. e is opposed to 
 the initial E.M.F. E ; and would, if acting alone, drive a current 
 opposed to that actually running. But in electro-dynamics, when 
 the current changes its direction, then attractions and repulsions 
 are interchanged. Hence, in all cases coming under this head, the 
 induced E.M.F. opposes the inducing movement. 
 
 If, in the simple case considered, the pole be absolutely free 
 to move, then it will move with increasing velocity until the in- 
 duced E.M.F. , which (see 4, note) is measured by 4^77/1, becomes 
 equal to E. As e approaches E in value the current C becomes 
 
 smaller : since C = ^-^. Hence the force acting on the pole be- 
 K. 
 
 comes smaller and smaller. For this reason it will take an infinite 
 time for e to become actually equal to E, though it may not take long 
 for it to become nearly equal. Supposing it to have become exactly 
 equal, then the current ceases ; there is no energy expended by the 
 battery, and, since the pole is by hypothesis perfectly * free ' to move, 
 it will continue revolving with constant velocity, and there will be no 
 work done. Things, therefore, will have come to a standstill, from 
 the conservation of energy point of view, but there is all the time an 
 electro-magnetic inductive action, since there is an induced E.M.F. e 
 equal to E. 
 
 (II.) Where positive mechanical work is done by an external 
 agent upon the electro -magnetic system. Where we alter the rela- 
 tive positions of poles or circuits so as to do positive work against 
 the electro-magnetic forces, there must be given to the electro- 
 magnetic system electrical energy equivalent to the work done. 
 This can only be, as was argued in 3, by the induced E.M.F. e 
 being in the same direction as the original E.M.F. E. Hence, the 
 induced E.M.F. tends to drive a current in the same direction as 
 that originally flowing. But it is owing to this original current 
 
 A A 
 
354 ELECTRICITY 
 
 CH. XXI. 
 
 that we are (by hypothesis) doing work against forces that oppose 
 the movement. Hence the induced E.M.F. is in such a direction 
 as to oppose the inducing movement. 
 
 Thus in the simple case of 3, if we whirl a pole about a wire 
 bearing a current, in the direction opposed to that in which it is 
 urged by the field due to the current, we do mechanical work per 
 second on the system ; and this appears as electrical energy eC 
 gained per second by the circuit. This induced E.M.F. e adds to 
 the current which is opposing the movement of the pole. There 
 is now given out in the circuit per second heat to the amount of 
 EC due to the battery, and also of eC due to the work done on 
 the system from outside. We feel resistance in moving the pole ; 
 and we can regard e C as a kind of equivalent ' heat of friction ' 
 given out per second in the circuit. 
 
 (III.) Where there was originally no current. We have seen 
 in 5 how there are E.M F.s induced by movements even when 
 there are initially no currents. 
 
 Considering the simple case of 3, it 'is easy to see in \vhat 
 direction the induced E.M.F. must be. 
 
 Let us whirl the pole about the wire, and let the induced 
 E.M.F. , and current arising from it, be in such a direction as to 
 oppose the movement. We then do work on the system, and gain 
 electrical energy in the system ; or conservation of energy is 
 maintained. But if we were to suppose the E.M.F. and current 
 to be induced in the opposite direction, i.e. so as to urge the pole 
 in the direction in which it is moving, we should then have elec- 
 trical energy both giving out heat in the circuit and doing external 
 work in urging the pole, and we should have created this energy 
 out of nothing ; since now no work is being done on the system. 
 And this is against conservation of energy. 
 
 Hence we conclude that the induced E.M.F. is such as to 
 oppose the inducing movement. 
 
 From considerations such as those given above we should 
 predict that -... 
 
 The currents which are induced in consequence of movements or 
 other changes in an electro-magnetic system are invariably in such a 
 direction that they tend to oppose these movements or changes. This 
 law was observed experimentally and enunciated by Lenz. It is 
 
CH. xxi. ELECTRO-MAGNETIC INDUCTION 355 
 
 called Lenz's law. The experiments that illustrate this law are 
 endless in number. We give here only a few. 
 
 Experiments to illustrate Lenz's /aw. In experiment (I.) of I we can, 
 without much difficulty, show that when the current in the primary A is started 
 there is induced a current in the secondary B in the opposite direction ; while, 
 when the current in A is broken, there is induced in B a current in the same 
 direction. In both cases, therefore, the induced current opposes the change, 
 for it will repel that in the primary while t,he latter is rising from zero to a 
 maximum, and will attract it while falling from a maximum to zero. Or, 
 again, the creation of a current in A has the effect of thrusting lines of force 
 into B in one or other direction, and the inverse induced current in B opposes 
 this by thrusting out lines in the opposite direction, while, when the current in 
 A ceases and the lines are withdrawn, the direct induced current in B creates 
 lines in the same direction, tending (as it were) to pull back those due to A. 
 
 (ii. ) So, again, it can be shown that in experiments (ii. ) and (iii. ) of I, 
 the induced current in B is such that the coil A or the magnet is repelled by B 
 when advanced, and attracted by it when withdrawn, respectively. Regarding 
 the coils as solenoids, and remembering the rule of Chapter XX. 3, it is easy 
 to verify this statement. Another way of stating the observed result is to say 
 that when more lines are thrust into the circuit B, the current induced is such 
 as to thrust opposed lines out ; while, when lines are withdrawn, the current 
 induced opposes the withdrawal by creating lines in the same direction. 
 
 (iii.) A circuit is arranged comprising a battery, resistances for regulating 
 the current, a single rectilinear wire B, and a delicate galvanometer. When 
 a wire A bearing a current is brought up to B, the decrease or increase in the 
 deflexion of the galvanometer will indicate a current induced in B in the oppo- 
 site direction to that flowing in A ; when A is withdrawn the galvanometer's 
 increase or decrease in deflexion indicates an induced current in the same 
 direction as in A. So, by Ampere's laws, the induced current is such as to 
 oppose the movement. 
 
 (iv. ) With the same apparatus we may rapidly pass a powerful magnet 
 pole across the wire B, without touching it. The current induced can be 
 shown to be such as tends, according to the law given in Chapter XVIII. 2 
 and 12, and Chapter XIX. 7, to oppose the inducing movement. 
 
 7. Constant Induced Currents. The question as to how we 
 can obtain a constant induced current is of great importance. 
 We shall, later on, discuss those machines by means of which we 
 can induce currents for the purpose of electric lighting, &c. ; and 
 we shall show how a practical constancy in the current induced 
 can be obtained. 
 
 In this place we propose only to show how an absolutely con- 
 stant induced current could theoretically be obtained. 
 
 A A 2 
 
356 ELECTRICITY CH. xxi. 
 
 (L) By revolution of a pole about a wire. When a pole n re- 
 volves with constant velocity a times per second about a wire, the 
 induced E.M.F. e is, as we have seen, constantly equal to 4 # TT p. 
 An arrangement in which the wire revolves with uniform velocity 
 about a pole, equivalent to the converse movement, is practicable, 
 but is not a very serviceable machine for obtaining induced cur- 
 rents. Since the resistance R of the wire is constant, it follows 
 
 that the current is constant, being equal to ----, or ~ 7r ^. 
 
 lx Jx 
 
 (II.) By the rail-and- slider arrangement. In the figure ARE 
 represents a conductor of resistance R, including a galvanometer. 
 A C and B D are thick strips of copper of no resistance that is 
 
 appreciable. C D is a cross piece, also of no resistance, sliding 
 in perfect contact with the two strips. The dots represent a uni- 
 form magnetic field in which we suppose the lines to run from 
 above perpendicularly down into the plane of the paper, i.e. into 
 the plane of the apparatus. 
 
 If the slider move with uniform velocity in the direction of the 
 arrow, we add to the circuit A C D B A a constant number of marked 
 lines of force per second; say a numbers. Then the induced E.M.F. 
 is measured by e = n ; and, since we suppose the resistance to lie 
 
 in the part A R B only, the induced current is equal to ^ or -**- 
 
 ix K. 
 
 and is constant. In the case given it will be such as to oppose 
 the introduction of more- lines down into the circuit. That is, it 
 will be such as to thrust lines up out of the circuit ; i.e. the 
 circuit will have a N polarity presented upwards, or the current 
 will flow in the direction C A B D, 
 
 If, other things remaining the same, we incline the field so 
 that ihe lines make an angle 6 with the perpendicular to the 
 
CH. xxi. ELECTRO-MAGNETIC INDUCTION 357 
 
 plane of the circuit, it is clear that now we add only ncosO marked 
 lines per second to the circuit. 
 
 Resistance expressed as a velocity. We have seen in Chapter XVI 1 1. 
 5 that, in the electro-magnetic system, R is of the dimensions of a 
 velocity. It happens that the above simple apparatus enables us to 
 show how R can in fact be measured by a velocity. (Compare also 
 Chapter XVIII. 4, note on method (ii.) of determining resistance.) 
 
 Let us assume for simplicity that the other quantities concerned 
 are each unity ; i.e. that the field is of unit strength and is perpendi- 
 cular to the plane of the apparatus, so that we have one line of force 
 piercing each i sq. cm. ; that the rail is of i cm. length, and that the 
 current is of unit strength. 
 
 The rail must move with such a velocity that, in the formula given 
 above, n may be numerically equal to R. This means that we must 
 add R sq. cms. to the area of the circuit each second ; or, since the rail 
 is i cm. long, this must move with a velocity expressed by the number 
 R, or with R cms. per second. 
 
 Hence, under the above ' unit conditions ' we can measure R by 
 the velocity of the rail required to give unit current. 
 
 The velocity that would thus measure the absolute unit of resistance 
 is that of i cm. per second, while io 9 cms. per second measures i ohm. 
 
 8. Changes that give Induced Currents. In considering the 
 question as to whether in any particular case there will be an 
 induced current, we have to remember two facts. 
 
 (a) That when a conductor cuts across lines of force there is 
 always an induced E.M.F. 
 
 (If) That when there is a change in the number of lines piercing 
 a circuit there is a resultant E.M.F. induced in the circuit. 
 
 We will consider several cases. 
 
 (I.) A circuit moving in a uniform field. If the circuit move 
 parallel to itself (in such a way that the direction of movement 
 causes the circuit to cut the lines of force), the number of marked 
 lines embraced will not alter ; and there will be no resultant E.M.F., 
 or current, induced. The two sides of the circuit, it is true, do cut 
 the lines ; and so, by principle (a), there is an E.M.F. induced in 
 each side. But these E.M.F.s are equal and opposed. There is 
 no resultant E.M.F. in the circuit, but the top and bottom of the 
 circuit, where the wire does not cut the lines of force, will be 
 maintained at different potentials, as could be demonstrated by 
 connecting them with a quadrant electrometer. 
 
35^ ELECTRICITY CH. xxi. 
 
 If, however, there be any movement of rotation, then the 
 number of marked lines embraced is altered, and there is an E.M.F. 
 induced. It is, in fact, a very usual way of obtaining induced 
 currents to rotate a circuit or coil in a field that is more or less 
 uniform. 
 
 (II.) Movement of a circuit in a non-uniform field. Where 
 the field is not uniform, movements of a circuit will in general 
 produce a change in the number of marked lines embraced, and so 
 there will be an E.M.F. induced. In certain cases the two move- 
 ments of translation and rotation respectively may give as a result 
 no change in the number of .lines embraced ; in which case there 
 is no E.M.F. induced in the circuit as a whole. 
 
 (III.) Movements of an incomplete circuit ; e.g. of a rectilinear 
 wire, In considering the movements of an incomplete circuit, 
 such, e.g., as a rectilinear piece of wire, we may adopt two courses. 
 We may consider it to form part of a closed circuit, the rest of 
 which is at an infinite distance, and is indefinitely remote from 
 the magnetic field in question ; or we may consider the piece of 
 wire by itself. It may be stated as a general law that there will 
 or will not be an E.M.F. induced according as the wire cuts or 
 does not cut the lines offeree, as strings are cut by a knife. 
 
 Note. If this were absolutely true we should never have an E.M.F. 
 induced when the wire either moved in its own direction, piercing the field end- 
 foremost as a needle, or if it moved in any way in a plane in which lay the 
 lines of force. Now, it is certainly true that in the latter case we never have 
 an induced current. But in the former case we might have a current ; for, the 
 wire might, so to speak, ' tunnel its way ' end-on along the axis into a cylindrical 
 system of lines of force ; the self-induction in a rectilinear current, mentioned 
 in the note to 10, being a case of this nature. 
 
 While, therefore, the statement that 
 
 When a conductor moves in a magnetic field there infill or will not be an 
 indticed current according as it does or does not cut the lines of force 
 is a good general rule, covering all cases of importance, still it is well to seek 
 for a rule that shall cover all cases. Now, such a rule can be found from 
 Lenz's law, if we remember the nature of the field given by a rectilinear cur- 
 rent, and so by a current of any shape. We may say that 
 
 When a conductor moves in a magnetic field there will or will not be an 
 indticed E. M. F. in it according as the field due to a current produced by 
 such an E.M.F. can or can not oppose the change in field due to the movement. 
 
 (IV. ) Cases where the conductor (or circuit] is stationary, the field 
 being altered. In such cases there is no obvious cutting of lines 
 
CH. xxi. ELECTRO-MAGNETIC INDUCTION 359 
 
 of force. But both theory and experiment tell us that all such 
 cases of alteration in the field about a conductor, or circuit, can 
 be regarded as equivalent to cases where the same change has 
 been produced by movements on the part of the conductor from 
 a weaker to a stronger part of the field, or conversely. We may, 
 therefore, consider that all cases included under this head have 
 been already discussed above. 
 
 9. Coefficient of Mutual Induction, or of Mutual Potential. 
 Let us consider two simple circuits, A and A', carrying currents 
 C and C' respectively. Each gives a magnetic field, and each is 
 placed in the field due to the other. If n and n are the number 
 of marked lines due to A' embraced by A, and due to A embraced 
 by A', respectively, then the potentials on each circuit due to the 
 other are Cn and C'ri , respectively. That is. it would require Cn 
 ergs to bring up A from infinity to its present position, A' remain- 
 ing stationary ; or Cn' ergs to bring up A', A remaining stationary. 
 Now, if A be moved to infinity, we do Cn ergs work ; and if A' 
 be moved after it into the same relative position as initially, we 
 do + Cn' ergs work. But, on the whole, since things are exactly 
 as they were, we must, by ' Conservation of energy,' have done no 
 work. Hence it follows that Cn and Cn' must be equal ; and, 
 therefore, two circuits exert on one another a mutual potential. 
 
 Again, the number n of marked lines of force due to A' that 
 pierce A are, cteteris paribus, directly proportional to the current 
 C' of A'; this following from the fact that in our system of units 
 we measure currents by the field-strength produced at constant 
 distance. And the number n 1 of marked lines that pierce A' are 
 in like manner directly proportional to C. 
 
 Hence, since we have shown the potential to be { mutual,' this 
 * mutual potential ' must be measured by some expression of the 
 form CC ' M\ where M depends upon the shapes and positions of, 
 and distance between, the two circuits, and not on the current- 
 strengths. 
 
 Thus we have Cn C'n' = CC'M. 
 
 If we make C and C' both unity, we find that M = n or n'. 
 Hence we see that when unit current flows in each circuit, the 
 number of marked lines due to the other, piercing each circuit 
 respectively, is the same. 
 
360 ELECTRICITY CH. xxi. 
 
 And we see, further, that the symbol M, whose meaning we 
 had not given exactly, represents this number. The reader will 
 notice that, for unit current, this number is, as we said, something 
 depending upon the shapes and positions of, and distance between, 
 the two circuits. 
 
 It only remains to state that M is called the coefficient of mutual 
 induction or of mutual potential. 
 
 Hence, the coefficient M of mutual induction, or of mutual poten- 
 tial, between two circuits is measured by the number of marked lines 
 due to either that are embraced by the other when the currents are 
 both unity. And when the currents are C and C' respectively, then 
 the mutual potential will be expressed by CC' M. 
 
 If we consider a current C sent through the one circuit, it will in con- 
 sequence send C M marked lines of force through the other. If it take 
 / seconds to establish the current (where t may be a small fraction of a 
 
 second) then there are lines added per second to the second cir- 
 cuit. Hence the induced E.M.F. e = 1VT . This shows us (i) the 
 
 reason for M being called the coefficient of mutual induction as well 
 as that of mutual potential ; and (2) how, in such an arrangement, the 
 magnitude of the induced E.M.F. e is inversely proportional to the 
 time taken to establish or destroy the primary or inducing current. 
 
 10. Self-Induction. The 'Extra Current,' Let us consider 
 a circuit comprising a coil of many turns and a battery; this circuit 
 being so arranged that it can be made or broken at will. 
 
 When the circuit is made the current does not rise to a maxi- 
 mum at once. If we consider only the positive current, or the 
 flow of + electricity from the 4- pole of the battery to the pole, 
 we find that it takes time to rise to a maximum ; and that in so 
 rising the turns of wire that are more remote from the + pole of 
 the battery always lag behind those that are nearer to it. Each 
 turn of wire, as the current in it increases, thrusts an increasing 
 number of marked lines through the adjacent turns of wire. It, 
 therefore, acts inductively on these, and, by Lenz's law, the induced 
 E.M.F. is in such a direction as to oppose the rise in current- 
 strength. There is, in fact, an inverse induced current. Thus the 
 coil, as a whole, offers an inductive obstruction to the rise in current 
 that is quite distinct from resistance ; and that will, as far as it is 
 
CH. xxi. ELECTRO-MAGNETIC INDUCTION 361 
 
 due to the above given cause, disappear if the coil be unwound 
 and laid out as a straight wire. 
 
 Note. In a straight wire there is also induction, but to a less degree. In 
 this case each bit of wire gives a field of circular lines of force, not only about 
 itself, but also, to a much smaller degree, about the wire in front and behind ; 
 and any increase in this field is also opposed by induction (see 8 (III.), 
 note}. 
 
 When the circuit is broken, induction in the inverse direction, 
 i.e. so as to oppose the cessation of the current, ensues ; and there 
 is an induced current added to the original current just as the 
 latter is ceasing. Now, the primary current can be broken much 
 more abruptly than it can be made; and hence, since e is inversely 
 proportional to the time taken to effect the change in field, the 
 E.M.F. of this direct induced current is much greater than is that of 
 the inverse induced current ; it may, indeed, be made very great 
 indeed. 
 
 This direct induced current that occurs in the coil when the 
 circuit is broken is called the extra current. 
 
 Experiments. (i.) In the figure B i; a coil in circuit with a battery whose 
 poles are E and E' ; this circuit can be broken at E. The two points A and C 
 are connected by a short circuit that includes a galvanometer G. 
 
 When this circuit is complete the current flows as shown, and the galvano- 
 meter is deflected as indicated. When the circuit is broken at E, it is easy to 
 see that a direct extra current in the coil would now pass round the galvano- 
 meter in the contrary direction to that in which it before passed, although in 
 the coil B it flows in the same direction as before. Hence the extra current 
 
362 ELECTRICITY CH. xxi. 
 
 would be indicated by an opposite direction of deflexion in G. In order to 
 distinguish this reverse swing from the fall back to zero, which would ensue 
 merely from the fact that the battery circuit is broken, it is necessary to keep 
 the needle at zero by means of stops placed so as to prevent the former deflexion, 
 but so as to permit of the second deflexion if such there be. Such a deflexion 
 is, in fact, observed ; and it demonstrates the existence of the extra current. 
 (The present writer has found that very marked results may be obtained with 
 two Leclanche cells as battery, an ordinary rough astatic galvanometer at G, 
 and a powerful electro-magnet instead of the simple coil B. ) 
 
 (ii. ) If there be in circuit a powerful electro-magnet and a sufficient battery, 
 and if the circuit be broken by a person holding in his moistened hands the 
 two ends of the circuit, a shock will be perceived. The high E.M.F. of the 
 induced extra current is sufficient to drive some current through the human 
 body that has suddenly been interposed in the circuit. 
 
 We see, then, that when a circuit is made, the current rises 
 slowly, and research has shown that its rise is oscillatory. It then 
 continues uniform as long as the battery is constant. When the 
 circuit is broken, there is a sudden leap in the magnitude owing 
 to the extra current ; and finally it ceases again in an oscillatory 
 manner, much more abruptly than it began. All this is very 
 readily exhibited by means of a curve, the abscissae measuring, 
 time and the ordinates magnitude of current. 
 
 n. Induced Currents of Higher Orders. 
 
 It is found, as might indeed have been predicted, that induced 
 currents will themselves act as inducing currents. 
 
 We may arrange a series of coils somewhat as follows. First, a 
 primary A, and round it a secondary B. B may then be in circuit 
 with a coil B' at a distance from A, so that currents induced in B will 
 circulate also in B' ; and round B'is placed another coil C upon which 
 B' can act inductively, while A is too remote to have any direct influ- 
 ence. We will use the words ' direct ' and ' inverse ' when the currents 
 are in the same direction as, or in the opposite direction to, the original 
 current in A respectively. 
 
 We find that when the current is made in A, we have an inverse 
 current induced in B and therefore passing in B'. This current, as it 
 rises in strength from zero to a maximum, induces in C an opposed 
 current, which will therefore be direct ; and, as it falls again to zero, 
 it induces in C a current in the same direction, which will therefore be 
 inverse. 
 
 So, when the current in A is broken, we have a direct current induced 
 in B ; and in C, an inverse followed by a direct current. 
 
CH. XXII. 
 
 CHAPTER XXII. 
 
 ARAGO'S DISC, RUHMKORFF's COIL, AND OTHER CASES OF 
 INDUCTION. 
 
 i, Induced Currents ('Eddy Currents') in Solid Metallic 
 Masses moving in a Magnetic Field, When a conducting mass 
 moves in a magnetic field currents are induced. Here, as always, 
 the currents are such as to oppose the movement ; work must be 
 done on the masses in order to move them, and we have developed 
 in the mass equivalent electric energy, which finally runs down 
 into the form of an equivalent of heat. 
 
 Such currents will, as a rule, run in eddies ; but if we connect 
 two points in the mass by a conducting wire, these two points 
 being so chosen as to be at different potentials, we thereby modify 
 a portion, at least, of the induced * eddy currents ' into a current 
 running round a definite circuit. 
 
 Experiments. (i. ) If a copper cube be caused to rotate between the poles 
 of a powerful electro-magnet while this latter is as yet ' un-made,' and if we 
 then ' make ' the electro-magnet by sending a current round it, the cube will 
 be visibly retarded or stopped in its movement. 
 
 (ii. ) A disc caused to rotate in the field will get perceptibly heated. 
 
 (iii. ) In 3 we shall see how a current may be ' collected ' from the disc. 
 
 (iv. ) When a person cuts through the field between the poles of a very 
 powerful electro-magnet with a copper knife, it will appear as though he were 
 cutting through soft cheese, so strong is the opposition due to the induced 
 currents. 
 
 2. Arago's Disc and Magnetic Needle There is one case 
 of the above that is of especial historic interest, and though in no 
 way peculiar, will be described at some length. In 1824 Arago 
 discovered the ' damping ' effect produced by the presence of 
 copper and of other conducting masses on magnetic needles 
 oscillating near them. If a needle oscillate very close over a 
 copper disc, and still more if it oscillate between two copper discs, 
 
364 ELECTRICITY CH. xxn. 
 
 it will very soon come to rest. This is due to the induction of 
 currents in the copper, these currents being such as to oppose that 
 motion of the needle which is the origin of the induction. 
 
 In the figure we have a copper disc caused to rotate with great 
 velocity under a magnetic needle ; a sheet of glass between the 
 two obviates any disturbance due to air-eddies caused by the 
 rotating disc. The needle is deflected in the direction of rotation 
 of the disc, and will, if the velocity be great enough, finally rotate 
 also. This motion of the needle is not difficult to explain. In 
 consequence of the rotation of the disc in the magnetic field due 
 to the needle, currents are induced in the former. These currents 
 are in such a direction that they oppose the relative motion of disc 
 
 and needle. They would be induced equally if the disc were 
 stationary and the needle rotated. We have thus a reaction 
 between needle and disc that tends to stop the relative motion ; 
 and so, if the needle be free to move, it will be urged round in the 
 same direction as the disc. If slits be cut radially these interfere 
 with the induced currents, and therefore with the actions described; 
 but if they be cut in circles whose centres lie on the common axis 
 of rotation of disc and of needle, their presence makes much less 
 difference. 
 
 3. Continuous Current Collected from Barlow's Wheel, 
 In the figure the dots represent a field of force, supposed to be 
 running down into the plane of the diagram. The circle is a 
 copper disc, revolving in the direction of the arrow. O B C A O 
 
CH. XXIT. ARAGO'S DISC AND RUHMKORFF'S COIL 365 
 
 is a circuit, of which OCA is a wire connected at O with the 
 axis of the disc, and having sliding contact with the edge of the 
 disc at A ; this circuit is completed by whatever radius of the disc 
 happens to lie between O and A. If a current from an external 
 source be sent in the direction A O B C A, the disc will rotate in 
 the direction of the arrow. We can consider the field as acting 
 always on the moveable radius O A, urging it (by the law given in 
 Chapter XIX. 7, and elsewhere) to the left. When so used, the 
 disc is called 'Barlow's wheel' If there be no external source of 
 current, but the disc be forcibly turned with the arrow, there will 
 be a current induced in the circuit in such a direction as to oppose 
 
 B 
 
 the motion, i.e. in the direction O A C B O ; the seat of the in- 
 duced E.M.F. being in the shifting radius O A, which is always 
 cutting the lines of force. The current thus induced gives lines 
 of force rising perpendicularly upward from the plane of the 
 diagram, opposed to those of the inducing field. 
 
 With a powerful electro-magnet this experiment is very easily 
 performed. 
 
 If the rotation be performed in the opposite direction, a very 
 remarkable result follows. The current is of course, in any case, 
 in opposite directions in the portions O A and B C, which we may 
 for simplicity suppose to be parallel. Hence the motion of the 
 disc (when this turns in the contrary direction to that of the 
 arrow), tending as it does always to move O A towards B C, will be 
 against the electro-dynamic repulsion of the parallel and opposed 
 currents in O A and B C respectively. This will give rise to in- 
 duction opposing the motion; i.e. the opposed currents in O A 
 
366 
 
 ELECTRICITY 
 
 CH. XXII. 
 
 and B C, or the current of the circuit, will increase in strength. 
 Thus, if a current be started in the circuit, and if then the external 
 field be caused to vanish, induction will continue, owing to the 
 action between the portions O A and B C. We thus have the 
 phenomenon of a current maintained solely by work done on a 
 system of copper conductors ; there being no external magnetic 
 field, no battery, and no electricity due to friction. 1 
 
 If the disc turn as in the figure, the inductive action between 
 B C and O A tends again to oppose the motion ; and in this case 
 the effect will be to lessen the current in the portions O A and 
 B C, i.e. the current of the circuit. 
 
 4. Induction in the Earth's Field. We can obtain induced 
 currents by rotation of a coil in the earth's field. In general, when 
 a coil so rotates the number of lines of force piercing it is varied, 
 and there will be E.M.F.s induced. 
 
 In the figure, S R is a coil of many turns of insulated wire, and 
 XY represents the direction of the earth's lines of force. The 
 coil is turned by means of a handle M. The whole is mounted 
 upon a stand in such a way that the axis of rotation n:ay lie in 
 any direction whatever. If the coil be initially in the position 
 shown, viz. perpendicular to the lines of force, then as it is turned 
 
 1 The apparatus as so used is called ' Sir W. Thomsons electric current 
 accumulator. ' 
 
CH. xxn. ARAGO'S DISC AND RUHMKORFF'S COIL 567 
 
 the number of lines piercing it will be diminished, becoming zero 
 when the coil presents its edge to the lines of force. As it is 
 further turned the lines will begin to pierce the other face, so that, 
 between the initial position shown and that which is 180 from it, 
 the number of lines piercing the one face of the coil will pass from 
 a + maximum, through zero, to a maximum. All this will 
 induce an E.M.F. and current in one direction. Thus, if the lines 
 offeree run from X to Y, this rotation through 180 causes, first, 
 a withdrawal of the -j- lines piercing the face that is initially 
 uppermost, and then an introduction of lines into this face. 
 Both these changes, which may, indeed, be regarded algebraically 
 as a continuous diminution in the -f lines piercing this face, are 
 opposed by the face in question becoming S in polarity ; for this 
 polarity opposes the weakening of the + field piercing the face, 
 and also opposes the introduction of lines. Hence, in this 
 case, the current will run clockwise. But as the coil continues to 
 rotate through the remaining 180 back to its initial position, 
 similar reasoning shows us that an opposite current will be induced. 
 Hence, as the coil rotates steadily, there are equal and opposite 
 currents induced each half-turn. To collect these into one 
 current that may be caused to deflect a galvanometer, &c., a com- 
 mutator is arranged at a. This is a simple contrivance, to be 
 described later in Chapter XXIII. 4, by means of which the two 
 ends of the coil interchange their connection with the two ends of 
 the external circuit exactly 'at the time when the current changes 
 its direction ; i.e. just when the coil is passing the position shown 
 in the figure. 
 
 Experiments. ^.*} It can be shown that the current is greatest when the 
 change per second in number of marked lines embraced, for the same speed of 
 rotation, is greatest. This will be when the axis of rotation is perpendicular 
 to the lines of force ; since then the lines pass from the maximum possible to 
 the minimum possible. 
 
 (ii. ) When the axis lies along the lines of force, there is no current. For 
 now no lines pierce the face, and there is no change produced by rotation, the 
 lines lying always in the plane of the coils. 
 
 (iii. ) The current is greater as the velocity of rotation is greater. 
 
 This apparatus would, theoretically at least, enable us to de- 
 termine the earth's elements at a place. For that direction of the 
 axis which gives us zero induced current on rotation, is the direc- 
 
368 ELECTRICITY 
 
 CH XXII. 
 
 tion of the earth's lines. And if the axis be now placed at right 
 angles to these, the magnitude of the induced current for a given 
 velocity of rotation would, if the * constants ' of the instrument are 
 known, give us the earth's field-strength. 
 
 5. Induction Coils ; General Plan. We have seen in 
 Chapter XXI. i how currents may be induced in a ' secondary ' 
 coil by the making and breaking of a current in another coil, 
 called the ' primary,' placed inside the former. We have further 
 seen how the magnitude of the induced E.M.F. depends mainly 
 upon three conditions (see Chapter XXI. 4), viz 
 
 (i.) The field-strength given by the primary circuit when the 
 current flows through it. 
 
 (ii.) The number of turns of wire in the secondary that are 
 pierced by the lines of force due to the primary. 
 
 (iii.) The suddenness with which the primary current is made 
 or broken. 
 
 In order to fulfil condition (i.), the primary is provided with a 
 soft iron core, and the current sent through the primary is strong. 
 Thus we make or destroy a powerful field each time we make or 
 break the primary current. 
 
 For condition (ii.), it is necessary to have very many turns of 
 wire in the secondary. . The wire, therefore, has to be very thin, 
 and we get a high E.M.F. but a great resistance. 
 
 For condition (iii.), there are special arrangements devised. 
 These will be given in 8 and elsewhere. 
 
 By paying proper attention to these three conditions, we may 
 obtain induced currents of very high E.M.F. but of very short 
 duration. Such currents give sparks, produce shocks, and pro- 
 duce other phenomena not obtainable from the primary current. 
 For certain purposes this modification of the energy of the 
 primary battery is desirable. 
 
 An arrangement consisting of a primary coil with its own iron 
 core, a secondary coil of many turns of wire, and a contrivance for 
 making and breaking the primary circuit, forms what is called an 
 induction coil. 
 
 If N be the total number of marked lines of force due to the 
 primary that pierce the m turns of the secondary, and if it take 
 /! and / 2 seconds respectively to break and to make the primary, 
 then we have 
 
CH. xxii. ARAGO'S DISC AND RUHMKORFF'S COIL 369 
 
 On breaking, the direct induced E.M.F. e L = N x m . 
 
 *i 
 
 On making, the inverse induced E.M.F. e 2 = >< -^. 
 
 By ' making ' or ' breaking ' we mean the increase from zero to 
 the maximum, and decrease from the maximum to zero, respec- 
 tively. The times / t and / 2 will be some small fraction of a second 
 of time. 
 
 6. Practical Difficulties to be Overcome. The main diffi- 
 culty encountered in the contriving of an induction coil which 
 shall give induced currents of high E.M.F., lies in fulfilling the 
 last of the above three conditions. 
 
 When the current is made there is an inverse current induced 
 in the primary itself, and this causes the rise to a maximum to be 
 relatively gradual. Hence the E.M.F. e<> of the inverse current 
 induced in the secondary will be relatively small, since the time 
 *> ( see 5) is relatively great. (We may here remind the reader 
 that when a coil has an iron core the inverse and direct currents of 
 self-induction are far greater than when there is no iron core, since 
 the field due to the core is added to that due to the current.) 
 This difficulty cannot be overcome j and hence the devisers of 
 induction coils have turned their attention entirely to rendering 
 the break of current very abrupt, or /, very small, and so 
 obtaining in the secondary a direct induced current of high 
 E.M.F. 
 
 When the primary current is broken, there is in the primary a 
 direct extra current due to self-induction, the presence of the 
 iron core making this effect greater. This direct extra current has 
 a high E.M.F., so high, indeed, that the current breaks disrup- 
 tively across the break just made in the circuit. Thus, when 
 we break the primary the extra current gives a spark at the 
 break ; and this both prolongs the current, or increases /,, and 
 therefore diminishes e lt and also may fuse and injure the metal 
 at the key by which the ciicuit is made and broken. 
 
 This difficulty is met and overcome by means of the condenser 
 described in 8. 
 
 Another obstacle in the way of abrupt break is the induction 
 of eddy current in the iron core ; these currents, of course, oppo- 
 ing the cessation of the primary current. This is obviated by 
 
 B B 
 
370 ELECTRICITY CH. xxn. 
 
 slitting the core longitudinally or by making it of a bundle of soft 
 iron wires insulated from one another. 
 
 Again, the iron core takes a certain time to become magnetised 
 and demagnetised. This is not such an important obstacle ; but 
 even this is nearly entirely obviated by means of the condenser, as 
 will be explained in 8. 
 
 Again, so great is the E.M.F. induced in the secondary that 
 fliere is danger of a spark penetrating the insulating coating of 
 this wire. This danger is obviated by a special method of 
 winding. 
 
 7. Ruhmkorff's Coil. Fig. i. represents a form of coil 
 called Ruhmkorff's coil, after the inventor. Here N and P are 
 
 FIG. i. 
 
 the terminals of the primary connected with the battery ; the 
 primary coil is not shown, but its core A, composed of a bundle 
 of soft iron wires insulated from each other so as to obviate eddy 
 currents, is seen slightly projecting from the coil ; C is an ar- 
 rangement for turning the current off, or on, permanently ; b is a 
 piece of metal called the hammer, which alternately is attracted up 
 to the iron core A, thereby breaking the current, and falls back by 
 its own weight, thus making the current again ; B is the secondary 
 coil ; p and/' are its terminals ; inside the stand K is the con- 
 denser, whose function will be described later. 
 
