-
UNIVERSITY Of I -X CALIFORNIA I
A TREATISE
ON
ELECTRICITY AND MAGNETISM
MAXWELL
VOL. II.
Honfcon MACMILLAN AND CO.
PUBLISHERS TO THE UNIVERSITY OF
Clareniron
A TREATISE
ON
ELECTRICITY AND MAGNETISM
BY
JAMES CLERK MAXWELL, M.A.
LLD. EDIN., F.R.SS. LONDON AND EDINBURGH
HONORARY FELLOW OP TRINITY COLLEGE,
AND PROFESSOR OF EXPERIMENTAL PHYSICS
IN THE UNIVERSITY OF CAMBRIDGE
VOL. II
AT THE CLARENDON PRESS
1873
[All rights reserved]
v.
.
J/VV*Wt
CONTENTS.
PART III.
MAGNETISM. CHAPTER I.
ELEMENTARY THEOEY OF MAGNETISM.
Art. Page
371. Properties of a magnet when acted on by the earth .. .. 1
372. Definition of the axis of the magnet and of the direction of
magnetic force 1
373. Action of magnets on one another. Law of magnetic force .. 2
374. Definition of magnetic units and their dimensions 3
375. Nature of the evidence for the law of magnetic force .. .. 4
376. Magnetism as a mathematical quantity 4
377. The quantities of the opposite kinds of magnetism in a magnet
are always exactly equal .* .. .; .. 4
378. Effects of breaking a magnet .. .. 5
379. A magnet is built up of particles each of which is a magnet .. 5
380. Theory of magnetic 'matter' 5
381. Magnetization is of the nature of a vector 7
382. Meaning of the term 'Magnetic Polarization' 8
383. Properties of a magnetic particle 8
384. Definitions of Magnetic Moment, Intensity of Magnetization,
and Components of Magnetization .. .; .. .. .. 8
385. Potential of a magnetized element of volume 9
386. Potential of a magnet of finite size. Two expressions for this
potential, corresponding respectively to the theory of polari zation, and to that of magnetic 'matter* 9
387. Investigation of the action of one magnetic particle on another 10
388. Particular cases 12
389. Potential energy of a magnet in any field of force 14
390. On the magnetic moment and axis of a magnet 15
812246
vi CONTENTS.
Art. Page
391. Expansion of the potential of a magnet in spherical harmonics 16
392. The centre of a magnet and the primary and secondary axes
through the centre 17
393. The north end of a magnet in this treatise is that which points
north, and the south end that which points south. Boreal magnetism is that which is supposed to exist near the north pole of the earth and the south end of a magnet. Austral magnetism is that which belongs to the south pole of the earth and the north end of a magnet. Austral magnetism is con sidered positive 19
394. The direction of magnetic force is that in which austral mag
netism tends to move, that is, from south to nortb, and this is the positive direction of magnetic lines of force. A magnet is said to be magnetized from its south end towards its north end.. 19
CHAPTER II.
MAGNETIC FORCE AND MAGNETIC INDUCTION.
395. Magnetic force defined with reference to the magnetic potential 21
396. Magnetic force in a cylindric cavity in a magnet uniformly
magnetized parallel to the axis of the cylinder 22
397. Application to any magnet 22
398. An elongated cylinder. — Magnetic force 23
399. A thin disk. — Magnetic induction 23
400. Relation between magnetic force, magnetic induction, and mag
netization 24
401. Line-integral of magnetic force, or magnetic potential .. .. 24
402. Surface-integral of magnetic induction 25
403. Solenoidal distribution of magnetic induction .. .. .. .. 26
404. Surfaces and tubes of magnetic induction 27
405. Vector-potential of magnetic induction 27
406. Relations between the scalar and the vector-potential .. .. 28
CHAPTER III.
PARTICULAR FORMS OF MAGNETS.
