Notes on Recent Researches in Electricity and Magnetism Intended as a Sequel to Professor Clerk-Maxwell’s Treatise on Electricity and Magnetism

8/9/2019 Notes on Recent Researches in Electricity and Magnetism Intended as a Sequel to Professor Clerk-Maxwells Treatis

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Title: Notes on Recent Researches in Electricity and Magnetism

Intended as a Sequel to Professor Clerk-Maxwell\s Treatise

on Electricity and Magnetism

Author: J. J. Thomson

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Language: English

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NOTES

ON

RECENT RESEARCHES IN

ELECTRICITY AND MAGNETISM

INTENDED AS A SEQUEL TO

PROFESSOR CLERK-MAXWELLS TREATISE

ON ELECTRICITY AND MAGNETISM

BY

J. J. THOMSON, M.A., F.R.S.Hon. Sc.D. Dublin

FELLOW OF TRINITY COLLEGEPROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE

Oxford

AT THE CLARENDON PRESS

1893


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PREFACE

In the twenty years which have elapsed since the first appearance of

Maxwells Treatise on Electricity and Magnetismgreat progress has beenmade in these sciences. This progress has been largelyperhaps it wouldnot be too much to say mainlydue to the influence of the views set forthin that Treatise, to the value of which it offers convincing testimony.

In the following work I have endeavoured to give an account of some re-cent electrical researches, experimental as well as theoretical, in the hopethat it may assist students to gain some acquaintance with the recent

progress of Electricity and yet retain Maxwells Treatise as the source fromwhich they learn the great principles of the science. I have adopted ex-clusively Maxwells theory, and have not attempted to discuss the conse-quences which would follow from any other view of electrical action. I haveassumed throughout the equations of the Electromagnetic Field given byMaxwell in the ninth chapter of the second volume of his Treatise.

Thefirst chapterof this work contains an account of a method of re-

garding the Electric Field, which is geometrical and physical rather thananalytical. I have been induced to dwell on this because I have found thatstudents, especially those who commence the subject after a long courseof mathematical studies, have a great tendency to regard the whole ofMaxwells theory as a matter of the solution of certain differential equa-tions, and to dispense with any attempt to form for themselves a mentalpicture of the physical processes which accompany the phenomena they areinvestigating. I think that this state of things is to be regretted, since itretards the progress of the science of Electricity and diminishes the valueof the mental training afforded by the study of that science.

In the first place, though no instrument of research is more powerfulthan Mathematical Analysis, which indeed is indispensable in many de-partments of Electricity, yet analysis works to the best advantage when


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PREFACE v

employed in developing the suggestions afforded by other and more physi-cal methods. One example of such a method, and one which is very closelyconnected with the initiation and development of Maxwells Theory, is thatof the tubes of force used by Faraday. Faraday interpreted all the lawsof Electrostatics in terms of his tubes, which served him in place of thesymbols of the mathematician, while in his hands the laws according towhich these tubes acted on each other served instead of the differentialequations satisfied by such symbols. The method of the tubes is distinctlyphysical, that of the symbols and differential equations is analytical.

The physical method has all the advantages in vividness which arisefrom the use of concrete quantities instead of abstract symbols to representthe state of the electric field; it is more easily wielded, and is thus moresuitable for obtaining rapidly the main features of any problem; when,however, the problem has to be worked out in all its details, the analyticalmethod is necessary.

In a research in any of the various fields of electricity we shall be actingin accordance with Bacons dictum that the best results are obtained when

a research begins with Physics and ends with Mathematics, if we use thephysical theory to, so to speak, make a general survey of the country, andwhen this has been done use the analytical method to lay down firm roadsalong the line indicated by the survey.

The use of a physical theory will help to correct the tendencywhich Ithink all who have had occasion to examine in Mathematical Physics willadmit is by no means uncommonto look on analytical processes as the

modern equivalents of the Philosophers Machine in the Grand Academy ofLagado, and to regard as the normal process of investigation in this subjectthe manipulation of a large number of symbols in the hope that every nowand then some valuable result may happen to drop out.

Then, again, I think that supplementing the mathematical theory byone of a more physical character makes the study of electricity more valu-able as a mental training for the student. Analysis is undoubtedly the

greatest thought-saving machine ever invented, but I confess I do not thinkit necessary or desirable to use artificial means to prevent students fromthinking too much. It frequently happens that more thought is required,and a more vivid idea of the essentials of a problem gained, by a roughsolution by a general method, than by a complete solution arrived at bythe most recent improvements in the higher analysis.


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PREFACE vi

The method of illustrating the properties of the electric field which Ihave given inChapter Ihas been devised so as to lead directly to the dis-tinctive feature in Maxwells Theory, that changes in the polarization ina dielectric produce magnetic effects analogous to those produced by con-duction currents. Other methods of viewing the processes in the ElectricField, which would be in accordance with Maxwells Theory, could, I haveno doubt, be devised; the question as to which particular method the stu-dent should adopt is however for many purposes of secondary importance,provided that he does adopt one, and acquires the habit of looking at theproblems with which he is occupied as much as possible from a physicalpoint of view.

It is no doubt true that these physical theories are liable to imply morethan is justified by the analytical theory they are used to illustrate. Thishowever is not important if we remember that the object of such theories issuggestion and not demonstration. Either Experiment or rigorous Analysismust always be the final Court of Appeal; it is the province of these physicaltheories to supply cases to be tried in such a court.

Chapter IIis devoted to the consideration of the discharge of electric-ity through gases; Chapter III contains an account of the application ofSchwarzs method of transformation to the solution of two-dimensionalproblems in Electrostatics. The rest of the book is chiefly occupied withthe consideration of the properties of alternating currents; the experimentsof Hertz and the development of electric lighting have made the use of thesecurrents, both for experimental and commercial purposes, much more gen-

eral than when Maxwells Treatise was written; and though the principleswhich govern the action of these currents are clearly laid down by Maxwell,they are not developed to the extent which the present importance of thesubject demands.

Chapter IV contains an investigation of the theory of such currentswhen the conductors in which they flow are cylindrical or spherical, whileinChapter Van account of Hertzs experiments on Electromagnetic Waves

is given. This Chapter also contains some investigations on the Electro-magnetic Theory of Light, especially on the scattering of light by smallmetallic particles; on reflection from metals; and on the rotation of theplane of polarization by reflection from a magnet. I regret that it was onlywhen this volume was passing through the press that I became acquaintedwith a valuable paper by Drude (Wiedemanns Annalen, 46, p. 353, 1892)


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PREFACE vii

on this subject.Chapter VI mainly consists of an account of Lord Rayleighs inves-

tigations on the laws according to which alternating currents distributethemselves among a network of conductors; while the last Chapter con-tains a discussion of the equations which hold when a dielectric is movingin a magnetic field, and some problems on the distribution of currents inrotating conductors.

I have not said anything about recent researches on Magnetic Induction,as a complete account of these in an easily accessible form is containedin Professor Ewings Treatise on Magnetic Induction in Iron and otherMetals.

I have again to thank Mr. Chree, Fellow of Kings College, Cambridge,for many most valuable suggestions, as well as for a very careful revisionof the proofs.

J. J. THOMSON.


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CONTENTS

CHAPTER I.ELECTRIC DISPLACEMENT AND FARADAY TUBES OF FORCE.