 As a special instance we may quote the dimensions of Spottiswoode's 
 large coil. This was constructed on the same principles as the above ; 
 but the arrangement for making the circuit was worked separately, 
 and was not the automatic hammer contrivance shown in the figure. 
 
CH. xxn. ARAGO'S DISC AND RUHMKORFF'S COIL 
 
 371 
 
 In this large coil the primary was a wire of about i inch diameter 
 and 660 yards in length, wound in about 1,350 turns. The secondary 
 was a wire of about ^~ inch diameter, and 280 miles in length. The 
 whole coil was 4 feet in length and 20 inches in diameter. 
 
 The maximum spark obtained was about 42 inches long. This coil 
 is the largest yet made. 
 
 In fig. ii. we have a section of a coil very similar to that of 
 fig. i., though the two do not correspond in all their details. 
 Initially the hammer b lies on 
 M, and the primary current 
 from the battery X passes by 
 P into the primary coil D, out 
 of this again to the metal piece 
 L, and so by the hammer b 
 through M and N to the battery 
 X again. By the passage of 
 the current the core A is mag- 
 netised, increasing many-fold 
 the field due to the primary 
 coil alone. This creation of a 
 field inside the secondary coil B 
 induces an inverse E.M.F. in 
 this coil ; and, if the terminals 
 //' be in contact or very near 
 to one another, an inverse 
 current will pass. This inverse 
 current is, as we have explained, of no very great E.M.F. 
 
 The moment that A is magnetised, the hammer b is attracted 
 upward, and thus the primary circuit is broken. This causes the 
 core A to be unmagnetised ; and, in consequence of the with- 
 drawal of the field, there is a direct E.M.F. induced in the 
 secondary coil B. This E.M.F. is high, since the break of the 
 primary is very abrupt, and /, (see 5) is very small. 
 
 We have not yet described the part played by the condenser 
 F F, as this matter deserves a section to itself. We may here re- 
 mark that L and N are connected by the route N M b L only, there 
 being no connection through q and s. 
 
 Caution. It may be well to warn the student that exposure to 
 shocks from the induced currents of an induction coil may be very 
 
 B B 2 
 
 FIG. ii. 
 
3/2 ELECTRICITY CH. xxn. 
 
 pamful or even fatal. He should not take shocks from any but small 
 coils, say those giving -inch sparks as their maximum. 
 
 As regards the winding of the secondary coil, we may say that this 
 wire is wound in sections, separated from one another by ebonite 
 partitions ; these sections being such that any two portions of the wire 
 which are at a great difference of potential at the make or break, are 
 separated from each other by one of these partitions. In fact, great 
 care is taken to prevent the coil being ruined by internal discharge. 
 
 The primary may be made and broken in many ways ; the ' ham- 
 mer' arrangement is only one of many forms. Thus the 'make- 
 and-break ' is sometimes effected between a platinum point and an 
 alloy of mercury and platinum, under the surface of alcohol ; altera- 
 tion of the surfaces due to the extra-current spark being thus obviated. 
 The speed of the make-and-break may be regulated by hand, by clock- 
 work, or in other ways. Thus the primary currents are more under 
 control, and the inverje and direct induced currents may be observed 
 separately. 
 
 8. The Part Played by the Condenser. Connected with the 
 metal piece L q are a series of sheets of tin-foil ; and connected 
 with NJ are another series. These two sets lie as shown in 
 the fig. ii. of the last section, being separated by sheets of waxed 
 paper or of other insulating material. The whole thus forms a 
 condenser of very large surface, the set connected with L forming 
 one plate, and that connected with N forming the other. Now 
 when the primary current is broken by the rise of b, an extra 
 current is self-induced in the primary. Were theie no condenser, 
 this would leap across the space between b and M in disruptive 
 discharge ; thus both prolonging the time /, of break, and 
 injuring the surfaces of contact. But with the condenser the 
 case is different. Before the extra current can give a spark across 
 the gap between b and M, it must raise the two large condenser 
 plates connected with b and M, to the necessary difference of 
 potential. These plates have a very large capacity, however. 
 Hence the extra current is employed in charging the condenser, 
 and does not give a spark across M b at all. 
 
 This is the main use of the condenser ; it prevents ' spark- 
 ing,' and thus permits of .an abrupt break to the primary. 
 
 Another use is as follows. While b is still in mid air between 
 M and A, the extra current not only charges the condenser, but 
 again rebounds (as it were) ; and, traversing the primary in the 
 
CH. xxii. ARAGO'S DISC AND RUHMKORFF'S COIL 373 
 
 contrary direction, helps to reduce A more abruptly to the neutral 
 state. 
 
 A further effect of the condenser is to lower the E.M.F. of 
 the inverse current that is induced in B on ' making ' the primary. 
 For the current has to charge the condenser ; and hence its rise 
 to a maximum is delayed. We have said that it is on the direct 
 induced current that we must depend for a high E.M.F. ; and we 
 shall show in n how it is an advantage to reduce as far as 
 possible the E.M.F. of the neglected inverse current. The con- 
 denser, therefore, will have* been of service in a third way. 
 
 9. Condition of the Secondary Circuit when Closed. 
 Referring to the formula of 5, we see that in the coil just 
 described the number N is the same for both induced currents, 
 but the time / 2 is greater than /*,. 
 
 From this it follows that the inverse induced E.M.F. e 2 is less 
 than the direct E.M.F. <?,, but exists for a longer time. Therefore 
 the inverse current is weaker than the direct, but lasts longer. 
 
 Now suppose the secondary circuit to be closed ; no air-space 
 left between the terminals p and p' of 7. We should predict 
 that for each make-and-break the total quantities of electricity 
 flowing in the two directions respectively will be equal. Various 
 experiments may be tried with the closed secondary circuit which 
 will illustrate the conditions of things that then obtains. 
 
 Experiments. (i. ) When we include in the closed secondary circuit a cell 
 consisting of a copper solution and two copper plates sufficiently large to 
 obviate polarisation and to give good conduction, there is zero resultant action. 
 Here the action depends upon quantity, not E.M.F. ; and the result indicates 
 that the quantities in the two currents are equal, and flow in opposite 
 directions. 
 
 If the plates are small, so that there is high resistance and polarisation, 
 there will be some slight action showing a predominance of the direct current 
 which has a higher E.M.F. 
 
 (ii.) If we use a cell of acidulated water and platinum electrodes, we have 
 results which vary with the size of these electrodes and with the conductivity 
 of the dilute acid. In general we have liberated at each electrode a mixture 
 of the two gases ; these will be in proportions which are nearly equivalent at 
 each pole, if the electrodes are large and the liquid very conducting, but will 
 show a predominance of the direct current if the converse conditions hold. 
 This is again due to the higher E.M.F. of the direct current. 
 
 (iii.) If we include in the circuit a galvanometer of sufficiently long coil, 
 we have a feeble action in favour of the direct current ; but this action is very 
 
374 ELECTRICITY . CH. xxn. 
 
 faint compared with that which obtains when (as will be explained in 10) we 
 cut off the inverse current altogether. 
 
 (iv. ) A person included in the circuit experiences shocks from both currents, 
 but mainly from the direct with its higher E.M.F. (see 7, caution). 
 
 10. Secondary Circuit with Air-Break, If now we separate 
 the terminals of the secondary circuit, gradually increasing the 
 distance between them, we find that less and less of the inverse 
 current breaks across the air-space ; until at last, by interposing a 
 sufficient interval, we have left only the direct current, which from 
 its high E.M.F. is able still to bridge over the gap with a disruptive 
 discharge. 
 
 Experiments. (i.) Interposing such an air-space in the circuit, we find 
 that electrolytic cells in the circuit exhibit decompositions that indicate a 
 powerful direct current. 
 
 (ii.) A galvanometer included in the broken circuit now shows a steady 
 deflexion due to the direct currents that rapidly succeed one another. 
 
 ii. Electrostatic Condition of the Open Secondary Termi- 
 nals. The Charging of Leyden Jars, Now let us suppose the 
 interval between the secondary terminals to be so great that not 
 even the direct induced current can strike across. 
 
 When the current in the primary is ' made,' the secondary 
 terminals pp' become of positive and negative potentials respec- 
 tively ; we may, e.g., suppose/ to become + and/' to become 
 But, as the E.M.F. of the inverse current is not great, so the. 
 electrostatic difference of potentials observed at/ and/' will not 
 be great. 
 
 When the primary current is 'broken,' the direct induced 
 current will cause/ to become , and/' to become + ; and the 
 electrostatic difference of potential between / and /' will now be 
 far greater than it was. 
 
 When, as with the vibrator or other high speed interruptor, the 
 direct and inverse currents follow one another rapidly, the termi- 
 nals / and /' will show a permanent potential difference, owing 
 to the difference in the E.M.F.s of the two currents. We shall 
 find/ to be , and/' to be + . This A V is of course not really 
 constant, but is continually rising and falling. When, however, 
 the currents follow one another with sufficient rapidity, the needle 
 of a quadrant electrometer will give a deflexion indicative of the 
 average A V between / and /'. 
 
CH. xxn. ARAGO'S DISC AND RUHMKORFF'S COIL 37$ 
 
 We saw in 8 that the condenser acted not only to increase the 
 E.M.F. of the direct, but also to decrease the E.M.F. of the 
 inverse, induced current Hence the condenser acts so as to in- 
 crease the permanent A V between the secondary terminals. 
 
 Coil used with Ley den jar. If the two coatings of a Leyden jar 
 are connected with m and n in the manner shown in the figure, 
 
 the discharge is modified in form. We do not now, as is the case 
 when no jar is used, have a discharge each time that the * break ' 
 is abrupt enough to give the E.M.F. required to overcome the air- 
 space m n. On the contrary, the jar is charged only in virtue of 
 the difference in E.M.F.s between the two currents; and the 
 discharge across m n takes place only when the jar is sufficiently 
 charged ; so that the period between two discharges depends upon 
 the capacity of the jar. The striking distance is very much 
 smaller than when no jar is used ; and, for some reason which 
 the present writer believes has not yet been made quite clear, it 
 depends further upon the capacity of the jar. 
 
 Charging a Leyden jar battery. The above given method may 
 be employed to charge a very large jar or battery. But where it 
 is desired to charge such a battery and to leave it charged, 
 another arrangement is adopted. Here the battery is charged by 
 means of the direct current alone, an air-space being interposed. 
 When the inner coating of the jars rises to a certain potential, 
 sparks will cease to strike across the air-space. The battery may 
 then be removed and its charge utilised. 
 
 The use of a * secondary condenser,' as it is called, causes the 
 
376 ELECTRICITY CH. xxn. 
 
 spark to be much denser ; the noise also becomes very much 
 greater. 
 
 12. Various Phenomena of the Secondary Discharge. 
 
 (I.) Luminous effects. When the discharge takes place under 
 ordinary atmospheric pressures, and without the use of a ' secon- 
 dary condenser ' (i.e. a Leyden jar connected with the secondary 
 terminals), it generally takes the form of a zigzag bluish-white 
 spark, a peculiar sharp cracking sound accompanying it Under 
 certain conditions this may be seen to be surrounded by a luminous 
 haze. Indeed, there appear to be two modes of discharge that 
 occur at the same time ; viz. the true spark, and a quieter form 
 resembling the 'brush discharge ' spoken of in Chapter VI. 15. 
 When the space is too great to admit of a spark passing, there 
 may be observed a phenomenon resembling the brush, or silent, 
 discharge that occurs under similar conditions with electric 
 machines. 
 
 The luminous effects occurring in high, and in low, vacua will 
 be discussed in 14, &c. 
 
 (II.) Chemical effects. When, by means of an air-break, the 
 inverse current is cut off and the direct only passes, we may 
 obtain all the electrolytical phenomena described in Chapter XII. 
 But there is another effect produced by the spark of the induction 
 coil, as also by the continued spark of the Holtz or other similar 
 machine. If the spark be passed for a considerable time through 
 mixtures of gases, there will in certain cases occur combinations 
 that rannot be obtained directly by any other means. Thus we 
 may in this way cause nitrogen to combine directly with oxygen. 
 
 (III.) Heating effects. The temperature of the spark, especially 
 when a secondary condenser is used, is very high. It is, however, 
 rather difficult to distinguish clearly between effects due to the 
 high temperature of the spark, and those due to the violent 
 mechanical disturbance that accompanies the disruptive electric 
 discharge. If, for example, we find the material of the terminal 
 carried off and vaporised, we cannot say that this is wholly due 
 to heat ; it may be partly, at least, what we may more fairly call a 
 mechanical action. 
 
 (IV.) Use with the spectroscope. It is found that the discharge 
 has the effect of carrying off and vaporising part of the material 
 of the terminals ; this being especially the case when the secon- 
 
CH. xxii. ARAGO'S DISC AND RUHMKORFF'S COIL 37/ 
 
 dary condenser is used. When the spark is observed with the aid 
 of the spectroscope, there will be observed the spectra peculiar to 
 the materials composing the terminals. 
 
 13. High and Low Vacua. 
 
 In the present Course there is no space to give an account of the 
 kinetic theory of gases. The student is therefore recommended to read 
 elsewhere enough on the subject to understand the following points. 
 
 (i.) The general nature of a gas. 
 
 (ii.) How, though the motion of a single molecule cannot be 
 followed, certain general statements may be made as to the average 
 condition of the molecules. 
 
 (iii.) What is meant by the mean velocity and the mean free path 
 of the molecules. 
 
 When we say that there is in a vessel an ordinary, or lout, vacuum, 
 we imply that the mean free path of the molecules is still of incon- 
 siderable length as compared with the dimensions of the vessel, 
 though, of course, much longer than under ordinary atmospheric 
 pressure. Thus, in low vacua, no group of molecules can move across 
 the tube, or from end to end of it, as a whole ; if such a group were to 
 start together in one direction, it would be broken up, and the mole- 
 cules composing it would be scattered and dispersed before any con- 
 siderable distance had been traversed. In fact, with respect to the 
 dimensions of the vessel, we may say that the rare gas that fills it 
 when there is a low vacuum presents the same perfect confusion of 
 movement on the part of the molecules that is presented by gas under 
 ordinary pressures, though, of course, there are much fewer molecules 
 per unit volume. 
 
 When we say that there is in a tube a high vacuum, sometimes 
 called a Crookes 1 vacmim, we imply that the mean free path is now of 
 a length equal to, or exceeding, the dimensions of the vessel. When 
 this is the case, a group of molecules started from one side or end of 
 the vessel will as a whole reach and impinge against the other side. 
 
 14. Discharge in Low Vacua. 
 
 (a) Vacuum tubes. For the purpose of exhibiting the phenomena 
 of discharge in low or high vacua, tubes or other vessels are prepared 
 in which the air or other gas has been reduced to the required degree 
 of rarity, a true vacuum being necessarily unattainable. These tubes 
 are provided with platinum terminals that pass through and are fused 
 into the glass. Usually there are two such terminals, one connected 
 with the +, and the other with the , pole of the source of 
 current. 
 
 ,3 Almost all that is stated in this and in the next section is true 
 
3/8 ELECTRICITY CH. xxn. 
 
 whatever be the source of the current, whether an induction coil, a 
 dynamo, a Holtz machine, or a voltaic battery. The E.M.F. of the 
 source must, however, be very great. "But in what follows we shall, 
 unless the contrary be stated, assume that the source is an induction 
 coil. 
 
 (y) Consider now that we begin to exhaust the air from one of 
 these tubes, the platinum terminals being connected with the 
 secondary terminals of the coil. When the pressure has fallen to 
 half an atmosphere or so, the discharge will have greatly altered its 
 character. Instead of the comparatively short loud-sounding spark, 
 we shall have a silent apparently continuous illumination of the whole 
 tube, and the distance across which the discharge will take place will 
 have become much extended. 
 
 At a pressure of I mm. or so the brilliant soft illumination is very 
 remarkable. 
 
 The difference between this and the ordinary spark reminds one 
 of, and is doubtless analogous to, the difference between the aurora 
 borealis and the lightning spark. 
 
 This glow, when examined, gives us the spectrum peculiar to the 
 rare gas that fills the tube. 
 
 (fi) Fluorescence phenomena. This discharge appears to be pecu- 
 liarly rich in the rays (mainly the invisible short ultra-violet waves) 
 that excite fluorescence. When the glass of the tube contains gtiinia 
 sulphate, uranium oxide, or other suitable body, the characteristic 
 fluorescent glow will be observed, in addition to the light due to the 
 gas in the tube. 
 
 () The discharge is disruptive. When two points are separated 
 by an air-space and sparks due to any source of current are striking 
 across, there is a constant average A V between the points. This is 
 the case in disruptive discharges. But when the points are separated 
 by a conductor, whether good or bad, the A V between the points varies 
 according to the E.M.F. of the source, and the distribution of resist- 
 ances in the circuit, as explained in Chapter XIII. n. In the 
 present case the discharge is thus shown to be disruptive. 
 
 () StricB or strata. Under certain conditions the discharge is 
 not continuous in appearance, but shows striae or strata of greater 
 luminosity separated by darker intervals. The figure here given 
 indicates the appearance presented by these. 
 
 The existence of these, their form, and their movements, depend 
 upon the rapidity of the make-m&-break of the primary, the steadiness 
 of the primary battery, and upon other conditions as yet not fully 
 understood. 
 
 All striae seem to have their origin at the + terminal. 
 
CH. XXII. 
 
 ARAGO'S DISC AND RUHMKORFF'S COIL 
 
 379 
 
 (?;) Behaviour in a magnetic field. When a long luminous 
 discharge in a wide vacuum vessel is exposed to a magnetic field, 
 it behaves as would a flexible and extensible conductor carrying a 
 
 FIG. i. 
 
 current. Thus, in fig. ii. we see how a horseshoe magnet acts upon 
 such a discharge ; the discharge is inflected. 
 
 By means of a more complex piece of apparatus the discharge can 
 be caused to rotate about a magnetic pole. 
 
 FIG. ii. 
 
 (&] Action of parallel discharges upon one another. 
 
 Dr. Crookes has recently verified experimentally that two parallel 
 discharges in a low vacuum obey Ampere's laws of parallel currents 
 given in Chapter XVIII. 16. 
 
 15. Discharge in High Vacua, 
 
 The remarks made in (a) and (?) of 14 apply also to the present 
 section. We now proceed to mention briefly the phenomena peculiar 
 to high vacua. To Dr. Crookes is due our knowledge of these inter- 
 esting and beautiful phenomena. 
 
 (a) Let us suppose that the tube in the figure has a negative pole 
 at the centre, and a positive pole at each end. 
 
3 80 ELECTRICITY CH. xxn. 
 
 When the exhaustion is what is usually called 'good 5 for a low 
 vacuum, say, for example, that the pressure is only half a millimetre 
 of mercury or about j^ of an atmosphere, there is observed on each 
 side of the negative electrode a dark space, this space being bounded 
 by a brilliant luminosity. Many experiments tend to show that the 
 extent of this dark space measures the mean free path of those mole- 
 
 FIG. i. 
 
 cules which are charged by contact with the pole and are then 
 repelled. It seems that such molecules are projected from the 
 pole normally to its surface ; they beat back the other molecules, and 
 at the surface over which this conflict is most energetic there is the 
 brilliant light as evidence of the energy. There is no dark space at 
 the positive pole. 
 
 (3) As the exhaustion proceeds, the dark space (or mean free path 
 of the molecules) increases, and at last the dark space extends right 
 across the vessel, and the molecules projected normally from the - 
 pole bombard the glass on the opposite side. It appears that all 
 ordinary glass thus bombarded exhibits a peculiar greenish phosphor- 
 escence. Many precious stones and minerals are rendered beautifully 
 luminous by this molecular bombardment. This projection takes 
 place from the pole only, and it proceeds straight from it, no matter 
 where the 4- pole may be. In the figure we see to the left the appear- 
 ance in a low vacuum, the discharge taking place between the poles. 
 To the right we see the phenomenon of a high vacuum ; the direction 
 of discharge^ from the pole being independent of the position of 
 the + pole. 
 
 (y) Deflexion by a magnet. It appears that the molecular stream 
 is deflected by a magnet, but not in the same way as was the 
 
CH. xxn. ARAGO'S DISC AND RUHMKORFF'S COIL 
 
 381 
 
 discharge in a low vacuum. In the first place the stream is per- 
 manently ^fleeted, not merely /^fleeted as was the low vacuum 
 
 FIG. ii. 
 
 discharge of 14 (?), fig. ii. This is shown here in fig. iii. In the 
 second place the deflexion is independent of the direction in which 
 
 FIG. iii. 
 
 the molecules are moving. To use Dr. Crookes' simile, the deflexion 
 may be compared to the action of a high wind on a stream of 
 
 UNIVERSITY OF CALIFOM 
 
382 ELECTRICITY CH. xxn. 
 
 .mitrailleuse bullets, being constant for both directions of movement of 
 the stream. This is not the case with low vacuum discharges. 
 
 (8) Rotation about a pole. So also we can obtain rotations about 
 the pole of a magnet ; but, unlike the case of low vacuum discharges, 
 this is independent of the direction of movement of the molecules. 
 
 (() Action of molecular streams upon one another. Dr. Crookes 
 found that two streams running in the same direction side by side 
 repel one another ; they appear to behave as mutually repelling simi- 
 larly charged particles, not as parallel currents. 
 
 Note. It has been supposed that two similarly charged particles 
 moving side by side have two actions on one another, the one being 
 an electrostatic repulsion, the other being an electro-magnetic attrac- 
 tion. When they have the ' ratio velocity ' (see Chapter XVI 1 1. 5), or 
 about the velocity of light, it is supposed that these two actions 
 balance one another ; when a smaller velocity, the electrostatic repul- 
 sion predominates ; when a greater velocity, the electro-magnetic 
 attraction predominates. But from the extreme difficulty there is in 
 conducting such experiments with any certain data as to velocities, 
 &c., there is still much that is uncertain and demanding further 
 investigation. 
 
CH. XXIII. 
 
 383 
 
 CHAPTER XXIII. 
 
 DYNAMO-ELECTRIC MACHINES. 
 
 In the brief sketch of ' Dynamos' here given, the author has been 
 guided mainly by the writings of Prof. S. P. Thompson. To these 
 writings the student is referred for fuller information on this subject, 
 
 To the same authority the author is indebted also for the principle 
 of the diagram given in Chapter XXIV. 7 ; and to the courtesy of 
 the publishers, Messrs. Spon & Co., for the use of the diagrams 
 given in 1 6, 1 8, 19, 21 of the present Chapter. 
 
 i. General Nature of a ' Dynamo.' We have seen in 
 Chapter XXI. that when a conductor cuts lines of force in a 
 magnetic field, there is an E.M.F. induced in this conductor; 
 and that when a coil is moved in a magnetic field in such a way 
 as to alter the number of lines of force piercing it, there is a 
 resultant E.M.F. induced in this coil. At the end of 4 in 
 Chapter XXL we indicated the manner in which this induced 
 E.M.F. may be calculated. 
 
 Those machines on which we expend mechanical work in 
 causing the necessary movements, and from which obtain elec- 
 trical energy, are called dynamo-electric machines, or more briefly 
 dynamos. 
 
 2. General Account of Induced Currents. Let the symbols 
 e, Nj, N 2 , /, and / have the same meaning as in Chapter XXI. 
 4, end. Then we have in general ........ . . 
 
 (N-N xm 
 
 If the resistance of each turn of wire be R, then the resistance 
 of the whole coil is m R. Hence, if the external resistance be r 
 we have by Ohm's law an induced current C measured by . * ". 
 
384 ELECTRICITY CH. xxm. 
 
 (N t - N 2 ) X m 
 
 _ 
 
 01 R + r m R + r 
 
 We have then a case similar to that of a series of voltaic cells 
 arranged in series. By the principles of Chapter XIII. 6 and 13, 
 we can determine whether it is more advantageous to arrange the 
 turns of wire in ' series, 7 or to arrange them * parallel' so as to be 
 equivalent to one turn of thicker wire, or to arrange them partly 
 in 'series' and partly 'parallel.' It all depends upon the relative 
 magnitude of R and r. 
 
 As an example we may consider the simple case of Chapter 
 XXII. 4. Let the area of the coil, measured in sq. cms., be S ; 
 let the number of turns of wire be m ; let the strength of the 
 earth's field be I ; and let the coil be completely rotated a times 
 per second. When the coil has its plane perpendicular to the lines 
 of force, there are m I S marked lines of force piercing the circuit ; 
 this is because there are m turns of wire, and because the 'number 
 of lines of force ' piercing each sq. cm. measure the field-strength, 
 as is explained in Chapter X. 13 and 14, and in Chapter XVII. 
 i. When the coil is edgeways to the lines, no lines pierce it. 
 When it has turned through 180, so as to be again perpendicular 
 to the lines of force, then m I S lines pierce it in the opposite 
 direction. Hence there is a total change of 2 m I S lines. When 
 it is turned through the remaining 180, the same change occurs. 
 For one complete revolution there is therefore a change of 4 ;;/ 1 S 
 lines. And in one second there will be a change of 4 a m I S 
 lines. Hence, if by means of a commutator the contrary E.M.F.s 
 induced in the two halves of a revolution be caused to give a 
 current, in a constant direction (see Chapter XXII. 4, and 4 of 
 the present Chapter), we have ........... '\\ 
 
 e = 4 a m I S 
 
 j r^ 4 a mIS 
 and C = -U- . 
 m R + r 
 
 The reader is recommended to return to the general remarks 
 of this and the last section when he has read the accounts of 
 several dynamos described in what follows. 
 
 We may add that though large E.M.F.s may be obtained by 
 means of dynamos, we cannot approach in magnitude the E.M.F.s 
 obtainable by means of induction coils. 
 
CH. XXIII. 
 
 DYNAMO-ELECTRIC MACHINES 
 
 385 
 
 3. Clark's Machine, The simplest form of dynamo is that 
 constructed on the pattern of the 'Clark's machine.' This has a 
 field given by permanent magnets, and is therefore pften called 
 a magneto machine. 
 
 In Clark's machine there is a permanent magnet A that gives 
 the field ; the strength of this field can be regulated by connect- 
 ing the two poles of the magnet com- 
 pletely, partially, or not at all, by a soft 
 iron keeper. In 'medical coils' this 
 regulator is an essential addition. In 
 fr nt f ^6 ma g net poles there 
 rotate two coils B and B' wound with 
 one continuous wire in such a way 
 that the E.M.F.s induced in the two 
 coils act in one direction along 
 the wire. These coils are pro- 
 vided with soft iron cores in 
 order to increase by magnetic 
 
 induction the number of marked lines of force piercing the coils. 
 
 Let B and B' start from the position shown. Then, as we 
 turn them through 180, we first withdraw the lines that pierce 
 them, and then introduce lines in the opposite direction. Hence 
 the E.M.F. in each coil acts in one direction through this 180 of 
 rotation ; and the coils are so wound that the E.M.F.s of B arid 
 B' act in one direction. Through the second 180 of rotation the 
 E.M.F.s are reversed ; but a commutator, described in 4, main- 
 tains the current in a constant direction in the external circuit. 
 
 The direction of current can easily be predicted from Lenz's 
 law ; for, as B leaves the N pole of the magnet and approaches 
 
 c c 
 
386 ELECTRICITY en. xxm. 
 
 the S pole, it will acquire on the side turned towards the magnet 
 a S polarity, or the current will there run ' clock-wise,' since this 
 will oppose the motion. 
 
 By giving a rapid motion and having many turns of wire we 
 increase the a and the m of 2, or increase the induced E.M.F. 
 We can thus get an E.M.F. large enough to give shocks and 
 small sparks. 
 
 4. The Simple Commutator. The figure here given repre- 
 sents a simple form of commutator. The one end of the wire is 
 soldered to the metal piece o, and the other to o ; these pieces 
 
 pass each nearly half-way round the axis J, but are insulated from 
 one another by two slits, one of which is seen in the figure. 
 The axis itself is of ivory, ebonite, or some other insulating 
 material. The circuit is completed by means of the metal springs 
 b and c. Just as the coils are passing the position in which their 
 E.M.F.s change direction, the metal springs b and c cross the two 
 slits and come into contact each with that metal piece (o or o') 
 that was just before in contact with the other spring. Thus the 
 external current is maintained constant in direction. 
 
 The other metal piece, shown in the figure, is for another 
 purpose that we need not describe. 
 
CH. xxiir. 
 
 DYNAMO-ELECTRIC MACHINES 387 
 
 5. Siemens's Armature. We may here mention that the 
 system of core and coil, in which the E.M.F. is induced, is called 
 an armature. In order to get the coil into a powerful field close 
 to the magnets, Siemens invented a very compact form of armature. 
 
 In this form the core is a long cylinder of soft iron in which 
 are two deep grooves cut longitudinally on opposite sides. The 
 wire is then wound longitudinally in the grooves. The general 
 appearance of this armature is shown in the figure. 
 
 6. The Self-Exciting Principle. It soon occurred to makers 
 that electro-magnets would give far more powerful fields than 
 could be obtained from permanent steel magnets. (We may here 
 mention that the magnets giving the field in which the armature 
 moves are called field-magnets.) 
 
 These electro-magnets could be excited by the whole or part 
 of the current given by the armature of the machine itself. 
 
 It is found that the (so-called) soft iron cores of the electro- 
 magnets always retain enough magnetism to give a current in the 
 armature when rotated. This current, or part of it, passing round the 
 field- magnet coils, increases the*magnetism of the cores, and there- 
 fore the strength of the field. In consequence of this the current 
 induced in the armatures becomes stronger. The reciprocal 
 action proceeds until the field-magnets acquire some maximum 
 strength depending on various conditions not specified here. 
 
 Mr. Ladd was the first to construct a ' self-exciting ' machine of 
 this sort. In his machine there were two armatures ; one served 
 to excite the field-magnets, and the other to give the external 
 current. 
 
 7. Continuous Current' Machines. In the machines de- 
 scribed above we do not obtain a continuous current. In the coils 
 there is in one half-revolution induced an E.M.F. that begins, 
 rises to a maximum, and then ceases ; in the other half-revolution 
 an E.M.F of similar rise and fall, but contrary to the former in 
 direction. This gives in the coils two currents in opposite direc- 
 tions, separated by an instant of zero current ; and, if there be a 
 
 c c 2 
 
388 ELECTRICITY CH. xxm. 
 
 commutator, this gives in the external circuit two currents for 
 each revolution, these currents being in the same direction, each 
 rising to a maximum and falling again, separated from each other 
 by an instant of zero current. 
 
 For many purposes it is desirable to have a current more con- 
 tinuous in nature. This can be effected in more than one way ; 
 we will describe very briefly one of these ways, and more at length 
 another. 
 
 (a) The first method is to have an armature in which there 
 are more than one pair of opposite coils, these pairs being arranged 
 round the circumference of a circle. Thus we may conceive of a 
 Clark's machine in which there are four, six, eight, or more pairs 
 of coils ; each pair is wound with one continuous wire whose 
 extremities terminate in two metallic segments fixed to opposite 
 sides of the axle. All these segments are insulated from one 
 another ; and therefore the circuit of each pair of coils is closed 
 only when they come into contact with the two metallic springs or 
 * brushes ' by means of which (as in the simple case of 4) the 
 circuit is completed through the external circuit. Each pair of 
 coils thus feeds the external circuit in turn ; and, by so arranging 
 the springs that they come into contact with one pair of segments 
 before they have quite broken contact with the preceding pair, 
 we can insure that the current will never fall to zero. It is, how- 
 ever, of an undulatory character. 
 
 (/3) The second method cannot be understood until some 
 description has been given of quite a different form of armature. 
 As this method is of great interest, we shall describe at some 
 length one form of machine that is constructed on this principle ; 
 the machine, viz., that is called the Gramme, after the name of 
 its inventor. His armature, however, is not quite original in con- 
 struction, being similar to an earlier form, invented in 1862 by 
 Prof. Pacinotti. 
 
 8. The ' Gramme.' Construction of Armature. In fig. i. 
 we give a general view of a hand-model of the Gramme machine. 
 Here we can see the ring-shaped armature turning between the 
 pole-pieces a and b of the steel magnet A ; the arrangements for 
 giving a rapid rotation to this armature ; and the collecting springs 
 ot brushes *r and 2, in contact with the axle on opposite sides of it, 
 by means of which a current is sent through the external circuit. 
 
CH. XXIII. 
 
 DYNAMO-ELECTRIC MACHINES 
 
 339 
 
 Fig. ii. shows us the construction of this armature. There is 
 a circular core composed of soft iron wires ; these are shown in 
 
 section. From the way in which 
 the armature rotates these wires 
 do not cut the lines of force ; 
 and thus induced 'eddy-currents,' 
 a cause of waste of energy, do 
 not occur as would be the case 
 were the core solid. 
 
 Round this core is wound 
 the insulated copper wire that 
 forms the coil in which the 
 E.M.F.s are induced. This wire 
 it must be carefully noted is 
 continuous-, and herein we have 
 a great difference between this kind of armature and that of 
 Clark or Siemens. In the upper part of fig. ii. we see this wire 
 
 FIG. ii. 
 
390 ELECTRICITY CH. xxm. 
 
 as it is actually wound ; and in the lower part of the figure there 
 are shown a few sections of the coil, to indicate the mode of 
 winding. 
 
 This wire is, as we said, continuous. But at regular and close 
 intervals the wire is brought out, is laid bare of insulating material, 
 is soldered to an insulated copper segment m n (a few only of 
 these copper segments are given in the figure), and is then led 
 back to continue the winding. From what we have said it is clear 
 that we have a continuous wire with which we can, by means of 
 the insulated segments m n, &c., make metallic contact at regular 
 and frequent intervals all round the ring. These segments are 
 arranged all round the axle, and it is with them that the brushes 
 of the external circuit make contact. In fig. ii. O represents the 
 solid body of the axle ; it is made of hard wood or other insulating 
 material, there being an inner core of metal, insulated from the 
 segments m n, to give necessary strength. 
 