407. Definition of a magnetic solenoid 31
408. Definition of a complex solenoid and expression for its potential
at any point 32
CONTENTS. Vll
Art. Page
409. The potential of a magnetic shell at any point is the product of
its strength multiplied by the solid angle its boundary sub tends at the point 32
410. Another method of proof 33
411. The potential at a point on the positive side of a shell of
strength <I> exceeds that on the nearest point on the negative
side by 477$ 34
412. Lamellar distribution of magnetism .. 34
413. Complex lamellar distribution 34
414. Potential of a solenoidal magnet 35
415. Potential of a lamellar magnet 35
416. Vector-potential of a lamellar magnet 36
417. On the solid angle subtended at a given point by a closed curve 36
418. The solid angle expressed by the length of a curve on the sphere 37
419. Solid angle found by two line-integrations 38
420. II expressed as a determinant 39
421. The solid angle is a cyclic function 40
422. Theory of the vector-potential of a closed curve 41
423. Potential energy of a magnetic shell placed in a magnetic field 42
CHAPTER IV.
INDUCED MAGNETIZATION.
424. When a body under the action of magnetic force becomes itself
magnetized the phenomenon is called magnetic induction .. 44
425. Magnetic induction in different substances 45
426. Definition of the coefficient of induced magnetization .. .. 47
427. Mathematical theory of magnetic induction. Poisson's method 47
428. Faraday's method 49
429. Case of a body surrounded by a magnetic medium 51
430. Poisson's physical theory of the cause of induced magnetism .. 53
CHAPTER V.
MAGNETIC PKOBLEMS.
431. Theory of a hollow spherical shell 56
432. Case when K. is large 58
433. When t = l 58
434. Corresponding case in two dimensions. Fig. XV 59
435. Case of a solid sphere, the coefficients of magnetization being
different in different directions 60
viii CONTENTS.
Art. Page
436. The nine coefficients reduced to six. Fig. XVI 61
437. Theory of an ellipsoid acted on by a uniform magnetic force .. 62
438. Cases of very flat and of very long ellipsoids 65
439. Statement of problems solved by Neumann, Kirchhoff and Green 67
440. Method of approximation to a solution of the general problem
when K is very small. Magnetic bodies tend towards places of most intense magnetic force, and diamagnetic bodies tend to places of weakest force 69
441. On ship's magnetism 70
CHAPTER VI.
WEBER'S THEORY OF MAGNETIC INDUCTION.
442. Experiments indicating a maximum of magnetization .. .. 74
443. Weber's mathematical theory of temporary magnetization .. 75
444. Modification of the theory to account for residual magnetization 79
445. Explanation of phenomena by the modified theory 81
446. Magnetization, demagnetization, and remagnetization .. .. 83
447. Effects of magnetization on the dimensions of the magnet .. 85
448. Experiments of Joule ' 86
CHAPTER VII.
MAGNETIC MEASUREMENTS.
449. Suspension of the magnet 88
450. Methods of observation by mirror and scale. Photographic
method 89
451. Principle of collimation employed in the Kew magnetometer .. 93
452. Determination of the axis of a magnet and of the direction of
the horizontal component of the magnetic force 94
453. Measurement of the moment of a magnet and of the intensity of
the horizontal component of magnetic force 97
454. Observations of deflexion 99
455. Method of tangents and method of sines 101
456. Observation of vibrations 102
457. Elimination of the effects of magnetic induction 105
458. Statical method of measuring the horizontal force 106
459. Bifilar suspension 107
460. System of observations in an observatory Ill
461. Observation of the dip-circle Ill
CONTENTS. IX
Art. Page
462. J. A. Broun's method of correction 115
463. Joule's suspension 115
464. Balance vertical force magnetometer 117
CHAPTER VIII.
TERRESTRIAL MAGNETISM.
465. Elements of the magnetic force 120
466. Combination of the results of the magnetic survey of a country 121
467. Deduction of the expansion of the magnetic potential of the
earth in spherical harmonics 123
468. Definition of the earth's magnetic poles. They are not at the
extremities of the magnetic axis. False poles. They do not exist on the earth's surface 123
469. Grauss' calculation of the 24 coefficients of the first four har
monics 124
470. Separation of external from internal causes of magnetic force .. 124
471. The solar and lunar variations 125
472. The periodic variations 125
473. The disturbances and their period of 11 years 126
474. Keflexions on magnetic investigations 126
PART IV.
ELECTROMAGNET ISM. CHAPTER I.
ELECTROMAGNETIC FORCE.