Art. Page

1. Electric displacement . . . . . . . . . . . . . . . . . 12. Faraday tubes . . . . . . . . . . . . . . . . . . . . 13. Unit Faraday tubes . . . . . . . . . . . . . . . . . . 2

4. Analogy with kinetic theory of gases . . . . . . . . . . . 35. Reasons for taking tubes of electrostatic induction as the unit 46. Energy in the electric field . . . . . . . . . . . . . . . 47. Behaviour of Faraday tubes in a conductor . . . . . . . . 58. Connection between electric displacement and Faraday tubes . 59. Rate of change of electric polarization expressed in terms of the

velocity of Faraday tubes . . . . . . . . . . . . . . 6

10. Momentum due to Faraday tubes . . . . . . . . . . . . 811. Electromotive intensity due to induction . . . . . . . . . 912. Velocity of Faraday tubes . . . . . . . . . . . . . . . 1013. Systems of tubes moving with different velocities . . . . . . 1114. Mechanical forces in the electric field . . . . . . . . . . . 1415. Magnetic force due to alteration in the dielectric polarization . 1516. Application of Faraday tubes to find the magnetic force due to a

moving charged sphere . . . . . . . . . . . . . . . 1517. Rotating electrified plates . . . . . . . . . . . . . . . 2318. Motion of tubes in a steady magnetic field . . . . . . . . 2819. Induction of currents due to changes in the magnetic field . . 3120. Induction due to the motion of the circuit . . . . . . . . . 32


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CONTENTS ix

Art. Page

21. Effect of soft iron in the field . . . . . . . . . . . . . . 3322. Permanent magnets . . . . . . . . . . . . . . . . . . 34

23. Steady current along a straight wire . . . . . . . . . . . 3524. Motion of tubes when the currents are rapidly alternating . . 3725. Discharge of a Leyden jar . . . . . . . . . . . . . . . 3726. Induced currents . . . . . . . . . . . . . . . . . . . 4027. Electromagnetic theory of light . . . . . . . . . . . . . 412832. Behaviour of tubes in conductors . . . . . . . . . . 424533. Galvanic cell . . . . . . . . . . . . . . . . . . . . 47

34. Metallic and electrolytic conduction . . . . . . . . . . . 48

CHAPTER II.

PASSAGE OF ELECTRICITY THROUGH GASES.

35. Introduction . . . . . . . . . . . . . . . . . . . . . 5236. Can the molecules of a gas be electrified? . . . . . . . . . 52

37. Hot gases . . . . . . . . . . . . . . . . . . . . . . 5338. Electric properties of flames . . . . . . . . . . . . . . 5639. Effect of ultra-violet light on the discharge . . . . . . . . 5640. Electrification by ultra-violet light . . . . . . . . . . . . 5841. Disintegration of the negative electrode . . . . . . . . . . 5842. Discharge of electricity from illuminated metals . . . . . . 5943. Discharge of electricity by glowing bodies . . . . . . . . . 61

44. Volta-potential . . . . . . . . . . . . . . . . . . . . 6245. Electrification by sun-light . . . . . . . . . . . . . . . 6446. Electric Strength of a gas . . . . . . . . . . . . . . . 6747. Effect of the nature of the electrodes on the spark length . . 6748. Effect of curvature of the electrodes on the spark length . . . 6749. Bailles experiments on the connection between potential differ-

ence and spark length . . . . . . . . . . . . . . . . 68

50. Liebigs on the same subject . . . . . . . . . . . . . . 7051. Potential difference expressed in terms of spark length . . . . 725253. Minimum potential difference required to produce a spark 72735461. Discharge when the field is not uniform . . . . . . . 75826265. Peaces experiments on the connection between pressure and

spark potential . . . . . . . . . . . . . . . . . 8385


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CONTENTS x

Art. Page

6668. Critical pressure . . . . . . . . . . . . . . . . . 88896971. Potential difference required to spark through various

gases . . . . . . . . . . . . . . . . . . . . . 89907276. Methods of producing electrodeless discharges . . . . . 919377. Appearance of such discharges . . . . . . . . . . . . . 947880. Critical pressure for such discharges . . . . . . . . . 959681. Difficulty of getting the discharge to pass across from gas to metal 968286. High conductivity of rarefied gases . . . . . . . . . 9710187. Discharge through a mixture of gases . . . . . . . . . . 1018893. Action of a magnet on the electrodeless discharges . . 10210594. Appearance of discharge when electrodes are used . . . . . 10695. Crookes theory of the dark space . . . . . . . . . . . . 10796. Length of dark space . . . . . . . . . . . . . . . . . 1089798. Negative glow . . . . . . . . . . . . . . . . . . . 10899103. Positive column and striations . . . . . . . . . 109112104107. Velocity of discharge along positive column . . . . 113116

108116. Negative rays . . . . . . . . . . . . . . . . 116121117. Mechanical effects produced by negative rays . . . . . . . 121118123. Shadows cast by negative rays . . . . . . . . . 123125124125. Relative magnitudes of time quantities in the discharge 125127126128. Action of a magnet on the discharge . . . . . . . 128129129. Action of a magnet on the negative glow . . . . . . . . . 129130133. Action of a magnet on the negative rays . . . . . 130135

134. Action of a magnet on the positive column . . . . . . . . 135135. Action of a magnet on the negative rays in very high vacua . 136136. Action of a magnet on the course of the discharge . . . . . 137137138. Action of a magnet on the striations . . . . . . 138139139147. Potential gradient along the discharge tube . . . . 139146148151. Effect of the strength of the current on the cathode fall 147150152155. Small potential difference sufficient to maintain current

when once started . . . . . . . . . . . . . . . 150152156162. Warburgs experiments on the cathode fall . . . . 152155163165. Potential gradient along positive column . . . . . . . 155166168. Discharge between electrodes placed close together . 156159169176. The arc discharge . . . . . . . . . . . . . . 159163


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CONTENTS xi

Art. Page

177178. Heat produced by the discharge . . . . . . . . 163164179182. Difference between effects at positive and negative elec-

trodes . . . . . . . . . . . . . . . . . . . . 165167183186. Lichtenbergs and Kundts dust figures . . . . . . 168169187193. Mechanical effects due to the discharge . . . . . 170173194201. Chemical action of the discharge . . . . . . . . 173177202. Phosphorescent glow due to the discharge . . . . . . . . 180203206. Discharge facilitated by rapid changes in the strength of the

field . . . . . . . . . . . . . . . . . . . . . 181184

207229. Theory of the discharge . . . . . . . . . . . . 184201

CHAPTER III.

CONJUGATE FUNCTIONS.

230233. Schwarz and Christoffels transformation . . . . . 203206234. Method of applying it to electrostatics . . . . . . . . . . 206

235. Distribution of electricity on a plate placed parallel to an infiniteplate . . . . . . . . . . . . . . . . . . . . . . 206236. Case of a plate between two infinite parallel plates . . . . . 211237. Correction for thickness of plate . . . . . . . . . . . . 213238. Case of one cube inside another . . . . . . . . . . . . 217239240. Cube over an infinite plate . . . . . . . . . . 220221241. Case of condenser with guard-ring when the slit is shallow . 221

242. Correction when guard-ring is not at the same potential as theplate . . . . . . . . . . . . . . . . . . . . . . 225

243. Case of condenser with guard-ring when the slit is deep . . . 227244. Correction when guard-ring is not at the same potential as the

plate . . . . . . . . . . . . . . . . . . . . . . 229245. Application of elliptic functions to problems in electrostatics . 231246. Capacity of a pile of plates . . . . . . . . . . . . . . 233

247. Capacity of a system of radial plates . . . . . . . . . . 235248. Finite plate at right angles to two infinite ones . . . . . . 237249. Two sets of parallel plates . . . . . . . . . . . . . . . 239250. Two sets of radial plates . . . . . . . . . . . . . . . 240251. Finite strip placed parallel to two infinite plates . . . . . . 241252. Two sets of parallel plates . . . . . . . . . . . . . . . 242


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CONTENTS xii

Art. Page

253. Two sets of radial plates . . . . . . . . . . . . . . . 244254. Limitation of problems solved . . . . . . . . . . . . . 244

CHAPTER IV.

ELECTRICAL WAVES AND OSCILLATIONS.