 9. The Gramme. The E.M.F.s Induced in the Ring 1 . Let 
 us now consider the condition of this ring-armature, with its con- 
 tinuous wire, when it is rotated between the poles a b of the 
 magnet ; and let us at present suppose that there are no springs 
 or brushes in contact with the segments. Were it not for the core, 
 the magnetic field would consist of lines of force running nearly 
 straight across from one pole-piece to the other. These lines 
 would pierce the coils of the armature, the number piercing each 
 coil depending upon the position of that coil. Thus, a coil which 
 is at the top of the ring depicted in 8, fig. i., or which is situated 
 equatorially, embraces the maximum of lines, supposing the field 
 to be uniform. As the ring turns, this coil embraces fewer and 
 fewer, until, when it lies edgeways to the lines or is situated 'axially,' 
 it embraces none. (If the field be not uniform this statement 
 must be modified ; but, in any case, the coil embraces zero lines 
 when situated axially.) As the ring turns still further, our coil 
 begins to embrace lines in the other direction ; and these reach a 
 maximum when it is at the bottom of the ring, i.e. when it is again 
 situated equatorially, 180 from its initial position. All this will 
 give an induced E.M.F. in one direction, as has been already ex- 
 plained in Chapter XXII. 4, and elsewhere. As the coil turns 
 through the other 180 back to its initial position, there will, it is 
 clear, be induced an E.M.F. in the opposite direction. 
 
CH. XXIII. 
 
 DYNAMO-ELECTRIC MACHINES 
 
 391 
 
 Thus, as the ring rotates, all the coils that are to the one side 
 of the equatorial diameter give an E.M.F. in the one direction, 
 and all those lying to the other side give an equal E.M.F. in the 
 opposite direction. The result will be that no current will flow, 
 but that the different parts of the wire, and so also the copper 
 segments connected with them, respectively, will be maintained at 
 different potentials. Thus, in the figure here given we shall have 
 
 FIG. i. 
 
 FIG. ii. 
 
 the wire at A and B, i.e. at the extremities of the equatorial dia- 
 meter, maintained at the greatest AV, the potential falling from 
 (say) A symmetrically along the wire, through each route, down to B. 
 
 Thus, though the individual coils assume all positions in turn, 
 the ring, as a whole, maintains a character that is constant as long 
 as there is constant field-strength and velocity of rotation. The 
 revolving ring has been not inaptly compared to a stationary 
 system of two equal batteries (see fig. ii.) set against one another ; 
 these batteries giving no current in their circuit, but maintaining 
 the poles A and B at a constant AV. We can evidently obtain a 
 current in an external circuit by connecting A and B. 
 
 The core. We have hardly mentioned the core in what we 
 have said. The fact is that the core only modifies the field ; its 
 presence does not affect the general theory of the ring as given 
 above. Its action is mainly to concentrate the field upon the 
 coils. The lines of force cannot now, to any considerable extent, 
 
392 ELECTRICITY CH. xxm. 
 
 get across the space inside the ring; they follow the core through 
 the coils. The presence of the core does not affect the two main 
 facts: (i) that when the coils lie axially they cut no lines- of force; 
 and (2) that the change in direction of the E.M.F. occurs as the 
 coils pass the equatorial position. 
 
 It is found that the soft iron wire changes its magnetism in 
 a time that is practically inappreciable ; and hence we can, if we 
 like, regard the core as stationary, and the coils as slipping round 
 upon it 
 
 (For a modification of the sense in which axially and equatori- 
 ally must be used when a current is running, see 12.) 
 
 10. The Gramme. The Collecting Brushes As in the 
 analogous case of the two opposed batteries, represented in 9, 
 fig. ii., so, in the case of the gramme-ring represented in fig. i., we 
 can obtain a current if we connect the parts A and B. 
 
 We have explained in 8 how the wire that forms the coil is, 
 at a series of points all round the ring, connected with insulated 
 copper segments that are fixed in the axis on which the ring turns. 
 If, then, there be metallic springs or brushes formed of wire, press- 
 ing against the axis at the extremities of the equatorial diameter 
 of the same, these will practically be in contact with each segment 
 of the wire coil in turn as it arrives at the positions A or B. In 
 the Gramme machine these collectors are brushes of metallic wire. 
 However great the .speed of revolution, these brushes will never 
 be jerked away from contact, as might a single spring ; and with 
 them it is easier to have contact always with two consecutive seg- 
 ments at once, and thus insure a current that is continuous, even 
 though undulatory. The wires of the external circuit are, of 
 course, attached to these collectors. The two halves of the ring 
 then combine, as would the two batteries acting 'parallel,' to send 
 a current through the external circuit. 
 
 n. Curve of Potential Round the Collecting- Axis, Let us 
 suppose the brushes to be detached, and the potential at different 
 positions round the axis to be examined. This may be done by 
 employing a quadrant electrometer, of which one pair of quadrants 
 are to earth, and the other pair connected with an insulated wire 
 brush ; this brush is applied to the axis at different points in suc- 
 cession. 
 
 We find a fall or rise in potential from segment to segment 
 
CH. XXITI. DYNAMO-ELECTRIC MACHINES 393 
 
 as we move from A to B or from B to A ; this fall, in properly 
 constructed machines, occurring symmetrically down both halves 
 of the axis. 
 
 This fall in potential is, of course, discontinuous, since the 
 segments are limited in number. But when the segments are very 
 numerous the fall in potential round the col- 
 lecting axis can be represented approximately 
 by a continuous curve. 
 
 In the accompanying diagram, the shaded 
 circle represents a section of the axis ; the dark 
 dots on this circle represent sections of the in- 
 sulated copper segments ; and the larger dotted 
 curve represents the fall in potential from 
 the upper extremity A of the equatorial diameter, both ways, 
 round to the lower extremity B of the same. If we draw a 
 radius from the centre of the axis, through the dot representing 
 any particular segment, to meet the outer curve, then the inter- 
 cept between this dot and the outer curve represents in magni- 
 tude the relative potential of the segment in question. (This 
 will indicate to the reader how the outer ' curve of potential ' is 
 plotted out.) 
 
 In a good machine this fall of potential should be regular, 
 The brushes should be in contact with the axis at the points of 
 maximum and minimum potential respectively. We shall find, 
 however, that, as soon as a current runs, these positions of maxi- 
 mum and minimum potential shift round the axis. 
 
 12. The * Lead ' that Occurs when a Current is Running. 
 
 First, let us suppose that the ring is revolving, but that, the 
 external circuit being not yet completed, there is no current 
 flowing. The two halves of the ring act merely to maintain two 
 points, e.g. A and B, at a certain maximum AV. If the lines of 
 force (see 9, fig. i.) run straight across in what we have called an 
 axial direction, then A and B will lie equatorialfy as shown. If, 
 however, the core of the ring take an appreciable time to gain 
 and lose its magnetism, the lines of force, and so also the line 
 A B joining the points of maximum AV, will be shifted round 
 to a greater or less extent. This question can be tested directly 
 by experiment. A brush makes contact with the axis at different 
 points in succession round its circumference. An insulated wire 
 
394 ELECTRICITY CH. xxnr. 
 
 connects this brush with one pair of quadrants of an electrometer, 
 the other pair of quadrants being to earth. It is stated that when 
 the positions A and B of maximum and minimum potential are 
 thus tested, the line A B is found to be practically equatorial, or 
 lies at right angles to the line joining the poles of the field-magnets, 
 whatever be the speed of revolution of the ring. Hence we 
 conclude that the core of the ring does not take any appreciable 
 time to change its magnetism as it rotates. 
 
 This view is supported by S. P. Thompson and others ; but there 
 are some who are not satisfied of its truth. 
 
 Secondly, let us suppose that the brushes are attached and that 
 an external current is running. If the direction of the current be 
 followed, according to the principles explained in Chapter XXL, it 
 will be seen that (see 9, fig. i.) the currents in the two halves 
 of the ring both act to make the core of a N polarity at B, 
 and of a S polarity at A, Now the field-magnets tend to 
 make the core of a N polarity at N' opposite to S, and of a 
 S polarity at S' opposite to N. Hence, on the whole the core 
 will acquire a N polarity at some point between N' and B, and a 
 S polarity at some point between S' and A, opposite to the former. 
 The effect will be as if the polarity of the core were shifted on in 
 the direction of movement of the ring. Hence the line A B, of 
 the points of highest and lowest potential, will also be shifted on 
 through an equal angle. And therefore the brushes must also be 
 shifted on. 
 
 This shifting on of the brushes in the direction of rotation is 
 called 'Lead? 
 
 According to the above view, which is supported by experi- 
 ments on the part of S. P. Thompson and others, the amount of 
 lead depends upon the ratio between the field-strength due to the 
 field-magnets, and the inductive action upon the core of the current 
 in the ring. It does not depend upon velocity of rotation, provided 
 that the current is constant ; whereas, were 'lead* due to residual 
 magnetism in the core, the amount of this lead would depend 
 upon the velocity of rotation though the current were constant. 
 
 13. Armatures Wound 'for E.M.F.' and 'for Current' 
 As with batteries, so with armatures ; we can construct them either 
 to give a high E.M.F., regardless of the resistance thereby un- 
 avoidably introduced, or to give a low resistance, thereby sacri- 
 
CH. xxin. DYNAMO-ELECTRIC MACHINES 395 
 
 ficing E.M.F. All depends upon the nature of the external 
 circuit (see Chapter XIII. 6 and 13). 
 
 For high E.M.F. we must have many turns of wire ; and 
 therefore it must be long and fine, or of high resistance. 
 
 For low internal resistance we must have few turns, and thick 
 wire. Having few turns, we have low E.M.F. (see 2). 
 
 Sometimes there are on the same ring, or on parallel rings, 
 two coils, these being capable of being joined either in series or 
 in parallel circuit. 
 
 14. The Siemens- Alteneck Armature. As we have stated in 
 Chapter XXI. 8, the origin of the induced E.M.F. must doubtless 
 be sought for in the cutting of the lines of force by the wire of 
 the coil. Now since, as can be experimentally shown, there are 
 hardly any lines of force that find their way across the inside of 
 the ring, it follows that the wire lying on the inside of the ring is 
 not cutting any lines, and is therefore passive as regards the pro- 
 ductiveness of an E.M.F. It is simply a dead resistance. In 
 the Siemens- Alteneck machine a drum replaces the ring, and the 
 wire is wound over this, there being therefore no wire, saving at 
 the ends of the drum, that does not cut lines of force. The long 
 shape of the drum further renders smaller the proportion of the 
 idle wire at the ends. 
 
 15. The Brush Machine. On a very different principle is 
 constructed the ' Brush ' machine ; the name being after that of 
 the inventors. The theory of this machine is so complicated that 
 we cannot give it here clearly and yet briefly.- We therefore refer 
 the reader to S. P. Thompson's work on ' Dynamo-Electric Ma- 
 chinery' (Spon & Co., London). 
 
 This machine gives very high E.M.F.s, and is therefore used 
 where such are required. 
 
 1 6. Magneto Machines, We will now discuss very briefly 
 the four most important types of machines, beginning with the 
 ' magneto machine* as the oldest and simplest form. The accom- 
 panying figure represents diagrammatically the typical ' magneto.' 
 In this form, of which the machine shown in fig. i. of 8 was an 
 example, the field is given by a permanent steel magnet. Such 
 fields are very constant, and therefore the E.M.F. of the current 
 will vary simply with the velocity of rotation. For purposes of 
 demonstration and explanation of the theory of dynamos, such a 
 
396 
 
 ELECTRICITY 
 
 CH. XXIII. 
 
 property is very useful. Hence the magneto is much used for 
 teaching purposes. 'The same property, coupled with the easy 
 
 regulation of the field by means 
 of a keeper (see 3), makes the 
 magneto useful also for medical 
 purposes. But in the arts and 
 in commercial undertakings it 
 is never used, as being relatively 
 feeble for its mass as compared 
 with any of the electro-magnet 
 forms. 
 
 Magnetos cannot be made 
 with certainty to be of the same 
 E. M.F. We can therefore couple 
 them in series only, such a series 
 of magnetos acting like one ma- 
 chine of the sum of their E.M.F.s 
 and of the sum of their resist- 
 ances. We cannot couple them 
 parallel, or * for current,' for the 
 machine of lower E.M.F. might 
 serve only as a branch circuit 
 through which the other machine 
 was driving a current. It would 
 act as a ' motor' (see Chapter XXIV.) and not as a driver of current. 
 We may add that when a magneto is run at constant speed 
 its E.M.F. is constant. Hence, in calculations we may deal with 
 such as if they were voltaic batteries of constant E.M,F. 
 
 1 7. Separately-Excited Machines. When the field-magnets 
 are electro-magnets ^ excited by a current from an independent 
 source, we have what is called a separately -excited machine. This 
 form differs very little in principle from a magneto, but the field 
 is much stronger, and we can vary its strength within very wide 
 limits. Hence we have great control over the E.M.F. 
 
 Such machines can be used in series or in parallel circuit. 
 An obvious disadvantage of this form is that we require a 
 separate source of current. Neither the magneto nor the sepa- 
 rately-excited machine can be constructed so as to be self-regu- 
 lating, whereas we shall see that some other forms can be so 
 constructed, these forms being compound. 
 
CH. XXIII. 
 
 DYNAMO-ELECTRIC MACHINES 
 
 397 
 
 1 8. Series-Excited Machines. In some machines, as was 
 stated at the end of 6, the current from the armature is led first 
 round ihe field-magnets and 
 then round the external circuit. |i||iBl 
 A machine thus constructed 
 is called a series -dynamo. 
 
 Let us suppose that in 
 such a machine the external 
 circuit is completed, and that 
 the armature is then run at 
 a certain velocity. There will 
 be initially a weak field, due 
 to the residual magnetism in 
 the cores of the field-magnets, 
 and this will give a slight 
 initial current in the armature. 
 This current, passing through 
 the coils of the field-magnets, 
 will make the field stronger; 
 this will induce a stronger 
 current in the armature, and 
 so on until some final, limiting 
 value is attained. This limit- 
 ing value of the current de- 
 pends upon 
 
 (i.) The construction and resistance of the armature, and upon 
 its core. 
 
 (ii.) The construction and resistance of the field-magnet coils, 
 and upon their cores. 
 
 (iii.) The velocity of rotation of the armature. 
 
 (iv.) The resistance of the external circuit. 
 
 Let us suppose that the resistance of the external circuit be 
 increased. This will, of course, produce directly a decrease of 
 current, and this in turn will weaken the field and cause a further 
 decrease. Finally, the current will settle down to some new value 
 depending upon the new conditions. 
 
 In the case, therefore, of a series-dynamo, any change in the 
 external resistance produces a greater change in the current than 
 would be the case with a magneto or separately-excited machine. 
 It is very ' sensitive ' to variations in external resistance. 
 
39* 
 
 ELECTRICITY 
 
 CH. XXIII, 
 
 If the object be to drive a constant current, whatever the 
 external resistance, the series-dynamo must certainly not be used ; 
 for (since the E.M.F. depends upon the current) the E.M.F. falls 
 just when it should become greater in order to drive the current 
 against an increased resistance, and rises just when, owing to 
 decreased resistance, a smaller E.M.F. is desired. 
 
 The series -dynamo is particularly suited to such cases as that 
 of Chapter XXV. 18, where there is an external circuit of many 
 parallel branches, of which there may be at one time many and 
 at another time only one in use, and where it is desired to keep 
 the current through any one branch constant, whether the others 
 be in use or not. For such a purpose we must have a current 
 that is greater or less according as there are more or fewer of the 
 branches in use. Now we know, by Chapter XIII. 5, that the 
 
 more branches there are in 
 use the smaller is the total 
 resistance. And we have seen 
 that the series-dynamo, more 
 than any other form of ma- 
 chine, will respond to de- 
 creased resistance by increase 
 of current. In such an ar- 
 rangement, therefore, it will 
 respond to demands made 
 upon it. 
 
 19. Shunt-Dynamos. 
 In a very important class of 
 machines there are two alter- 
 native paths for the current 
 to pursue when it leaves the 
 collecting brushes. The one 
 path is through the coils of 
 the field-magnets, which are 
 made long and of relatively 
 high resistance ; the other path 
 is through the external cir- 
 cuit. If there be introduced 
 
 into the external circuit more resistance or an opposed E.M.F., 
 this will throw a larger proportion of the current, round the 
 
CH. xxm. DYNAMO-ELECTRIC MACHINES 399 
 
 field-magnet, and will increase the E.M.F. of the machine. 
 Hence, this method of construction is particularly suited to 
 cases where it is desired to keep a constant current under varia- 
 tions in the external circuit; for the E.M.F. of the machine rises 
 as opposition in the external circuit increases, and falls as it de- 
 creases. Such a machine is suited to the case of, e.g., electric 
 lamps arranged in series, where it is desired to maintain a constant 
 current, no matter whether one or more lamps be thrown in. It 
 is entirely unsuited to the demands of the case mentioned in 18, 
 viz. that of an external circuit composed of many parallel branches. 
 The converse is true of the series-dynamo. 
 
 20. Other Methods of Winding. In 18 and 19 we have 
 indicated for what purposes the series-, and shunt-, dynamos are 
 respectively adapted. It is found, however, that neither the one 
 nor the other give with sufficient exactness the result demanded. 
 It is found necessary to combine different methods of winding. 
 In some machines we have a combination of series-excitation with 
 separate-excitation ; in some, series with shunt; in others again, 
 shunt with separate ; and so on. The choice of method of winding 
 is determined by the nature of the work that the machine is desired 
 to perform, and upon the sort of ' constancy ' that is demanded of it. 
 
 21. Alternate Current Machines. For certain purposes, 
 chiefly for electric lighting (see Chapter XXV.), it is not necessary, 
 nor even advisable, to obtain currents that flow in a constant 
 direction. Machines that give alternating currents are called 
 alternate current machines. 
 
 If in the Clark or Siemens's armature (see 3 and 5) there be 
 on the axis two metallic rings, insulated from one another, with 
 which the collecting springs are continually in contact, and if the 
 two ends of the armature coil be soldered to these respectively, 
 then we shall have in the external circuit the same alternating 
 currents that occur in the armature. 
 
 In fact, it may be stated as a general principle that in most 
 machines, excepting always those in which the armature is con- 
 structed on the same principle as is the Gramme armature, the 
 induced currents are naturally alternate, and require the addition 
 of a commutator if it be desired to obtain in the external circuit 
 currents in one constant direction. As a rule, such commutation 
 introduces waste of energy from extra-currents and sparks produced. 
 
4OO 
 
 ELECTRICITY 
 
 CH. XXIII. 
 
 It may also be stated, as generally true, that it is easier to 
 construct alternate current machines which shall give a very high 
 E.M.F. than it is to construct machines which shall give a current 
 of high E.M.F. in a constant direction. 
 
 The accompanying figure illustrates the principle of more than 
 one form of alternate current machines. A series of coils provided 
 with soft iron cores is arranged on the circumference of a wheel, 
 
 their axes being parallel to the axis of rotation. Three of these 
 coils are diagrammatically represented in the figure. 
 
 These coils rotate between the opposed poles of a series ot 
 electro-magnets, arranged round the circumferences of two other 
 wheels, as indicated in the figure. The large dotted arrow repre- 
 sents the direction in which the circle of coils is rotating. The 
 small arrows show the direction in which the current flows in 
 each as it leaves the one pair of electro-magnets and approaches 
 
CH. xxin. DYNAMO-ELECTRIC MACHINES 401 
 
 the next pair. When the coils pass the next pairs of electro- 
 magnets respectively, the current in each is reversed. The method 
 of winding is such that the E.M.F.s of the coils act in one and 
 the same direction along the wire. The final terminals of the wire 
 are soldered to two insulated rings on the axis, against which the 
 two collectors are pressed respectively. (Sometimes the current 
 from a few coils is ' commutated,' and is used to excite the field- 
 magnets ; in such a case we have practically two machines in one.) 
 
 22. The Ferranti Alternate Current Machine. 
 
 In this machine the armature is composed of flat 'strip' copper 
 arranged in folds, so as to present an appearance somewhat similar to 
 that of a star-fish. 
 
 Here we have no iron cores ; but, owing to the extreme thinness 
 of this flat, star-shaped armature, the two sets of field-magnets can be 
 placed so close to one another that the field cut by the copper-strip of 
 the armature is exceedingly strong. 
 
 Neither have we, in a strict sense, any ' coils ; ' the copper-strip 
 being arranged in open folds. Hence we cannot here use the expres- 
 sion alteration in number of lines of force embraced by the coils. We 
 must fall back upon the principle that, as we have already stated, 
 underlies the above-quoted condition for induction ; this more funda- 
 mental principle being that there is an E.M.F. induced in any con- 
 ductor when it cuts lines of force. 
 
 In the present case things are so arranged that the E.M.F.s, in- 
 duced in the several folds that are cutting lines of force, act in one and 
 the same direction. These E.M.F.s alternate in direction as each fold 
 approaches a pair of field-magnets, and leaves the same, respectively. 
 
 D D 
 
402 ELECTRICITY CH. xxiv, 
 
 CHAPTER XXIV. 
 
 DYNAMOS AND MOTORS. 
 
 i. Magnetos, or Separately-Excited Machines, as Motors. 
 
 Let us consider what will happen if a current from an external 
 source be run through the armature of any machine in which the 
 magnetic field is constant, i.e. through the armature of a magneto 
 or of a separately-excited machine. 
 
 A reference to the figures of Chapter XXIII. 3, 5, 8, and 9, 
 shows us that action will take place between the field- magnets 
 on the one hand, and the armatures (mainly the iron cores of these) 
 on the other ; this reaction causing rotation of the armatures, 
 since they only are moveable. 
 
 Either by tracing carefully the direction of polarity given in 
 each case to the armature by the external current, or by the ap- 
 plication of the arguments given in Chapter XXI. 6, and 
 summed up in Lenz's law, we can see that the rotation will be of 
 such a nature that an E.M.F. is induced in the armature opposed 
 to the E.M.F. of the external current. If the armature move with 
 little friction, and be not constrained to do any work in turning, 
 it will move faster and faster as the action continues. The faster 
 it rotates the greater will be the induced E.M.F. opposed to the 
 exciting current, and the smaller will this current consequently 
 become. The limit of velocity possible is reached when the in- 
 duced E.M.F. equals that of the exciting current, so that this is 
 reduced to zero ; this limit can never be reached in practice 
 owing to friction. If the exciting current be reversed in direction, 
 so will the rotation. A dynamo thus caused to turn by the 
 passage of an external current through its armature is called an 
 electro-motor. Any dynamo can be employed as a motor, but as 
 a rule the best form for a motor is not the same as the best form 
 for a driver of current. 
 
 Two ' magnetos ' coupled. If we connect the terminals of two 
 
 ' 
 
CH. xxiv. DYNAMOS AND MOTORS 403 
 
 magnetos, such as the Grammes drawn in 8, and if one machine 
 be worked, the other will turn also in consequence of the current 
 driven through its armature. The two machines will act, at least 
 as regards direction of rotation, as would two wheels connected by 
 an endless strap passing round them. A galvanometer in the 
 circuit will indicate the changes in the current that ensue when 
 that machine which is for the time acting as motor is held fast, 
 allowed to turn slowly against friction, or allowed to run freely, 
 respectively. [Compare with 4.] 
 
 Magneto and secondary cell. If we connect a Gramme, or 
 other magneto, with a Faure's 'accumulator' or * storage cell' an 
 interesting experiment may be tried. Let us charge the cell by 
 means of the machine, and let us then leave off turning this latter. 
 The Faure's cell will now drive a current reverse in direction 
 to that by which it was charged. This will pass through the 
 Gramme, and will cause it to turn as a motor ; the direction of 
 rotation being, of course, such as to oppose the current from the 
 Faure's cell. Now such a direction of rotation must be the same 
 as that in which the Gramme turned when it was charging the 
 cell. Or we shall see the Gramme, acting now as a motor, con- 
 tinuing to turn in the same direction as that in which it was 
 turned when it acted as a driver of current. 
 
 2. Series-Dynamos as Motors. Inasmuch as the magnetic 
 field in the series-dynamo changes sign when the current through 
 the armature does so [for the field-magnets are here in one 
 circuit with the armature] it follows that a series-dynamo when 
 driven as a motor will turn in one direction only. We can readily 
 determine whether this direction will be with the brushes i.e. in 
 the direction in which the machine is intended to turn, that in 
 which the segments on the axis do not meet the points of the col- 
 lecting brushes ; or whether the direction will be against the 
 brushes i.e. such that the segments on the axis meet the points of 
 the brushes. Imagine the machine to be turning with the 
 brushes as a driver of current, and now imagine an external cur- 
 rent to be driven through the machine contrary in direction to 
 that which was driven by the machine. Were the field constant 
 in direction, the machine would turn in the same direction as 
 before, since such rotation would oppose the external exciting 
 current. But the field is reversed by this reverse exciting current. 
 
 D D 2 
 
404 ELECTRICITY CH. xxiv. 
 
 Hence the dynamo, acting now as a motor, will be driven in a 
 reverse direction against the brushes. If we reverse the exciting 
 current, we reverse the current in the armature and also the field ; 
 and hence the machine runs still in the same direction, i.e. against 
 the brushes. 
 
 Similar remarks apply to the shunt- dynamo when used as a 
 motor. 
 
 3. General Remarks on Dynamos and Motors. In what 
 follows we shall speak of that machine which is driving the current 
 as the dynamo, and that which is being driven by the current as 
 the electro-motor or motor. This latter is generally of a different 
 construction from the former, though not differing in principle. 
 We shall use the following symbols. E = the E.M.F. of the 
 dynamo ; e the reverse E.M.F. induced in the motor when this 
 is- turned ; C = the current in the circuit ; R the resistance of 
 this whole circuit, including the armatures of both dynamo and 
 motor ; C the current that runs when the motor is not albwed 
 to turn, or when e is zero. 
 
 By Ohm's law it follows that C = |-, while C = ^Llll. 
 
 -K _K 
 
 To give the above symbols a clear meaning, and to simplify 
 the reasoning that will, follow in 4 and 5, we shall, unless the 
 contrary be stated, assume 
 
 (i.) That the dynamo and motor are both either magnetos or 
 separately- excited machines, so that in both of them the E.M.F. 
 depends solely on the velocity of rotation of the armatures. 
 
 (ii.) That in the simple (i.e. not branched nor shunted) circuit 
 composed of the two armatures and the external connecting wires, 
 R is constant ; the speed of rotation not affecting appreciably the 
 resistance .occurring where the brushes make contact with the 
 segments on the axes. 
 
 (iii.) That the speed of rotation of the driving dynamo, and 
 therefore (since we have assumed its field to be constant) the 
 driving E.M.F. E, is constant. 
 
 When the driver is a series-dynamo, matters are not so simple. 
 As the reverse E.M.F. of the motor increases in magnitude the 
 current falls off, and, therefore, so will the field and E.M.F. of 
 the driver. Any accidental retardation in the dynamo may now 
 
CH. XXIT. DYNAMOS AND MOTORS 405 
 
 so lower E that it becomes less than e. The current will in such 
 a case be reversed, and so will the field and E.M.F. E of the 
 dynamo. Thus, when a series-dynamo is used as driver, not 
 only are calculations less simple on account of the variability of 
 E, but also we may have to encounter sudden reversals of action. 
 
 In practice such reversals can be obviated by means of pieces 
 of apparatus that interfere automatically to set matters right 
 whenever the current begins to be reversed. 
 
 With a shunt-dynamo matters are still more complex. For 
 here we have to consider not only the changes in E, but also the 
 complex nature of R due to the branched circuit that is open to 
 the current. 
 
 Hence, as stated above, we shall keep to the simple case where 
 dynamos and motors both have constant fields. 
 
 Referring to Chapter XV. 9, mutatis mutandis, it is plain 
 that the following statements are true. 
 
 E C = work per second, or activity, expended on the 
 
 dynamo. 
 e C = work per second, or activity, expended against the 
 
 reverse E.M.F. of the motor. 
 (E e) C = C 2 R = work per second, or activity, expended in 
 
 developing heat in the circuit. 
 
 4. Formulae for Activity, &c. Maximum Activity. Let us 
 suppose that a dynamo of a constant E.M.F. E is driving a motor, 
 and let the symbols have the same meaning as in 3. Instead of 
 work per second we shall use the proper expression, activity. If E 
 and e be given in volts, and R in ohms, then the activity is given 
 in watts. We refer the reader to Chapter XV. 2 and 3 for the 
 relations between watts, ergs per second, and English horse-power. 
 
 The steam-engine expends a certain activity on the dynamo. 
 In practice a small portion of this has its equivalent in heat 
 evolved per second owing to friction. We shall, however, dis- 
 regard this at present, and shall assume that it has its equivalent 
 entirely in the electrical activity E C. 
 
 We may, as we did before in the case of a voltaic battery, think of 
 our dynamo as a machine in which the electricity is pumped up through 
 a certain AV measured by E, and thus acquires potential energy. In 
 the voltaic cell we expended chemical energy in order to gain this 
 
406 ELECTRICITY en. xxiv e 
 
 electrical energy ; in the case of the dynamo it is mechanical energy 
 that is so expended. 
 
 This activity E C is again transformed ; it appears as (a) heat- 
 activity measured by (E e) C, or C 2 R, since C = ~ e , to- 
 gether with (b) the activity e C expended against the reverse E.M.F. 
 of the motor. 
 
 This division of the expended activity E C is the only one 
 which would agree with Joule's law, and with the fact that to 
 drive a current C against an E.M.F. e must require activity to the 
 amount of e C ; conservation of energy being also satisfied. 
 
 From the nature of an electro-motor it is clear that the activity 
 e C can only appear as mechanical activity ; either kinetic energy 
 gained per second, or work done per second against some force. 
 We here neglect jriction in the motor. 
 
 Now let us consider what will happen if we start with the motor 
 at rest, and let it gradually increase in velocity until e comes to be 
 as nearly equal to E as possible. 
 
 Tf 
 
 (1) Motor at rest, e is zero. We now have C = , or the 
 
 R 
 
 current is at a maximum. Hence the activity E C expended on 
 the dynamo is at a maximum. But it all appears as heat-activity 
 evolved in the circuit ; it gives us E C , or C 2 R, heat-activity 
 (measured in watts). 
 
 (2) Motor moving slowly, e is small. Now we have C = ^-^, 
 
 JR. 
 
 or C < C . The activity expended on the dynamo is also less 
 than before, being now E C, which is < E C . Of this we expend 
 e C on the motor, while (E e) C appears as heat-activity. 
 
 (3) Motor moving rapidly, e nearly equal to E. In this case C 
 
 is very small, since it equals ~ . The activity EC expended 
 
 _K 
 
 on the dynamo is also very small, or the dynamo is ' easy to turn.' 
 Of this activity the part e C spent on the motor is the greater, and 
 the part (E e) C wasted in heat is the smaller. 
 
 Thus by letting the motor run very fast, which would be effected 
 by giving it very little work per second to do, we can convert into 
 useful activity a very large percentage of that expended upon the 
 dynamo, and waste in heat a relatively small amount ; but at the 
 
CH. xxiv. DYNAMOS AND MOTORS 407 
 
 same time we must notice that the activity is very small altogether. 
 We are doing work without much waste, but the work is 'light.' 
 
 (4) Motor moving so that e E, (approximately). In the limit 
 we can conceive of e becoming finally equal to E. Under these 
 
 conditions the current is zero, since ^ now equals zero. So 
 
 JX 
 
 that at the moment when we attain the result of having no waste of 
 activity in heat, inasmuch as E e now is zero, and of converting 
 all the activity expended upon the dynamo into activity used in 
 the motor, at this limiting moment the current becomes zero arid 
 the activity itself is zero. 
 
 Thus in case (i) the useful activity e C is zero because e is zero; 
 while in case (4) the same is zero because C has become zero. 
 
 (5) It is of interest to discover at what value of e and C the 
 useful activity e C is-at a maximum. 
 
 This is easily found by an ordinary algebraical device. 
 We wish to find when e C is at a maximum. 
 
 Now eC - ^ ~ % and is at a maximum when e (E e) is so. 
 We may write e (E - e) in the form 
 
 (?)' -(-I)'- - 
 
 Here each term is positive in sign, since each is a square. Hence 
 the whole is maximum when the last term is zero, i.e. when e = . 
 
 We see then that the work per second done on the motor is 
 greatest when this motor is allowed to run at such a rate that the 
 
 E.M.F. e is equal to 1 E. When this is so, we have 
 
 C m = ^=A^ = 1. 1 = I C . We have also (E - e) C m = e C m ; 
 
 or we use only half the total activity expended on the dynamo, 
 the other half being wasted on heat. 
 
 If then we wish to get through some work at the greatest pos- 
 sible rate, we cannot do it economically ; we must waste half our 
 expended work. 
 
 The reader will observe that these results are, mutatis mutandis, 
 the same as those of Chapter XV. 7 (/3). [See also Chapter XIII. 
 
 13-1 
 
408 ELECTRICITY CH. xxiv. 
 
 5. Efficiency, The ratio that the useful activity bears to the 
 total activity expended is called the efficiency. 
 
 Thus it follows from this definition that . . . :* . . .. . . 
 
 1 The efficiency is measured by ^-,, or by =,.' . 
 
 ii, C il 
 
 Thus in case (i) of 4 the efficiency is zero. As e increases, 
 so does the efficiency. When the work is maximum, this occur- 
 ring when e = E as was shown, the efficiency is -. In the limit 
 2 2 
 
 the efficiency reaches its maximum value of unity first when e = E, 
 or when current and activity vanish together. 
 
 The reader must distinguish very carefully between maximum 
 activity and maximum efficiency. We do most work per second 
 
 when the efficiency is but - ; and when this latter approaches the 
 
 economic ideal of unity ^ then the former approaches zero. 
 
 6. Representative Curves. We have seen that as the motor 
 increases in velocity from zero to its limiting value, so that e rises 
 from zero to a value approximating to that of E, we have the 
 following variations. 
 
 (i.) The total activity falls from its initial and maximum value 
 E C to its final zero value. 
 
 (ii.) The useful activity rises from zero to its maximum value 
 
 of - E x - C , or of -, and then falls to zero again. 
 
 22 4 
 
 (iii.) The efficiency rises from zero to its limiting value of 
 unity. 
 
 Now all these changes can be very conveniently presented to 
 the eye in one view by means of the graphical method, of which 
 we have already given examples in Chapters XIII. and XVI. 
 
 In researches on the behaviour of dynamos, Dr. Hopkinson, 
 Marcel Deprez, and others have made much use of the graphic 
 method. It has been, found possible to construct for each machine 
 a curve, determined by a few experimental data ; and then from 
 this curve to predict by geometric methods the behaviour of the 
 said machine under varied conditions. 
 
 Such curves are called characteristic curves ; their construction 
 was first suggested by Dr. Hopkinson. 
 
DYNAMOS AND MOTORS 
 
 409 
 
 7. S. P. Thompson's Diagrams, Professor S. P. Thompson 
 has devised a very simple form of diagram in which the results of 
 4 and 5 are presented to the eye. 
 
 In the diagrams here given we have a figure similar to that of 
 Euclid, Book II. 4. 
 
 A B C D is a square ; the length A D of its side represents in 
 magnitude the constant E.M.F. E of the dynamo ; the vari ble 
 
 A K 
 
 part A K represents on the same scale the variable reverse E.M.F. e 
 of the motor, so that the remainder K B represents (E e). 
 