475. Orsted's discovery of the action of an electric current on a
magnet 128
476. The space near an electric current is a magnetic field .. .. 128
477. Action of a vertical current on a magnet 129
478. Proof that the force due to a straight current of indefinitely
great length varies inversely as the distance 129
479. Electromagnetic measure of the current 130
X CONTENTS.
Art. Page
480. Potential function due to a straight current. It is a function
of many values 130
481. The action of this current compared with that of a magnetic
shell having an infinite straight edge and extending on one side of this edge to infinity 131
482. A small circuit acts at a great distance like a magnet .. .. 131
483. Deduction from this of the action of a closed circuit of any form
and size on any point not in the current itself 131
484. Comparison between the circuit and a magnetic shell .. .. 132
485. Magnetic potential of a closed circuit 133
486. Conditions of continuous rotation of a magnet about a current 133
487. Form of the magnetic equipotential surfaces due to a closed
circuit. Fig. XVIII 134
488. Mutual action between any system of magnets and a closed
current 135
489. Reaction on the circuit 135
490. Force acting on a wire carrying a current and placed in the
magnetic field 136
491. Theory of electromagnetic rotations .. .. 138
492. Action of one electric circuit on the whole or any portion of
another 139
493. Our method of investigation is that of Faraday 140
494. Illustration of the method applied to parallel currents .. .. 140
495. Dimensions of the unit of current 141
496. The wire is urged from the side on which its magnetic action
strengthens the magnetic force and towards the side on which
it opposes it 141
497. Action of an infinite straight current on any current in its
plane .. • 142
498. Statement of the laws of electromagnetic force. Magnetic force
due to a current 142
499. Generality of these laws .. 143
500. Force acting on a circuit placed in the magnetic field .. ..144
501. Electromagnetic force is a mechanical force acting on the con
ductor, not on the electric current itself 144
CHAPTER II.
MUTUAL ACTION OF ELECTRIC CURRENTS.
502. Ampere's investigation of the law of force between the elements
of electric currents .. 146
CONTENTS. xi
Art. Page
503. His method of experimenting 146
504. Ampere's balance 147
505. Ampere's first experiment. Equal and opposite currents neu
tralize each other 147
506. Second experiment. A crooked conductor is equivalent to a
straight one carrying the same current ..148
507. Third experiment. The action of a closed current as an ele
ment of another current is perpendicular to that element .. 148
508. Fourth experiment. Equal currents in systems geometrically
similar produce equal forces 149
509. In all of these experiments the acting current is a closed one .. 151
510. Both circuits may, however, for mathematical purposes be con
ceived as consisting of elementary portions, and the action
of the circuits as the resultant of the action of these elements 151
511. Necessary form of the relations between two elementary portions
of lines 151
512. The geometrical quantities which determine their relative posi
tion 152
513. Form of the components of their mutual action 153
514. Kesolution of these in three directions, parallel, respectively, to
the line joining them and to the elements themselves .. .. 154
515. General expression for the action of a finite current on the ele
ment of another 154
516. Condition furnished by Ampere's third case of equilibrium .. 155
517. Theory of the directrix and the determinants of electrodynamic
action 156
518. Expression of the determinants in terms of the components
of the vector-potential of the current 157
519. The part of the force which is indeterminate can be expressed
as the space-variation of a potential 157
520. Complete expression for the action between two finite currents 158
521. Mutual potential of two closed currents 158
522. Appropriateness of quaternions in this investigation .. .. 158
523. Determination of the form of the functions by Ampere's fourth
case of equilibrium 159
524. The electrodynamic and electromagnetic units of currents .. 159
525. Final expressions for electromagnetic force between two ele
ments 160
526. Four different admissible forms of the theory 160
527. Of these Ampere's is to be preferred 161
xii CONTENTS.
CHAPTER III.
INDUCTION OF ELECTRIC CUEEENTS.