255. Scope of the chapter . . . . . . . . . . . . . . . . . 246256. General equations . . . . . . . . . . . . . . . . . . 246257. Alternating currents in two dimensions . . . . . . . . . 248

258. Case when rate of alternation is very rapid . . . . . . . . 254259260. Periodic currents along cylindrical conductors . . . . . 257261. Value of Bessels functions for very large or very small values of

the variable . . . . . . . . . . . . . . . . . . . . 258262. Propagation of electric waves along wires . . . . . . . . 259263264. Slowly alternating currents . . . . . . . . . . 266268265. Expansion ofxJ0(x)/J

0

0(x) . . . . . . . . . . . . . . 269

266. Moderately rapid alternating currents . . . . . . . . . . 272267. Very rapidly alternating currents . . . . . . . . . . . . 274268. Currents confined to a thin skin . . . . . . . . . . . . 275269. Magnetic force in dielectric . . . . . . . . . . . . . . 277270. Transmission of disturbances along wires . . . . . . . . . 279271. Relation between external electromotive force and current . . 283272. Impedance and self-induction . . . . . . . . . . . . . 287

273274. Values of these when alternations are rapid . . . . 289290275276. Flat conductors . . . . . . . . . . . . . . . . . 291277. Mechanical force between flat conductors . . . . . . . . 296278. Propagation of longitudinal magnetic waves along wires . . . 297279. Case when the alternations are very rapid . . . . . . . . 301280. Poyntings theorem . . . . . . . . . . . . . . . . . 303281. Expression for rate of heat production in a wire . . . . . . 309

282. Heat produced by slowly varying current . . . . . . . . . 310283284. Heat produced by rapidly varying currents . . . . . . 312285. Heat in a transformer due to Foucault currents when the rate of

alternation is slow . . . . . . . . . . . . . . . . . 313286. When the rate of alternation is rapid . . . . . . . . . . 317287. Heat produced in a tube . . . . . . . . . . . . . . . 318


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CONTENTS xiii

Art. Page

288. Vibrations of electrical systems . . . . . . . . . . . . 323289. Oscillations on two spheres connected by a wire . . . . . . 324

290. Condition that electrical system should oscillate . . . . . . 325291. Time of oscillation of a condenser . . . . . . . . . . . 326292. Experiments on electrical oscillations . . . . . . . . . . 328293297. General investigation of time of vibration of a condenser

. . . . . . . . . . . . . . . . . . . . . . . 329336298299. Vibrations along wires in multiple arc . . . . . . 337340300. Time of oscillations on a cylindrical cavity . . . . . . . . 340

301. On a metal cylinder surrounded by a dielectric . . . . . . 343302. State of the field round the cylinder . . . . . . . . . . . 346303. Decay of currents in a metal cylinder . . . . . . . . . . 348304305. When the lines of magnetic force are parallel to the axis of

the cylinder . . . . . . . . . . . . . . . . . . 350352306307. When the lines of force are at right angles to the axis 353356308. Electrical oscillations on spheres . . . . . . . . . . . . 358

309. Properties of the functions SandE . . . . . . . . . . . 360310. General solution . . . . . . . . . . . . . . . . . . 363311. Equation giving the periods of vibration . . . . . . . . . 365312. Case of the first harmonic distribution . . . . . . . . . . 366313. Second and third harmonics . . . . . . . . . . . . . . 369314. Field round vibrating sphere . . . . . . . . . . . . . . 369315. Vibration of two concentric spheres . . . . . . . . . . . 370

316. When the radii of the spheres are nearly equal . . . . . . 373317. Decay of currents in spheres . . . . . . . . . . . . . . 375318. Rate of decay when the currents flow in meridional planes . . 378319320. Effect of radial currents in the sphere . . . . . . 381382321. Currents induced in a sphere by the annihilation of a uniform

magnetic field . . . . . . . . . . . . . . . . . . . 382322. Magnetic effects of these currents when the sphere is not made

of iron . . . . . . . . . . . . . . . . . . . . . . 385323. When the sphere is made of iron . . . . . . . . . . . . 386


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CONTENTS xiv

CHAPTER V.

ELECTROMAGNETIC WAVES.

Art. Page

324. Hertzs experiments . . . . . . . . . . . . . . . . . 387325327. Hertzs vibrator . . . . . . . . . . . . . . . 387388328. The resonator . . . . . . . . . . . . . . . . . . . 390329. Effect of altering the position of the air gap . . . . . . . 390330331. Explanation of these effects . . . . . . . . . . . . 391332. Resonance . . . . . . . . . . . . . . . . . . . . . 393333335. Rate of decay of the vibrations . . . . . . . . . 394396

336339. Reflection of waves from a metal plate . . . . . . 397398340342. Sarasins and De la Rives experiments . . . . . . 398402343. Parabolic mirrors . . . . . . . . . . . . . . . . . . 402344346. Electric screening . . . . . . . . . . . . . . 403404347. Refraction of electromagnetic waves . . . . . . . . . . . 404348. Angle of polarization . . . . . . . . . . . . . . . . . 404349350. Theory of reflection of electromagnetic waves by a dielectric

. . . . . . . . . . . . . . . . . . . . . . . 405409351. Reflection of these waves from and transmission through a thin

metal plate . . . . . . . . . . . . . . . . . . . . 413352354. Reflection of light from metals . . . . . . . . . 416418355. Table of refractive indices of metals . . . . . . . . . . . 419356. Inadequacy of the theory of metallic reflection . . . . . . 420357. Magnetic properties of iron for light waves . . . . . . . . 421358. Transmission of light through thin films . . . . . . . . . 422359360. Reflection of electromagnetic waves from a grating . 424427361368. Scattering of these waves by a wire . . . . . . . 427436369. Scattering of light by metal spheres . . . . . . . . . . . 437370. Lambs theorem . . . . . . . . . . . . . . . . . . . 438371. Expressions for magnetic force and electric polarization . . . 440

372. Polarization in plane wave expressed in terms of spherical har-monics . . . . . . . . . . . . . . . . . . . . . . 442373376. Scattering of a plane wave by a sphere of any size . 444447377. Scattering by a small sphere . . . . . . . . . . . . . . 448378. Direction in which the scattered light vanishes . . . . . . 450379384. Hertzs experiments on waves along wires . . . . 452457


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CONTENTS xv

Art. Page

385. Sarasins and De la Rives experiments on waves along wires . 460390392. Comparison of specific inductive capacity with refractive

index . . . . . . . . . . . . . . . . . . . . 469472393401. Experiments to determine the velocity of electromagneticwaves through various dielectrics . . . . . . . . . 473483

402. Effects produced by a magnetic field on light . . . . . . . 484403. Kerrs experiments . . . . . . . . . . . . . . . . . 484404. Oblique reflection from a magnetic pole . . . . . . . . . 485405. Reflection from tangentially magnetized iron . . . . . . . 486406. Kundts experiments on films . . . . . . . . . . . . . 487407. Transverse electromotive intensity . . . . . . . . . . . 487408. Hall effect . . . . . . . . . . . . . . . . . . . . . 488409414. Theory of rotation of plane of polarization by reflection from

a magnet . . . . . . . . . . . . . . . . . . . 491503415416. Passage of light through thin films in a magnetic field 506510

CHAPTER VI.DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS.

417418. Very rapidly alternating currents distribute themselves soas to make the Kinetic Energy a minimum . . . . . . . 512

419. Experiments to illustrate this . . . . . . . . . . . . . 513420. Distribution of alternating currents between two wires in parallel 514421. Self-induction and impedance of the wires . . . . . . . . 518422. Case of any number of wires in parallel . . . . . . . . . 520423426. General case of any number of circuits . . . . . . 522526427428. Case of co-axial solenoids . . . . . . . . . . . 527529429. Wheatstones bridge with alternating currents . . . . . . 530430432. Combination of self-induction and capacity . . . . 532534432*. Effect of two adjacent vibrators on each others periods . . 535

CHAPTER VII.

ELECTROMOTIVE INTENSITY IN MOVING BODIES.

433. Equations of electromotive intensity for moving bodies . . . 538434439. Sphere rotating in a symmetrical magnetic field . . 539546440. Propagation of light through a moving dielectric . . . . . . 547


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CONTENTS xvi

Art. Page

441. Currents induced in a sphere rotating in an unsymmetrical field 551442. Special case when the field is uniform . . . . . . . . . . 555

443. Case when the rotation is very rapid . . . . . . . . . . 556444. Magnetic force outside the sphere . . . . . . . . . . . 559445. Couples and Forces on the Rotating Sphere . . . . . . . 559446. The magnetic force is tangential when the rotation is rapid . 561447. Force on the sphere . . . . . . . . . . . . . . . . . 561448. Solution of the previous case gives that of a sphere at rest in an

alternating field . . . . . . . . . . . . . . . . . . 562

Appendix.

The Electrolysis of Steam . . . . . . . . . . . . . . . 564

ADDITIONS AND CORRECTIONS.

Page65.For further remarks on electrification by incandescent bodies seeAppendix, p.575.

,, 119. E. Wiedemann and Ebert have shown (Wied. Ann. 46, p. 158,1892) that the repulsion between two pencils of negative raysis due to the influence which the presence of one cathode exertson the emission of rays from a neighbouring cathode.

,, 170. Dewar (Proc. Roy. Soc. 33, p. 262, 1882) has shown that theinterior of the gaseous envelope of the electric arc always showsa fixed pressure amounting to about that due to a millimetreof water above that of the surrounding atmosphere.

,, 178. For 90 C. read 100 C.


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NOTES ON

ELECTRICITY AND MAGNETISM.


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CHAPTER I.