 Now we know that C = ~ e , where R is constant. We may, 
 
 _K 
 
 therefore, for purposes of comparison, neglect the constant R ; and 
 may say that the line K B represents in magnitude the current C 
 also. 
 
 Hence, in the general case of fig. i. we have the magnitudes 
 of the several activities represented by the areas of corresponding 
 rectangles, viz 
 
 (the total activity expended, i.e. E C) by the rectangle A H ; 
 
 (the useful activity, i.e. e C) by the rectangle A M or H L ; 
 
 (the activity wasted in heat, i.e. C 2 R or (E - e) . C) by the square K H ; 
 
 (the efficiency, i.e. ] by the ratio of areas - ; 
 \ E/ A H 
 
 or by the ratio of lines , 
 
 The four diagrams correspond, fig. i. to the general case, fig. ii. to 
 the case where e zero, fig. iii. to the case of maximum work, and 
 fig. iv. to the case where e is approaching E in magnitude respec- 
 tively. Further explanation is not needed. 
 
 8. Transmission of ' Power ' from a Distance It often 
 happens that there is a source of energy at one place, while it is 
 desired to utilise energy at another. Such is the case, as a rule, 
 where the source of energy is wind or water ; or where a powerful 
 central engine is to be used for the work required over a range of 
 
410 ELECTRICITY CH. xxiv. 
 
 workshops. When activity is transmitted from the source to the 
 place, more or less distant, at which it is to be utilised, we have 
 what is called transmission of power. 
 
 For very short distances such transmission is effected by means 
 of levers or toothed wheels ; for longer distances by means of 
 chains or flexible straps ; for still longer distances by means of 
 water, as in the gold-washing operations of California. Of late 
 years such distant transmission has been effected by means of 
 electric currents. 
 
 The question naturally arose, Would it not be necessary to 
 have very thick, and therefore very expensive, conductors,' if it 
 were desired to transmit activity of any considerable magnitude ? 
 It was (the writer believes) Sir W. Thomson who first pointed out 
 that this would not be necessary, if the transmission were effected 
 under proper conditions. 
 
 Suppose, for example, it were desired to transmit activity from 
 the Falls of Niagara to the city of New York. At the Falls we 
 might have a number of dynamos worked by means of turbines or 
 other contrivances, while at New York might be stationed the 
 motors to be driven by the current. Now the activity utilised in 
 New York would be measured bye C, while the current C is given 
 
 by ~ e . Hence we could have a large activity utilised, while the 
 R 
 
 current to be carried by the conductor remained small, provided 
 that E and e were nearly equal and were very large. 
 
 We must therefore have at the one end a dynamo that gives a 
 large E.M.F. E, and at the other a motor running so fast as to 
 give a reverse E.M.F. e that is nearly as large as E. That this 
 may be the case, the motor must be connected by a 'geared-down' 
 arrangement with the machine that it is intended to drive ; so that 
 the motor may run very fast while the machine moves at the usual 
 more moderate speed. 
 
 In such an arrangement the current would be small, and hence 
 the transmitting wire need not be of large diameter. 
 
 It is interesting to notice that, in the above case, if the motor were 
 
 Tf 
 
 stopped the wire would be exposed to the large current and would 
 
 R 
 
 very likely be fused by it ; and if the dynamo were suddenly stopped, 
 the wire might be fused by the current to which it would be for a 
 
CH. xxiv. DYNAMOS AND MOTORS 41 
 
 moment exposed, the motor maintaining for a moment its velocity 
 owing to the inertia of the revolving masses. 
 
 9. Electric Railways and Tram-Cars; Telpherage; &c, 
 
 Electro-motors can be used for locomotive, purposes in more than 
 one way ; but at present such applications have been but little carried 
 out into practice. 
 
 Electric tram-cars. -These are usually driven by means of Faure's 
 accumulators that are carried in the car itself. These cells supply a 
 current to an electro-motor, and this, through the intermediency of 
 suitable geared-down connections, drives the car. The motor moves 
 rapidly ; the driving wheels of the car move with relative slowness. 
 
 Electric screw-launches. Such trial boats as have been made as 
 yet are driven either by means of accumulators, or by means of 
 'primary' bichromate batteries ; these supplying a current to the motor 
 tha't drives the screw. 
 
 Electric railways. Lines of railway have been devised on which 
 the train would be driven by motors, the current being conducted to 
 them, from the centres at which the dynamos are being driven, by 
 means of the rails themselves. Thus, stationary steam-engines supply 
 mechanical energy to the train in all parts of its journey, through the 
 intermediency of the current. 
 
 The whole distance can be so broken up into sections, and the 
 electrical connections can be made in such a way, that the current is 
 supplied to a train only when other trains are at a safe distance. In 
 fact, collisions could thus be automatically prevented, breaks automatic- 
 ally applied, and risk of accidents reduced to the ' unavoidable 
 minimum.' 
 
 Telpherage. The idea of propelling light trains along single over- 
 head rails was first conceived by Professor Fleeming Jenkin. Such 
 trains might, e.g., serve to transport material from mines or quarries 
 over fields to the works for which such material might be destined, or 
 for any similar purpose. This automatic transport has been called 
 Telpherage. The reader will perceive that it differs only in detail, 
 not at all in principle, from the railway system mentioned above. 
 
 The practical application of Telpherage is due to Professors 
 Fleeming Jenkin, Ayrton, and Perry, who acted in concert in this 
 matter. 
 
 In October 1885, there was opened in Sussex the first Telpher line 
 instituted in this country. It serves to convey clay from the pits over 
 meadows (where a tramway would have been inconvenient or injurous) 
 to the railroad ; and it was adopted, by the Cement Company who 
 own it, as the most suitable method of transport. 
 
412 ELECTRICITY CH. xxiv. 
 
 (For further details as to this line the reader is referred to ' Nature, 
 Vol. xxxiii. p. 13, whence this brief notice has been derived.) 
 
 10. Distribution of Potential in the Circuit of a Dynamo 
 and Motor. In what follows we shall speak of a dynamo of con- 
 stant E.M.F. E, and a motor of reverse E.M.F. e. The reader 
 can easily apply the same methods to the similar case of voltaic 
 piles and electrolytic cells (see Chapter XV.. 9). 
 
 As in Chapter XIII. so here, the ordinates represent potentials ; 
 the abscissas represent resistances ; and the whole diagram ex- 
 hibits the fall of potential, that occurs in obedience to Ohm's law, 
 throughout the circuit. 
 
 We have for convenience represented the E.M.F. E of the 
 dynamo as a AV occurring abruptly at one point in the armature. 
 This is not the case ; for the E.M.F. E is the sum of a number of 
 E.M.F.s occurring in the different turns of wire. But in what 
 follows, this simple assumption will not lead us to any erroneous 
 results ; and without it we could have no such simple diagrams 
 as those here given. 
 
 (i.) Dynamo ; open circuit. In fig. i. the points A and B re- 
 present the terminals (or brushes) of the dynamo ; A x and y B 
 
 together measure the internal resist- 
 ance RI of the dynamo ; the ordinate 
 xy measures the E.M.F. E. 
 
 Since there is no current, it is clear 
 that Ax and jyB are horizontal, or 
 that the points A and B have a differ- 
 ence of ordinate measured by xy. 
 In other words, the E.M.F. E may 
 be measured by the AV of the ter- 
 minals A and B when the circuit is 
 open. 
 
 (ii.) Dynamo ; closed circuit ; no other E.M.F. in the circuit. 
 Now let the circuit be closed through an external wire whose re- 
 sistance is measured by the abscissa b a' of fig. ii. Here A and 
 A' are really one point, the circuit being complete. The lines AJC 
 and y A' make an angle </> with the axis O r such that C is pro- 
 portional to tan <, or to ^- i , as explained in Chapter XIII. 
 
 It is clear that in this case the points A and B have a difference 
 
CH. XXIV. 
 
 DYNAMOS AND MOTORS 
 
 413 
 
 of ordinates that is .less than the ordinate xy. This means that 
 the AV between the terminals will not measure the total E.M.F. 
 E of the dynamo, but will be less than this latter ; there being 
 some fall of potential down the internal resistance R b according 
 
 Direction of current 
 
 a' 
 
 FIG. ii. 
 
 to Ohm's law. We may conveniently designate the AV between 
 A and B by the symbol E B . It will not be hard to find what 
 
 relation E B bears to the total E.M.F. E. 
 
 Let Rj be the internal resistance ab, and R 2 the external re- 
 sistance ba' . 
 
 Then, since there is in the whole circuit a total fall of potential 
 measured by E, there will in the portion R t be a fall measured 
 
 by ^ . E, by Ohm's law. This expression can be written 
 
 TT> . 
 
 also as R! C, since C = - 
 
 hR 2 * 
 Hence, E falls short of E by the amount R^ C ; or . " . . . 
 
 We can therefore find out the value of the E.M.F. of a dynamo 
 (supposing it not to be shunt-wound) when it is running and when 
 the circuit is closed, by finding the AV between its terminals and 
 by adding to this the product of its resistance into the current. 
 If measures be made in amperes, ohms, and volts, there will be no 
 'constants ' involved in these expressions. 
 
 (iii.) Dynamo and motor ; a reverse E.M.F. In fig. iii. we 
 have the case of a circuit that comprises a dynamo and motor. 
 A and B are the terminals of the dynamo, C and D those of the 
 motor. The resistance R t of the former is measured by the 
 
414 
 
 ELECTRICITY 
 
 CH. XXIV. 
 
 abscissa ab, and the resistance R 2 of the latter by ca. The re- 
 maining abscissae bc&!\&da' together represent the resistance r 
 of the connecting wires. A x, y z, and w A', are of course parallel. 
 
 Direction of current 
 
 c d 
 
 FIG. iii. 
 
 The ordinate xy represents the E.M.F. F of the dynamo, 
 while zw represents the reverse E.M.F. e of the motor. The 
 
 current is given by the expression J ~~ e - ; or, in the dia- 
 
 - 
 ! + -K-2 + r 
 
 "" w 
 
 
 gram, it is proportional to 
 
 Using the same notation as in (ii.) above, we may say that 
 
 the AV E B between the terminals A B is related to E by the 
 equation ................... 
 
 E = E* + R! C. 
 
 In a similar way we have for the AV E D between the terminals 
 of the motor the relation .............. 
 
 E^ = e + R 2 C, 
 
 since here the fall of potential due to Ohm's law is added to the 
 fall measured by e. 
 
 The reader should notice the fact, at first sight somewhat para- 
 doxical, that in the dynamo the E.M.F. (represented for convenience 
 by the abrupt rise xy} appears to be opposed to the current ; while in 
 the motor the E.M.F. (similarly represented by zw] appears to be 
 acting with the current. 
 
 These apparent difficulties disappear upon a little consideration. 
 
 It is the work done on the dynamo that keeps, as it were, pumping 
 
CH. xxiv. DYNAMOS AND MOTORS 415 
 
 up the electricity from x to y\ and it is the advantage thus gained that 
 enables the current to flow in the direction indicated by the arrow. 
 Hence the driving E.M.F. E is represented suitably in the diagram 
 by the rise x y. 
 
 In the case of the motor we may observe that if the current is to 
 do work, it must be by being, as it were, let down an f electrical hill.' 
 Hence zw must occur in the opposite direction to that in which xy 
 occurs ; the current must fall down z w. Moreover, such a fall as 2 w 
 rightly represents a reverse E.M.F., or one which opposes the current, 
 for the following reason. The current is proportional to tan $ (see 
 Chapter XIII.) ; and hence the current is diminished, if the inclina- 
 tion of the lines Ax, yz, and wA', to the axis Or be diminished. 
 Now the fall zw does diminish the angle <, as is easily seen if we 
 compare fig. iii. with a figure in which z w is removed, xy and a a' 
 remaining unaltered. 
 
 Hence e is rightly represented by the fall ziv. 
 
 ii. Work done per Second upon a Dynamo as Related to 
 the Velocity v of Rotation. 
 
 The mechanical work per second or activity expended in driving 
 a dynamo is, if we neglect friction, measured in watts by the product 
 EC. 
 
 The manner in which the magnitude of this product depends upon 
 the velocity v of rotation of the armature varies according to the nature 
 of the dynamo. 
 
 (i.) Case of magneto, or other constant-field, machine. Here we 
 have, by the formula of Chapter XXIII. 2, that E is proportional to 
 77 ; and hence that, when there is no other E.M.F. in the circuit, C is 
 also proportional to v. 
 
 Therefore the work is proportional to ir. 
 
 (ii.) Case of series, or other vary ing-field, machines. In these the 
 E.M.F. is first affected directly by v, and then is affected also in- 
 directly, inasmuch as increase of current increases the field-strength. 
 
 Hence the work is proportional to some higher power of v than 
 the second power. Since however the increase in field-strength, cor- 
 responding to a given increase in the current, depends upon the 
 degree to which the cores of the field-magnets are already saturated, 
 it is not possible to express in any simple and yet exact manner the 
 relation between the activity E C and the velocity of rotation v. 
 
416 ELECTRICITY CH. xxv. 
 
 CHAPTER XXV. 
 
 VARIOUS APPLICATIONS OF ELECTRICITY ; TELEGRAPHS, 
 TELEPHONES, MICROPHONES, ELECTRIC LIGHTING. 
 
 i. Introductory, In the present Chapter we shall describe, 
 but necessarily in a very brief manner, various applications of 
 ' electricity ' ; these applications being for convenience divided 
 into groups, as indicated in the heading of the Chapter. From 
 the nature of the subject there will not be, as there was in most 
 of the previous Chapters, any order or progress from group to 
 group. The whole forms a somewhat miscellaneous collection, 
 illustrative of various principles explained earlier in the Course. 
 
 TELEGRAPHS AND ELECTRIC SIGNALLING. 
 
 2. General Principle of Telegraphy, Let us consider two 
 stations A and B, more or less remote from one another, con- 
 nected by an insulated wire, the circuit being completed either 
 through a return wire, or by means of large metallic plates buried 
 in the soil, through the earth itself. It is clear that from either 
 station can be sent currents that will pass through the other 
 station. And if each station be provided, not only with a battery 
 and with a suitable instrument for making or breaking current, 
 but also with a receiving instrument on which the current acts to 
 deflect a needle, sound a bell, or in other ways attract attention, 
 it is clear that we have the means of exchanging signals and 
 messages between the two stations. 
 
 Note on the l return wire.' We may remark that as a rule a return wire is 
 not employed, the circuit being completed by means of large metallic plates 
 buried in the earth, as is seen in the diagram to 4. It is not necessary to 
 suppose that the current does actually return through the earth. For the 
 plates will be kept at what will be approximately zero potential (or the same 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 
 
 417 
 
 potential), since each has a large surface in contact with a practically un- 
 limited conductor. And, as long as each end of the line is kept at zerv 
 potential (or the same potential), so long will the battery send a current 
 through it. 
 
 (i) The electric bell. The accompanying figure represents a 
 very simple form of signalling instrument : it is similar in principle 
 to the automatic make- and -break de- 
 scribed in Chapter XXII. 7. 
 
 When in the position shown here 
 (see fig. i.) the circuit is complete. But 
 as soon as a current passes, the electro- 
 magnet attracts the iron keeper a, 
 causing the hammer to strike the bell , 
 and the circuit is broken at the place 
 where the spring C touched the keeper 
 a. The current ceases, and with it the 
 magnetism. Hence the keeper a is no 
 longer attracted ; and so, acted upon 
 by the spring to which it is attached, 
 it flies back again and makes contact 
 once more with C. The process is 
 then repeated. As long, therefore, 
 as the current is kept on the line, so 
 long will the hammer vibrate and the 
 bell sound. This instrument is used only for attracting attention, 
 not for messages. 
 
 FIG. i. 
 
 FIG. ii. 
 
 ^2) The electric sounder. This instrument (see fig. ii.) is very 
 simple in construction. The current from the sending station passes 
 
 E E 
 
418 
 
 ELECTRICITY 
 
 through the coils of the electro-magnet, and the iron keeper 
 a is attracted downward. This causes the other end of the 
 lever to hit with a sharp sound the screw d. When the current 
 ceases, the spring S pulls the lever down again, and this hits 
 the lower screw e. When the current is made and broken 
 very rapidly, these two sounds occur almost simultaneously, as a 
 sharp double ' click.' If the current be not broken again at once, 
 
 FIG. 
 
 the two sounds are separated by a corresponding interval, and 
 the sound is more deliberate. These two sounds can be readily 
 distinguished ; and by suitable combinations of these we can, as 
 will be explained in 3, represent all the letters of the alphabet, 
 and so form a code for messages. 
 
 It is to be noticed that if there be passing between the point 
 d and the lever a strip of paper, moved by clock-work at a uniform 
 
CH. xxv, , VARIOUS APPLICATIONS OF ELECTRICITY 
 
 419 
 
 rate, then each sharp make-and-break of current will cause a dot 
 to be made on the strip ; while each slower make-and-break will 
 keep the point pressed against the paper sufficiently long to make 
 a dash on it. When the signals are thus to be printed, the in- 
 strument is modified, the 
 lever being provided with 
 a pointed style, or with a 
 small sharp-edged wheel, 
 and d being replaced by 
 a smooth roller. 
 
 (3) The needle tele- 
 graph. -By sending cur- 
 rents round any form of 
 galvanometer in opposite 
 directions, we can give to 
 the needle deflexions in 
 opposite directions respec- 
 tively. On this principle 
 is constructed the com- 
 mon ' needie telegraph/ 
 of which one form is 
 shown in fig. iii. Here 
 the coils and the needle 
 are both vertical, and 
 the motion of the needle 
 is limited by two ivory 
 stops. In fig. iv. we see 
 at the base of the instru- 
 ment the commutating key by means of which the current can 
 be made or broken, and sent in either direction. The current 
 passes first through the sending instrument, deflecting its needle 
 in one or the other direction. It passes thence through the line 
 wire, and affects in a corresponding manner the needle of the 
 receiving instrument. 
 
 By means of a suitable code both this and the last instrument 
 are used to transmit verbal messages. 
 
 3. Telegraphic Alphabets, In all those instruments that 
 are in extensive use there are only two elementary signals : the dot, 
 to which we agree to consider a deflexion to the left to be the 
 
 FIG. 
 
ELECTRICITY 
 
 CH xxv. 
 
 corresponding signal in the needle instrument ; and the dash, to 
 which corresponds the deflexion to the right of the needle. 
 
 The signals answering to the letters of the alphabet are various 
 combinations of the above elements, the only principle observed 
 being that simple signals shall represent constant recurring letters. 
 
 
 SDTGIE 
 
 
 
 sixcra 
 
 PRINTING. 
 
 :NEEDLE. 
 
 
 PHTNTCTG. 
 
 TSTEDIE. 
 
 A 
 
 y 
 
 
 N 
 
 A 
 
 B 
 
 /XV 
 
 
 
 
 I/I 
 
 c 
 
 AA 
 
 
 P ^_ 
 
 Jk 
 
 D . 
 
 Ax 
 
 
 Q , 
 
 IIJ 
 
 E - 
 
 i 
 
 
 R 
 
 vA 
 
 1 
 
 
 
 
 
 F 
 
 \\A 
 
 
 S 
 
 \NN 
 
 G 
 
 /A 
 
 
 T 
 
 / 
 
 H --,,-- 
 
 NX\\ 
 
 
 TJ 
 
 VN/ 
 
 I -- 
 
 \\ 
 
 
 V 
 
 XNS/ 
 
 J 
 
 V/// 
 
 
 W 
 
 S// 
 
 K 
 
 u 
 
 
 X 
 
 AN/ 
 
 L 
 
 x/xr 
 
 
 Y 
 
 A// 
 
 M 
 
 II 
 
 
 Z 
 
 /As 
 
 This system is called, from its inventor, the Morse code. The 
 reader will notice how readily this code, intended originally for 
 dots and dashes in the Morse instrument described in 5, adapts 
 itself to other methods of signalling ; as by the needle instrument, 
 by the voice, by flashes of light, by flags, &c. 
 
 4. The Needle System of Telegraphy. We will now de- 
 scribe briefly the simplest manner in which two stations may be 
 united on the needle system. In the figure, A and B represent 
 the two needle instruments (see 2, fig. iii.) at -the two stations 
 respectively ; jthe batteries are represented as usual. E and E' are 
 large plates buried in the earth, serving to keep the two ends of 
 the line at the zero potential ; K and K' are the two .commutating 
 keys by means of which each station can send a current in either 
 
xxv. VARIOUS APPLICATIONS OF ELECTRICITY 
 
 421 
 
 direction through the other station ; the line wire (shortened out 
 of all proportion, of course) is seen at the top of the diagram. 
 
 Action of the commutator key. The terminals of the battery 
 are connected with two strips of metal, marked white and black 
 respectively in our diagram. These are insulated from each other. 
 The pieces / and r are two keys of metal, that are kept pressed 
 against the upper (or white] strip by means of a spring. In this, 
 A B 
 
 which is the normal condition of things, there is a complete circuit 
 through A, the upper strip, the line, B, the other upper strip, the 
 earth plate E', the other earth plate E, and so to A again. The 
 batteries are cut out of the circuit ; since the lower (or black) strips, 
 and with them the negative poles of the batteries, are insulated. 
 
 Now let the key / be depressed ; thus leaving the upper, and 
 touching the lower, strip. The connections make it evident that 
 this will send a current along the .line to B, through E' and E, 
 and back to the pole of the battery by A and /. If the other 
 key be depressed, the current will pass in the other direction. 
 
 If similar 'keys be depressed at both stations simultaneously 
 the circuit will be broken, and no message can be sent. 
 
 5. The Morse System. The Morse instrument is in all 
 essentials the sounder of 2, fig. ii. ; but it is modified so as to print 
 dots or dashes on a strip of paper. 
 
 In the figure we show the simplest manner of connecting two 
 stations by means of Morse instruments. The two Morses are at 
 M and M' ; K and K' are simple keys, not commutators ; G and 
 G' are galvanometers to indicate to the sender whether or no the 
 current actually passes, or whether the line is stopped ; this being 
 
422 ELECTRICITY CH. xxv. 
 
 needed, inasmuch as the current does not pass through the sender's 
 Morse. In the position shown, the keys, kept in place by a spring, 
 make contact at 3 and 4, so that each station is ready to receive 
 a message. On either key being depressed a current is sent along 
 
 the line and through the other instrument ; so that dots or dashes 
 will there be printed, according to the duration of the contact 
 made by the sending key. 
 
 If both keys be depressed at once no message can pass. A 
 preliminary signal is usually sent, so that the receiving clerk may 
 set going the clock-work that moves the paper. 
 
 The Morse is often worked by sound, the ear soon acquiring 
 the power of reading off the message in this way. 
 
 6. Relays. On long lines the current will usually be too 
 feeble to work the printing Morse. In such cases a relay is em- 
 ployed. This is an arrangement consisting of a local battery in 
 circuit with the Morse, and a key that can make or break this 
 circuit ; this key being worked by the feeble line current through 
 the intermediency of an electro-magnet. 
 
 In the figure on the next page the feeble line current works 
 the electro-magnet R ; this acts upon the soft iron piece a, and by 
 means of the lever makes contact at i, 2 ; the local circuit is thus 
 completed, and the Morse S is worked by the strong local current. 
 
 7. Transmission through Cables, under Water. When the 
 line connecting two places has to be laid under water, it then 
 consists of one or more copper wires thickly coated with some 
 insulating material ; round this again is generally twisted stout 
 iron wire in order to give the strength necessary, and this is coated 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 423 
 
 with some waterproof material. The whole is called a cable ; the 
 messages conveyed by it are sometimes called cablegrams. 
 
 Difficulties occurring in transmission by cable. It was soon 
 found that it was not possible to send, through a cable of any 
 considerable length, the clearly separated dots and dashes required 
 by the Morse receiver. The sharp tap on the key at the sending 
 end is represented at the receiving end by a feeble current that 
 slowly rises to a maximum and again slowly sinks to zero. With 
 such currents we cannot use any of the instruments described 
 
 above. Had not the Thomson's reflecting galvanometer, or some 
 similar instrument, been invented, cablegraphy would hardly have 
 been practicable. With this instrument, however, the reading of 
 a message is easy. The principle of the reflected beam of light 
 gives the means of magnifying the motion to any desired extent. 
 The needle is of very small mass and possesses but little moment 
 of inertia about its axis of suspension ; and hence its deflexion will 
 follow, with but an inappreciable lag, the varying undulatory 
 current. And the extreme delicacy of the instrument enables very 
 
424 ELECTRICITY CH. xxv. 
 
 small currents to be perceived. Finally, the small mass of the 
 needle, its method of suspension, and the command of it that is 
 given by the controlling magnet, renders it possible to use a 
 modified form of this instrument on board ship during the laying 
 of the line. 
 
 Reading the messages. On a land line we are able to give to 
 the needle very rapid deflexion to right and to left of the zero 
 mark. But on a cable the currents last so long that it is not 
 practicable to wait for the dying away of one current before sending 
 another. Hence the currents overlap one another, and allowance 
 has to be made for this in reading the message. The clerk at the 
 receiving end watches the spot of light, and observes, not its move- 
 ments to right or left of the zero mark in the middle of the scale, 
 but its jerks to right or left wherever it happens to be. In fact, a 
 shifting zero mark is used. 
 
 Cause of retardation of signals. The origin of the alteration of 
 the sharp signals sent, into the slow and gradual undulations 
 received, lies mainly in the fact that the cable forms a long con- 
 denser of very great capacity. 
 
 In the case of an overland wire, the sharp making of contact 
 effected by striking the key is followed by the passage along the 
 wire of what we may call a tide of electricity ; this tide having a 
 relatively abrupt front, as kas the well-known ' bore ' that is some- 
 times seen on a tidal river. The receiving instrument is affected, 
 almost instantaneously, with the full force of the current. But in 
 the case of the cable matters are very different. The tide of 
 electricity rushes into the cable, but as it proceeds it is to a great 
 extent detained to charge electrostatically the cable ; this being a 
 condenser in which the wire and the sea-water form two coatings 
 separated by the insulating di-electric. Thus the rise to a maxi- 
 mum occurs very gradually, as would the rise of tide in a stream 
 out of which ran many side channels that must be filled by the 
 tide before it can advance up the river. (We need not say that 
 the analogies here given are very incomplete.) 
 
 Hence at the receiving instrument the current rises very 
 slowly to a maximum. 
 
 8. Earth Currents. Condenser System of Working, It 
 is found that currents are continually flowing from one point of 
 the earth to another, these currents being due to thermo-electric 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 425 
 
 or to other unknown causes. When the two earth plates of a 
 telegraphic line happen to be buried where the earth is not at one 
 potential, a current will flow along the line wire. The greater the 
 A V of the two places at which the plates are sunk, the stronger 
 will be the E.M.F. driving the current along the line. During 
 those disturbances of the earth's magnetic field which are known 
 as magnetic storms, these earth currents become very strong, and 
 may entirely over-ride and nullify the ordinary currents by which 
 the messages are transmitted ; such ' storms ' occur, for example, 
 during brilliant displays of the aurora borealis, and during any 
 noticeable changes in the surface of the sun. 
 
 These earth currents give great trouble on long lines, and 
 especially in long cables. There are two methods of meeting the 
 difficulty. 
 
 (i.) The return wire method. Where it is practicable, a return 
 wire may be used instead of an ' earth.' If the clerks find the 
 earth current to be so strong on any particular day as to give 
 trouble, they may temporarily complete the circuit by means of 
 any wire that does not happen to be in use on that day. 
 
 (ii.) The condenser system. The diagram here given represents 
 the principle of the condenser method. In this system the line is 
 
 not continuous, but the line from the sending station is connected 
 with one condenser plate, while from the other plate passes away 
 the line to the receiving station. In the simple arrangement shown 
 in the figure, a depression of the key would send a charge into the 
 condenser plate a. This would call up an equal and opposite 
 charge in the plate b ; and the effect upon the instrument G would 
 
426 ELECTRICITY CIT. xxv. 
 
 be the same as though the current sent had passed on. This 
 action of course ceases when the condenser is charged, but, as 
 this latter is always of very great capacity, the current due to the 
 electrostatic induction lasts as long as is necessary. 
 
 When the key is again released, the plate a is discharged to 
 earth ; and hence there will be a discharge of b also, and therefore 
 a reverse current through the instrument G. 
 
 In practice there is a condenser at each end, the cable being 
 totally insulated. There is double induction, but, by the electro- 
 static principles discussed in Chapter X. and earlier, the instrument 
 at the receiving end will be influenced by currents passing in the 
 same direction as if they had come direct from the sending station. 
 
 The condensers employed consist usually of many sheets of 
 tin-foil separated by insulating sheets of paraffined paper ; the 
 sheets of tin-foil being connected up into two alternate sets, 
 answering to the two plates of a condenser. 
 
 9. Insulation of Wires. If we cut connection with the 
 earth at one end of a line, and send a current into the wire at the 
 other end, a galvanometer in the line will settle down to some 
 steady deflexion. In fact, the c insulated ' line wire leaks to earth 
 along the whole distance ; the insulators acting as conductors of 
 very great resistance. The longer the line, the more is the leak- 
 age and the lower is the resistance offered by the insulators collec- 
 tively. It is usual to fix upon some definite resistance (in ohms) 
 as the resistance per mile that ought to be offered by any given 
 line of wire. If, without any special cause, such as fogs, rain, or 
 snow, the insulation fall below this standard, then it is suspected 
 that some accidental connection with earth has been made. 
 
 .io. Duplex Telegraphy. In 4 and 5 we noticed that it 
 was not possible to send a message from each station to the other 
 respectively at one and the same time. Could this be done, it is 
 evident that we could get twice the use out of a line. 
 
 We can readily devise a method in which each clerk could 
 transmit his own message through his own instrument ; the 
 problem to be solved is much harder, viz. how the message sent 
 from A shall be indicated by the instrument at B only, while that 
 sent from B shall be indicated at A only. 
 
 This problem has been solved in two ways, one of which we 
 will briefly describe. 
 
en. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 427 
 
 (i.) The differential method. In the figure, E and E' represent 
 the two Morse instruments (or relays), the electro-magnet in each 
 being wound with two equal coils. The line from the key K 
 branches at a ; one path is through the one coil of E, through 
 the line /, through the one coil of the receiving instrument E', 
 and so to earth ; the other path is through the second coil of E, 
 
 through an adjustable resistance box r, and so direct to the other 
 terminal of the battery. The same may be said with respect to 
 the paths from the other key K'. 
 
 In order to work this system it is necessary that the two paths 
 from either key offer equal resistances. In this case, since there 
 are in each instrument two equal and opposite coils, the depression 
 of the key K alone will not affect E, but will affect E'; while the 
 depression of K' alone will not affect E', but will affect E. In 
 the diagram the keys are shown in their normal position. This fact 
 enables us to test for the desired equality of resistances ; r and r 1 
 being adjusted until the desired end is attained. 
 
 It is to be noticed that in the action on the Morse instruments or 
 relays we need not consider the direction of the current. 
 
 Let us now suppose that, while A is sending a message to B* 
 by depressing the key K, the key K' is also depressed. This will 
 destroy the balance previously existing. Through the one coil of 
 E there will now flow less current than before, or zero current, or 
 even a reverse current; this depending upon the relative E.M.F.s 
 of the two opposed batteries. So long, then, as K and K' are 
 both depressed, the instrument E is worked by this difference of 
 
428 ELECTRICITY CH. xxv. 
 
 currents in its two coils ; and, if K be released, E will be worked 
 directly by the current from? B alone. Hence, under both condi- 
 tions E will attract its armature, or release it, in obedience to the 
 movements of the key K' at the station B. The same holds good 
 with respect to the instrument E'. 
 
 Thus, by means of a system in which checks to the currents 
 sent are recorded as currents received, the problem of duplex 
 telegraphy was solved. 
 
 (ii.) The WheatstonJ s bridge method. In another method a 
 somewhat different principle is employed. Here each instrument 
 is placed in a ' bridge,' and is unaffected as long as the extremi- 
 ties of the bridge are at the same potential. When matters are 
 properly adjusted, this state of equilibrium at the one station A is 
 destroyed by the depression of the key at the other station B; and 
 this is the case whether the key at A be worked or no. Hence 
 duplex working is rendered possible. 
 
 TELEPHONES. 
 
 ii. Telephones. Introductory. Telephones are instruments 
 by means of which it is possible to transmit between two stations, 
 more or less remote from each other, musical notes or other 
 sounds, and even articulate speech ; an instrument at the one end 
 is spoken to or sung to, and the instrument at the other end gives 
 out the words spoken or the tune sung. [The same result can be 
 obtained for comparatively short distances by means of speaking 
 tubes ; but to such a system, which is merely a case of reflexion 
 of sound, the word telephone is not applied.] 
 
 The only telephones much used are those in which the trans- 
 mission is effected electrically. There are mechanical telephones, 
 sometimes used, in which mechanical impulses are transmitted along 
 the connecting wire ; but in electric telephones it is an undulatory 
 electric current that passes. 
 
 In all electric telephones we have a transmitter into which the 
 message is spoken, and a receiver which again utters the message : 
 these two instruments may be identical or different in form. 
 
 In some telephones the speaking of the messages causes an 
 undulatory current to spring into existence : and this again causes 
 the receiver to utter the message. In such, there is no external 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 429 
 
 source of current needed. In other telephones there is an external 
 source of current, and the speaking of the messages causes this 
 previously -existing current to become undulatory. 
 
 12. The Bell Telephone. The most complete form of tele- 
 phone, for general use, is that invented by Graham Bell, called, after 
 his name, the Bell telephone. In this instrument no external source of 
 current is needed, and it can act either as receiver or as transmitter. 
 
 In the figure we have a section of the instrument ; from this 
 the construction can be explained. E is the mouthpiece into which 
 we speak; M is a bar-magnet of about four inches long; B is a 
 long coil of very fine insulated wire surrounding the pole of the 
 magnet; C C are the terminal screws; and finally, D is a vibrating 
 membrane made of soft iron and very thin, When we speak into 
 
 the mouthpiece, the soft iron diaphragm vibrates in exact accord 
 with the air-waves impinging upon it. This acts inductively upon 
 the steel magnet, and alters the number of marked lines of force 
 piercing the coil of fine long wire. This gives rise to induced 
 currents in the coil; these will be feeble, but of high E.M.F. 
 Hence, when we speak into the transmitting instrument, we cause 
 undulatory currents to pass along the line wire, these undulations 
 answering to the air vibrations, and therefore answering to the 
 words spoken. 
 