Art. Page
528. Faraday's discovery. Nature of his methods 162
529. The method of this treatise founded on that of Faraday .. .. 163
530. Phenomena of magneto-electric induction 164
531. General law of induction of currents 166
532. Illustrations of the direction of induced currents .. *. .. 166
533. Induction by the motion of the earth 167
534. The electromotive force due to induction does not depend on
the material of the conductor 168
535. It has no tendency to move the conductor 168
536. Felici's experiments on the laws of induction 168
537. Use of the galvanometer to determine the time-integral of the
electromotive force 170
538. Conjugate positions of two coils 171
539. Mathematical expression for the total current of induction .. 172
540. Faraday's conception of an electrotonic state 173
541. His method of stating the laws of induction with reference to
the lines of magnetic force 174
542. The law of Lenz, and Neumann's theory of induction .. .. 176
543. Helmholtz's deduction of induction from the mechanical action
of currents by the principle of conservation of energy .. .. 176
544. Thomson's application of the same principle 178
545. Weber's contributions to electrical science 178
CHAPTER IV.
INDUCTION OF A CUEEENT ON ITSELF.
546. Shock given by an electromagnet 180
547. Apparent momentum of electricity 180
548. Difference between this case and that of a tube containing a
current of water 181
549. If there is momentum it is not that of the moving electricity .. 181
550. Nevertheless the phenomena are exactly analogous to those of
momentum 181
551. An electric current has energy, which may be called electro-
kinetic energy 182
552. This leads us to form a dynamical theory of electric currents .. 182
CONTENTS. xiii CHAPTER V.
GENERAL EQUATIONS OF DYNAMICS.
Art. Page
553. Lagrange's method furnishes appropriate ideas for the study of
the higher dynamical sciences 184
554. These ideas must be translated from mathematical into dy
namical language 184
555. Degrees of freedom of a connected system 185
556. Generalized meaning of velocity 186
557. Generalized meaning of force , .. ..186
558. Generalized meaning of momentum and impulse ,. ,. .. 186
559. Work done by a small impulse .. ., 187
560. Kinetic energy in terms of momenta, (Tp) .. .. ,. .. 188
561. Hamilton's equations of motion .. .. , 189
562. Kinetic energy in terms of the velocities and momenta, (Tp,j) .. 190
563. Kinetic energy in terms of velocities, (T^) ,, ., .. .. 191
564. Relations between Tp and T^, p and q 191
565. Moments and products of inertia and mobility .. .. ,. 192
566. Necessary conditions which these coefficients must satisfy .. 193
567. Relation between mathematical, dynamical, and electrical ideas 193
CHAPTER VI.
APPLICATION OF DYNAMICS TO ELECTROMAGNETISM.
568. The electric current possesses energy 195
569. The current is a kinetic phenomenon 195
570. Work done by electromotive force 196
571. The most general expression for the kinetic energy of a system
including electric currents ., .. .. 197
572. The electrical variables do not appear in this expression .. .. 198
573. Mechanical force acting on a conductor 198
574. The part depending on products of ordinary velocities and
strengths of currents does not exist 200
575. Another experimental test , ,, ., .. 202
576. Discussion of the electromotive force 204
577. If terms involving products of velocities and currents existed
they would introduce electromotive forces, which are not ob served ,. ,. ,. 204
CHAPTER VII.
ELECTROKINETICS.
578. The electrokinetic energy of a system of linear circuits .. .. 206
579. Electromotive force in each circuit . . 207
xiv CONTENTS.
Art. Page
580. Electromagnetic force 208
581. Case of two circuits 208
582. Theory of induced currents 209
583. Mechanical action between the circuits 210
584. All the phenomena of the mutual action of two circuits depend
on a single quantity, the potential of the two circuits .. .. 210
CHAPTER VIII.
EXPLOBATION OF THE FIELD BY MEANS OF THE SECONDARY CIRCUIT.
585. The electrokinetic momentum of the secondary circuit .. .. 211
586. Expressed as a line-integral 211
587. Any system of contiguous circuits is equivalent to the circuit
formed by their exterior boundary 212
588. Electrokinetic momentum expressed as a surface -integral .. .212
589. A crooked portion of a circuit equivalent to a straight portion 213
590. Electrokinetic momentum at a point expressed as a vector, Ql .. 214
591. Its relation to the magnetic induction, 3B. Equations (A) .. 214
592. Justification of these names 215
593. Conventions with respect to the signs of translations and rota
tions 216
594. Theory of a sliding piece 217
595. Electromotive force due to the motion of a conductor .. .. 218
596. Electromagnetic force on the sliding piece ..218
597. Four definitions of a line of magnetic induction 219
598. General equations of electromotive force, (B) 219
599. Analysis of the electromotive force 222
600. The general equations referred to moving axes 223
601. The motion of the axes changes nothing but the apparent value
of the electric potential 224
602. Electromagnetic force on a conductor 224
603. Electromagnetic force on an element of a conducting body.
Equations (C) 226
CHAPTER IX.