ELECTRIC DISPLACEMENT AND FARADAY TUBES OF FORCE.

1.] The influence which the notation and ideas of the fluid theory

of electricity have ever since their introduction exerted over the scienceof Electricity and Magnetism, is a striking illustration of the benefits con-ferred upon this science by a concrete representation or construibar vorstel-lung of the symbols, which in the Mathematical Theory of Electricity de-fine the state of the electric field. Indeed the services which the old fluidtheory has rendered to Electricity by providing a language in which thefacts of the science can be clearly and briefly expressed can hardly be over-

rated. A descriptive theory of this kind does more than serve as a vehiclefor the clear expression of well-known results, it often renders importantservices by suggesting the possibility of the existence of new phenomena.

The descriptive hypothesis, that of displacement in a dielectric, used byMaxwell to illustrate his mathematical theory, seems to have been foundby many readers neither so simple nor so easy of comprehension as the oldfluid theory; indeed this seems to have been one of the chief reasons why

his views did not sooner meet with the general acceptance they have sincereceived. As many students find the conception of displacement difficult, Iventure to give an alternative method of regarding the processes occurringin the electric field, which I have often found useful and which is, from amathematical point of view, equivalent to Maxwells Theory.

2.] This method is based on the conception, introduced by Faraday, oftubes of electric force, or rather of electrostatic induction. Faraday, as iswell known, used these tubes as the language in which to express the phe-nomena of the electric field. Thus it was by their tendency to contract, andthe lateral repulsion which similar tubes exert on each other, that he ex-plained the mechanical forces between electrified bodies, while the influenceof the medium on these tubes was on his view indicated by the existenceof specific inductive capacity in dielectrics. Although the language which


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4.] ELECTRIC DISPLACEMENT AND FARADAY TUBES OF FORCE. 3

diminishing strength, the case is however different if we regard these tubesof force as being no longer merely a form of mathematical expression, butas real physical quantities having definite sizes and shapes. If we takethis view, we naturally regard the tubes as being all of the same strength,and we shall see reasons for believing that this strength is such that whenthey terminate on a conductor there is at the end of the tube a chargeof negative electricity equal to that which in the theory of electrolysis weassociate with an atom of a monovalent element such as chlorine.

This strength of the unit tubes is adopted because the phenomena ofelectrolysis show that it is a natural unit, and that fractional parts of thisunit do not exist, at any rate in electricity that has passed through anelectrolyte. We shall assume in this chapter that in all electrical processes,and not merely in electrolysis, fractional parts of this unit do not exist.

The Faraday tubes either form closed circuits or else begin and end onatoms, all tubes that are not closed being tubes that stretch in the etheralong lines either straight or curved from one atom to another. When thelength of the tube connecting two atoms is comparable with the distance

between the atoms in a molecule, the atoms are said to be in chemicalcombination; when the tube connecting the atoms is very much longerthan this, the atoms are said to be chemically free.

The property of the Faraday tubes of always forming closed circuits orelse having their ends on atoms may be illustrated by the similar propertypossessed by tubes of vortex motion in a frictionless fluid, these tubeseither form closed circuits or have their ends on the boundary of the liquid

in which the vortex motion takes place.The Faraday tubes may be supposed to be scattered throughout space,

and not merely confined to places where there is a finite electromotiveintensity, the absence of this intensity being due not to the absence of theFaraday tubes, but to the want of arrangement among such as are present:the electromotive intensity at any place being thus a measure, not of thewhole number of tubes at that place, but of the excess of the number

pointing in the direction of the electromotive intensity over the number ofthose pointing in the opposite direction.

4.] In this chapter we shall endeavour to show that the various phenom-ena of the electromagnetic field may all be interpreted as due to the motionof the Faraday tubes, or to changes in their position or shape. Thus, fromour point of view, this method of looking at electrical phenomena may be


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regarded as forming a kind of molecular theory of Electricity, the Faradaytubes taking the place of the molecules in the Kinetic Theory of Gases: theobject of the method being to explain the phenomena of the electric fieldas due to the motion of these tubes, just as it is the object of the KineticTheory of Gases to explain the properties of a gas as due to the motion ofits molecules.

These tubes also resemble the molecules of a gas in another respect, aswe regard them as incapable of destruction or creation.

5.] It may be asked at the outset, why we have taken the tubes ofelectrostatic induction as our molecules, so to speak, rather than the tubesof magnetic induction? The answer to this question is, that the evidenceafforded by the phenomena which accompany the passage of electricitythrough liquids and gases shows that molecular structure has an exceed-ingly close connection with tubes of electrostatic induction, much closerthan we have any reason to believe it has with tubes of magnetic induc-tion. The choice of the tubes of electrostatic induction as our moleculesseems thus to be the one which affords us the greatest facilities for ex-

plaining those electrical phenomena in which matter as well as the ether isinvolved.6.] Let us consider for a moment on this view the origin of the energy

in the electrostatic and electromagnetic fields. We suppose that associatedwith the Faraday tubes there is a distribution of velocity of the ether bothin the tubes themselves and in the space surrounding them. Thus wemay have rotation in the ether inside and around the tubes even when

the tubes themselves have no translatory velocity, the kinetic energy dueto this motion constituting the potential energy of the electrostatic field:while when the tubes themselves are in motion we have super-added tothis another distribution of velocity whose energy constitutes that of themagnetic field.

The energy we have considered so far is in the ether, but when a tubefalls on an atom it may modify the internal motion of the atom and thus

affect its energy. Thus, in addition to the kinetic energy of the ether arisingfrom the electric field, there may also be in the atoms some energy arisingfrom the same cause and due to the alteration of the internal motion of theatoms produced by the incidence of the Faraday tubes. If the change in theenergy of an atom produced by the incidence of a Faraday tube is differentfor atoms of different substances, if it is not the same, for example, for an


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atom of hydrogen as for one of chlorine, then the energy of a number ofmolecules of hydrochloric acid would depend upon whether the Faradaytubes started from the hydrogen and ended on the chlorine or vice versa.Since the energy in the molecules thus depends upon the disposition ofthe tubes in the molecule, there will be a tendency to make all the tubesstart from the hydrogen and end on the chlorine or vice versa, accordingas the first or second of these arrangements makes the difference betweenthe kinetic and potential energies a maximum. In other words, there will,in the language of the ordinary theory of electricity, be a tendency for allthe atoms of hydrogen to be charged with electricity of one sign, while allthe atoms of chlorine are charged with equal amounts of electricity of theopposite sign.

The result of the different effects on the energy of the atom produced bythe incidence of a Faraday tube will be the same as if the atoms of differentsubstances attracted electricity with different degrees of intensity: this hasbeen shown by v. Helmholtz to be sufficient to account for contact andfrictional electricity. It also, as we shall see inChapter II, accounts for

some of the eff

ects observed when electricity passes from a gas to a metalor vice versa.7.] The Faraday tubes when they reach a conductor shrink to molecular

dimensions. We shall consider the processes by which this is effected atthe end of this chapter, and in the meantime proceed to discuss the effectsproduced by these tubes when moving through a dielectric.

8.] In order to be able to fix the state of the electric field at any

point of a dielectric, we shall introduce a quantity which we shall call thepolarization of the dielectric, and which while mathematically identicalwith Maxwells displacement has a different physical interpretation. Thepolarization is defined as follows: LetAandBbe two neighbouring pointsin the dielectric, let a plane whose area is unity be drawn between thesepoints and at right angles to the line joining them, then the polarization inthe direction AB is the excess of the number of the Faraday tubes which

pass through the unit area from the sideAto the side B over those whichpass through the same area from the side B to the side A. In a dielectricother than air we imagine the unit area to be placed in a narrow crevasse cutout of the dielectric, the sides of the crevasse being perpendicular to AB.The polarization is evidently a vector quantity and may be resolved intocomponents in the same way as a force or a velocity; we shall denote the


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components parallel to the axes ofx, y, zby the letters f, g, h; these aremathematically identical with the quantities which Maxwell denotes by thesame letters, their physical interpretation however is different.

9.] We shall now investigate the rate of change of the components of thepolarization in a dielectric. Since the Faraday tubes in such a medium canneither be created nor destroyed, a change in the number passing throughany fixed area must be due to the motion or deformation of the tubes. Weshall suppose, in the first place, that the tubes at one place are all movingwith the same velocity. Let u, v, w be the components of the velocitiesof these tubes at any point, then the change in f, the number of tubespassing at the point x, y, z, through unit area at right angles to the axisofx, will be due to three causes. The first of these is the motion of thetubes from another part of the field up to the area under consideration;the second is the spreading out or concentration of the tubes due to theirrelative motion; and the third is the alteration in the direction of the tubesdue to the same cause.