 These currents arrive at the receiving instrument, and there 
 pass round the coil. This alters the magnetism of the magnet's 
 pole, and this again causes the iron diaphragm to vibrate. Finally, 
 this iron diaphragm, vibrating in exact accord with that of the 
 transmitter, will cause the air to vibrate, and, when the ear is held 
 close to the instrument, the words of the message will be heard. 
 
43 ELECTRICITY CH, xxv. 
 
 It is to be noticed that the vibrations of the diaphragm of the 
 transmitter to and fro will give rise to induced currents in opposite 
 directions respectively ; and that in the receiver these currents will 
 strengthen or weaken the magnet pole, and therefore attract further 
 or release somewhat the diaphragm of the transmitter respectively. 
 Hence, the to and fro vibrations of the one diaphragm will be re- 
 produced in the other. 
 
 Sensitiveness of the telephone. This telephone is of remarkable 
 sensitiveness. By it we can detect currents too faint to affect any 
 ordinary galvanometer, provided that the currents are undulatory, 
 or at least subject to abrupt variations in strength. Hence, in 
 many cases the telephone can be used as a very delicate galvano- 
 scope. 
 
 If a telephone (protected by a bridge when necessary) be in a 
 secondary circuit such as that of a RuhmkorrTs coil, the note that 
 it emits will indicate the number of induced currents passing per 
 second; or the number of makes-and-breaks per second in the 
 primary circuit. It has been used in investigations regarding the 
 stratifications that appear in vacuum tubes; and it has thrown light 
 upon the relation that these bear to the number per second of 
 induced currents, and also to the steadiness of the contact maker 
 in the primary. 
 
 In the circuit of a ' Gramme,' the telephone shows plainly, by 
 .emitting a note, that the current is really undulatory, though for 
 many purposes it acts as if continuous. 
 
 Induction disturbances on telephonic lines. The great sensitive- 
 ness of telephones is a source of difficulty in their practical use. 
 If the return be made through the earth, conversation cannot as a 
 rule be carried on at all. For, what with earth currents and leak- 
 age from other circuits, the instrument emits a continuous bubbling 
 or frying sound that drowns the faint 'speech.' Return wires are 
 therefore always used. But even then there is much induction if 
 the wires run anywhere near ordinary telegraph wires ; and it re- 
 quires a very special method of laying the telephone wires to 
 reduce the Babel of sounds, due to these induced currents, to 
 comparative silence. 
 
 Details as to the precautions taken will be found in more 
 technical works. 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 431 
 
 13. Telephones with External Source of Current, The 
 Bell telephone is a wonderful instrument, certainly ; it acts as 
 transmitter or receiver, and it requires no battery to work it. But 
 it must be remembered that the currents transmitted by it are 
 very small indeed, and that consequently the message is delivered 
 by the receiving instrument in a very faint way. 
 
 To obviate this difficulty, telephones have been invented which 
 make use of strong currents driven by an external source, such as 
 a Leclanche cell. 
 
 Edison's carbon transmitter is an example of this class of 
 telephone. This instrument has a mouthpiece and a vibrating 
 metallic diaphragm, but no magnet or coil. At the back of the 
 diaphragm is a button of carbon, this resting against a piece of 
 metal. As the diaphragm vibrates the carbon will make better or 
 worse contact with the pieces of metal touching it on the two 
 sides ; better contact when it is pressed harder, worse when it is 
 somewhat released. Now the two pieces of metal and the carbon 
 between them form part of the circuit of a Leclanche or other 
 cell. Hence, as the diaphragm vibrates, the current flowing in 
 this circuit will vary in strength, the variations corresponding 
 exactly to the original sound vibrations. Now this current forms 
 the primary to a small induction coil, the secondary wire of which 
 is in circuit with the line to the distant station. No current will 
 flow in the secondary, and therefore none through the line, so 
 long as the primary is steady. But when some one speaks to the 
 transmitter and thus causes variations to occur in the primary, 
 then will there be currents induced in the secondary, in exact 
 accord with the air vibrations of the speech uttered to the trans- 
 mitter. These currents traverse the line to the distant station, 
 and the receiver (which we may suppose, e.g., to be a Bell telephone) 
 will reproduce the original speech. 
 
 14. Microphones. Professor Hughes found that when in a 
 circuit there is a loose contact, or still better a group of loose 
 contacts, the current is exceedingly sensitive to even very slight 
 mechanical disturbances, such, e.g., as those produced by sound- 
 waves impinging upon the loose contact or upon the stand sup- 
 porting the same. Undulations are produced in the current that 
 is passing, and these undulations are so exactly in accord with 
 the mechanical vibrations to which they are due, that a telephone 
 
432 ELECTRICITY CH. xxv. 
 
 included in the circuit will reproduce with greater or less distinct- 
 ness the sounds spoken near the loose contact. 
 
 An instrument constructed on the principle and for the object 
 indicated above is called a microphone. 
 
 It is found that loose contacts of carbon give the most striking 
 results, and so in the usual form of instrument a carbon rod 
 rests loosely between two other carbon pieces. If a fly walk over 
 the stand of this instrument, the faint jarring due to its tread 
 produces undulations in the current passing, and a telephone in 
 the circuit will give out what we may, not however very exactly, 
 call the ' sound ' of the fly's tread. 
 
 15. Properties of Selenium. The Photophone, It is well 
 known that in general the electrical resistance of bodies depends 
 upon their temperature, and varies with it either inversely or 
 directly. 
 
 There seems no reason to doubt but that whenever radiant 
 energy (whether this consist of rays whose chief action is heating, 
 or of rays remarkable chiefly for the impressions of light that they 
 produce when they impinge upon the retina) impinges upon a 
 body, the molecules of that body are to a greater or less degree 
 affected ; and that consequently the electrical resistance of the 
 body is affected. 
 
 Now it is found that the body selenium, at least when in one 
 of its several physical conditions, is affected in this way to a very 
 remarkable degree. Even when the longer waves, those whose 
 action is par excellence ' heating,' are sifted out of the ray, so that 
 the radiation left is remarkable mainly for its action upon the 
 retina and for its chemical action, still the effect produced upon 
 selenium is very great. Hence one often hears of ' the action of 
 light on selenium.' In order to exhibit or use this variability in 
 conductivity, we must construct what is called a selenium cell. In 
 one form of this there are two separate spirals of wire wound 
 round a cylinder of hard wood or of other insulating material. 
 These spirals are separated from each other, but run parallel to, 
 and close to, one another through all their length. The one spiral 
 is attached to the one end of the battery circuit, the other spiral to 
 the other. So that the circuit is, as far as we have described it, 
 broken, but the two halves run close to each other for a considerable 
 length and in a compact manner. The space between the two 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 433 
 
 spirals is now filled up with selenium, which is melted on. We 
 thus have the circuit completed by what is equivalent to a strip of 
 selenium of very small length for the current to traverse, but 
 of very great width. Thus the resistance is small, while the 
 action of the radiant energy upon the selenium ought to produce 
 a relatively great effect upon the current, there being such an 
 extent of selenium to be acted upon. 
 
 With the help of such a 'selenium cell' a piece of apparatus, called 
 a photophone, has been constructed, by means of which sounds, and 
 even articulate speech, can be transmitted from a distance. It is so 
 contrived that, on speaking to the transmitting instrument, a beam of 
 light reflected from it to the receiving station varies its intensity in 
 exact accordance with the vibrations due to the voice. The beam 
 falls on a selenium cell, which is in a circuit with a battery and a 
 telephone. The resistance of the cell varies with the variations in the 
 beam, and so also does the current. Hence the current at the 
 receiving station varies in accordance with the vibrations due to the 
 voice at the transmitting station, and the telephone thus reproduces 
 the sounds uttered. 
 
 ELECTRIC LIGHTING. 
 
 1 6. General Account of Electric Lighting. Illumination by 
 means of electricity is effected in two main ways. We will give 
 a brief preliminary description of each of these, and will then 
 proceed to some detail. 
 
 (i.) Incandescent lighting. In the one system a thin conducting 
 filament is heated to whiteness by the passage of a current. The 
 filament is usually or always enclosed in a vacuous glass vessel, 
 and hence is not consumed. Here we have simply a case of 
 Joule's law : if the resistance be R and the current be C, we have 
 given out per second radiant energy measured in watts by C 2 R. 
 It is necessary to make the filament small and highly resisting, 
 so that the radiant energy may be intense in quality, i.e. so that 
 the temperature may be very high. 
 
 (ii.) Arc lighting. If we have a battery of high E.M.F. and a 
 circuit of not too great resistance (forty Bunsen's cells of fair size 
 will fulfil these conditions), and the current be passed through two 
 rods of hard carbon which first touch one another, the point of 
 contact will soon glow and attain a dazzling brilliancy ; and if now 
 
 F F 
 
434 
 
 ELECTRICITY 
 
 the carbons be slightly separated, the current will continue to flow 
 across the interval in the intensely dazzling continuous discharge 
 known as the electric arc. The illumination seems to be due to 
 the intense ignition of a stream of carbon particles that are con- 
 tinually transported from the + to the terminal. The arc can 
 be deflected by a draught of air, or by being acted upon by a 
 magnetic field. Its length depends upon the current-strength ; i.e. 
 upon the E.M.F. of the battery and the total resistance in the 
 circuit. 
 
 It is found that the arc is the seat of a reverse E.M.F. e, 
 against which the current is driven by the battery, and that also 
 it offers resistance R as does a conductor. Hence the radiant 
 energy evolved per second in the arc is expressed in waits by the 
 sum of the two activities C e and C 2 R. Here then we have not a 
 simple case of Joule's law. 
 
 17. The Incandescent 'Lamp.' The substance chosen for 
 the 'filament' spoken of in 16 (i.) is now invariably carbon. 
 This substance possesses the great advan- 
 tages of being practically infusible and of 
 being a very good radiator. 
 
 The main difficulty encountered in its 
 use was that of obtaining a filament that 
 should be at once thin enough, and at the 
 same time tough and not liable to break. 
 Edison found the suitable material in the 
 prepared fibres of a certain kind of bamboo. 
 Swan and others acted upon cotton thread 
 or fine strips of card with strong sulphuric 
 acid, and then carbonised the tough threads 
 thus obtained. 
 
 In the figure is shown a Swan lamp. 
 It will be observed that, in order to give a 
 greater amount of light within the same space, the filament is 
 given a twist into one or more spirals. Near the ends the 
 filament is thicker ; this is to obviate any great heating near 
 the points of support, as this would lead to the breakage of the 
 filament. 
 
 If the globe be filled before exhaustion with a hydrocarbon gas 
 instead of with air, the slight residue that always exists even in 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 435 
 
 a good * vacuum ' will act rather to strengthen the filament by 
 deposition of carbon than to weaken it by combustion. 
 
 It is found that the carbon filament is slowly dispersed and 
 otherwise weakened by continued use. Each lamp has, in fact, a 
 definite ' life ' that is usually expressed in ' hours of use with so 
 many amperes j or in ' ampere-hours.' 
 
 ' Candles per horse-power^ in incandescent lamps. It is of 
 great interest and importance to know the average relations be- 
 tween the candle-power of an incandescent lamp, and the horse- 
 power expended upon the lamp. 
 
 We know that the activity in watts expended upon the lamp 
 is C 2 R, where C is the current in amperes and R is the resist- 
 
 C 2 R 
 
 ance of the lamp in ohms. Or it is measured by -'^ (approx.) 
 
 in English horse-power. The unit by which we usually measure 
 candle-power is very arbitrary and somewhat uncertain ; it is the 
 illumination given by a wax-candle of a certain form in which 
 120 grains of wax are consumed per hour. It is fairly evident 
 that the desired relation must be established by experiment, and 
 cannot be calculated. 
 
 In 'Nature,' vol. xxiv. p. 270, may be found a series of 
 results obtained by Sir W. Thomson and Mr. J. T. Bottomley. 
 It is found that if we wish to obtain such a degree of ignition 
 only as shall give the lamp a chance of a fairly long life, we must 
 be content with from 200 to 250 candles per horse-power. It 
 must be noticed that we obtain, for the same horse-power ex- 
 pended, better luminous effects if we employ high temperatures. 
 The total radiant energy emitted each second from several lamps 
 moderately heated may equal that emitted from one lamp intensely 
 heated ; but the radiation from the former may be useless from 
 the luminous point of view. We should therefore raise the lamp 
 to as high a temperature as is compatible with its safety. From 
 what we have said, then, it is evident that the expression C 2 R 
 does not give us a measure of the light, but only of the total 
 radiant energy, emitted per second by the lamp. 
 
 The resistance of a lamp varies. As the temperature of carbon 
 rises v its resistance decreases. Hence the resistance R of the 
 
 F F 2 
 
436 ELECTRICITY 
 
 CH. XXV. 
 
 lamp is not constant. In order to calculate the watts or horse- 
 fiouer expended on the lamp when any definite current C is 
 running, it is necessary to measure the difference of potential 
 EB existing between the terminals of the lamp at the time. The 
 expression C x Eg gives us the required number of watts without 
 error. We can measure C by means of a galvanometer of low 
 resistance included in the circuit, and E B by means of a potential 
 galvanometer of high resistance (see Chapter XIV. n) con- 
 nected up with the terminals of the lamp. 
 
 1 8. Arrangement of Incandescent Lamps in Parallel. The 
 simplest manner of arranging a number of incandescent lamps 
 of equal resistance is to connect them in parallel with two 
 stout conductors that are connected with the two terminals of the 
 dynamo respectively. The lamps are of equal resistance, and 
 possess no reverse E.M.F. Hence the current between the two 
 stout conductors (which we may add are called leads) is divided 
 equally between the lamps. If one lamp be turned off, the 
 current that now flows is divided between the others. 
 
 Referring to the case of * m equal branches,' discussed in 
 Chapter XIII. 9, II., let us suppose that here the resistance of 
 each lamp is R; that of the rest of the circuit, including the 
 dynamo, is negligible ; and that the E.M.F. is E. Then, when 
 the m lamps are arranged in parallel, the equivalent resistance p 
 
 will equal 5 and the current will be C= = ; . Now the 
 
 !R R 
 
 /;/ 
 
 current flowing in each will be C, or will be . Therefore, 
 
 m R 
 
 under the not quite attainable conditions assumed, viz. of zero 
 resistance in circuit and in dynamo, the turning off or on of any 
 lamps would not alter the current flowing through each of the 
 remaining lamps ; for, if m lamps were left, we should have for 
 
 each lamp a current that is -J-th of , or should have as before. 
 
 m R R 
 
 We may put the matter somewhat differently. In order to 
 supply each lamp of resistance R with a constant current C, it is 
 necessary that the terminals of each lamp be kept at a constant 
 AV measured by E, where E = C R. Now the stout conductors 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 437 
 
 between which the incandescent lamps are slung have, we assume, 
 zero resistance, and are therefore each at one potential throughout 
 its length. We shall then attain our end if we keep these copper 
 leads at the required SV measured by E. But, by Ohm's law, if 
 we have zero resistance in the dynamo and leads, it follows that 
 the whole fall of potential in the circuit will occur across the 
 lamps, and none will occur through the rest of the circuit. That 
 is, the leads will be maintained at the full AV measured by E, 
 and the current through each lamp will be constant. 
 
 In practice, however, we are not able to construct an armature, 
 though we can make leads, of zero resistance. If the armature 
 have a resistance B, while the resistance of m lamps in parallel is 
 
 R, then the total resistance of the circuit is B + R. Hence 
 m m 
 
 by Ohm's law there will be through the armature a fall of potential 
 
 T> 
 
 measured by . E, while the AV between the leads will 
 
 B + z - R 
 
 m 
 
 -'-R 
 
 be measured by the remaining fall m _ . E. Now as m in- 
 
 B + lR 
 m 
 
 creases this expression is not constant, but decreases. Hence, in 
 practice it will not do to supply a set of lamps in parallel from 
 a machine of constant E.M.F.^L for the current in each lamp 
 will decrease as the number of lamps increases. We must have a 
 dynamo in which E increases somewhat when m increases, and 
 conversely. This condition is fulfilled by some dynamo of 
 the series nature. For, as m (i.e. the number of lamps) increases, 
 
 the total resistance which is measured by B + R will decrease ; 
 
 m 
 
 hence C increases, and therefore [from the nature of a series- 
 dynamo] E will also increase. 
 
 19. The Economy cf Incandescent Lighting. Supposing 
 that we have 100 lamps, each of 50 ohms resistance when hot, ar- 
 ranged in parallel. This will give an equivalent resistance of but 
 5 ohm. (We will neglect the resistance of the leads.) If we wish 
 to use 90 per cent, of our energy in the lamps, and waste only 10 
 
438 ELECTRICITY 
 
 CH. XXV. 
 
 per cent, indicating the dynamo, then by Joule's law the resistance 
 of the dynamo must bear to that of the lamps the ratio of i to 9. 
 That is, the resistance of the dynamo must be but '05 ohms. 
 
 Hence, for incandescent lighting we must have dynamos in 
 which the current passes only through very thick wire, dynamos 
 of very low resistance. If the current demanded by each lamp 
 be i "2 amperes, then we must maintain the leads at a AV 
 measured by 60 volts ; since by Ohm's law we have . . . .& 
 
 E = C . R = 1*2 x 50 = 60, for each lamp. 
 
 Again, this AV will be T 9 o of the whole fall of potential given 
 by the dynamo ; or the E.M.F. demanded of the dynamo must 
 be y x 60 = 66 volts. We use activity measured by (i'2) 2 x -5 
 watts ; and we waste activity measured by (i'2) 2 x '05 watts. 
 
 If, as we have done, we neglect all other waste, then we shall 
 
 require ^ 2 ' x ' 5 + 5j horse -power to do work at the rate here 
 746 
 
 required. 
 
 The relation between activity and candle-power is not an 
 exact one, since the latter depends upon the quality of the radiant 
 energy emitted per second as well as upon its quantity. But it 
 will give some idea of the relations between the two if we mention 
 that according to Mr. Swan (' Nature,' vol. xxvi. p. 358) we can 
 obtain at least 200 candle-power for each horse-power of activity 
 evolved in the lamp (see also 17). 
 
 Another authority states that, allowing for 10 per cent, waste 
 in heating the armature, we can run nine twenty-candle Swan 
 lamps per horse-power. This estimation agrees practically with 
 the last, since in that no deduction was made for waste in the 
 circuit. 
 
 We must further, if we aim at economy, run the lamps at 
 as high a temperature as is consistent with their safety ; as was 
 explained in 17. 
 
 20. The Arc Lamp. In the figure is shown a simple form of 
 arc lamp, such as is now used only for lecture-room purposes. 
 Initially the two carbons, made of hard gas coke or of some other 
 very dense form of carbon, are in contact. When, however, the 
 current is well established, by which time the carbons at the point 
 of contact will have been raised to a very intense white heat, a small 
 
CH. XXT. VARIOUS APPLICATIONS OF ELECTRICITY 439 
 
 electro-magnet that is in the circuit acts upon a lever ; and this, 
 by means of a clutch that acts by friction, raises the rod bearing 
 the upper carbon. Thus the carbons are slightly separated and 
 the arc is established. If by 
 the using up of the carbons the 
 space between becomes too 
 great and the current becomes 
 enfeebled, the electro-magnet's 
 hold on the lever and clutch 
 becomes relaxed ; and the latter 
 allows the upper carbon rod to 
 slide down nearer to the lower. 
 When the current again rises 
 to its proper value, this slipping 
 movement is checked by means 
 of the electro-magnet. If the 
 arc be entirely broken, it can 
 be restored only by the car- 
 bons coming again into actual 
 contact. 
 
 Appearance of the arc. 
 With the naked eye the arc 
 cannot be studied ; the extre- 
 mities of the carbons and the 
 arc itself presenting a dazzling 
 liquid brightness that, with the consequent irradiation effect, baffles 
 analysis. Through dark glass we can, however, study the arc; 
 and we then see that the source of the light lies in the extremities 
 of the carbons, whence the arc springs across, and in the arc itself. 
 This latter presents a flame-like appearance. 
 
 Mechanical transport of carbon. When thus examined, or 
 when a magnified image is thrown upon the screen by means of a 
 lantern, it is seen that one carbon becomes hollow after some time. 
 By actual measurements of weight it seems that it is the + carbon 
 that wastes most, there being a continual transport of particles 
 from it across to the carbon ; and that the hollow appearance 
 of this latter is due to the fact that this stream of transported par- 
 ticles is deposited upon it in a crater-like manner. Of course 
 both carbons are also consumed slowly when the arc is exposed to 
 
443 ELECTRICITY CH. xxv. 
 
 the action of the air. The transference from + to carbon is 
 therefore best shown when the arc is used in vacuo. 
 
 This mechanical transport is obviated by the use of alternate 
 current machines. 
 
 Temperature oj the arc. According to some experiments ot 
 M. Rosetti, of Padua, the + carbon attains the higher tempera- 
 ture. These temperatures vary when the current varies. The 
 experimentalist in question believes that we can regard the 
 carbon to attain 2,500 C, and the + carbon 3,200 C, at 
 least. 
 
 Definition of equivalent resistance. If under any given condi- 
 tions of dri\mg-.M.jF. E, &c., we remove the arc and substitute 
 such a resistance p that the current remains unaltered, then p is 
 called the equivalent resistance of the arc. The reader will re- 
 member (see 1 6) that the arc offers a reverse E.M.F. as well as a 
 resistance. Hence the expression ' equivalent resistance ' has not the 
 simple meaning of Chapter XIII. ; its exact meaning will be dis- 
 cussed in 22. It must carefully be noticed that we do not here 
 in the least imply that under all conditions we could substitute p 
 for the lamp without altering the current ; but only that under the 
 given conditions the one is equivalent to the other. 
 
 21. The Reverse E.M.F. of the Arc, If by means of some 
 form of quadrant electrometer we examine the two carbons after 
 the arc has been established, we find a large AV existing between 
 them, varying in general from 25 to 55 volts. 
 
 Now if we consider any two points in a circuit, there will be a 
 certain AV between them. If r a be the resistance between these 
 two points, V a the AV between them, and C the current, and if 
 there be no source of JZ.M.F. between these points, then by 
 Ohm's law we have V a = r a . C. In such a case we get double of 
 V a if we choose points such that the resistance between them is 
 double of r at e.g. if on a uniform wire we take two points sepa- 
 rated by double the former distance, C being constant. 
 
 But in the case of the arc we find that the AV is nothing like 
 doubled if, when C is maintained constant, we double the length 
 of the arc. In fact it is found that the AV between the carbons, 
 or V M consists of two parts ; the one part is a reverse E.M.F. e M 
 while the other part follows Ohm's law and is measured by r a C, 
 where r is the true resistance of the arc. M. Edlund, in his ex- 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 441 
 
 periments, found that the total A V between the carbons was thus 
 composed. We may then write .......... 
 
 understanding that of the two constituents of V a by far the larger 
 is^ a . 
 
 M. Edlund even detected the reverse E.M.F. e a as existing for 
 a fraction of a second after the current had ceased to flow. 
 
 It is worthy of remark that we have here a somewhat new case 
 of transformation of energy. We expend activity to the amount 
 of C . e aj against the reverse E.M.F. e a ; and we get an equivalent, 
 not in chemical activity as in an electrolytic cell, nor in me- 
 chanical activity as in a motor, but in heat activity. 
 
 22. 'Equivalent Resistance' in an Arc lamp. Consider 
 the terminals A and B of an arc lamp, the AV between these 
 terminals being V fl . 
 
 Now if E be the E.M.F. of the dynamo, the fall V a occurs in 
 the lamp, and the remaining fall (E V fl ) occurs in the rest of 
 the circuit. If we remove the lamp from between its terminals 
 and insert the equivalent resistance (.^"definition in 20), the AV 
 V a must remain unaltered, otherwise the fall of potential through 
 the rest of the circuit, and therefore the current, would be altered, 
 which would be contrary to the definition of equivalent resist- 
 ance. 
 
 In the first case, therefore, we have ........ ,. 
 
 V a = e a + r a C; 
 and in the second case we have ............ 
 
 Whence p = = 
 
 Now it is found that V a does not vary in proportion to the 
 current C, but that, within certain limits of current, it remains 
 faiily constant. Hence, as we hinted in 20, p is not a constant 
 at all, but it varies inversely as. C approximately, within certain 
 limits of C. 
 
 Example. One form of arc lamp used in the Brush system 
 
442 ELECTRICITY CH. xxv u 
 
 requires ten amperes current. The AV between its poles, or V M 
 is about fifty volts. Here, then, we have ': 4 
 
 p = ^=-5 ohms. 
 10 
 
 Hence, on the assumption that a current of ten amperes is to 
 be maintained, we may in our calculations substitute five ohms 
 resistance for each lamp. 
 
 23. Series Arrangement of Arc Lamps. Since the reverse 
 E.M.F. of a lamp varies from zero, when the carbons are in con- 
 tact, to fifty volts or so when the lamp is in full power, it would be 
 impossible to arrange arc lamps in parallel. We could not insure 
 any equality of distribution even for a moment, and the system 
 would be eminently unstable. 
 
 We must, therefore, arrange the lamps in series, so as to insure 
 the same current passing through each. Even then there are 
 difficulties that have been overcome by means of many very 
 ingenious regulating contrivances. 
 
 We must consequently employ machines of very high E.M.F. ; 
 and, if we desire to avoid the waste that ensues if the armatures 
 have a high resistance, we must obtain this high E.M.F. in part 
 at least by exceedingly rapid rotation of the armature (see formula, 
 Chapter XXIII. 2). It is usual to employ alternate current 
 machines, as these can be readily constructed to give high E.M.F.s, 
 and as we thereby waste the carbons equally. 
 
 As an example let us suppose that we wish to light twenty Brush 
 lamps in series. With the necessary current of ten amperes we 
 may count each lamp as equivalent to five ohms. Hence, if E be 
 the required E.M.F., we must by Ohm's law have . . . . >,..> 
 
 E == C R = 10 x (5 x 20 + R') Tolts, 
 
 where R' is the resistance offered by the dynamo and the rest 
 of the circuit. If R' be about 10 per cent, of the total resistance, 
 we must have 
 
 ( I00 + } = 
 \ 9 / 
 
 E = 10 
 
 9 / 9 
 
 Dangers from currents of high E.M.F. This explains the 
 dangerous character of the currents employed in arc lighting as 
 compared with those employed in incandescent lighting. If a 
 person touch with one hand a wire conveying a current at such a 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 443 
 
 high E.M.F. as the above, he may form a branch circuit in conse- 
 quence of leakage to earth ; and at this high E.M.F. a fatal 
 current may pass through him. The effects of alternate currents 
 on the human frame are far more serious than are those of con- 
 tinuous currents. Of course contact with two hands is even 
 more dangerous. Instant death has occurred in both these ways. 
 There are of course fire risks, due mainly or entirely to the igni- 
 tion of wires by the passage of successive currents. 
 
 For an account of the various means by which arc lamps are 
 regulated, we must refer the reader to more technical works. To 
 such works we must also refer him for descriptions and figures 
 of other forms of electric lamps, as \he Jab loch koff candle, Jamins 
 lamp, Werdermanris lamp, the Sun lamp, &c. 
 
 24, Further Remarks on the Use of Arc Lamps. 
 
 For the reason given before, viz. that it is only the radiant energy 
 of high temperature that is useful for illumination, arc lamps are more 
 economical than incandescent lamps ; but of course this holds only 
 when the areas to be illuminated are sufficiently great to demand 
 such powerful centres of illumination. Small arc lamps are not 
 economical, nor indeed can they be readily kept steady ; hence for 
 houses the incandescent is the only practicable light. On the same 
 principle it is more economical, though often not at all convenient, to 
 expend our horse-power on one large lamp than on several small ones. 
 One obvious consideration is as follows : that as we must have arc 
 lamps in series, and as two smaller lamps in series will give a 
 combined reverse E.M.F. that is very much greater than that of one 
 large lamp, the same current that will maintain the latter in full action 
 may be totally unfit to maintain the former. 
 
 In order to insure economy in working it is, as a rule, better to 
 construct dynamos specially for the particular work for which they are 
 intended, and not to use the same dynamo for a large and for a small 
 number of lamps in series. 
 
 MORE RECENT MEASURING INSTRUMENTS. 
 
 25. Current-meters or Ammeters. As was observed at the 
 end of Chapter XVII., practical measuring instruments must be 
 independent of the earth's magnetic field, and must not be 
 affected by such magnetic disturbances as are of usual occur- 
 rence. 
 
444 ELECTRICITY CH. xxv. 
 
 We require two classes of such instruments. Firstly, the 
 necessarily more delicate ' standard instruments,' by means of 
 which we can from time to time test and calibrate the ordinary 
 * working instruments ' ; and secondly, a rougher and more portable 
 type of instrument, whose construction can be varied to suit 
 various requirements. 
 
 Standard instruments. Our standard instrument should be 
 delicate, uninfluenced by external magnetic disturbances, and 
 capable of measuring large or small currents. Moreover its de- 
 flexions should be proportional to the currents causing them ; 
 and its ' constant ' should not change perceptibly in (say) a few 
 months. 
 
 All these requirements are met very fairly indeed by a modi- 
 fied form of a permanent-magnet instrument, an ' improved 
 Deprez-d' Arsonvale galvanometer. ' 
 
 In this instrument a coil of fine wire, wound on a copper 
 frame for damping purposes (see 27), is suspended between the 
 poles of a permanent horse-shoe magnet whose lines of force lie 
 more or less in the plane of the coil. The current is conveyed to 
 and from the coil by means of very fine wires [Professors Ayrton 
 and Perry use very fine flat wires of phosphor-bronze] which 
 give practically no torsion-couple for usual deflexions of the coil. 
 Embedded in the coil, and lying along the lines of force when 
 the coil is in its zero position, are a number of small needles of 
 soft iron. 
 
 When a current passes through the coil, the field acts on the 
 coil with a deflecting couple tending to set it at right angles to its 
 'zero-position' ; and it also acts on the soft iron needles with a 
 couple that tends to restore the needles and coil to their zero- 
 position ; and the final deflexion, indicated, as shown on p. 284, 
 by lamp, minor, and scale, is such as gives equilibrium between 
 these two couples. The particular value of getting rid of a tor- 
 sional restoring couple, and substituting for it a magnetic restoring 
 couple, is that even considerable changes in the magnetism of 
 the permanent magnet produce no change in the value of the 
 deflexions ; since both deflecting and restoring couples are affected 
 alike. 
 
 The galvanometer is usually ' shunted ' off the main circuit ; 
 and, by varying suitably the resistance of the shunt and that in 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 445 
 
 the galvanometer circuit, we may arrange the instrument so as to 
 measure currents of any strength in the main circuit [see Chapter 
 XIII. 10]. The instrument is calibrated by electro-chemical 
 means ; details of such calibration will be found in practical text- 
 books. 
 
 It then serves to caliorate other instruments ; these instru- 
 ments being placed in the main circuit, currents of various strengths 
 measured by the [shunted] standard instrument being passed 
 through them, and their readings being noted. 
 
 Ammeters in which soft iron is used. In a very large number 
 of ammeters the field due to the current causes movement in a 
 piece of soft iron, and this causes an index to move. The de- 
 flecting moment so caused is balanced usually by a torsional 
 couple, or by the moment of a 'weight' about an axis. In most 
 of such instruments there may be very large errors due to resi- 
 dual magnetism, if currents of very different magnitudes be 
 measured soon after one another. Such errors can best be 
 obviated by having the iron used both very soft and very thin. 
 
 Solenoidal Ammeters. More accurate, though usually less port- 
 able and more expensive, are instruments in which current passes 
 through two coils, one of which is movable. Such an instrument 
 was alluded to in Chapter XVII. 15. Here the action is between 
 the two coils only ; there is no soft iron, and therefore no errors 
 due to residual magnetism. 
 
 Sir W. Thomson has devised accurate instruments based upon 
 this principle. 
 
 ' Deflexional ' and ' zero' instruments. We may for some 
 purposes divide measuring instruments into two classes : 
 
 (i. ) those in which we observe the deflexion of an index, 
 
 (ii.) and those in which a measured force or couple brings the 
 index back to zero. 
 
 Concerning (i.) it may be said that they enable us to see at 
 once the magnitude of the current [or other quantity to be 
 measured], and to observe changes in it. But the range of 
 movement of the index, and therefore the range of measurement, 
 is usually not great ; and the divisions on the scale are usually 
 not very proportional to the current [&c.j, since the arm of the 
 couple acting on the coil or piece of soft iron usually changes 
 owing to its movement in the field. 
 
446 ELECTRICITY CH. xxv. 
 
 Concerning (ii.) it may be said that we get a far larger range of 
 measurement by a suitable choice (e.g.) of torsion wire ; and, as 
 there is no movement in the field, the deflecting couple, and 
 therefore also the restoring couple, is proportional to the current 
 or to the square of the current. But it is inconvenient not to be 
 able to observe at once changes in the current. 
 
 By means of a ' magnifying spring ' Professors Ayrton and 
 Perry have contrived to make deflexional instruments in which 
 the index has a long range of movement, and yet the piece of soft 
 iron moves but very slightly. Thus these instruments possess in 
 a large measure the two main characteristic advantages of the 
 ' zero type ' of instrument. 
 
 26. Voltmeters. Most voltmeters are practically simply 
 ammeters in which the resistance is very great as compared with 
 the resistance lying between the points whose A V we wish to 
 measure [see Chapter XIV. n]. It is clear that if the resist- 
 ance of the voltmeter changes, the A V between the terminals 
 that answers to a certain deflexion will change also. This is a 
 source of error in all such voltmeters. 
 
 The Cardew Voltmeter. If we have a wire so shielded that it 
 is not subject to variable currents of air, then its rise in tempera- 
 ture and consequent elongation will depend only upon the magni- 
 tude of the current traversing it ; that is, upon the A V existing 
 between its terminals 
 
 On this principle Cardew constructed his ' heated wire volt- 
 meter.' One of its great advantages is that it can be used for 
 alternating currents also. 
 
 [It is evident that if the wire is to change in temperature 
 readily, and to heat and cool rapidly, it must be very fine, and 
 therefore be of high resistance. Hence the Cardew instrument, 
 based upon the above principle, is adapted for a voltmeter, but not 
 for an ammeter]. 
 
 By another ingenious application of their ' magnifying spring ' 
 Professors Ayrton and Perry have contrived to produce an instru- 
 ment of this type that is of much smaller size and greater sensi- 
 tiveness, and in which the waste of electric energy is much less. 
 
 27. < Damping ' in Ammeters and Voltmeters. It is usually 
 of considerable importance that the index of the instrument 
 should rapidly come to rest 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 447 
 
 The method of 'damping' by means of induced eddy- cur- 
 rents [see Chapter XXII. 2] is often employed. Thus ordinary 
 needle-galvanometers usually have copper plates under the needles; 
 in some instruments the needle is of such a shape that it can 
 be entirely enclosed in a copper cylinder ; and, in the Deprez- 
 d'Arsonvale instrument, described in 25 of this chapter, the 
 movable coil is wound on a copper frame that moves with it in 
 the magnetic field. In some of their more recent instruments 
 Professors Ayrton and Perry have made use of the ' viscosity of 
 air ' ; and, by attaching to the index a vane fringed with fine 
 feathers, have succeeded in getting a very dead-beat action. 
 