GENERAL EQUATIONS.
604. Recapitulation 227
605. Equations of magnetization, (D) 228
606. Relation between magnetic force and electric currents .. •• 229
607. Equations of electric currents, (E) 230
608. Equations of electric displacement, (F) 232
CONTENTS. xv
Art. Page
609. Equations of electric conductivity, (G) 232
610. Equations of total currents, (H) 232
611. Currents in terms of electromotive force, (I) .. .. .. .. 233
612. Volume-density of free electricity, (J) 233
613. Surface-density of free electricity, (K) 233
614. Equations of magnetic permeability, (L) 233
615. Ampere's theory of magnets 234
616. Electric currents in terms of electrokinetic momentum .. .. 234
617. Vector-potential of electric currents 236
618. Quaternion expressions for electromagnetic quantities .. .. 236
619. Quaternion equations of the electromagnetic field 237
CHAPTER X.
DIMENSIONS OF ELECTKIC UNITS.
620. Two systems of units .. .. 239
621. The twelve primary quantities 239
622. Fifteen relations among these quantities 240
623. Dimensions in terms of [e] and [m] 241
624. Reciprocal properties of the two systems 241
625. The electrostatic and the electromagnetic systems 241
626. Dimensions of the 12 quantities in the two systems .. .. 242
627. The six derived units 243
628. The ratio of the corresponding units in the two systems .. 243
629. Practical system of electric units. Table of practical units .. 244
CHAPTER XI.
ENERGY AND STRESS.
630. The electrostatic energy expressed in terms of the free electri
city and the potential 246
631. The electrostatic energy expressed in terms of the electromotive
force and the electric displacement 246
632. Magnetic energy in terms of magnetization and magnetic force 247
633. Magnetic energy in terms of the square of the magnetic force .. 247
634. Electrokinetic energy in terms of electric momentum and electric
current 248
635. Electrokinetic energy in terms of magnetic induction and mag
netic force 248
636. Method of this treatise 249
637. Magnetic energy and electrokinetic energy compared .. .. 249
638. Magnetic energy reduced to electrokinetic energy 250
xvi CONTENTS.
Art. Page
639. The force acting on a particle of a substance due to its magnet
ization 251
640. Electromagnetic force due to an electric current passing through
it 252
641. Explanation of these forces by the hypothesis of stress in a
medium 253
642. General character of the stress required to produce the pheno
mena 255
643. When there is no magnetization the stress is a tension in the
direction of the lines of magnetic force, combined with a pressure in all directions at right angles to these lines, the
magnitude of the tension and pressure being — — • ^2, where •$
O7T
is the magnetic force 256
644. Force acting on a conductor carrying a current 257
645. Theory of stress in a medium as stated by Faraday .. .. 257
646. Numerical value of magnetic tension 258
CHAPTER XII.
CURRENT-SHEETS.