Let 1fbe the change in fdue to the first cause, then in consequence

of the motion of the tubes, the tubes which at the time t + tpass throughthe unit area will be those which at the time twere at the point

x ut, y vt, z wt,hence 1fwill be given by the equation

1f= u

df

dx+ v

df

dy+ w

df

dzt.

In consequence of the motion of the tubes relatively to one another,those which at the timetpassed through unit area at right angles to xwillat the time t +tbe spread over an area

1 +t

dv

dy+

dw

dz

;

thus 2f, the change in fdue to this cause, will be given by the equation

2f= f

1 +t

dv

dy+

dw

dz

f,

or 2f= tf

dv

dy+

dw

dz

.


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In consequence of the deflection of the tubes due to the relative motionof their parts some of those which at the time t were at right angles tothe axis of x will at the time t+t have a component along it. Thus,for example, the tubes which at the time t were parallel to y will after a

timethas elapsed be twisted towards the axis ofxthrough an angletdudy

,

similarly those parallel tozwill be twisted through an angletdu

dz towards

the axis ofx in the time t; hence 3f, the change in fdue to this cause,will be given by the equation

3f=t

g

du

dy + h

du

dz

.

Hence iffis the total change in f in the time t, since

f=1f+2f+3f,

we have

f= udf

dx+ v

df

dy+ w

df

dz f

dv

dy+

dw

dz + g du

dy+ h

du

dz t,

which may be written as

df

dt =

d

dy(ug vf) d

dz(wf uh) u

df

dx+

dg

dy+

dh

dz

. (1)

If is the density of the free electricity, then since by the definition ofArt. 8the surface integral of the normal polarization taken over any closed

surface must be equal to the quantity of electricity inside that surface, itfollows that

= df

dx+

dg

dy+

dh

dz,

hence equation (1) may be written

Similarly

df

dt

+u= d

dy

(ug

vf)

d

dz

(wf

uh).

dg

dt + v =

d

dz(vh wg) d

dx(ug vf),

dh

dt + w=

d

dx(wf uh) d

dy(vh wg).

(2)


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ifa,b,c are the components of the magnetic induction parallel to x,y,z.

Similarly V =ha f c,W =f b

ga.

) (5)

Thus the momentum per unit volume in the dielectric, which is due tothe motion of the tubes, is at right angles to the polarization and to themagnetic induction, the magnitude of the momentum being equal to theproduct of the polarization and the component of the magnetic inductionat right angles to it. We may regard each tube as having a momentumproportional to the intensity of the component of the magnetic induction

at right angles to the direction of the tube. It is interesting to notice thatthe components of the momentum in the field as given by equations (5)are proportional to the amounts of energy transferred in unit time acrossunit planes at right angles to the axes of x, y, z in Poyntings theoryof the transfer of energy in the electromagnetic field (Phil. Trans. 1884,Part II. p. 343); hence the direction in which the energy in Poyntingstheory is supposed to move is the same as the direction of the momentum

determined by the preceding investigation.11.] The electromotive intensities parallel tox,y,zdue to the motion of

the tubes are the differential coefficients of the kinetic energy with regard tof,g,hrespectively, hence we obtain the following expressions forX,Y,Zthe components of the electromotive intensity,

X=wb vc,

Y =uc wa,Z=va ub.

(6)Thus the direction of the electromotive intensity due to the motion of

the tubes is at right angles both to the magnetic induction and to thedirection of motion of the tubes.

From equations (6) we get

dZdydY

dz =v da

dy+ w da

dz udb

dy+ dc

dz

+ a

dv

dy+

dw

dz

b du

dy cdu

dz.


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But since the equation

da

dx+

db

dy+

dc

dz = 0

holds, as we shall subsequently show, on the view we have taken of themagnetic force as well as on the ordinary view, we have

dZ

dydY

dz =u

da

dx+ v

da

dy+ w

da

dz+ a

dv

dy+

dw

dz

bdu

dy cdu

dz.

The right-hand side of this investigation is by the reasoning given in

Art. 9 equal to

da

dt

, the rate of diminution in the number of lines of

magnetic induction passing through unit area at right angles to the axisofx: hence we have

Similarly

dZ

dydY

dz = da

dt.

dX

dzdZ

dx = db

dt,

dYdxdX

dy = dc

dt.

(7)

Now by Stokes theoremR(X dx + Y dy+ Z dz)

taken round a closed circuit is equal to

ZZldZ

dydY

dz

+ mdX

dzdZ

dx

+ ndY

dxdX

dy

dS,

where l, m,nare the direction-cosines of the normal to a surface S whichis entirely bounded by the closed circuit. Substituting the preceding valuesfordZ/dy dY/dz, &c., we see that the line integral of the electromotiveintensity round a closed circuit is equal to the rate of diminution in thenumber of lines of magnetic induction passing through the circuit. Hence

the preceding view of the origin of magnetic force leads to Faradays rulefor the induction of currents by the alteration of the magnetic field.

12.] When the electromotive intensity is entirely due to the motion ofthe tubes in an isotropic medium whose specific inductive capacity is K,


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we have

f= K

4X

= K

4 {wb vc},and since

b= 4{f w hu}, c= 4{gu f v},we have f=K{f(u2 + v2 + w2) u(f u + gv + hw)};similarly g=K{g(u2 + v2 + w2) v(f u + gv + hw)},

h=K{h(u2

+ v2

+ w2

) w(f u + gv + hw)},hence f u + gv + hw= 0,

and therefore u2 + v2 + w2 = 1K

.

Hence when the electromotive intensity is entirely due to the motionof the tubes, the tubes move at right angles to themselves with the ve-locity 1/

K, which is the velocity with which light travels through the

dielectric. In this case the momentum is parallel to the direction of motion,and the electromotive intensity is in the direction of the polarization. Inthis case the polarization, the direction of motion and the magnetic force,are mutually at right angles; their relative disposition is shown inFig. 1.

Collecting the preceding results, we see that when a Faraday tube is inmotion it is accompanied by (1) a magnetic force right angles to the tubeand to the direction in which it is moving, (2) a momentum at right angles

to the tube and to the magnetic induction, (3) an electromotive intensityat right angles to the direction of motion and to the magnetic induction;this always tends to make the tube set itself at right angles to the directionin which it is moving. Thus in an isotropic medium in which there is nofree electricity and consequently no electromotive intensities except thosewhich arise from the motion of the tubes, the tubes set themselves at rightangles to the direction of motion.

13.] We have hitherto only considered the case when the tubes at anyone place in a dielectric are moving with a common velocity. We canhowever without difficulty extend these results to the case when we havedifferent sets of tubes moving with different velocities.

Let us suppose that we have the tubesf1,g1,h1, moving with a velocitywhose components areu1,v1,w1, while the tubes f2,g2,h2 move with the


30/616


Fig. 1.

velocitiesu2,v2,w2, and so on. Then the rate of increase in the number oftubes which pass through unit area at right angles to the axis ofx is, bythe same reasoning as before,

d

dy P(ug vf) d

dz P(wf uh)

P(u).

Hence we see as before that the tubes may be regarded as producing amagnetic force whose components , , are given by the equations

= 4P

(vh wg),= 4

P(wf uh),

= 4P

(ug vf).

(8)

The Kinetic energy per unit volume, T, due to the motion of these

tubes is given by the equation

T =

8{2 +2 +2},

or

T = 2hP

(vh wg)2 + P(wf uh)2 + P(ug vf)2i .


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Thus dT/du1, the momentum per unit volume parallel toxdue to thetube with suffix 1, is equal to

4g1 P(ug vf) h1 P(wf uh) ,=g1c h1b,

where a, b, c are the components of the magnetic induction.Thus U, V, W, the components of the momentum per unit volume

parallel to the axes ofx, y,zrespectively, are given by the equations

U=c

Pg b

Ph,

V =aPh cP f,W =bP f aP g.

(9)

Thus when we have a number of tubes moving about in the electricfield the resultant momentum at any point is perpendicular both to theresultant magnetic induction and to the resultant polarization, and is equalto the product of these two quantities into the sine of the angle betweenthem.