 28. Wattmeters. It is often desirable to measure the watts 
 expended in some particular portion of a conductor, as e.g. in an 
 incandescent lamp. We might measure separately the AV in 
 volts between the extremities of the conductor, and the current in 
 amperes flowing through it, by means of voltmeters and ammeters 
 respectively. But instruments have been devised which will give 
 us the watts more directly. 
 
 As it is necessary to see such instruments in order to under- 
 stand them thoroughly, we here give merely a representative 
 diagram, illustrating the principle. 
 
 Let L represent an incandescent lamp, and A B its terminals ; 
 let Eg be the AV between its terminals, and C the current 
 flowing through it. F is a coil of fine wire of relatively very high 
 resistance ; so that the current C' that flows through it is insigni- 
 ficant as compared with C. Of course C is proportional to E, 
 assuming that the resistance of F does not alter perceptibly. 
 
 E is a coil cf stout wire, through which passes the whole cur- 
 rent (C+C). 
 
 Kl 1 1 1 1 _ _ 
 
 OF 
 
448 ELECTRICITY CH. xxv. 
 
 In reality, the coils F and E are part of one and the same 
 instrument ; and one of the two, say F, is movable. If the 
 instrument is of the 'zero type' we can bring back F to its zero- 
 position by a measured force or torsion-couple. The electro- 
 magnetic force that we thus balance and measure is of course 
 proportional to the product C x (C + C') ; that is, very nearly to 
 C x C', since C' is relatively small so that (C ) 2 can be neglected. 
 But C' is proportional to E^. Hence our measured force or 
 torsion-couple is proportional to the watts, E^ x C, expended in 
 the lamp. 
 
 Mr. Swinbourne has devised a means of eliminating the small 
 error mentioned above. 
 
 As regards the fine coil F, the wattmeter is subject to error 
 due to change of resistance with changing temperature. 
 
 29. Integrating Instruments. Coulomb-meters. Though 
 it is generally recognised that it is energy, or (quantity) x (difference 
 of potential), that should be measured and paid for, and not 
 quantity alone, yet instruments designed to measure the total 
 quantity supplied with varying current-strength are worth noticing. 
 We can, however, give but a brief description of one type only. 
 Of course if the A V between the * leads,' where they enter the 
 house, remains constant, then the total quantity supplied will be 
 proportional to the total energy supplied ; and this may very well 
 be the case where the house is supplied for incandescent lighting 
 only, since for this the A V should be constant. 
 
 Dr. Arorfs 'Pendulum' Coulomb-meter. This instrument is 
 in detail somewhat complicated ; but the principle of its action, 
 which is all that we can give here, is easily explained. A clock, 
 that under ordinary conditions keeps accurate time, has its bob 
 composed mainly or partly of a. permanent magnet of very hard 
 steel. The current passes round a solenoid of thick wire, so 
 placed as to give a vertical magnetic field symmetrical with respect 
 to the pendulum-bob when this latter is in its lowest position. 
 The direction of the current and the polarity of the bob are such 
 that when the current passes there is a force directed vertically 
 upwards, acting on the bob. 
 
 When the current passes, then, the practical effect is to change 
 the gravitational constant ^ into some slightly smaller value (gk\ 
 The upward magnetic force acting on the bob must be very small 
 
CH. xxv. VARIOUS APPLICATIONS OF ELECTRICITY 449 
 
 as compared with its weight ; or k must be very small as compared 
 with g. And the field must not be so strong as to alter the 
 strength of the magnet. When these two conditions hold it can 
 be shown that the increase of the time T of a complete oscillation 
 [due to the weakening of the total downward pull on the bob] is 
 (approximately) in direct proportion to the current-strength. 
 
 Hence the amount of 'time lost' by the clock during each 
 minute is also directly proportional to the average current-strength 
 during that minute. And so, finally, the total 'time lost ' by the 
 clock during the day will be a measure ot the total quantity of 
 electricity that has passed during that day. [In infinitesimal nota- 
 tion it will measure C.dt.] 
 
 30. Integrating Instruments. Energy-meters It is, how- 
 ever, a far more important matter to measure total energy supplied. 
 Professors Ayrton and Perry have devised an instrument by means 
 of which this may be done. In general respects it is like the 
 instrument last described ; but the bob of the pendulum is a coil 
 of fine wire of relatively very high resistance [see Chapter XIV. 
 n]. The terminals of this coil are connected with the two 
 ' leads ' by which the electrical energy to be measured is being 
 supplied ; and thus at each moment the current C' passing 
 through this fine coil is directly proportional to the A V between 
 these ' leads.' The main current C, in comparison with which C 
 is negligible owing to the ' relatively very high resistance ' of the 
 fine-wire bob, passes through two fixed coils of very thick wire 
 situated one on each side of the bob when this latter is in its 
 lowest position. 
 
 Again, as in 29, we have the practical effect that the gravita- 
 tional constant g is altered into (gk). But in the present case the 
 value of k is proportional to the product CC', and not to C alone ; 
 
 that is, is proportional to the product * * .* 
 
 (current) x ( A V between the leads) ; 
 
 or to the watts supplied to the house at each moment. Hence 
 the ' loss of time ' during each minute is a measure of the watts 
 supplied during that minute ; and the total ' loss of time ' in the 
 day is a measure of the total electric energy supplied during the 
 
 day. [In infinitesimal notation it will measure I W. dt, where W 
 
 represents the watts at the time /.] 
 
 G G 
 
450 ELECTRICITY CH. xxv. 
 
 31. Testing of Coulomb-meters and Energy-meters. By 
 
 means of ammeters and voltmeters it is not difficult to see how the 
 instruments just described can be tested and their constants 
 found. But a description of the manner in which this is done in 
 practice would be out of place here indeed, the student should 
 go through a laboratory course if he wishes to understand the 
 details of the use and calibration of measuring instruments. 
 
 DISTRIBUTION BY ALTERNATING CURRENTS AND 
 1 TRANSFORMERS.' 
 
 32. General Principle of the System. One of the most 
 important problems in electrical engineering is how to supply a 
 district with electrical energy from a central station, without either 
 an undue percentage of waste in the ' leads ' from having too thin 
 wires, or undue expense from having these too massive. 
 
 Now, in Chapter XXIV. 8 we saw that power could be 
 transmitted from a distance along thin leads with economy and 
 on a sufficiently large scale, provided that we used a high ' driving 
 E.M.F,' in the dynamo, and also a high ' reverse E.M.F. ' in the 
 motor. And, again, we could attain the same desired result in the 
 case of incandescent lamps if only their total resistance were very 
 great as compared with the resistance of the leads. As, however, 
 incandescent lamps are worked 'in parallel,' their combined resist- 
 ance will not be relatively high if the leads be long and thin. 
 
 The system named above aims at solving the problem in what 
 we may call a third way ; and we will briefly describe the principle 
 of this system, it being now of considerable importance. 
 
 Let us suppose that there are upon the same soft iron core 
 two coils of wire ; one composed of many turns of very fine wire, 
 and the other of fewer turns of very thick wire ; and let an 
 alternating current be sent through the fine-wire coil. 
 
 Then we have an instrument that resembles a Ruhmkorft's 
 coil, in which the long fine coil is chosen *s> primary, and the short 
 thick coil as secondary. Instead, therefore, of obtaining small cur- 
 rents of high E.M.F. by means of large currents of low E.M.F., 
 we conversely obtain large currents of low E.M.F. by means of 
 small currents of high E.M.F. 
 
 When the secondary circuit is closed there will be [by mutual 
 
CH, xxv. VARIOUS APPLICATIONS OF ELECTRICITY 45 1 
 
 induction] a reverse E.M.F. induced in the primary. Thus if E 
 be the E.M.F. of the dynamo, and the reverse E.M.F. in the 
 
 primary be <?, while the resistance of the whole primary circuit is 
 
 -Q e 
 
 R, there will be a current, C, measured by -, and this may be 
 
 JK. 
 
 very small. The energy given each second to the secondary cir- 
 cuit will be C e waffs. Thus we have 
 
 /C E watts supplied by dynamo, 
 
 I C (Ee) watts wasted in heat, 
 
 \ C e waffs usefully expended in induction on the secondary, 
 
 and - will measure the efficiency. 
 I E 
 
 Note. In all the above we have wished only to give the principle of the 
 action. Hence we have used the simple symbols C, E, and e, as if we were 
 dealing with steady currents. To deal properly with alternating currents is a 
 matter for a more advanced book than the present. 
 
 It is found that instruments of this kind, called for obvious 
 reasons ' Transformers,' can be so constructed as to give a high 
 efficiency, and also [which is a different matter] plenty of current in 
 the secondary circuit. We can thus, for reasons referred to in 
 Chapter XXIV. 8, use ' leads ' of thin wire and small cost 
 without much percentage waste. The central station supplies the 
 whole circuit with an alternating current of high E.M.F. ; and 
 each house can have a transformer, suited to its requirements, 
 from the secondary of which it draws its supply. 
 
 GG 2 
 
QUESTIONS AND EXAMPLES. 
 
 CHAPTER I. 
 
 1. It is sometimes found that two magnetic poles, which at some distance 
 apart repel one another, will attract one another when they are brought nearer. 
 Account for this. 
 
 2. Explain the statement that ' repulsion of the pole of a magnetic needle 
 is the only certain test that the body presented to it is a magnet. ' 
 
 3. You are required to convert the blade of a pen-knife into a magnet, 
 having \h& point for its &?2<M-seeking, or , pole. How would you do this ? 
 
 4. How could you show experimentally that in the case of action between 
 a magnet and a piece of iron the law that ' action and reaction are equal and 
 opposite ' holds good ? 
 
 5. How would you treat a needle if you wished to give it a north -seeking 
 pole at each end, and a wtt&i-seeking pole at the centre? 
 
 6. Give any reason that occurs to you to account for the fact that white-hot 
 iron exhibits no magnetic properties. 
 
 7. A number of soft iron needles are floating in a vertical position on small 
 and separate bits of cork in a vessel of water ; they are very close to one another. 
 A powerful magnetic pole is held above the group. Mention, and explain, 
 any vertical and horizontal movements that will take place. 
 
 CHAPTER II. 
 
 1. A stone traverses 2 kilometres in i^ mimites. What is its velocity in 
 C.G.S. units? 
 
 2. In a certain moving body there is observed a steady increase of velocity ; 
 the velocity, added each 3 minutes, being a velocity of 30 metres per 20 
 seconds. What is this acceleration in C.G.S. units ? 
 
 3. A mass of 2^ kilogrammes loses, each 2.\ mimites, a velocity of 300 
 metres per 2 hours. Express in dynes the force acting. 
 
 4. Two forces, of 30 dynes and 40 dynes respectively, act on a material 
 particle at right angles to one another. Find the magnitude of their resultant, 
 and the angle that it makes with the direction of the smaller force. 
 
 5. A force of 5000 dynes is resolved into others acting at right angles to 
 one another. One of these is 3000 dynes. Find the other component, and 
 the angle that it makes with the original force. 
 
- XXIV ELECTRICITY 
 
 6. One couple has an arm of I metre, and each force is 600 dynes. Another 
 couple in an opposite direction has each force 20 dynes. What must be its 
 arm in order that it may just balance the former ? 
 
 7. In a magnetic needle, the pole has a strength of 20 C.G.S. units, and 
 the length between the poles is 12 cms. What is the magnetic moment of 
 the needle ? 
 
 8. In a certain magnetic field a pole of strength 13 is urged with 20 dynes 
 force. What is the strength of the field ? 
 
 9. A magnetic field has strength 12. In it is placed a needle of magnetic 
 moment 10, so as to be at right angles to the lines of force. With what couple 
 is the needle urged ? Explain the italicised expressions. 
 
 10. A pole of strength 20 is placed in a magnetic field in which another 
 pole of strength 6 was urged with 13 dynes force. With what force is the 
 former pole urged ? And what is the component of this force in a direction 
 making 30 with the lines of force of the field ? 
 
 CHAPTER III. 
 
 1. In a field of strength 20 is placed a magnetic needle of pole-strength 6, 
 and length 8 cms., lying at right angles to the line of force. W 7 hat is the 
 magnitude of the couple with which the field acts upon the needle ? 
 
 2. In the same field is placed a needle of magnetic moment 15, making 
 an angle of 30 with the lines of force'. What is the couple in this case ? * 
 
 3. There is a soft iron mast in a certain ship, and the ship's compass is 
 placed between the foot of the mast and the stern of the vessel. Describe in 
 general terms how the presence of the mast will affect the indications of the 
 needle in each of the following cases. 
 
 (i. ) At the magnetic equator, when the ship lies N and S, E and W, S and 
 N, and N W and S E respectively. 
 
 (ii. ) In latitudes about 45 north, for the same positions of the ship re- 
 spectively. 
 
 (iii.) At the north magnetic pole. 
 
 Hence, make a comprehensive general statement as to the general nature 
 of the disturbances due to such a vertical iron mast. 
 
 4. Discuss in a similar manner the action, on the needle, of a horizontal 
 beam of soft iron running right across the vessel between the compass and 
 the bows. 
 
 5. Explain, with formulae also, how to use the torsion balance so as to com- 
 pare two magnetic fields ; assuming the magnetic moment of the needle to 
 remain constant. Point out clearly the importance of this assumption. 
 
 6. How would you use the torsion balance to compare the magnetic moments 
 of two needles ; assuming that each can be in turn attached to the thread of the 
 same torsion balance ? Use formulae also. 
 
 7. What considerations justify the statement that it is a more complicated 
 matter to compare the magnetic moments of two needles by the method of 
 oscillations than by the method of (6) ? 
 
QUESTIONS AND EXAMPLES XXV 
 
 8. A small magnetic needle, placed at a certain distance from the pole of a 
 magnet, is found to make 12 oscillations per minute ; the lines of force due to 
 the magnet having, in the neighbourhood of the small needle, the same direction 
 as those due to the earth. When in the same position with respect to the pole 
 of another magnet that gives just twice as strong a field as the former, it makes 
 15 oscillations per mimtte. ' How many oscillations per minute would it make 
 in the earth's field alone ? , 
 
 9 A needle makes 9 oscillations per minute in a certain magnetic field. 
 How many will it make when re-magnetised so that its magnetic moment is 
 half as great again as before ? 
 
 10. Assume that in a certain room the earth's field has betn neutralised, 
 and that there a small needle makes 20 oscillations per minute when 4 inches 
 distant from the pole of a magnet. How many will it make when 3 inches 
 distant ? [Neglect here the other pole of the magnet.] 
 
 n. Two poles of strength + 10 and 8 respectively are placed 20 centi- 
 metres apart. Find in dynes the stress between them ; interpreting the sign. 
 
 12. If it be desired for the purpose of experiment to neutralise as far as 
 possible the earth's field about a needle, how would you do it ? And how 
 would you test whether or no the desired result had been attained ? 
 
 13. How would a truly-balanced dip-needle behave under the following 
 conditions ? Its plane of swing is in the magnetic meridian ; and a bar-magnet 
 is placed in a horizontal position symmetrically on one side of it, with the 
 <?rM-seeking pole pointing north and the .ww^-seeking pole pointing south. 
 
 14. A needle, oscillating at two different places, makes 80 oscillations in 3 
 minutes, and 100 oscillations in 4 minutes ', respectively, when acted upon by 
 the earth's horizontal field only. Compare the horizontal fields at the two 
 places respectively. What assumption with respect to the needle do we here 
 make ? 
 
 15. Given that the earth's horizontal field has a strength measured by -5, 
 and that the dip is 30, at a certain place ; required, (i. ) the total field-strength, 
 (ii. ) the vertical component, at that place. 
 
 1 6. A needle vibrates in a horizontal plane under the earth's horizontal 
 field, and makes 16 oscillations per minute. How many oscillations per minute 
 would it make where the horizontal field is stronger in the ratio of 9 to 4 ? 
 
 CHAPTER IV. 
 
 1. A person is seen to bring a rod near to a magnetic needle, and this 
 latter is thereby deflected. What further experiment would you direct him to 
 perform in order to determine whether the observed action is magnetic or 
 electrical ? 
 
 2. Some mercury contained in a dry glass vessel is connected by a wire 
 with a gold-leaf electroscope. The end of a dry glass rod (unexcited) is 
 gently dipped into the mercury and is then suddenly withdrawn. Describe 
 and explain what happens in the two cases respectively. 
 
 3. A soap-bubble is blown at the end of a narrow insulated tube, and the 
 
XXvi ELECTRICITY 
 
 other end of this tube is then closed. Explain what occurs when the whole is 
 charged with electricity. 
 
 4. Discuss the statement that 'repulsion is the only certain sign of 
 electrification.' 
 
 5. Suggest any experiments tending to prove that an electric charge resides 
 on the surface only of an electrified conductor. 
 
 6. Given two charged bodies, and that it is required to determine whether 
 or no their charges are equal and of opposite signs. Explain why it is an 
 accurate test to drop them both into a hollow conducting vessel placed upon 
 the plate of the gold-leaf electroscope, and not an accurate test to lay the two 
 bodies on the plate. 
 
 7. A small insulated conducting sphere charged with 4 units of + electricity 
 is at a distance of 2 cms. from an identical sphere charged with 2 units of 
 electricity. What is the stress between them, in dynesi And what will be 
 the stress if they are made to touch one another and are then removed to the 
 same distance apart ? 
 
 8. How would you show that a charge + Q induces on surrounding 
 earth-connected surfaces an equal and opposite Q ? 
 
 9. A sheet of tin-foil is rolled upon an ebonite rod and is electrified so that 
 a pair of pith-balls, suspended from the foil by cotton threads, diverge widely. 
 On unrolling the foil it is observed that the balls diverge less widely. Explain 
 this action by reference to definite electrical principles. 
 
 10. A charged conductor is placed, insulated, inside a hollow conductor 
 also insulated. The latter is put to earth for a moment and is then re-insulated 
 When the charged conductor is removed, in what condition is the hollow con 
 ductor left ? Suggest any further experiment needed to prove the truth of your 
 statement. 
 
 11. You are given a charged and insulated body A; a hollow insulated 
 conducting vessel B ; and a larger hollow insulated conductor C. The vessel 
 B can be placed entirely inside C without touching it, and A can be placed 
 inside B without touching it. Show how it is possible to give to C any multiple 
 of the charge on A, and this without bringing A into contact with anything. 
 
 (Note. The methods of 16 are to be followed.) 
 
 CHAPTER V. 
 
 1. How is the distribution on an insulated conductor affected by .... 
 (i. ) The shape of the conductor ? 
 
 (ii. ) The position of surrounding bodies? 
 
 Is the potential of the conductor affected by these conditions; and, if so, 
 in what way ? 
 
 2. A sphere of gun-cotton and a metal sphere of much larger radius, both 
 uncharged and insulated, are rubbed together and then separated. What will 
 be the condition of the two respectively, both as regards quantity of charge and 
 as regards potential ? 
 
QUESTIONS AND EXAMPLES xxvii 
 
 3. (Note. In the following it will be well to draw first a figure of the 
 arrangement described.) 
 
 Two thin gold leaves are suspended within an ordinary glass flask A, by 
 a wire which is fitted air-tight in the neck of the flask and is provided with a 
 metal knob outside. The flask is placed in a larger metal vessel B which is 
 connected with earth ; the vessel B reaches up to the level of the top of the, 
 flask, but the knob projects above B. A wire-gauze cylinder C, closed at one 
 end, is inverted over the flask, resting on its shoulders ; this gauze cylinder is 
 insulated from B and from the knob, and the gold leaves in A hang down 
 below the level of the lower edge of C. State and explain what will take 
 place 
 
 (i.) when a + electrified glass rod is held over the system ; 
 
 (ii.) when water is then poured into B until it finally touches the lower 
 edge of C. 
 
 4. Explain, using a diagram, under what circumstances a conductor may 
 be said to be ' isolated.' 
 
 5. Explain, with a diagram, why the ' capacity ' of such a conductor [con- 
 sidered alone] increases when it is brought near to the walls of a room. 
 
 CHAPTER VI. 
 
 1. A thin sheet of ebonite, coated on its upper side with tin-foil extending 
 to within some distance from the edges, is laid upon a conducting table. The 
 tin-foil is struck with dry flannel, and some pith-balls are dropped upon it. 
 A person then lifts the ebonite from the table, holding it by its edges only. 
 Explain what- happens at each stage in this experiment. 
 
 2. A person holds a charged Leyden jar by its outer coating, and brings 
 its knob into contact with the knob of a similar uncharged jar which another 
 person holds by its outer coating. Describe and explain what happens. 
 
 3. A jar is charged -fly while its outer coating is to earth. It is then 
 placed upon an insulating stand, and its outer coating is charged + . Discuss, 
 in a general manner, the final condition of the jar. 
 
 CHAPTER X. 
 
 1. Two conductors, A and B, are insulated and are placed near to one 
 another. 
 
 (i. ) A is charged to a potential + V. What will be the potential condi- 
 tion of B ? 
 
 (ii. ) If A be charged to a sufficiently high potential it is observed that at 
 last a spark will pass between A and B. What does this fact prove with^ 
 respect to the rise of potential of B during the charging of A ? 
 
 (Note. We here assume that the passage of a spark indicates the attain- 
 ment of a certain difference of potential between A and B. ) 
 
 2. Two small pith-balls are each of mass I gramme. They are suspended 
 from the same point by silk fibres each 10 centimetres long, and of negligible 
 
XXV111 ELECTRICITY 
 
 mass. They are then electrified so that each string is inclined 30 to the 
 vertical. How many units of electricity must each ball possess ? 
 
 {Note. Take moments about the point of suspension, equating the elec- 
 trical moment and the gravitational moment, g being assumed to be 981.) 
 
 3. Two small balls, each of mass I gramme, are suspended from the same 
 point by silk fibres, each of which is 490^ centimetres in length. Taking to 
 be 981, show that when each ball is charged with a unit of electricity they will 
 diverge to a distance of I centimetre from each other. 
 
 (Note. This example is taken from Professor Garnett's lectures.) 
 
 4. Three insulated conducting spheres, A, B, and C, whose radii are re- 
 spectively i, 2, and 3 centimetres, are so charged that their respective poten- 
 tials are 3, 2, and I units of potential. They are then connected by wires of 
 negligible capacity. What is the total quantity of charge, what is the total 
 capacity, and what the final common potential of the system ? 
 
 5. A small insulated charged sphere is placed near the knob of an electro- 
 scope, this latter being previously uncharged. Explain why the gold leaves 
 diverge. What will happen if we interpose 
 
 (i.) an ebonite plate ; 
 
 (ii. ) an insulated and uncharged brass plate ; 
 
 (iii.) an un-insulated brass plate, 
 
 respectively, between the charged sphere and the electroscope ? 
 
 6. Find the capacity of an insulated sphere that is 10 centimetres in 
 diameter, it being surrounded by a concentric sphere io - 2 centimetres in 
 internal diameter. 
 
 7. Find in absolute electrostatic units the capacities of 
 
 (i. ) an isolated sphere of 10 centimetres radius ; 
 
 (ii.) the same when the ' large room,' in which the sphere is placed, is filled 
 with shellac (specific inductive capacity <r) ; 
 
 (iii. ) a spherical condenser in which the two radii are 10 centimetres and 
 1 1 centimetres respectively, air being the dielectric ; 
 
 (iv.) a cylindrical Leyden jar in which the radius is 5 centimetres, the height 
 of the tin-foil 20 centimetres, the thickness of glass -I centimetre, and the 
 specific inductive capacity of the glass is 1 76. (The jar is not coated with tin- 
 foil at the bottom ; only round the side.) 
 
 8. A Leyden jar has a coated surface of 100 square centimetres, the specific 
 inductive capacity of the glass is i '9, and the thickness is '2 centimetres. What 
 would be the diameter of an isolated metal sphere of the same capacity as this 
 jar? 
 
 9. A condenser consists of two parallel plates. When there is air between 
 them, the one plate A is put to earth, and the other B is charged to a poten- 
 tial V. Next the plate A is also insulated, and a sheet of ebonite is introduced 
 between the two plates. How is the potential of each plate affected ? 
 
 10. A sphere is charged, and is placed in the middle of a 'large' room. 
 The lines of force and equipotential surfaces are marked out in the usual 
 manner. The room is now filled with a medium whose specific inductive 
 capacity is greater than that of the air or other medium that previously filled 
 
QUESTIONS AND EXAMPLES xxix 
 
 the room. Discuss and explain any change in the system of lines offeree and 
 equipotential surfaces ; these being marked out on the same principle as 
 before. 
 
 n. Two small conductors are charged; and, when placed a certain dis- 
 tance apart in air, they repel each other with a certain force. They are then 
 immersed in anhydrous paraffin oil, having their charges and distance apart 
 unaltered. Explain in a general way what alteration will be observed in the 
 stress between them. 
 
 12. The capacity of a conductor is K, and it is charged with Q units of 
 electricity. How many ergs work were done in charging it ? 
 
 13. An insulated conducting sphere of given radius receives a given charge. 
 Another sphere of equal radius, uncharged, is made to touch the former, and 
 is then removed. Show, in symbols, how to find the energy of discharge 
 attending the contact between them. 
 
 14. Discuss the case of the last section when the radii of the spheres are 
 unequal. 
 
 15. What main alterations must be made in the wording of 9-16 in 
 order that the results there given may apply also to the magnetic fields ? 
 
 1 6. What will be the magnetic potential of a point situate at 10 centimetres 
 from the + pole, and 15 centimetres from the pole, of a bar-magnet, the 
 pole-strength of this magnet being 40 in absolute measure ? 
 
 CHAPTER XL 
 
 1. A zinc vessel having a hole at the bottom is filled with finely powdered 
 copper. Underneath this vessel is an insulated copper cylinder open at both 
 ends, but provided internally with a metal tongue that reaches across from 
 side to side near the centre of the cylinder. The former vessel is put to 
 earth, and the powdered copper drops from the opening, strikes the metal 
 tongue of the cylinder, and then falls clear to the ground. What will be the 
 condition of the insulated cylinder when the action has proceeded for some 
 time? [Assume the view of 4.] 
 
 2. A plate of pure zinc is immersed in dilute sulphuric acid. A little 
 copper-sulphate is then mixed with and dissolved in the dilute acid. It is 
 observed that copper is instantly deposited upon the zinc, and that after this 
 hydrogen is given off freely. Of what nature is this latter action ? Explain ; 
 comparing with any similar action with which you are acquainted. 
 
 3. Into a platinum dish is poured some acidified solution of an antimony 
 salt, and then a piece of zinc is dropped in. The platinum is found to acquire 
 a coating of metallic antimony. Explain this action. 
 
 4. In Question 2, what will be the action if more copper-sulphate solution 
 be poured in ? 
 
 5. In Question 3, what will be observed if there be plenty of acid, but 
 very little antimony salt, in the solution ? 
 
 6. Explain, with a figure, how you would arrange matters if you wished to 
 
XXX ELECTRICITY 
 
 try whether an electric current could be obtained from two given liquids and a 
 single given metal. 
 
 7. Under what circumstances does a ' constant ' battery become polarised 
 and fall off in power ? 
 
 CHAPTER XII. 
 
 1. A copper dish contains an unacidulated solution of copper-sulphate, 
 and in it is suspended a plate of lead so as not to touch the copper. The dish 
 is connected with the pole of a battery, and the lead with its + pole. 
 
 (i. ) What changes take place ? 
 
 (ii. ) What happens when the battery is removed, and the copper dish is 
 connected by a wire with the lead plate ? 
 
 2. The poles of a single DanielPs cell are connected with two platinum 
 plates immersed in dilute sulphuric acid, and a galvanometer is included in 
 the circuit. What is observed (i. ) the moment that the circuit is completed ; 
 (ii. ) and after a time ? If we remove the DanielPs cell and complete the 
 circuit without it, what is observed ? 
 
 3. In the last question vhat difference does it make if we employ two 
 Daniell's cells arranged in series ? 
 
 4. If a ' storage battery ' be charged by a dynamo running at full speed, 
 and if then the speed of the dynamo (and therefore its electro-motive force) fall 
 off, the connections remaining unaltered, what may happen ? 
 
 5. How would you prepare a secondary battery to give an E.M.F. greater 
 than that of the primary battery employed in the charging ? 
 
 6. How would you construct a secondary battery if you required a current 
 at a moderate E. M. F. but to last a long time ? 
 
 7. Compare the action and theory of a secondary battery and of a Leyden 
 jar respectively. 
 
 8. Given a number of uncharged secondary cells capable of giving an 
 E.M.F. of i voit, a large primary cell of E.M.F. of I -5 volt, and an electro- 
 lytic cell whose reverse E.M.F. exceeds 1*6 volt, show how you can decom- 
 pose the electrolyte in this cell. 
 
 9. Criticise the common expression e storage of electricity ' as applied (i. ) 
 to the Leyden jar, (ii.) to the secondary cell, respectively. Give any analogies 
 that may occur to you, applicable to the two cases respectively. 
 
 10. Discuss whether there is or is not a true distinction to be made between 
 primary and secondary batteries. 
 
 CHAPTER XIII. 
 
 Preliminary notes. (\,~) In what follows we shall occasionally use the fol- 
 lowing units, of which a further account will be given in Chapter XIV. ; 
 viz. the ohm for resistance, the volt for E.M.F., and the ampere for current. 
 They are connected with one another (by Ohm's law) in the following very 
 simple manner. 
 
 Current in amflres = E.M.R in volts 
 resistance in olutis 
 
QUESTIONS AND EXAMPLES XXXI 
 
 (ii. ) In all graphical examples we may, without causing any error in the 
 results obtained, consider the E.M.F. of a cell to be represented by an abrupt 
 rise in potential occurring at one place in the cell. 
 
 If a volt and an ohm are represented by one and the same length marked 
 off on the axes of ordinates and of abscissae respectively, then the value of 
 the current in amperes is given by tan a, where a is the angle made by the line 
 representing the fall of potential with the axis of abscissae. 
 
 For most purposes, unless the contrary is requested, the student may repre- 
 sent the total E. M. F. in the circuit by a single abrupt rise of potential occurring 
 at one place in the circuit. 
 
 1. A cell has a tangent galvanometer of zero resistance, and nothing else, 
 in its circuit ; the deviation is 30. When 9 more similar cells are added in 
 series, the deflexion remains unaltered. Show how this experiment tends to 
 confirm Ohm's law. 
 
 2. Two cells, of equal E.M.F. but of unequal resistances, are joined up in 
 series, but with opposed poles, and a galvanometer is included in the circuit. 
 Explain in words and in symbols your reasons for stating that a current will or 
 will not pass. 
 
 3. A circuit includes a battery, some wire, and an electrolytic cell consisting 
 of dilute acid in which are immersed platinum electrodes. Discuss the applica- 
 tion of Ohm's law to such a case. 
 
 4. What will happen if 
 
 (i. ) one point only in a battery circuit, 
 
 (ii.) two points simultaneously 
 
 be put to earth respectively ? Discuss whether or no two points could be found 
 which might be put to earth simultaneously without the current being affected. 
 
 (Note. This question is perhaps answered most easily by the help of the 
 graphical method.) 
 
 5. A cell of E.M.F. e } gives a deflexion 0, ; and gives a (smaller) deflexion 
 2 when a resistance r^ is added to the circuit. Another cell of E. M. F. e., is 
 substituted for the first, and the resistance is so adjusted that we have the 
 deflexion 0, again ; it is then found necessary to add a resistance r 2 in order 
 to reduce the deflexion to 2 . Show that , : <? 2 = r\ : r. z . 
 
 6. Given 12 voltaic cells, each of an internal resistance 15 ohms, explain 
 in how many different ways they may all be (conveniently) arranged. Which 
 arrangement will give the greatest current when the external resistance is 
 20 ohms ? 
 
 7. A single Bunsen's cell was found to give the same current through a 
 galvanometer as did two DanielPs cells joined in series. If now the Bunsen's 
 cell and the Daniell's battery be included in the same circuit, but with their 
 E.M.F. s opposed, will there necessarily be no current? Explain in words 
 and in symbols. 
 
 8. There is a circular ring of uniform wire. Two points in it, 135 apart, 
 are connected with the opposite poles of a battery. Compare the currents in 
 the two segments respectively. 
 
XXXli ELECTRICITY 
 
 9. In the last case, if the resistance of the whole ring be taken as unity, 
 what is the ' equivalent resistance ' between the two points A and B ? 
 
 10. What is the best arrangement of 6 cells, each of ohm resistance, 
 when the external resistance is 2 ohms ? 
 
 11. In what sort of cases can two different arrangements of a given number 
 of cells give the greatest current possible through a fixed external resistance ? 
 
 12. Compare the two currents obtained in the following cases respectively. 
 (i.) Two cells, each of 3 ohms resistance, are joined in series, and the 
 
 circuit is completed through a wire having 20 ohms resistance. 
 
 (ii. ) The same two cells are. joined in parallel, and the circuit is completed 
 through the same wire, but this time doubled on itself. 
 
 13. If you wished to have a bichromate battery that would have a small 
 internal resistance, and would yield a moderate current for a relatively long 
 time, what instructions would you give to the makers ? 
 
 14. It is required to drive a current through a fixed external resistance r, 
 and there are at your disposal an indefinite number of Leclanche cells, each of 
 E.M.F. E and resistance B. What is the limiting value of the current that you 
 can drive through r by means of any number of these eel's all arranged in series ? 
 
 15. If in the last question you may arrange the cells in any way you please, 
 is there a limit to the magnitude of the current ? Explain fully. 
 
 16. Given an external resistance r that is very small as compared with the 
 resistance B of a single cell, and a number of cells. Would you, in order to 
 obtain the greatest possible current, arrange all your cells in parallel, however 
 many cells there may be ? 
 
 17. In tlie last case find the limiting value of the current obtainable by 
 coupling an unlimited number of the cells in parallel. 
 
 18. There are two cells A and B, of E.M.F.s E A and E B respectively, 
 where EA is greater than E B . They are placed in circuit with a tangent 
 galvanometer (see Chapter XVII. 5). When they are opposed, they give a 
 deflexion a./ ; and when they drive the current in the same direction they give 
 a deflexion o,. Find the ratio EA '. E B . 
 
 (Note. fat the following ' graphical ' exercises, see the note at the begin- 
 ning of these questions on Chapter XIII.) 
 
 19. There is a cell of E.M.F. 2 volts, and internal resistance I ohm. If 
 the external resistance be 2 ohms, exhibit graphically the condition of the 
 circuit when closed. 
 