647. Definition of a current-sheet 259
648. Current-function 259
649. Electric potential , 260
650. Theory of steady currents 260
651. Case of uniform conductivity 260
652. Magnetic action of a current-sheet with closed currents .. .. 261
653. Magnetic potential due to a current-sheet 262
654. Induction of currents in a sheet of infinite conductivity .. .. 262
655. Such a sheet is impervious to magnetic action 263
656. Theory of a plane current-sheet 263
657. The magnetic functions expressed as derivatives of a single
function 264
658. Action of a variable magnetic system on the sheet 266
659. When there is no external action the currents decay, and their
magnetic action diminishes as if the sheet had moved off with constant velocity R 267
660. The currents, excited by the instantaneous introduction of a
magnetic system, produce an effect equivalent to an image of that system 267
661. This image moves away from its original position with velo
city R 268
662. Trail of images formed by a magnetic system in continuous
motion . 268
CONTENTS. xvn
Art. Page
663. Mathematical expression for the effect of the induced currents 269
664. Case of the uniform motion of a magnetic pole 269
665. Value of the force acting on the magnetic pole 270
666. Case of curvilinear motion 271
667. Case of motion near the edge of the sheet .. .. ..-'.', 271
668. Theory of Arago's rotating disk 271
669. Trail of images in the form of a helix 274
670. Spherical current-sheets 275
671. The vector- potential 276
672. To produce a field of constant magnetic force within a spherical
shell 277
673. To produce a constant force on a suspended coil 278
674. Currents parallel to a plane 278
675. A plane electric circuit. A spherical shell. An ellipsoidal
shell 279
676. A solenoid 280
677. A long solenoid 281
678. Force near the ends 282
679. A pair of induction coils 282
680. Proper thickness of wire 283
G81. An endless solenoid 284
CHAPTER XIII.
PAKALLEL CURRENTS.
682. Cylindrical conductors 286
683. The external magnetic action of a cylindric wire depends only
on the whole current through it .. 287
684. The vector-potential 288
685. Kinetic energy of the current 288
686. Repulsion between the direct and the return current .. .. 289
687. Tension of the wires. Ampere's experiment ,. 289
688. Self-induction of a wire doubled on itself 290
689. Currents of varying intensity in a cylindric wire 291
690. Relation between the electromotive force and the total current 292
691. Geometrical mean distance of two figures in a plane .. ,. 294
692. Particular cases 294
693. Application of the method to a coil of insulated wires .. .. 296
CHAPTER XIV.
CIRCULAR CURRENTS.
694. Potential due to a spherical bowl 299
695. Solid angle subtended by a circle at any point 301
VOL. II. b
xviii CONTENTS.
Art. Page
696. Potential energy of two circular currents 302
697. Moment of the couple acting between two coils 303
698. Values of Q? 303
699. Attraction between two parallel circular currents 304
700. Calculation of the coefficients for a coil of finite section .. .. 304
701. Potential of two parallel circles expressed by elliptic integrals 305
702. Lines of force round a circular current. Fig. XVIII .. .. 307
703. Differential equation of the potential of two circles 307
704. Approximation when the circles are very near one another .. 309
705. Further approximation 310
706. Coil of maximum self-induction 311
CHAPTER XV.
ELECTROMAGNETIC INSTRUMENTS.
707. Standard galvanometers and sensitive galvanometers .. .. 313
708. Construction of a standard coil 314
709. Mathematical theory of the galvanometer 315
710. Principle of the tangent galvanometer and the sine galvano
meter 316
711. Galvanometer with a single coil 316
712. Gaugain's eccentric suspension 317
713. Helmholtz's double coil. Fig. XIX 318
714. Galvanometer with four coils 319
715. Galvanometer with three coils 319
716. Proper thickness of the wire of a galvanometer 321
717. Sensitive galvanometers 322
718. Theory of the galvanometer of greatest sensibility 322
719. Law of thickness of the wire 323
720. Galvanometer with wire of uniform thickness 325
721. Suspended coils. Mode of suspension 326
722. Thomson's sensitive coil 326
723. Determination of magnetic force by means of suspended coil
and tangent galvanometer 327
724. Thomson's suspended coil and galvanometer combined .. .. 328
725. Weber's electrodynamometer 328
726. Joule's current -weigher 332"
727. Suction of solenoids 333
728. Uniform force normal to suspended coil 333
729. Electrodynamometer with torsion-arm 334
CONTENTS. xix CHAPTER XVI.
ELECTROMAGNETIC OBSERVATIONS.
Art. Page
730. Observation of vibrations , ;. 335
731. Motion in a logarithmic spiral 336
732. Eectilinear oscillations in a resisting medium 337
733. Values of successive elongations 338
734. Data and qusesita 338
735. Position of equilibrium determined from three successive elon
gations 338
736. Determination of the logarithmic decrement 339
737. When to stop the experiment 339
738. Determination of the time of vibration from three transits .. 339
739. Two series of observations 340
740. Correction for amplitude and for damping 341
741. Dead beat galvanometer 341
742. To measure a constant current with the galvanometer .. .. 342
743. Best angle of deflexion of a tangent galvanometer 343
744. Best method of introducing the current 343
745. Measurement of a current by the first elongation 344
746. To make a series of observations on a constant current .. .. 345
747. Method of multiplication for feeble currents 345
748. Measurement of a transient current by first elongation .. .. 346
749. Correction for damping 347
750. Series of observations. Zurilckwerfungs methode 348
751. Method of multiplication 350
CHAPTER XVII.