The electromotive intensities X, Y, Z parallel to the axes of x, y, zrespectively are equal to the mean values ofdT/df, dT/dg, dT/dh, hencewe have

X=bw cv,Y =cu aw,Z=av bu;

(10)

where a bar placed over any quantity indicates that the mean value of thatquantity is to be taken.Thus when a system of Faraday tubes is in motion, the electromotive

intensity is at right angles both to the resultant magnetic induction and tothe mean velocity of the tubes, and is equal in magnitude to the productof these two quantities into the sine of the angle between them.

We see from the preceding equations that there may be a resultant

magnetic force due to the motion of the positive tubes in one direction andthe negative ones in the opposite, without either resultant momentum orelectromotive intensity; for if there are as many positive as negative tubespassing through each unit area so that there is no resultant polarization,there will, by equations (9), be no resultant momentum, while if the numberof tubes moving in one direction is the same as the number moving in the


32/616


opposite, equations (10) show that there will be no resultant electromotiveintensity due to the motion of the tubes. We thus see that when themagnetic field is steady the motion of the Faraday tubes in the field willbe a kind of shearing of the positive past the negative tubes; the positivetubes moving in one direction and the negative at an equal rate in theopposite. When, however, the field is not in a steady state this ceases tobe the case, and then the electromotive intensities due to induction aredeveloped.

Mechanical Forces in the Field.

14.] The momentum parallel to xper unit volume of the medium, dueto the motion of the Faraday tubes, is by equation (9)

cP

g bPh;thus the momentum parallel to x which enters a portion of the mediumbounded by the closed surface S in unit time is equal to

RRcP

g(lu + mv+ nw) bPh(lu + mv+ nw) dS,where dS is an element of the surface and l, m, n the direction-cosines ofits inwardly directed normal.

If the surface S is so small that the external magnetic field may beregarded as constant over it, the expression may be written as

c RRP g(lu + mv+ nw) dS b RRPh(lu + mv+ nw) dS.Now

RRPg(lu + mv+ nw) dS,

andRRP

h(lu + mv+ nw) dS,

are the number of Faraday tubes parallel to y and z respectively whichenter the element in unit time, that is, they are the volume integrals of thecomponents qand r of the current parallel to y and zrespectively: if themedium surrounded by S is a dielectric this is a polarization current, if

it is a conductor it is a conduction current. Thus the momentum parallelto x communicated in unit time to unit volume of the medium, in otherwords the force parallel toxacting on unit volume of the medium, is equalto

cq br;


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similarly the forces parallel to y andzare respectively

and

ar cp,bp

aq.

) (11)

When the medium is a conductor these are the ordinary expressions forthe components of the force per unit volume of the conductor when it iscarrying a current in a magnetic field.

When, as in the above investigation, we regard the force on a conductorcarrying a current as due to the communication to the conductor of themomentum of the Faraday tubes which enter the conductor, the origin of

the force between two currents will be very much the same as that of theattraction between two bodies on Le Sages theory of gravitation. Thus,for example, if we have two parallel currents AandB flowing in the samedirection, then ifAis to the left ofB more tubes will enter Afrom the leftthan from the right, because some of those which would have come fromthe right ifB had been absent will be absorbed by B , thus in unit time themomentum having the direction left to right which enters A will exceed

that having the opposite direction; thus A will tend to move towards theright, that is towards B, while for a similar reason B will tend to movetowardsA.

15.] We have thus seen that the hypothesis of Faraday tubes in motionexplains the properties and leads to the ordinary equations of the electro-magnetic field. This hypothesis has the advantage of indicating very clearlywhy polarization and conduction currents produce similar mechanical and

magnetic effects. For the mechanical effects and the magnetic forces at anypoint in the field are due to the motion of the Faraday tubes at that point,and any alteration in the polarization involves motion of these tubes justas much as does an ordinary conduction current.

16.] We shall now proceed to illustrate this method of regarding elec-trical phenomena by applying it to the consideration of some simple cases.We shall begin with the case which suggested the method; that of a charged

sphere moving uniformly through the dielectric. Let us suppose the chargeon the sphere iseand that it is moving with velocitywparallel to the axisofz. Faraday tubes start from the sphere and are carried along with it asit moves through the dielectric; since these tubes are moving they will, aswe have seen, produce a magnetic field. We shall suppose that the systemhas settled down into a steady state, so that the sphere and its tubes are


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all moving with the same velocity w. Let f, g, h be the components ofthe polarization at any point, , , those of the magnetic force. Theexpressions for X, Y, Z, the components of the electromotive intensity,will consist of two parts, one due to the motion of the Faraday tubes andgiven by equations (6), the other due to the distribution of these tubes andderivable from a potential ; we thus have, if the magnetic permeability isunity,

X= wddx

,

Y = wddy

,

Z= ddz

.

(12)

By equations (4)

= 4gw,= 4f w,

= 0.IfK is the specific inductive capacity of the medium, we have

X=4

Kf, Y =

4

Kg, Z=

4

Kh.

Since the magnetic permeability of the dielectric is taken as unity, wemay put 1/K=V2, whereVis the velocity of light through the dielectric.

Making these substitutions for the magnetic force and the electromotiveintensity, equations (12) become

4f(V2 w2) = ddx

,

4g(V2 w2) = ddy

,

4hV2

= d

dz;

and since df

dx+

dg

dy+

dh

dz = 0,

we get d2

dx2 +

d2

dy2 +

V2 w2V2

d2

dz2 = 0, (13)


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or putting z0 = V

{V2 w2} 12z,

equation (13) becomes

d2dx2

+d2

dy2 +d

2

dz02 = 0,

a solution of which is

= A

{x2 + y2 + z02}12

=

Ax2 + y2 +

V2

V2 w2 z2

12 . (14)To find Awe notice that the normal polarization over any sphere con-

centric with the moving one must equale, the charge on the sphere; henceifa is the radius of the moving sphere,

ZZnxa

f+ya

g+ za

ho

dS=e.

Substituting forf,g,h their values, we find

Aa

4(V2 w2)ZZ

dSx2 + y2 +

V2

V2 w2 z2

32

=e,

or A2(V2 w2)

Z

0

sin dsin2 +

V2

V2 w2cos2

32

=e.

The integral, ifV > w, is equal to

2{V2 w2} 12V

;

hence A=eV{V2 w2} 12 ,


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so that f= e

4

V

{V2 w2} 12x

x2 + y2 + V2

V2 w2 z2

32

,

g= e4

V{V2 w2} 12 y

x2 + y2 + V2

V2 w2 z2

32

,

h= e

4

V

{V2 w2} 12z

x2 + y2 + V2

V2 w2 z2

32

.

(15)

Thus fx

= gy

= hz

.

The Faraday tubes are radial and the resultant polarization varies in-versely as

r2 1 + w2

V2

w2

cos2 32

,

whereris the distance of the point from the centre, and the angle whichrmakes with the direction of motion of the sphere. We see from this resultthat the polarization is greatest where = /2, least where = 0; theFaraday tubes thus leave the poles of the sphere and tend to congregate atthe equator. This arises from the tendency of these tubes to set themselvesat right angles to the direction in which they are moving. The surfacedensity of the electricity on the moving sphere varies inversely as

1 + w2

V2 w2cos2

32

,

it is thus a maximum at the equator and a minimum at the poles.The components,,of the magnetic force are given by the equations

= 4wg = eV w{V2

w2}

12

y

x2 + y2 + V

2

V2 w2 z2 32 ,

= 4wf= eV w

{V2 w2} 12x

x2 + y2 + V2

V2 w2 z2

32

,

= 0.

(16)


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These expressions as well as (15) were obtained by Mr. Heaviside byanother method in the Phil. Mag. for April, 1889.

Thus the lines of magnetic force are circles with their centres in andtheir planes at right angles to the axis ofz. Whenw is so small thatw2/V2

may be neglected, the preceding equations take the simpler forms

f= e

4

x

r3, g=

e

4

y

r3, h=

e

4

z

r3,

= ewyr3

, =ewx

r3 .

(See J. J. Thomson On the Electric and Magnetic Effects produced bythe Motion of Electrified Bodies, Phil. Mag. April, 1881.)