 20. There is a circuit of 12 ohms total resistance, this including a battery 
 of 6 volts E.M.F. The resistance between the points A and B is 2 ohms. 
 Find graphically the difference of potential between A and B. 
 
 21. In the last case the points A and B are joined also by a second wire 
 of '6 ohm resistance. Find algebraically .' 
 
 (i. ) the equivalent resistance between A and B ; 
 
 (ii.} the alteration in total current produced by the introduction of the 
 second wire ; 
 
 (iii. ) the difference of potential between A and B. 
 
QUESTIONS AND EXAMPLES XXXl'ii 
 
 22. There are arranged in series six cells each of E.M.F. 2 volts and of 
 internal resistance I ohm. Represent graphically the condition of the circuit 
 
 (i. ) considering the whole E.M.F. to occur abruptly at one point ; 
 (ii. ) considering the E.M.F. of each cell to occur abruptly at the point 
 where the copper terminal is soldered to the zinc plate of each cell. 
 
 23. A battery has a total E.M.F. of 20 volts, and an internal resistance of 
 12 oluns. It is in circuit with a storage battery of reverse E.M.F. of 4 volts, 
 and resistance 2 ohms. The resistance of the rest of the circuit is 3 ohms. 
 Considering the total E.M.F., i.e. the algebraic sum of the E.M.F.s, to occur 
 abruptly at one point, represent graphically the condition of the closed circuit. 
 
 CHAPTER XIV. 
 
 1. If the conductivity of copper is to that of aluminium as 2-3 : I, and 
 their specific gravities are as 3*3 : I, show what would be the ratio between 
 the masses of two wires made of these two metals respectively, which for the 
 same length offered the same resistance. 
 
 2. To measure the resistance of sulphuric acid a "Wheatstone's bridge is 
 sometimes employed ; but the galvanometer is then replaced by a telephone 
 (see Chapter XXV.), and the battery is replaced by an induction coil or other 
 arrangement giving rapidly alternating currents. Explain the object aimed at 
 in these modifications, and state how you would conduct the experiment 
 indicated. 
 
 3. A Smee's cell is joined up with a tangent galvanometer (see Chap- 
 ter XVII. 5) of negligible resistance ; the rest of the external circuit having 
 also negligible resistance. A deflexion of is observed where tan 9 = 2-328. 
 When resistances of 775 ohm and of 2-325 ohms are successively (not 
 simultaneously) introduced into the circuit, the corresponding angles of 
 deflexion had tangents equal to -966 and -445 respectively. From these data 
 calculate two values for the internal resistance B of the cell, to four places of 
 decimals ; showing that these two results agree approximately. 
 
 4. The density of silver is 10-5, and the resistance of a silver wire I metre 
 in length and of I gramme mass is -1689 ohm. Find the specific resistance 
 of silver in ohms. 
 
 (Note. This means the resistance in ohms of a piece of silver I centimetre 
 long and I square centimetre cross section ; the current passing between two 
 opposite faces.) 
 
 5. Two separate pounds of copper of the same quality are drawn out into 
 uniform wires, the one twice the length of the other. Find the ratio between 
 their resistances. 
 
 6. A current of not less than '016 ampere is to be sent through an external 
 resistance of 360 ohms. What is the smallest number of Leclanche cells, each 
 with E.M.F. 1-4 volts and resistance 15 ohms, by which this can be effected ? 
 What would be the maximum strength of current obtainable if double this 
 number of cells were used? 
 
XXXIV ELECTRICITY 
 
 7. The zinc pole ol a Darnell's cell being joined to the platinum pole of a 
 Grove's, the other poles are connected up with a tangent galvanometer, and 
 the current is found to be '5665 ampere. The zinc pole of the former is now- 
 connected with the zinc pole of the latter, the two positive poles are connected 
 with the two terminals of the galvanometer, and the current is found to be 
 0875 ampere. The resistances and E. M.F.s being supposed constant, find 
 the ratio between the E. M.F.s of the two elements. 
 
 8. An iron wire -24 centimetre in diameter has a resistance of 7 '8 ohms 
 per mile. What is the diameter of another wire of the same material which 
 for the same length has a resistance of 12-87 ohms'*. 
 
 9. How would you use Wheatstone's bridge to ascertain the nature of the 
 effect produced by rise of temperature on the specific resistance of any 
 material ? 
 
 1C. How could you further ascertain the percentage change in resistance 
 for rises in temperature from o to 10 C, 10 to 20 C, &c. ? 
 
 II. The internal resistance of a cell is '5 ohm. When a tangent-galvano- 
 meter (see Chapter XVII. 5) of I ohm resistance is coupled with it by 
 wires of negligible resistance, the deflexion is 15. What will be the deflexion 
 if the connecting wires have 1*5 ohms resistance ? 
 
 (Note. A* table of ' tangents' must be used.) 
 
 CHAPTER XV. 
 
 1. I observe some zinc dissolving in dilute acid, and I notice that the 
 amount of heat calculated as due is not evolved. Explain how this may very 
 well be the case, and yet the law of ' Conservation of energy ' be unbroken. 
 
 2. From noticing the heat-energy evolved when a current passes through 
 a fine wire, should you be justified in speaking of ' electricity ' as a ' source of 
 energy ' ? Explain clearly. 
 
 3. A current of 130 amperes flows through an armature whose resistance 
 is -5 ohm. How many horse-power are wasted in the armature ? 
 
 4. Assuming that one horse-power is equivalent to 746 volt-amperes, and 
 also to 33,000 foot-pounds per minute ; and assuming that 1390 foot-pounds are 
 equivalent to a. pound-degree- Centigrade unit of heat ; find the number of pounds 
 of water that can be heated through i C. by the heat emitted in I hour by a 
 lamp through which a current of I ampere flows, the difference of potential 
 between the terminals of the lamp being 60 volts. 
 
 5. Two cells have internal resistances each equal to 3 ohms. In one case 
 they are connected in series by a wire of 3 ohms resistance. In another case 
 they are joined in parallel by the same wire. Compare the total heat evolved 
 per second in the two cases respectively, and show in each case its distribution 
 betwee* cells and wire. 
 
 6. A given wire, forming the whole external circuit of a battery, has re- 
 sistance of 3 ohms. You are provided with 8 cells, each of I ohm resistance 
 and E.M.F. 2 volts. How will you arrange the cells so as to make the wire 
 as hot as possible ? How many calories will then be evolved per second in the 
 
QUESTIONS AND EXAMPLES XXXV 
 
 wire ? And what -work per second (in ergs per second} will be the equivalent 
 of this ? 
 
 7. If you wish to heat a platinum wire, at a distance from the battery, to 
 as high a temperature as possible, what sort of connecting wires will you use, 
 and why ? And what arrangement of battery-cells will you adopt ? 
 
 8. If in the last case the insulation of the conducting wires was very im- 
 perfect, show whether it would be better to increase the number of cells arranged 
 in series, or the number arranged in parallel ; supposing that you have some 
 additional cells at your disposal. 
 
 9. A silver wire is joined end to end with an iron wire which is of the same 
 length, but of double the diameter and of six times the specific resistance of 
 the former. The whole is put into circuit with a battery, and when the 
 current has passed for a certain time it is found that the total heat evolved in 
 the two wires is 45 calories. How is this shared between the two wires ? 
 
 10. The difference of potential between two points in a circuit is 50 volts, 
 the resistance between them is I ohm, and the current is 10 amperes. Show 
 that there must be a reverse E.M.F. between the two points, and find its value. 
 
 11. In the last example, (i.) what is the work per second (in watts] done 
 against this reverse E.M.F. ; and (ii.) what is the heat per second (in calories 
 per second] evolved between the two points ? 
 
 12. In a certain circuit there is a battery of E.M.F. 20 volts, and resistance 
 2 ohms ; a wire of resistance I ohm ; and an electrolytic cell of resistance 
 I ohm, and reverse E.M.F. 2 volts. Find 
 
 (i. ) the watts expended by the battery ; 
 
 (ii. ) the calories per second evolved in the battery, in the wire, and in the 
 electrolytic cell, respectively ; 
 
 (iii. ) the work per second (in watts'], of a chemical nature, spent on the 
 electrolytic cell. 
 
 CHAPTER XVII. 
 
 1. If you have a sine-galvanometer unprovided with shunts, and a tangent 
 galvanometer, which would you use for the measurement of very feeble 
 currents, and which for the measurement of very strong currents ? Explain 
 your answer. 
 
 2. What if the sine-galvanometer be provided with shunts and compensa- 
 ting resistances ? 
 
 3. Given a delicate galvanometer and a current of unknown strength, how 
 would you so proceed in your measurement of this current as to obviate any 
 risk of injury to the galvanometer due to too great strength of current passing 
 through the galvanometer ? 
 
 4. You are given a set of constant cells ; a galvanometer whose law of 
 deflexion is complex and unknown ; a Wheatstone's rheostat ; and a tangent 
 galvanometer. Show how you could graduate the first-named galvanometer 
 so as to make it afterwards available for measuring, and not merely indicating, 
 currents. 
 
 H H 
 
XXXvi ELECTRICITY 
 
 5. Suppose that you have thus calibrated your galvanometer up to a con- 
 siderable angle of deflexion, and that you desire to use it for larger currents. 
 Show how you can do this if you are provided with a shunt whose resistance 
 bears a known ratio to the resistance of the galvanometer. 
 
 6. If you do not know this ratio, show how you can find it by the use of 
 the above apparatus. 
 
 7. How would you determine the ' compensating resistance ' to be inserted 
 when the shunt is used ? 
 
 8. In the above series of experiments (4-7), does it matter whether the 
 E. M. F. of the battery is constant or no ? Explain your answer. 
 
 (Note. Questions 4-8 were suggested by, or taken from, the electrical 
 course described in Worthington's ' Physical Laboratory Practice.') 
 
 CHAPTER XVIII. 
 
 1. A single turn of wire in the form of a circle of 10 centimetres radius is 
 placed in the plane of the magnetic meridian, and two turns of wire of 20 
 centimetres radius are placed with their common plane and centre coinciding 
 with those of the first circle. A small magnet is suspended at the common 
 centre in the plane of the coils. The same current is now .passed in one 
 direction through the single wire, and in the other direction through the 
 double wire, respectively. What effect is produced on the needle ? Explain 
 in words, and prove your answer in symbols. 
 
 2. A current is passed along a permanent magnet of hard steel, magnetised 
 in the direction of its length. Give reasons for arguing that the current will 
 take a spiral course along the bar, and show whether the spiral will form a 
 right-handed or a left-handed screw. 
 
 3. A current is passed along a bar of perfectly soft iron. Give reasons for 
 arguing that the molecules will now form closed magnetic circuits, which' are 
 circles having their planes perpendicular to the direction of the current. And 
 show in what direction the north- seeking and .wz^-seeking poles of each 
 molecule will point. 
 
 4. A circular coil of wire is suspended from some height, with its plane 
 vertical, so that it can both swing and rotate, and a current is passed through 
 it. Discuss the action of the pole of a magnet on the coil in various cases. 
 
 5. If it were true that the earth's magnetism is due to currents traversing 
 the earth's surface, show what would be their general direction. 
 
 6. An elastic spiral of wire hangs so that its lower end just dips into a 
 vessel of mercury. Describe and explain what happens when the top of the 
 spiral is connected with the one pole of a battery, and the mercury is connected 
 with the other pole of the same. 
 
 7. A coil of 50 turns of wire has a mean radius of 20 centimetres^ and a 
 current of 6 amperes traverses it. Find, in absolute C.G.S. units, the strength 
 of the magnetic field produced at the centre of the coil. 
 
 8. If. in the last example, the coil be placed in the plane of the magnetic 
 
QUESTIONS AND EXAMPLES xxxvii 
 
 meridian, and if the earth's horizontal field be of strength 4 2 in absolute 
 measure, find the tangent of the angle of deflexion of a small needle balanced 
 (so as to move horizontally) at the centre of the circle. 
 
 9. There is a circle of one turn of wire. A current of C absolute units 
 traverses it. What is the ' strength ' p t of the equivalent magnetic shell ? 
 
 10. A circular coil consists of n turns of wire, and is I centimetre thick, 
 and the current is C. If the equivalent magnetic shell were also I centimetre 
 thick, what would be the surface density p of its magnetism ? 
 
 11. In a similar case the thickness of the coil is not given ; but you are 
 told that the wire is coiled at the rate of n turns per I centimetre of thickness. 
 If the thickness of the equivalent magnetic shell be the same as that of the 
 coil, what must be its surface density p of magnetism ? 
 
 CHAPTER XX. 
 
 1. \Vhatarethebestmaterialsrespectivelyfor 
 
 (i. ) magnetic needles ? 
 
 (ii. ) the cores of electro-magnets ? 
 
 (iii. ) the wire forming the coils of an electro-magnet ? Explain each 
 answer. 
 
 2. If a solenoid be made of wire carrying a current C in absolute measure, 
 there being n turns of wire per i centimetre of length, what would be the 
 surface density p of magnetism on the faces of the equivalent bar-magnet ? 
 (See Chapter XVIII. Question ii above.} 
 
 3. It is generally assumed, as sufficiently accurate when the helix is long 
 as compared with its diameter, that the field-strength throughout the interior 
 of the helix is constant and is measured by 4 TT n C. What will be the surface- 
 density p of magnetism at the ends of an iron core inserted in the helix, the 
 * coefficient of magnetisation ' being k ? 
 
 4. If the helix and core have a common cross-section of A square centi- 
 metres, what is * 
 
 (i. ) The pole-strength of the helix alone? 
 (ii. ) That of helix and core together ? 
 
 5. Applying the results of Chapter X. 14, 15, &c., to the case of 
 magnetism, and assuming that all the resultant ' marked lines of force ' proceed 
 outwards from the face of a magnet, show what will be the number of re- 
 sultant ' marked lines ' due . 
 
 (i. ) To the pole of the helix alone ; 
 (ii. ) To the pole of the iron core alone. 
 
 CHAPTER XXL 
 
 i. If two wires, attached to the terminals of a Grove's battery of three or 
 four cells, be held in the hands, and be alternately placed in the hand and 
 separated, nothing is felt. But if a large electro-magnet be in the circuit, then 
 a shock is felt when contact is broken. Account for this. 
 
 H H a 
 
Xxxviii ELECTRICITY 
 
 2. A bar of perfectly soft iron is thrust into the interior of a coil of wire 
 whose terminals are connected through a galvanometer. An induced current 
 is observed. Explain how this may be. 
 
 3. Could the coil and bar be placed in such a position that the above action 
 might nearly or entirely disappear ? Explain fully. 
 
 4. A plane rectangular iron frame is placed vertically so that it faces due 
 'magnetic north.' It then is overthrown, falling northward into a horizontal 
 position. Discuss, in a general manner, what occurs. 
 
 5. In the last example assume some exact values, in absolute C.G.S. units, 
 for . .' V -. - .' 
 
 (i. ) the area of the frame ; 
 (ii. ) the dip and strength of the earth's field ; 
 (iii. ) the time occupied in the fall ; 
 and from these data deduce the average induced E. M.F. 
 
 6. A current flows through a wire, and a magnetic pole is consequently 
 urged into rotation round the wire. If the battery be now removed and be 
 replaced by a delicate galvanometer, and if the pole be moved by hand in the 
 same direction as before, what will be observed ? Of what laws is this an 
 example ? 
 
 CHAPTER XXII. 
 
 1. In constructing an induction coil, what special contrivances must -be 
 introduced in .order to obtain long sparks ? 
 
 2. An insulated wire is coiled round a bar of copper, and a current is 
 momentarily passed through the wire. What effects are produced ? 
 
 3. In the last case it is found that, while the current is flowing, there is 
 some resistance felt to the withdrawal of the copper bar from within the coil. 
 To what general law may this phenomenon be referred ? 
 
 4. A wire is coiled round a bar-magnet, and a bar of soft iron is brought 
 near to one pole of the magnet. Explain whether or no there will be a current 
 induced in the coil, and, if there be such a current, show in what direction 
 it will run. 
 
 5. In passing a current from an induction coil through a * vacuum-tube ' 
 the current passes in one direction only. How is this ? And how can you 
 determine what is this direction of current ? (Give two methods, at least.) 
 
 6. The number of oscillations made by a given needle in one and the same 
 place, in a given time, is affected by placing a copper plate under it. Explain 
 this, and point out what practical us.e can be made of the fact. 
 
 7. The poles of a voltaic battery are connected with two mercury-cups. 
 These cups are connected successively by ...., 
 
 (i. ) a long straight wire ; 
 
 (ii. ) 'the same wire arranged in a close spiral, the wire being covered with 
 some insulating material ; 
 
 (iii. ) the same wire coiled round a soft iron core. 
 
 Describe and discuss what happens in each case when the circuit is broken. 
 
QUESTIONS AND EXAMPLES xxxix 
 
 8. In the automatic ' make-and-break ' spiral, described in Question 6 of 
 Chapter XVIII. p. 458, discuss the effect of introducing a copper core that 
 does not touch the spiral. 
 
 9. In the same case discuss the effect of a. similar soft iron core. 
 
 10. In the same case discuss the effect of a similar core consisting of hard 
 steel magnetised longitudinally, with one or the other pole respectively at the 
 lower end. 
 
 CHAPTERS XXIII., XXIV., and XXV. 
 
 1. How should a dynamo, intended for electro-plating work, be constructed ? 
 Explain your answer. 
 
 2. Explain in what cases it would be dangerous for a person 
 
 (i. ) to touch with one hand only the wire in circuit with a dynamo ; 
 
 (ii. ) to touch the same at two different points with the two hands simul- 
 taneously. 
 
 3. The heat supplied per second to a boiler is equivalent to 750 kilogram- 
 metres of work. One-tenth of this is usefully employed in driving a dynamo, 
 the efficiency of which is 75. The current drives a motor that is also of effi- 
 ciency 75> an< ^ ~io per cent, of the energy of the dynamo's current is lost in the 
 form of heat. Find the work per second of the motor in kilogram-metres per 
 second. 
 
 4. A dynamo has a constant E.M.F. of 200 volts, and is in circuit with a 
 motor, the total resistance of the whole circuit being 6 ohms. Find .... 
 
 (i.) the current when the motor is at rest ; 
 
 (ii. ) the current, activity used in the motor, total activity, and activity 
 wasted in heat, when the motor is run at such a rate that there is 'maximum 
 work ' per second done ; 
 
 (iii. ) the current, total activity, and activity used in the motor, when this 
 is run at such a rate that the efficiency is 90 per cent. 
 
 5. It is desired to measure the resistance of a certain incandescent lamp 
 when the carbon is rendered white-hot by a current of I -3 amperes. It is found 
 that then there is a difference of potential of 65 volts between the terminals of 
 the lamp. What is the required resistance ? 
 
 6. A cable a mile long, whose ends are A and B, has a total resistance of 
 3-59 ohms. It is put into water, and a small hole is pierced to the core at a 
 certain point. When the end B is insulated and the resistance is tested from 
 the end A, this is found to be 2'8i ohms. When A is insulated, the resistance 
 tested from the end B is found to be 276 ohms. Find the distance in yards 
 of the puncture from the end A, and its resistance in ohms* 
 
xl ELECTRICITY 
 
 ANSWERS TO QUESTIONS. 
 
 CHAPTER I. 
 (4) Float the two on a piece of cork, on water. 
 
 CHAPTER II. 
 
 (I) 2222?. 
 
 (2) |. 
 
 (3) 
 
 (4) <p dynes-, I 
 
 ? = tan - 1 f . 
 
 (5) 
 
 (6) 30 metres. 
 
 (7) 240. 
 
 (8) 
 
 (9) 120. 
 
 (10) 43</> 
 
 'nes ; 2i 
 
 (5) 4000 dynes ; 6 = tan ~ T 
 
 CHAPTER III. 
 
 (I) 960. (2) 150. 
 
 (6) Deflect each to same angle with lines of earth's 'horizontal field.' 
 
 (7) The moments of inertia of the needles enter into the formulae of 
 calculation. 
 
 (8) -v/63 = 8, nearly. (9) n, nearly. (10) 26-6. 
 (n) '2 dynes ; there is attraction. 
 
 (12) By means of a large magnet suitably placed. 
 
 (14) Ratio is 256 : 225 ; magnetic moment of needle assumed constant. 
 
 (15) _L; _L,. (16)24. 
 
 CHAPTER IV. 
 
 (7) Attraction, 2 dynes ; repulsion, '25 dyne. 
 
 CHAPTER V. 
 
 (3) (i.) A and C are at a + potential, B at a zero potential, and the leaves 
 diverge with +. (ii. ) BC all at zero potential, and the leaves hang together. 
 
 CHAPTER VI. 
 
 (3) Both outside and inside are raised in potential, their difference of 
 potential remaining (practically) constant. 
 
ANSWERS TO QUESTIONS xli 
 
 CHAPTER X. 
 
 (i) B is at a + potential, but lower than A's. That, as we charge A, the 
 potential of B rises more slowly than does that of A. 
 
 (a) Q* = te 
 <3 
 
 (4) Q = 10 ; total K = 6 ; common potential = I '6. 
 
 (5) Divergence increases ; divergence decreases ; leaves collapse. 
 
 (6) K = 255. (7) 10 ; 10 a ; no ; 880. 
 
 (8) Diameter = Z5 centimetres. 
 
 (9) That of A rises, that of B falls, somewhat. 
 
 (10) The force on a + unit at a given distance will be less; hence th& 
 * marked lines ' will be fewer, and the equipotential surfaces further apart. 
 
 Q2 
 
 11 I ) It will decrease. (12) -t- ergs. 
 
 (15) Merely substitute magnetic unit pole for electric unit quantity, and take 
 infinity, not ' earth,' as region of sero potential. 
 
 (16) ii 
 
 CHAPTER XL 
 
 (1) Its potential will have fallen. 
 
 (2) Copper is deposited by chemical substitution, and then the whole acts 
 as a closed cell. 
 
 (3) We have a closed cell, and the nascent hydrogen sets free antimony on 
 the platinum plate. 
 
 (4) More copper is set free by the nascent hydrogen. 
 
 (5) Hydrogen will be set free from the platinum plate as in an ordinary 
 closed cell. 
 
 (7) If the current be too * dense,' i.e. too great per unit area of the plates. 
 
 CHAPTER XII. 
 
 (1) (i.) Copper deposited on copper dish, lead dioxide on the lead plate, 
 (ii.) A back current, and a return (at least partial] to the initial condition. 
 
 (2) A deflexion, and then a return to zero. A deflexion the other way, and 
 a return to zero. 
 
 (3) A continued deflexion. 
 
 (5) By suitable changes in method of coupling. 
 
 (8) Same method as in Question 5. 
 
 CHAPTER XIII. 
 
 (i) Easily proved by formula. 
 
 (5) Use formula for current in each case. 
 
 (6) In six main ways. 4 in series and 3 in parallel. 
 
xlii ELECTRICITY 
 
 (7) Depends on the respective internal resistances, as well as on E.M.F.s. 
 
 (8) Ratio is 3 : 5. (9) 15. 
 
 (10) Either 6 in series, or 3 in series and 2 in parallel. 
 
 (12) , and , amperes respectively. (14) Limit is C = -. 
 
 J 3 : 3 B 
 
 (15) There is no limit. Prove by formula. 
 
 p 
 
 (16) No. Prove by general formula. (17) Limit is . 
 
 r 
 
 , ON EA tan a, + tan a n 
 
 (18) _* = --. (20) i volt. 
 
 E B tan a, - tan a 2 
 
 (21) (i.) -5 ohm ; (ii. ) from *5 ampere to f ampere ; (iii. ) f volt. 
 
 CHAPTER XIV. 
 
 ^ Mass of copper . _ 3-3 
 Mass ot aluminium 2 '3 
 
 (3) The two answers are B = '5496, and B -= -5494, ohms respectively. 
 
 (4) i -608 + io fi , nearly. (5) Ratio is 4 : i. 
 
 (6) 5 cells in series ; -027 amperes, nearly. 
 
 (7) E.M.F.s have ratio of 479 : 654. (8) -187 centimetre, nearly. 
 
 (11) 7 37'- 
 
 CHAPTER XV. 
 (3) 11-32 h. p., nearly. (4) 115 /Af. nearly. 
 
 (5) (i.) Heat in first case : Heat in second case = 1& E- : - l6 E 2 = 2 i 
 
 3 3 
 
 (ii.) In the first case 3 of the heat is internal and external, while in the 
 second case is internal and external. 
 
 (6) 4 in series, 2 in parallel ; 1-8432 calories per second-, 77,414,400 ergs 
 per second. 
 
 (8) In parallel. ( 9 ) Heat in silver wire = 2_ 
 
 Heat in iron wire 3' 
 (lo) Reverse E.M.F. = 40 volts. 
 ( 1 1 ) 400 watts ; 24 calories per second. 
 
 (12) (i.) 90 -watts\ (ii.) 972, 4-86, and 4-86 calories per second respec- 
 tively ; (iii. ) 9 watts. 
 
 CHAPTER XVIII. 
 
 (I) The needle is unaffected. (4) Consider the equivalent magnetic shell. 
 
 (5) From E to W. 
 
 (6) It will vibrate vertically, making and breaking contact alternately. 
 
 (7) 9-425, nearly. (8) tan = 47-124, nearly. (9) Strength = pt = C. 
 
ANSWERS TO QUESTIONS xliii 
 
 ( 10) p x i = n C, or p = n C. 
 
 ( 1 1 ) Let t be the common thickness, expressed in centimetres, of coil and 
 shell. Then p t = t n C, or p = n C. 
 
 CHAPTER XX. 
 
 (2) + n C and n C at the two ends respectively. 
 
 (3) p = ^irn C x k. (4) (i.) n C A ; (ii. ) nC A + 
 
 (5) (i.)47rwCA; (ii. ) 4 TT * pole- strength = (4*)- x 
 
 CHAPTER XXI. 
 
 (2) Consider the earth's field and the magnetisation of the soft iron due to 
 this. 
 
 (6) This is an example of Lenz's law. 
 
 CHAPTER XXII. 
 
 (2) Eddy currents arise. (3) Induction ; Lenz's law. 
 
 (4) By magnetic induction the field due to the magnet is altered. 
 
 (6) Eddy currents arise, and Lenz's law holds. 
 
 (7) (i. ) Hardly any extra current; (ii. ) decided extra current ; (iii.) still 
 greater extra current. 
 
 (8) Slower vibration. Compare with Question 6. 
 
 (9) Still slower vibration. 
 
 (10) Observe that with suitable pole downwards the vibrations may even 
 be stopped altogether. 
 
 CHAPTERS XXIII. , XXIV., and XXV. 
 
 (1) Small internal resistance, low E.M.F. 
 
 (2) Much depends on whether' or no the dynamo can ' leak ' to earth. 
 
 (3) 37'96 kilogram-metres per second. 
 
 (4) (i-) 33i? amperes ; (ii.) i6f amperes, l666| watts, 3333^ watts, l666| 
 watts ; (iii.) 3^- amperes, 666| watts, 600 watts. 
 
 (5) 50 ohms. (6) 892 yards, nearly; '99 ohm. 
 
INDEX. 
 
 ACC 
 
 A CCELERATION, dimensions of, 13 
 t x Accumulator, electric current, 366 
 
 wa*er-dropping, 102 
 Activity, unit of, 239, 298 
 
 in the voltaic circuit, 240 et seq. 
 
 in the circuit of a dynamo and motor, 
 405 et seq. 
 
 in cases of induction, 347 et seq. 
 jEpinus's condenser, 76 
 
 experiments with, 78 
 
 Air, its specific inductive capacity taken 
 
 as unity," 121 
 Alphabet, Morse, 420 
 Alternate discharge, 83 
 Amalgam, electrical, 46 
 Amalgamation of zinc plate in galvanic 
 
 cell, 173 
 Amber, earliest observations of its electric 
 
 properties, 42 
 Ammeter, or current galvanometer, 291 
 
 calibration of, 293 
 
 different kinds of, 443 
 - ' damping ' in, 446 
 
 Ammonium chloride, electrolysis of, 185 
 Ampere, the practical unit of current, 191, 
 
 r:22, 298 
 Ampere's discovery of the equivalence of a 
 
 magnetic shell and an electric circuit, 306 
 
 laws of the actions between currents, 
 311 ; between elements of currents, 313 
 
 rule for ascertaining directions of lines 
 offeree, 273 
 
 theory of magnetism, 326 
 Angles, solid, 304, 322 
 Anions, 184 
 
 Anode, 183 
 
 Arago's disc and magnetic needle. 363 
 
 Arc, electric, 434 
 
 appearance of, 439 
 
 reverse E.M. F. of, 440 
 
 lamp, 438 
 
 Arc-lighting, series arrangement of lamps 
 
 in, 442 
 
 dangers of, 442 
 
 Armatures of electrostatic induction 
 
 machines, 104 et seq. 
 
 of dynamos. 385 ct seq. 
 
 Clark's, 386 
 
 Siemens's, 387 
 
 Gramme, 389 
 
 Siemens-Alteneck, 395 
 
 Brush, 395 
 
 BUN 
 
 Armatures wound 'for E.M.F.' and 'for 
 
 current," 394 
 
 Aron, coulomb-meter of, 4^8 
 Astatic system of needles for galvane- 
 
 meters, 281 
 Atmosphere, electrostatic potential at a 
 
 point in the, 117 
 
 ' Attracted-disc ' electrometer, 153 
 Attraction and repulsion in magnetism, 2 ; 
 
 in electrostatics, 43, 49 
 occurring in magnetic and diamag- 
 
 netic phenomena, 330, 334 
 Aurora borealis [set: references given ia 
 
 Preface, p. viii.] 
 
 Axis, magnetic, definition of, 29 
 correction for error in the, 33, 36 
 
 BA. ohm, 222 
 Barlow's wheel, 364 
 
 Battery cell, general view of, 171, 196, 
 238 
 
 various forms of, 177 et seq. 
 
 secondary or ' storage,' 196 et seq. 
 
 the economy of, 247 
 
 resistance of, 230 _ 
 
 methods of coupling, 180, 207, 219 
 
 arrangements of, with respect to heat 
 
 evolved, 245 
 
 constant, 177 
 
 molecular interchanges in, when cir- 
 cuit is closed, 187 
 standard, 177 
 
 Bell, electric, 417 
 
 Bell telephone, 429 
 
 Bichromate cell, 177 
 
 Bismuth is diamagnetic, 329 
 
 Bismuth-antimony thermo-electric pile, 
 
 254 
 
 cross, experiment with, 204 
 
 Boats, electro-motor, 411 
 
 Bohnenberger's electroscope, indication of 
 
 differences of potential by, 153 
 'Bound,' the term, as applied to electric 
 
 charges, 81 
 Boxes, resistance, 223 
 Bridge, Wheatstone's, 226 
 Brush discharge, 99 
 Brushes, collecting, of Gramme machine, 
 
 39 2 
 
 Brush machine, 395 
 Bunsen's cell, 178 
 
xlvi 
 
 INDEX 
 
 CABLE, telegraphic, 422 
 Cablegram, 423 
 
 Calcium chloride used to keep electro- 
 scopes dry, 43 
 Calibration of ammeters and volt-meters, 
 
 2 93. 
 