The moving sphere thus produces the same magnetic field as an elementof current at the centre of the sphere parallel tozwhose moment is equaltoew. When as a limiting case V =w, that is when the sphere is movingwith the velocity of light, we see from equations (15) and (16) that thepolarization and magnetic force vanish except when z= 0 when they areinfinite. The equatorial plane is thus the seat of infinite magnetic force and

polarization, while the rest of the field is absolutely devoid of either. Itought to be noticed that in this case all the Faraday tubes have arrangedthemselves so as to be at right angles to the direction in which they aremoving.

We shall now consider the momentum in the dielectric due to the motionof the Faraday tubes. Since the dielectric is non-magnetic the componentsU,V0, Wof this are by equations (9) given by the following expressions:

U= h = e24

V2wV2 w2

xzx2 + y2 + V

2

V2w2z23 ,

V0 = h = e2

4

V2w

V2 w2yz

x2 + y2 + V2

V2w2z23 ,

W = f

g=

e2

4

V2w

V2

w2

(x2 + y2)x2 + y2 +

V2

V2w2 z23 .

(17)

The resultant momentum at any point is thus at right angles to theradius and to the magnetic force; it is therefore in the plane through theradius and the direction of motion and at right angles to the former. Themagnitude of the resultant momentum per unit volume at a point at a


38/616


distance r from the centre of the sphere, and where the radius makes anangle with the direction of motion, is

e2w

4

V2

V2

w2

1

r4

sin 1 + w

2

V2w2cos2 3 .

Thus the momentum vanishes along the line of motion of the sphere,where the Faraday tubes are moving parallel to themselves, and continuallyincreases towards the equator as the tubes get to point more and more atright angles to their direction of motion.

The resultant momentum in the whole of the dielectric is evidently

parallel to the direction of motion; its magnitudeIis given by the equation

I=e2w

4

V2

V2 w2Z

a

Z 0

Z 20

sin2 r2 dr sin d d

r4

1 + w2

V2 w2cos2

3

=e2w

a

V2

V2 w2Z 1

0

sin2 d(cos )

1 + w2V2 w2

cos2 3 ,

or putting w

{V2 w2} 12cos = tan ,

we see that

I= e2V2

a{V2 w2}12Z tan

1 w

{V2w2}12

0

cos2 1V2

w2sin2 d;

or if tan1 w

{V2 w2} 12=,

I= e2

2a

V2

{V2 w2} 12

1 1

4

V2

w2

+ 1

2sin 2

1 + 1

4

V2

w2cos2

.

Thus the momentum of the sphere and dielectric parallel tozismw +I,

where mis the mass of the sphere; so that the effect of the charge will beto increase the apparent mass of the sphere by I/w or by

12

e2

a

V2

w{V2 w2} 12

1 1

4

V2

w2

+ 1

2sin 2

1 + 1

4

V2

w2cos 2

.


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When the velocity of the sphere is very small compared to that of light,

= w

V

1 + 1

6

w2

V2

approximately, and the apparent increase in the mass of the sphere is2

3

e2

a.

When in the limitw =Vthe increase in mass is infinite, thus a chargedsphere moving with the velocity of light behaves as if its mass were infinite,its velocity therefore will remain constant, in other words it is impossible

to increase the velocity of a charged body moving through the dielectricbeyond that of light.The kinetic energy per unit volume of the dielectric is

1

8(2 +2),

and hence by equations (16) and (17) it is equal to

w

2 W;

thus the total kinetic energy in the dielectric is equal to

12

wI ,

that is to

e2

4a

w V2

{V2 w2}1

21

14

V2

w2 + 12sin 21 + 14 V

2

w2

cos 2 .We shall now proceed to investigate the mechanical forces acting on the

sphere when it is moving parallel to the axis ofz in a uniform magneticfield in which the magnetic force is everywhere parallel to the axis ofxandequal toH.

IfU, V0,Ware the components of the momentum,

U=gc hb,V0 =ha f c,W =f b ga.

In this casec= 0, b= , a=+ H,


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where and have the values given in equations (16).The momentum transmitted in unit time across the surface of a sphere

concentric with the moving one has for components

ZZ wUcos dS,

ZZ wV0 cos dS,

ZZ wWcos dS,

the integration being extended over the surface of the sphere. Substitutingthe values ofU,V0,W, we see that the first and third of these expressionsvanish, while the second reduces to

e

4

V Hw

{V2

w2}12 ZZ

cos2 dS

r2

1 + w2

V2 w2cos2 32 ,

or 12

eVHw

{V2 w2} 12

Z 0

cos2 sin d1 +

w2

V2 w2cos2

32

,

which is equal to

eHwV{V2 w2} 12

((V2 w2) 32

V w2

V2 w2

w2

32

log

V + w

V w

12)

,

or to eH(V2 w2)

w

1 1

2

V

wlog

V + w

V w

.

When w/V is very small this expression reduces to

13 eHw.

This is the rate at which momentum is communicated to the sphere, inother words it is the force on the sphere; hence the force on the chargedsphere coincides in direction with the force on an element of current parallelto the axis ofz, but the magnitude of the force on the moving sphere is onlyone-third that of the force on an element of current alongzwhose momentisew. By the moment of an element of current we mean the product of theintensity of the current and the length of the element. Whenw=V, thatis when the sphere moves through the magnetic field with the velocity oflight, we see from the preceding expression that the force acting upon itvanishes.

We can get a general idea of the origin of the mechanical force on themoving sphere if we remember that the uniform magnetic field is (Art. 13)


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due to the motion of Faraday tubes, the positive tubes moving in one direc-tion, the negative ones in the opposite, and that in their motion throughthe field these tubes have to traverse the sphere. The momentum dueto these tubes when they enter the sphere is proportional to the magneticforce at the place where they enter the sphere, while their momentum whenthey leave the sphere is proportional to the magnetic force at the place ofdeparture. Now the magnetic forces at these places will be different, be-cause on one side of the sphere the magnetic force arising from its ownmotion will increase the original magnetic field, while on the other side itwill diminish it. Thus by their passage across the sphere the tubes will havegained or lost a certain amount of momentum; this will have been takenfrom or given to the sphere, which will thus be subject to a mechanicalforce.

Rotating Electrified Plates.

17.] The magnetic effects due to electrified bodies in motion aremore conveniently examined experimentally by means of electrified rotat-ing plates than by moving electrified spheres. The latter have, as far as Iknow, not been used in any experiments on electro-convection, while mostinteresting experiments with rotating plates have been made by Rowland(Berichte d. Berl. Acad. 1876, p. 211), Rowland and Hutchinson (Phil.Mag. 27, p. 445, 1889), Rontgen (Wied. Ann. 35, p. 264, 1888; 40, p. 93,1890), Himstedt (Wied. Ann. 38, p. 560, 1889). The general plan of theseexperiments is as follows: an air condenser with circular parallel platesis made to rotate about an axis through the centres of the plates and atright angles to their planes. To prevent induced currents being producedby the rotation of the plates in the earths magnetic field, radial divisionsfilled with insulating material are made in the plates. When the platesare charged and set in rotation a magnetic field is found to exist in theirneighbourhood similar to that which would be produced by electric cur-

rents flowing in concentric circular paths in the plates of the condenser,the centres of these circles being the points where the axis of rotation cutsthe plates.

Let us now consider how these magnetic forces are produced. Faradaytubes at right angles to the plates pass from one plate to the other. Weshall suppose when the condenser is rotating as a rigid body these tubes


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move as if they were rigidly connected with it. Then, taking the axis ofrotation as the axis of z, the component velocities of a tube at a pointwhose coordinates are x, y are respectivelyy and x, where is theangular velocity with which the plates are rotating.

If these were the only Faraday tubes in motion the components ,, of the magnetic force would by equations (4) be given by the equations

= 4x,

= 4y, (18)

= 0,

where (= h) is the surface-density of the electricity on either plate. Thesevalues for the components of the magnetic force do not however satisfy therelation

d

dx+

d

dy +

d

dz = 0,

which must be satisfied since the value of1

8ZZZ (2 +2 +2) dx dy dz

must, in a medium whose magnetic permeability is unity, be stationary forall values of, , which give assigned values to the currents, that is to

d

dx d

dy,

d

dzd

dx,

d

dy d

dz.

For let 0, 0, 0 be any particular values of the components of themagnetic force which satisfy the assigned conditions, then the most generalvalues of these components are expressed by the equations

=0+d

dx,

=0+d

dy,

=0+

d

dz,

where is an arbitrary function ofx, y, z.Then if RRR

(2 +2 +2) dxdydz

is stationary,

RRR( + + ) dx dy dz= 0. (19)


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Let the variations in , , be due to the increment of by an arbitraryfunction , then

=d

dx

, =d

dy

, =d

dz

.

Substituting these values for,,, and integrating by parts, equa-tion (19) becomesZZ

( dy dz+dz dx +dxdy)

ZZZ d

dx

+d

dy

+d

dz dxdydz= 0,

and therefore since is arbitrary

d

dx+

d

dy +

d

dz = 0.

The values of , , given by equation (18) cannot therefore be thecomplete expressions for the magnetic force, and since we regard all mag-

netic force as due to the motion of Faraday tubes, it follows that the tubeswhich connect the positive to the negative charges on the plates of thecondenser cannot be the only tubes in the field which are in motion; themotion of these tubes must set in motion the closed tubes which, Art. 2,exist in their neighbourhood. The motion of the closed tubes will producea magnetic field in which the forces can be derived from a magnetic po-tential . When we include the magnetic field due to the motion of these

closed tubes, we have

= 4xddx

=d0

dx,

= 4y ddy

=d0

dy,

=

d

dz

=d0

dz

,

if0 = 2(x2 + y2) ;

and since d

dx+

d

dy +

d

dz = 0,

we have d20

dx2 +

d20

dy2 +

d20

dz2 = 0.


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The question now arises, does the motion of the tubes which connectthe positive and negative electrifications on the plates only set those closedtubes in motion which are between the plates of the condenser, or does itaffect the tubes outside as well? Let us examine the consequences of thefirst hypothesis. In this case, since the Faraday tubes outside the condenserare at rest, the magnetic force will vanish except between the plates ofthe condenser; it follows, however, from the properties of the magneticpotential that it must vanish inside as well, so that no magnetic force atall would be produced by the rotation of the plates. As this is contrary tothe result of Rowlands experiments, the Faraday tubes stretching betweenthe plates must by their rotation set in motion tubes extending far awayfrom the region between the plates. The motion of these closed tubes musthowever be consistent with the condition that the magnetic force parallelto the plates due to the motion of the tubes must be continuous. Let usconsider for a moment the radial magnetic force due to the closed tubes:this may arise either from the rotation round the axis of tubes which passthrough the plates, or from the motion at right angles to the plates of

tubes parallel to them. In the first case, the velocity tangential to theplates of the tubes must be continuous, otherwise the tubes would break,and since the tangential velocity is continuous, the radial magnetic forcedue to the motion of these tubes will be continuous also. In the secondcase, the product of the normal velocity of the tubes and their numberper unit volume must be the same on the two sides of a plate, otherwisethere would be an accumulation of these tubes in the plate. The product

of the normal velocity into the number of the tubes is, however, equal tothe tangential magnetic force due to the motion of the closed tubes, sothat this must be continuous.

The open tubes which stretch from the positive electricity on one plateto the negative on the other will, however, by their motion produce adiscontinuity in the radial magnetic force, since these tubes stop at theplates, and do not pass through them. The radial magnetic force at a

point due to these tubes is 4r, where r is the distance of the pointfrom the axis of rotation. The conditions to determine the magnetic fieldare thus, (1) that except in the substance of the plates there must be amagnetic potential satisfying Laplaces equation, and (2) that at eitherplate the discontinuity in the radial magnetic force must be 4r, where is the surface-density of the electricity on the plates. These conditions


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are, however, exactly those which determine the magnetic force producedby a system of electric currents circulating in circles in the plates of thecondenser, the intensity of the currents at a distance r from the axis ofrotation beingrfor the positive and

rfor the negative plate. Hence

the magnetic force due to rotating the plates will be the same as thatproduced by this distribution of electric currents.

This conclusion seems to be confirmed by the results of the experi-ments of Rowland and Hutchinson (Phil. Mag. 27, p. 445, 1889), as usingthis hypothesis they found a tolerably accurate value of v, the ratio ofthe electromagnetic to the electrostatic unit of electricity, by means ofexperiments on a rotating plate.

We can see by similar reasoning that if only one of the plates is rotating,the other being at rest, the magnetic effect will be the same as that due toa system of electric currents circulating in the rotating plate, the intensityof the current at a distance r from the axis being r.

Some interesting experiments have been made by Rontgen (Wied. Ann.35, p. 264, 1888), in which, while the plates of the condenser were at rest,

a glass disc parallel to the plates and situated between them was set inrapid rotation and was found to produce a magnetic field. The rotation ofthe disc must thus have set in motion the Faraday tubes passing throughit, and these in turn have affected closed tubes extending into the regionbeyond the condenser.

Experiments of this kind seem to open up a field of enquiry which willthrow light upon a question which at present is one of the most obscure

in electricity: that of the relation between the velocities of the dielectricand of the Faraday tubes passing through it. This question is one of greatimportance in the Electro-magnetic Theory of Light, as but little progresscan be made in the Theory of Aberration until we have got an answer toit. Another question which we have not touched upon, but which is veryimportant in this connexion, is whether the motion of the Faraday tubesthrough ether devoid of matter would produce magnetic force, or whether

for this purpose it is necessary that the tubes should pass across ordinarymatter as well as ether. The point may be illustrated by the following case.Suppose we have a plate of glass between two parallel charged plates rigidlyelectrified, whether uniformly or otherwise, and that the whole system isset in rotation and moves like a rigid body, then, is or is not the motion ofthe system accompanied by magnetic force? Here the Faraday tubes move


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through the ether (assuming that the velocity of the ether is not the sameas that of the glass), but do not move relatively to the glass.

The motion of the tubes through the dielectricwill be requisite for theproduction of magnetic effects if we suppose that there are no closed tubesin the electric field, and that all the tubes connect portions of ordinarymatter. The recognition of closed tubes in the ether seems to be desirablein the present state of our electrical knowledge, as unless we acknowledgethe existence of such tubes we have to suppose that light being an electro-magnetic phenomenon cannot traverse a region wholly devoid of ordinarymatter, and further that the existence of magnetic force depends upon thepresence of such matter in the field.

A Steady Magnetic Field.

18.] Magnetic force on the theory we are now discussing is due tothe motion of the Faraday tubes. When the magnetic field is variable thepresence of these tubes is rendered evident by the existence of electromotiveintensities in the field: when however the field is steady, we have no directelectrical evidence of the presence of these tubes, and their disposition andvelocities have to be deduced from the equations developed in the precedingpages. We shall now proceed to examine this very important case more indetail.

In a steady magnetic field in which there is no free electricity the Fara-day tubes must be closed, exception being made of course of the shorttubes which connect together the atoms in the molecules present in thefield. Since in such a field there is no electromotive intensity, there mustpass through each unit area of the field the same number of positive asof negative tubes, that is, there must be as many tubes pointing in onedirection as the opposite. These tubes will (Art. 12) place themselves soas to be at right angles both to the direction in which they are movingand to the magnetic force. The distribution of the Faraday tubes and the

directions in which they are moving cannot be determined solely from themagnetic force; but for the purpose of forming a clear conception of theway in which the magnetic force may be produced, we shall suppose thatthe positive tubes are moving with the velocity of light in one direction, thenegative tubes with an equal velocity in the opposite, and that at any pointthe direction of a tube, its velocity, and the magnetic force are mutually


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at right angles.In a steady magnetic field surfaces of equal potential exist which cut

the lines of magnetic force at right angles, so that since both the Faradaytubes and the directions in which they are moving are at right angles tothe lines of magnetic force we may suppose that the Faraday tubes formclosed curves on the equipotential surfaces, a tube always remaining onone equipotential surface and moving along it at right angles to itself.

We shall now consider the motion of these tubes in a very simple mag-netic field: that surrounding an infinitely long circular cylinder whose axisis taken as the axis ofz, and which is uniformly magnetized at right anglesto its axis and parallel to the axis ofx.

The magnetic potential inside the cylinder is equal to

Hx,

where H is the magnetic force inside the cylinder.The potential outside the cylinder, ifa is the radius of the cylinder, is

equal to

Ha2 cos

r ,

where ris the distance from the axis of the cylinder of the point at which thepotential is reckoned, and the azimuth ofr measured from the directionof magnetization. Thus inside the cylinder the equipotential surfaces areplanes at right angles to the direction of magnetization, while outside theyare a

Documents

Notes on Recent Researches in Electricity and Magnetism Intended as a Sequel to Professor Clerk-Maxwell’s Treatise on Electricity and Magnetism