 Calorie, unit of heat so called, 239 
 Capacity, electrical, 68, 74 
 
 formulae for, 142, 146, 148 
 
 Carbon plates of constant battery-cells, 181 
 
 rods of arc lamp, action of current on, 439 
 respective temperatures of, 440 
 
 transmitter, Edison's telephone, 431 
 
 Card piercing by electric discharge, 92 
 
 Cardew, volt-meter of, 446 
 
 Cascade arrangement of Leyden jars, 88 
 
 energy of discharge in the, 152 
 
 Casella, dip circle of, 37 
 
 Cavendish's method of finding specific in- 
 ductive capacities, 122 
 
 Cells [see under Battery-cells, Thermo- 
 cells, and Electrolysis] 
 
 Centimetre, the unit of length. 12 
 
 Centimetre-gramme-second (C. G. S.) sys- 
 tem, 12 
 
 Charge, electric, distribution of, 55, 56 
 
 density of, 55 
 
 nature of, 90, 161, 341 
 
 - density of, distinguished from poten- 
 tial, 144 
 
 energy of, 150 
 
 sign of, distinguished from sign of 
 
 potential, 74 
 
 residual, 91 
 
 bound and free, 81 
 
 exemplified by alternate discharge, 
 
 83 
 
 * Chemical ' theory of voltaic cell, 170 
 
 Circuit, closed, action of a pole on, when 
 circuit is complete, 308 ; incomplete, 310 
 
 closed, laws regarding, 201 
 
 magnetic field due to, 271 et seq. 
 
 equivalent to a magnetic shell, 306 
 
 closure of, 174 
 
 fall of potential through the, 215 
 
 primary, making and breaking, 368 
 
 secondary, of induction coil, its condi- 
 tion when closed, 373 ; with air break, 
 
 Circuits, movement of, general law, 318 ; 
 when the field is uniform, 318 ; when the 
 field is not uniform, 318 ; when the cir- 
 cuit is incomplete, 319 
 
 divided, 211 
 
 magnetic potentials due to, 316, 322 
 
 potential on, due to magnetic field, 317, 
 3 2 3 
 
 mutual potential of two, 359 
 
 Clarke's dynamo-, or magneto-, electric 
 
 machine, 385 
 Clark's (Latimer) standard cell, 177 
 
 potentiometer, 234 
 Clouds, electrical potential of, 117 
 Coefficient of mutual induction, or mutual 
 
 potential, of two circuits, 359 
 w of magnetisation, 329 
 Coils, induction, 344, 368 et seq. 
 caution respecting their use, 371 
 
 resistance, 222 
 
 Combination, heats of, as connected with 
 E.M.F.s, 249 
 
 CUR 
 
 Commutator, simple, 386 
 
 key in needle system of telegraphy, 
 421 
 
 Compass, ordinary, 29 
 
 mariner's, 29 
 
 Condenser, yEpinus's, experiments with, 78 
 Condensers, 76 
 comparison with an isolated body, 83 
 
 capacities of, formulae, 149 
 
 ' investigated by ballistic galvano- 
 meter, 287 
 
 investigated by other methods, 123, 
 
 124 
 
 charge of, dependent on the nature of 
 the dielectric, 83, 121 et seq. 
 
 of induction coil, 372 
 
 used in secondary circuit of induction 
 coil, 375 
 
 use for obviating effects of earth cur- 
 rents in cables, 425 
 
 spherical, the capacity of, 145 
 
 plate, the capacity of, 148 
 Conductivity, 206 
 
 Conductor, perfect, definition of, 70 
 
 electrical capacity of, 68 
 
 hollow, acting as an electrostatic screen, 
 *39 
 
 Conductors and non-conductors, 44 
 Conductors of an electric current, 183 
 
 alterations in capacity of, 74 
 
 distribution of electric charge on, 55 ; 
 according to their shape, 55 
 
 specific resistances ot, 235 
 
 are opaque, 341 
 
 lightning. 113 
 
 for transmitting power from dynamos, 
 410 
 
 Cone, distribution of charge on, 56 
 Contact potential series for metals, 165 
 
 E.M.F.s, 267, 269 
 
 ' Contact ' theory voltaic cell, 170 
 
 Copper-plating, 193 
 
 Copper sulphate, electrolysis of, 185 
 
 Cores of electromagnets, 333 
 
 Coulomb, the practical unit of electric 
 
 quantity, 191, 298 
 Coulomb-meters, 448 
 Coulomb's torsion balance, 20 et seq. 
 
 uses in magnetism, 21 et seq. 
 
 uses in electrostatics, 50 
 
 determinations of distribution on a disc, 
 56 
 
 Couple, on a magnetic needle in a uniform 
 
 field, 16 
 Couples, 14 
 
 Couronne des tasses^ 169 
 Crookes s high vacua ; experiments in, 377 
 
 et ?eq. f 
 
 experiment in a low vacuum, 379 
 Cube, distribution of charge on, 55 
 Current, 202 
 
 follows closure of circuit, 174 
 
 energy of the, 240 
 
 heating effects of, 182, 241 
 
 effects of, on unequally heated metallic 
 conductors, 264 
 
 electro-magnetic unit of, 297 
 
 element of a, 295 
 
 law of its action on a magnetic pole, 296 
 
 reaction of a pole on, 308 
 
 elements of, actions between 313 
 
INDEX 
 
 xlvii 
 
 CUR 
 
 Current, making and breaking, in induc- 
 tion coil, 368 
 
 magnetic actions of, 221, 271, 294 
 
 instruments for measurement of, 276 
 et seq. 
 
 meters for household distribution, 443 
 
 accumulator, 366 
 
 practical unit of, 222, 298 
 
 circular, field due to a, 274, 302 
 
 continuous, collected from Barlow's 
 wheel, 364 
 
 - dynamos furnishing a, 387 
 obtained by induction, 355 
 
 the ' extra,' 360 
 
 maximum with a given number of cells, 
 219 
 
 Currents, action of, on magnetic poles, 
 271, 294, etc. 
 
 attraction and repulsion between, 311, 
 321 
 
 continuous rotations of, 313, 322 
 
 alternate, dynamo-electric machines 
 giving, 399 
 
 - induced, 343 et seq. 
 
 under \\hat conditions obtained, 357 
 
 disturbances caused by, in telephone 
 
 lines, 430 
 
 acting as inducing currents, 362 
 
 constant, 355 
 
 direction ot, 352 
 
 in conducting masses; 'eddy 
 
 currents,' 363 
 
 local, in impure zinc plate, 175 
 
 mobile, action of the earth's field on, 311 
 
 sinuous, action of on a magnetic pole, 307 
 Curves, magnetic, 17 
 
 of work, dynamo-electric, 408 
 Cylinder, distribution of electric charge 
 
 on, 56, 143 
 
 capacity of, 149 
 
 divided, induction experiment with, 54 
 
 electric machine, 60 
 
 Cylinders, magnetic and diamagnetic, 
 setting of in magnetic fields, 330 
 
 ' T^AMPING/observationof, by Arago, 
 J .363 
 
 applied to ammeters and volt-meters, 446 
 Daniell's cell, 179 
 
 molecular interchanges in, on closing 
 
 circuit, 187 
 Declination, magnetic, 30 
 
 method of measuring, 31 
 Declinometer, 32 
 Decomposition of electrolytes, 184 
 
 primary and secondary, 188 
 
 simultaneous, 189 ; exemplifying Fara- 
 day's laws of electrolysis, 190 
 
 ot water [see Water] 
 
 Deflexions, method of, as used in mag- 
 netism, 38 
 
 comparison of, to measure resistance, 
 225 
 
 Degradation of energy, 238 
 De la Rive's floating battery, 306 
 Diamagnetic bodies, list of, 334 
 Diamagnetism, relative, 335 
 
 whether there is absolute, 336 
 Dielectric, nature of -a, 83, 162, 341 
 strain in a, 90 
 
 5 forms of, 98, 377 et seq 
 
 jndary circuit of induction coil, 
 
 ELE 
 
 Dielectrics are the seat of electrostatic 
 potential energy, 161 
 
 specific inductive capacities of, 121 et 
 seq. 
 
 Dimensions of units, 12, 301 
 Dip of magnetic needle, 30 
 
 how to find, 36 
 Dip-circle, 37 
 
 Disc, distribution of charge on, 55 
 Discharge, alternate, 63 
 
 by Bunsen's flame, 48 
 
 mechanical eftecis of, 92 
 
 magnetic effects of, 92 
 
 heating effects of, 93 
 
 induction effects of, 94 
 
 duration of, 96 
 
 energy of, 150 ; in the cascade arrange- 
 ment of Leyden jars, 152 
 
 examples in, 159 
 
 various forms of, 98, 377 et seq 
 
 in secor 
 376 et seq. 
 
 Distribution of electric charge, 55 
 
 laws of, illustrated by the ice-pail, 57 
 
 from the potential point of view, 143 
 
 investigated by aid of electrometer, 158 
 
 by alternating currents and ' trans- 
 formers,' 450 
 
 Dots and dashes, Morse signals, 420 
 apparatus for printing, 421 
 
 Dynamo, work done upon, in relation to 
 velocity of rotation, 415 
 
 Dynamos, various lorms of, 385 et seq. 
 
 activuy expended upon, 405 
 
 as motors, 402 
 
 distribution of potential in circuit of, 
 412 
 
 general remarks on, 404 
 Dyne, definition of, 13 
 
 EARTH, magnetic field of, modified 
 by the presence of iron, 30 
 
 magnetic elements of the, 30 ; how to 
 measure, 31, 37 
 
 action of, on a magnetic needle, 28 
 Earth's field, observation of changes irr 
 
 the, 40 
 action of, on mobile currents, 311 
 
 total field resolved into components, 34 
 
 magnetism, 28 
 Ebonite, a j?o >d 'electric. '43 
 
 how to prevent it deteriorating, 63 
 Eddies in the ether about lines of mag- 
 netic force ('!) 339, 342 _ 
 
 Eddy currents due to induction, 363 
 
 Edison's telephone, 431 
 
 Eel, electric, 169 
 
 Efficiency, dynamo-electric, 408 
 
 Electricity, atmospheric, 113 
 
 distribution of, on conductors, 55 
 
 electrostatic induction machines for, 
 100 et seq 
 
 ' fluid ' theories of, 48 
 
 frictional, machines for, 60 
 
 measurable as a quantity, 65 
 
 specific heat of, 265 
 
 practical applications of, 416 et seq. 
 
 statical, phenomena of, 42 et seq. 
 
 ' current ' ; in what sense to be under- 
 stood, 163 
 
xlviii 
 
 INDEX 
 
 ELE 
 
 Electricity, suggested identity of, with the 
 ether, 341 
 
 positive and negative, 45 ; produced 
 simultaneously and in equal quantities, 
 46, 47 
 
 Electrics and non-electrics, 45 
 Electrification of persons, 64 
 Electrodes, 184 
 
 polarisation of the, 173, 194 
 Electro-dynamometer, 291 
 Electro-etching, 19* 
 Electrolysis, experiments in, 184 
 
 Faraday's laws of, 189 
 
 industrial application of, 191 
 
 nature of, 186 
 
 Electrolyte, molecular interchanges in, 
 
 during passage of current, 187 
 Electrolytes, 183 
 
 Electro-magnetic theory of light, 340 
 Electro-magnets, 33 3 
 
 paramagnetic and diamagnetic, ex- 
 periments with, 334 
 
 Electrometer, measuring E.M.F. of open 
 
 cell by, 232 
 Electrometers, 153 
 
 attracted-disc lorm, 1=13 
 
 quadrant, Sir W. Thomson's, 155 et 
 seg. 
 
 uses of, 158 
 
 Electromotive force [see E.M.F.]. 201 
 Electro-motor, distribution of potential in 
 
 circuit of, 414 
 Electro-motors, 402 
 
 general remarks on, 404 
 Electronegative, the term, 184 
 Electrophori, continuous-acting, looet seg. 
 Electrophorus, 59 
 
 Electro-plating, 191 
 Electropositive, the term, 184 
 Electroscope, condensing, 98 
 
 gold-leaf, 43 
 
 charged by induction, 54, 73 
 Electroscopes, differences of potential in- 
 dicated by, 153 
 
 Electrostatic fields offeree, 69, 131 
 * induction machines, 100 et seg. 
 Electrostatics, duality of phenomena of, 42 
 
 the three laws of, 49 ; expressed in one 
 formula, 52 
 
 'Elliott '-pattern quadrant electrometer, 
 
 *5<S 
 
 Elmo's (St.) fire, 114 
 E.M.F. affected by closure of circuit, 178, 
 
 232 
 
 dependent upon the nature of the cell, 
 202, 250 
 
 induced, general expression for, 349 
 
 measurement of: by electrometer with 
 open cell, 232 
 
 by volt-meter galvanometers, 233 
 
 by method of opposition, 234 
 
 by the potentiometer, 234 
 
 of thermo-cell, 254, 267 
 
 exhibited diagrammatically, 259 et 
 Iff. 
 
 in thermo-cell, sources of, 267 
 
 relation of to difference of potential, 202 
 
 reverse, due to polarisation, 176, 194, 
 196, 246 
 
 reverse, in a thermo-cell, 263 
 
 FOR 
 
 E.M.F., reverse, of an electro-motor, 402 
 of arc lamp, 440 
 
 of currents in arc-lighting, 442 
 
 practical unit of, 221, 298 
 
 absolute electro magnetic unit of, 297 
 E.M.F.s, table of, for different cells, 237 ; 
 
 explaine_d, 2 o 
 in relation to heats of combination, 249 
 
 induced in Gramme-armature, 391 
 Energy, chemical-potential, 238 
 
 conservation of, 128 
 
 in electrostatic discharges, 150 
 
 degradation of, 238 
 
 instrument for measuring, 449 
 
 - of an electrostatic charge resides in the 
 dielectric, 161 
 
 of battery-cell, transformations of, 238 
 
 of electric current, 240 
 
 per second [see ' Activity '], 239, 298 
 
 kinetic, 128 
 
 potential, 128 
 
 radiant, 341 
 
 Engravings reproduced by electrolysis, 193 
 Equator, magnetic, 31 
 Equivalents, electro-chemical, 191 
 
 table of, 192 
 Erg, definition of, 127 
 Ether, the, 340 
 
 electrostatic phenomena possibly due to 
 strains in, 341 
 
 FARADAY'S e'ectroscopic experiment 
 in an insulated ro<.m, 52 
 
 adoption of the chemical view of vol- 
 taic electricity. 170 
 
 discovery of induced currents, 343 
 of the rotation .of a plane polarised 
 
 ray in a magnetic field, 337 
 
 ice-pail, 57 
 
 laws of electrolysis, 189 
 
 method of finding specific inductive 
 capacities, 124 
 
 Faure's accumulator, 200 
 
 experiment with Gramme machine, 
 43 
 
 Ferranti alternate-current machine. 401 
 Ferric chloride solution, paramagnetic in 
 
 vacuo, 336 
 Field, electrostatic, lines of force in, 69 
 
 magnetic, 15 
 
 earth's total, resolved into com- 
 ponents, 34 
 
 relation of strength of, to current- 
 strength, 275 
 
 magnetic and diamagnetic bodies in, 
 328-331 
 
 due to electric currents, 271, 294 et 
 
 seg. 
 Field-strength, electromagnetic, 272 
 
 electrostatic, 135 
 
 magnetic, 15 
 
 Fluid, electric, the term, 48 
 
 Filings, iron, used in experiments, 3, 18, 
 
 272 
 Force on a positive unit of electrification 
 
 acting perpendicularly to a conducting 
 
 surface, 144 
 
 coen ive, the term, 6 
 
 between two magnetic poles, 26 
 
LOWER DSYISiON 
 
 INDEX 
 
 xlix 
 
 FOR 
 
 INS 
 
 Force, dimensions of, formula, 13 
 
 electromotive [see E.M.F.], 201 
 
 electrostatic fields of, general considera- 
 tion of, 161 
 
 electrostatic, laws of, 49, 51 
 
 fields of, considered from the potential 
 point of view, 131 
 
 lines of magnetic, shown by iron filings, 
 18 
 
 - ---- due to the earth, 30 
 
 due to a current, 272, 294, etc. 
 - -- due to a solenoid, 327 
 
 due to a hollow cylindrical 
 magnet, 328 
 
 mapped out, 136, 272 
 ---- 1- and directions of, 273 
 
 round a wire bearing an electro- 
 static discharge, 93 
 
 lines of, in an electrostatic field, 69 ; 
 mapped out, 136 
 
 - perpendicular to equipotential 
 surfaces, 134 
 
 magnetic, laws of, 26 
 
 tubes of, 138 
 
 Forces, parallelogram of, 13 
 
 Franklin, kite experiment of, 113 
 
 ' Free," the term as applied to electric 
 
 charges, 81 
 
 Frictional-electric machines, 60 
 Frog, Galvani's experiment with a, 164 
 
 GALVANI'S experiment, 164 
 Galvanometer, 173 
 
 simple form of, 275 
 
 for practical or commercial use, 293 
 
 resistance of a, 329 
 
 use of the shunt in, 214 
 
 astatic, 280 
 
 ballistic,_286 
 
 differential, 285 
 
 graded current, 290 
 potential, 289 
 
 long-coil and short-coil, 292 
 
 mirror, 283 
 
 multiplying, 280 
 
 sine, 279 
 
 tangent, 276 
 
 volt-meter, 233, 289 
 Gas-battery, 195 
 
 Gaugain's tangent galvanometer, 278 
 Glass, why used for Leyden jars, 86 
 
 specific inductive capacity of, 121 
 
 rod, experiments with, 43 
 Gold-leaf electroscope, 43 
 
 charged by induction, 54, 73 
 
 ces of p 
 by, 153 
 
 indication of differences of potential 
 
 Gordon's experiments on specific inductive 
 
 capacities, 126 
 Gramme, or mass unit, 12 
 Gramme machine, 388 
 Gravitation-potential, the term, 131 
 Grothiiss's hypothesis, 186 
 Grove's gas-battery, 195 
 cell, 179 
 Gunpowder, ignition of, by electric dis- 
 
 charge, 93 
 Guthrie (Pr 
 
 (Prof.), experiment in magnetic 
 induction attributed to, 8 
 
 Gutta-percha, specific inductive capacity 
 
 of, 121 
 Gymnotus, 169 
 
 HEAT, absorption and evolution of, at 
 junctions of thermo-electric pair, 263 
 
 accompanying electric current, 182 
 
 as affecting magnetisation, 6 
 
 effect on magnets, 7 
 
 converted into electric energy by thermo- 
 cell, 252 
 
 distribution of, in the circuit, 244 
 
 of electric discharge, 93, 150 
 
 evolved in a circuit, relation of, to 
 resistance and current-strength, 241 
 
 evolution of, according to cell arrange- 
 ments, 245 
 
 evolved in a circuit, formulae for measur- 
 ing* 243 
 
 tendency of energy to take the form of, 
 238. 
 
 unit of, 239 
 
 Heats of combination in relation to 
 
 E.M.Fs, 249 
 Helmholtz's modification of Gaugain's 
 
 tangent galvanometer, 278 
 Holtz induction machine, 108 
 Hughe-^'s investigation of the molecular 
 
 theory of magnetism, 4 
 
 invention of the microphone, 431 
 Hydrogen, evolution of, fiom voltaic cell, 
 
 J 73 
 
 film on copper plate of voltaic cell, 
 *75 
 
 means of getting rid of, 176 
 
 Hydrostatics, analogies from, 70, 151, 202 
 
 ICE-PAIL, Faraday's, 57 
 Incandescence, electric lightin^ji>y, 
 
 43 <l -i 
 bwan s lamp, 434 
 
 Inclination, magnetic, 30 
 
 how to find, 36 
 India-rubber, specific inductive capacity 
 
 Of, 121 
 
 Induction, phenomena of, accompanying 
 
 an electrostatic discharge, 94 
 Induction coils, 368 
 
 electromagnetic, 343 et seq. 
 
 velocity of propagation of, 342. 
 
 without in.tial current, 351 
 
 mutual, coefficient of, 359 
 
 self, 360 
 
 in the earth's field, 366 
 
 disturbance of telephone lines by, 430" 
 
 electrostatic, 52 
 
 illustrated by experiments with the 
 
 ice-pail, 57 
 examined from a 'potential ' point of 
 
 view, 70 
 machines, 100 
 
 magnetic, 7 
 
 along lines of force, 19, 331 
 
 its influence on magnetic measure- 
 ments, 28 
 
 Insulating st nds, 53 
 
 stool, 64. 
 
INDEX 
 
 INS 
 
 NEE 
 
 Insulators, 162, 183, 341 
 
 Integrating instruments for measurements, 
 
 448 
 Intensity of the earth's magnetic field, 30 ; 
 
 how to determine, 37 
 
 Ion, electro-chemical equivalent of, 191, 249 
 Ions, 184 : migration of the, 187 
 Iron is paramagnetic, 329 
 
 filings used in experiments, 3, 18, 272 
 
 white-hot, not attracted by a magnet, 7 
 
 as core of an ' electro-magnet,' 333 
 
 magnetisation of, i, 8, 333 
 
 in ships, how to render its magnetism 
 symmetrical, 30 
 
 soft, non-persistence of magnetism in, 6 
 acting as a ' magnetic screen,' 8 
 
 Isoclinic lines, 31 
 Isogonic line*, 31 
 ' Isolated,' meaning of, in electrostatics, 74, 
 
 142 
 
 Isolated bodies, formula for capacities of, 
 149 
 
 {ENKIN (Fleeming), his invention of 
 Telpherage, 411 
 >ule's law and Joule effect, 241, 263 
 
 ~17 ATHODE, 183 
 
 J\_ Keepers, magnet, 9 
 
 Kerr's electro-optical discoveries, 340 
 
 Key, Morse, 421 
 
 irchhoff's laws, 216 
 ite, Franklin's experiment with the, 113 
 
 ADD'S 'self-exciting' dynamo, 387 
 Lamellar distribution of magnetism, 
 
 T 
 La 
 
 c 
 c, 438 
 
 es arrangement of, 442 
 
 iSKdescent, 434 
 
 arrangement of, in parallel, 436 
 'Lead,' in Gramme machine, 393 
 -the metal) used in secondary batteries, 
 
 197 
 
 Leclanch^'s cell, 179 
 Lenz's law, 352 
 
 Level, electrical [see Potential], 66 
 Leyden jars, 85 
 
 energy of discharge of, 151 
 -- cascade arrangement of, 88 
 
 energy of discharge in the, 152 
 
 spark, 99 
 
 -- nature of the charge in, 90 
 
 used with induction coil, 375 
 Light, electro-magnetic theory of, 340 
 
 rotation of plane of polarisation of, 
 
 Lighting, electric, two systems of, 433 
 Lightning-conductors, 113 
 
 flashes, length of, 116 
 
 sheet, forked, and globe, 120 
 Lines, isoclinic and isogonic, 31 
 
 _ offeree [see Force] 
 
 Liquids, specific resistances of, 236 
 Laclestone, magnetisation of iron or steel 
 
 by, i ( 
 Lupton's tables of constants, 236 
 
 MACHINES, electric, 60 
 Magnet, controlling, used with 
 galvanometers, 282 
 
 cylindrical, constitution of, 324 
 
 long thin, constitution of, 2 
 
 completion of molecular circuit in, 9 
 Magnetic moments of needlts, i-, 26, 40 
 
 shell, 303 
 
 Magnetisation of steel or iron bars, i 
 
 coefficient of, 329 
 
 methods of, 10 
 
 by electrostatic discharge, 92 
 Magnetism, Ampere's theory of, 326 
 
 duality of phenomena of, 42 
 
 earth's, 28 
 
 evident, distribution of, along a bar- 
 magnet, 23 
 
 induced, 7 
 
 of iron ships, 30 
 
 lamellar distribution of, 303 
 
 laws of, 26 ; proved by torsion balance 
 and by method of oscillation, 27 
 
 molecular theory of, 3 
 
 relative, 335 
 
 affected by heat, 7 
 
 temporary and residual, 6 
 Magnetometer, 40 
 Magneto machines, 395 
 
 as motors, 402 
 Magnets, how to make, i, 10 
 
 result of breaking, 3 
 
 use of keepers for, 9 
 
 field-, 387 
 
 hollow, contrasted with solenoid, 327 
 Mance's method of measuring resistance of 
 
 a battery-cell, 231 
 Mariner's compass, 29 
 Mass, as distinguished from weight, 13 
 Maximum current with a given battery, 219 
 
 work with a motor. 405 
 
 Maxwell's (Clerk) electro-magnetic theory 
 
 of light, 340 
 Mechanics, preliminary knowledge of, 
 
 needed, 12, 128 
 
 Megohm, multiple of ohm, 2^5 
 Mercury, use in amalgamating zinc plate 
 
 of galvanic cell, 173 
 
 cups for connecting wires, 175 
 Meridian, magnetic, plane of the, 30 
 Metals, contact potential series for, 165 
 
 thermo-electric cells composed of, 252 
 et seg_ 
 
 specific resistances of, 236 
 
 Meters for ' total current ' supplied, 443 
 Microhm, fraction of ohm, 235 
 Microphones, 431 
 Molecular theory of magnetism, 3 
 Molecules, interchanges between, in elec- 
 trolysis, 187 
 
 Moment, magnetic, of a needle, 15 
 how measured, 17, 26, 40 
 Moments of single forces and couples, 14 
 Morse alphabet, 420 
 
 system of telegraphy, 421 
 Motor [see Electro-motor] 
 Mutual potential of two circuits, 359 
 
 N 
 
 EEDLE, magnetic, action of the earth 
 
 on a, 28 
 
INDEX 
 
 1. 
 
 NEE 
 
 RES 
 
 Needle, inagnetic, fracture experiment 
 
 with, 2 
 
 axis of a, 29 
 moment of, 15, 26, 40 
 
 telegraph, 419 
 
 system of telegraphy, 420 
 Needles, astatic system of, 281 
 Neutrality, electrostatic, 47 
 
 magnetic, 4 
 
 Nobili, thermo-pile of, 254 
 ' North pole,' caution as t<9 the use of the 
 term, 2 
 
 OERSTED'S experiment to show mag- 
 netic field about a current, 274 
 Ohm, practical unit of resistance so called, 
 
 221, 298 
 
 Ohm's law, 201, 203 
 
 - example of its application, 207, ^19 
 
 graphic representation -of, 209 
 
 Kirchhoff's extension of, 216 
 Opposition method of measuring E.M.F. 
 
 of open cell, 234 
 
 resistance of battery cell, 230 
 Oscillations, method of, 24 
 
 used to compare two magnetic 
 fields, 25 
 
 used to compare two magnetic 
 moments, 26 
 
 used to prove magnetic ' law of 
 inverse squares,' 27 
 
 used in measuring -earth's mag- 
 netic field-strength absolutely, 38 
 
 PARAFFIN, specific inductive capacity 
 Of, 121 
 
 anhydrous, use with ebonite, 63 
 
 ' Paramagnetic ' means same as ' magnetic,' 
 
 334 
 
 Paramagnetic bodies, list of, 334 
 Peltier effect in thermo-cells, 263 
 theory of, 268 
 
 E.M.F.S, 267 
 
 electrometer, 77 
 Photophone, 433 
 Pile, Volta's, 167 
 
 Zamboni's, 168 
 
 dryness of, 168 
 Pith-ball, experiments with, 43 
 Plane of the magnetic meridian, 30 
 Planters secondary cell, 197 
 
 Plate electric machines, 63 
 
 condenser, 147 
 
 Plumbic acetate, electrolysis of, 186 
 Polarisation in voltaic cells, 175 
 
 methods of obviating, 176 
 
 of electrodes, 194 
 
 -*- plane of, of light, rotated in a magnetic 
 
 field, 337 
 Polarity, magnetic, 2 
 
 of molecules, 4 
 
 Pole, magnetic, action of, upon a closed 
 
 circuit, 308 
 reaction of, on an element of current, 
 
 308. 
 
 unit magnetic, 14 
 
 Poles, magnetic, law of variation of force 
 between, 26 
 
 Poles, magnetic, mutual attraction and 
 repulsion of, 2 
 
 action of currents on, 294 
 
 action of element of current on, 296 
 
 positive and negative, 15 
 
 potentials on, 316, 323 
 
 of the voltaic cell, 172 
 Pole-strength, ' dimensions ' of, 300, 301 
 Potassium hydrate, electrolysis of, 185 
 Potential, electrostatic, 66, 131 
 
 at a point in the atmosphere, 117 
 
 methods of measuring, 118 
 
 results of observations on, 119 
 
 contact series, 165 
 
 curve of, round the axis of the 
 
 Gramme armature, 392 
 
 difference of, 66, 131 
 
 distribution of, in the circuit of a 
 
 dynamo and motor, 412 
 
 how to measure by work, 67, 131 
 
 as distinguished from electrostatic 
 
 density, 144 
 
 measurement o, 131, 153 
 
 rate of change of, 135 
 
 of an isolated sphere, 141 
 
 of thunder-clouds, 115 
 
 fall of, through the circuit, 21$ 
 
 sign of, distinguished from sign of 
 
 charge, 74 
 
 magnetic, due to circuits, 316, 322 
 due to magnetic shells, 304 
 
 mutual, of two circuits, 359 
 Potentiometer, 234 
 Poundal, the term, 13 
 Power, transmission of, from a distance^L 
 
 409 
 Proof-plane", 52 
 
 QUADRANT electrometer, formula 
 for, 157 
 
 potential of atmosphere me 
 
 119 
 uses of, 158 
 
 RAILWAYS, electric, 411 
 Rainfall after a lightning flash, 117 
 Bay, plane polarised, rotated in a mag- 
 netic field, 337 
 Relay, the, 422 
 
 Replenisher, electrostatic inductive, 104 
 Resinous electricity, the term, 46 
 Resistance, 202 
 
 on what conditions dependent, 204 
 
 unit of, 221, 298 
 
 measurement of, by method of substitu- 
 tion, 224 
 
 by comparison of deflexions, 225 
 
 by Wheatstone's bridge, 226 et 
 
 seq. _ 
 
 equivalent, of electric arc, 440, 441 
 of a 'multiple circuit,' 206 
 
 expressed as a velocity, 357 
 
 of a battery-cell, 230 
 
 of a galvanometer, 229 
 
 specific, 235 
 
 coils, 222 
 
 boxes, 222 
 
 Resistances, table of specific, 236 
 
Hi 
 
 INDEX 
 
 RES 
 
 Resistances compared by differential gal- 
 vanometer, 286 
 
 Rheostat, 224 
 
 Riess, his observation of electrostatic dis- 
 tribution on a cube, 55 
 
 Rigidity, molecular; same as 'coercive 
 force,' 6 
 
 Ring-armature, 388 
 
 Ruhmkorflf's coil, 370 
 
 SATURATION, magnetic, 6 
 of cores of electro-magnets, 333 
 Screens between two electrified systems, 
 
 58, 139 
 
 Second, or time unit, 12 
 Secondary battery-cells, 196, 247 
 Selenium cell, 432 
 Series-dynamos, 397 
 
 as motors, 403 
 
 Series, electrostatic, 165 
 
 thermo-electric, 254 
 
 I Shellac, specific inductive capacity of, 121 
 Shells, magnetic, equivalence of electric 
 circuit to, 306 
 fields and potentials due to, 302, 304, 
 322 
 potentials on, 317, 323 
 it Ships, iron, how to render their magnetism 
 
 symmetrical, 30 
 Shocks, ' return,' 115 
 
 from induction coils, caution, 371 
 Shunt-dynamos, 398 
 
 .mts, 214 
 mens's armature, 387 
 mens-Alteneck armature, 395 
 Silver-plating, 194 
 Sine galvanometer, 279 
 Smee's cell, 177 
 
 a, cannot decompose water, 248, 251 
 Sol^Mhd, ideal and practical, 324 
 
 dV n of, with and without iron core, 
 
 resemblance of, to a solid cylindrical 
 unagnet, 326 
 
 contrasted with hollow cylindrical mag- 
 net, 327 
 
 Solenoidal ammeters, 445 
 Sounder, telegraphic, 417 
 Spark, electric, duration of, 96 
 
 various forms of, 99, 376 et seq. 
 
 passage of, the term, 92 
 
 of induction coil, use with spectroscope, 
 
 37 6 37 8 
 
 Spark-board, Wheatstone's, 96 
 Sparks, electric, discharged from persons, 
 
 64 
 
 Specific inductive capacities, 121 etseq., 147 
 resistance, 235, 236 
 Sphere, isolated, distribution of electric 
 
 charge on, 55 
 
 electrostatic capacity of, 142 
 
 potential of, 142 
 
 Spheres of different radii, relation between 
 
 potential and charge of, 143 
 Stands, insulating, 53 
 Steel, hard, duration of magnetism in. 6 ; 
 
 why used for magnets. 10 
 
 at a white-heat not attracted by a mag- 
 net, 7 
 
 TOR 
 
 Steel, magnetisation of, i 
 
 by electrostatic discharge, 92 
 
 Stool, insulating, 64 
 Storage cells, 196, 247 
 Storms, magnetic, 31 
 
 atmospheric signs of, 425 
 Substitution, method of, applied to 
 
 measurement of resistance, 225 
 Sulphate of sodium, electrolysis of, 185 
 Sulphur, specific inductive capacity of, 121 
 Sulphuric acid, action of, in the voltaic 
 
 cell, 173 
 
 use of, in decomposition of water, 184 
 
 Surfaces, equipotential, electrostatic, 69, 
 
 *33 
 Swan lamp, 434 
 
 T AIT'S thermo-electric diagram and 
 formula, 259 262 
 
 Tangent galvanometer, 276 et seq. 
 Telegraphy, general principle of, 416 
 
 alphabet used in, 419 
 
 interferenceof earth currents with, 424 
 
 duplex, 426 
 
 Morse system of, 421 
 
 needle system of, 420 
 
 submarine, 422 
 Telephone, of Graham Bell, 429 
 
 Edison's carbon transmitter, 431 
 Telephones, 428 
 
 worked by an external battery, 431 
 Telpherage, 411 
 
 Theory, electro-magnetic, of light, 341 
 
 of thermo-cells, 265 et seq. 
 
 of Peltier and Thomson effects, 268 
 
 of electro-magnetic induction, 346 et seg, 
 Thermo-cell, theory of, 265 
 Thermo-cells, 252 
 
 Peltier effect in, 263 
 Thermo-diagrams, 259, 262 
 Thermo-electric neutral point, 257 
 
 power, 256 
 
 series, 254 
 Thermo-pile, 254 
 
 Thompson's diagrams, showing activities 
 
 of dynamo and motor, 409 
 Thomson effect, 264 
 - theory of, 268 
 
 (Sir W.), his electric current accumu- 
 lator, 366 note 
 
 graded current galvanometer, 
 290 
 
 graded potential galvanometer 
 
 289 
 
 guard-ring for the attracted, 
 disc electrometer, 154 
 
 method of measuring galvano- 
 meter resistance, 229 
 
 mirror galvanometer, 283 
 
 quadrant electrometer, 155 
 
 reflecting galvanometer, impor- 
 tant use of, in submarine telegraphy, 423 
 
 inductive replenisher, 104 
 water-dropping accumulator, 
 
 102 
 
 Thunder-clouds, high potential of, 115 
 
 Toision balance, Coulomb's, 20 
 
 strength of two magnetic poles com- 
 pared by, ?.? 
 
INDEX 
 
 TOR 
 
 Torsion balance, distribution of evident 
 magnetism observed by, 23 
 
 laws of magnetism proved by, 27 
 
 laws of electrostatics proved by, 50 
 
 Tram-cars, electric, 411 
 
 'Transformers,' or instruments of distribu- 
 tion, 451 
 
 Tubes of force, 138, 139 
 
 UNIT magnetic field, 15 
 - jar, 86 
 
 magnetic pole, 14 
 
 Units, electro-magnetic, absolute and 
 
 practical, 221, 297, 298 
 
 determination of, 299 
 
 and electrostatic ; ' dimensions ' of, 
 
 301 
 
 electrostatic, 66, 69, 131, 142, 301 
 
 mechanical, 12, 127 et seq. 
 
 VACUO, diamagnetic bodies in, 336 
 Vacuum tubes, phenomena of dis- 
 charge in, 377 et seq. 
 Vane, electric, 63 
 
 Variations of earth's magnetic elements, 31 
 Varley's induction machine, 103 
 Velocity, dimensions of, formula, 13 
 Velocity ' ratio ' ; connecting electrostatic 
 
 and electromagnetic units, 300, 301 
 Vitreous electricity, the term, 46 
 Volt, the practical unit of E.M.F., so 
 
 called, 221, 297 
 
 Volta, views and experiments of, 165 
 Volta's cell, 169 ; theory of, 171 
 
 nature of chemical action in, 173 
 
 pile, 167 
 
 Volt-meter galvanometers, 233, 289, 446 
 
 calibration of, 293 
 
 - 'damping' in, 446 
 Voss induction machine, 105 
 
 ZIN 
 
 WATER, decomposition of, r8 4 
 not possible by a single 
 Smee's cell, 248, 251 
 
 dropping accumulator, 102 
 
 molecular interchanges in, during de- 
 composition, 187 
 
 Watt, practical unit of activity so called, 
 
 240 
 
 Watt-meters, 447 
 Weber's electro-dynamometer, 291 
 Weight as distinguished from mass, 13 
 Wheatstone's bridge, 226 
 
 slide form of, 228 
 
 resistance-box form, 229 
 
 use of, in duplex telegraphy, 428 
 
 rheostat, 224 
 
 spark-board, 96 
 Winter's electric machine, 63 
 
 Wire, size of, used in making electro-mag- 
 nets, 333 
 
 network, as a lightning protector, 115 
 
 return, in telegraphy, 416 
 
 use of, against earth currents, 
 
 425 
 Wires, conducting ; heat evolved in, 243 
 
 et seq. 
 
 resistance of, 204 
 
 for resistance coils, 222 
 
 telegraphic, insulation of, 426 
 Work, unit of, 127 
 
 examples of, 128 
 
 in relation to force overcome, 129 
 
 formulas for, in dynamo-electric ma- 
 chines, 405 
 
 maximum, with a motor, 405 
 
 per second, or activity, 239, etc. 
 
 ZINC, solution of, in the voltaic cell, 
 173 
 
 heat evolved during, 182, 249 
 
 impure, local currents in, 175 
 
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