ELECTRICAL MEASUREMENT OF COEFFICIENTS OF INDUCTION.
752. Electrical measurement sometimes more accurate than direct
measurement 352
753. Determination of G^ 353
754. Determination of gl 354
755. Determination of the mutual induction of two coils .. .. 354
756. Determination of the self-induction of a coil 356
757. Comparison of the self-induction of two coils 357
CHAPTER XVIII.
DETERMINATION OF RESISTANCE IN ELECTROMAGNETIC MEASURE.
758. Definition of resistance 358
759. Kirchhoff's method 358
XX CONTENTS.
Art. Page
760. Weber's method by transient currents 360
761. His method of observation 361
762. Weber's method by damping 361
763. Thomson's method by a revolving coil 364
764. Mathematical theory of the revolving coil ..- 364
765. Calculation of the resistance 365
766. Corrections 366
767. Joule's calorimetric method 367
CHAPTER XIX.
COMPARISON OF ELECTROSTATIC WITH ELECTROMAGNETIC UNITS.
768. Nature and importance of the investigation 368
769. The ratio of the units is a velocity 369
770. Current by convection 370
771. Weber and Kohlrausch's method 370
772. Thomson's method by separate electrometer and electrodyna-
mometer 372
773. Maxwell's method by combined electrometer and electrodyna-
mometer 372
774. Electromagnetic measurement of the capacity of a condenser.
Jenkin's method 373
775. Method by an intermittent current 374
776. Condenser and Wippe as an arm of Wheatstone's bridge .. 375
777. Correction when the action is too rapid 376
778. Capacity of a condenser compared with the self-induction of a
coil 377
779. Coil and condenser combined 379
780. Electrostatic measure of resistance compared with its electro
magnetic measure 382
CHAPTER XX.
ELECTROMAGNETIC THEORY OF LIGHT.
781. Comparison of the properties of the electromagnetic medium
with those of the medium in the undulatory theory of light 383
782. Energy of light during its propagation 384
783. Equation of propagation of an electromagnetic disturbance .. 384
784. Solution when the medium is a non-conductor 386
785. Characteristics of wave-propagation 386
786. Velocity of propagation of electromagnetic disturbances .. .. 387
787. Comparison of this velocity with that of light 387
CONTENTS. xxi
Art. Page
788. The specific inductive capacity of a dielectric is the square of
its index of refraction 388
789. Comparison of these quantities in the case of paraffin .. .. 388
790. Theory of plane waves 389
791. The electric displacement and the magnetic disturbance are in
the plane* of the wave-front, and perpendicular to each other 390
792. Energy and stress during radiation 391
793. Pressure exerted by light .. .. 391
794. Equations of motion in a crystallized medium 392
795. Propagation of plane waves •,. .. 393
796. Only two waves are propagated 393
797. The theory agrees with that of Fresnel 394
798. Relation between electric conductivity and opacity .. .. 394
799. Comparison with facts 395
800. Transparent metals 395
801. Solution of the equations when the medium is a conductor .. 395
802. Case of an infinite medium, the initial state being given .. 396
803. Characteristics of diffusion 397
804. Disturbance of the electromagnetic field when a current begins
to flow 397
805. Rapid approximation to an ultimate state 398
CHAPTER XXI.
MAGNETIC ACTION ON LIGHT.
806. Possible forms of the relation between magnetism and light .. 399
807. The rotation of the plane of polarization by magnetic action .. 400
808. The laws of the phenomena 400
809. Verdet's discovery of negative rotation in ferromagnetic media 400
810. Rotation produced by quartz, turpentine, &c., independently of
magnetism 401
811. Kinematical analysis of the phenomena 402
812. The velocity of a circularly-polarized ray is different according
to its direction of rotation , 402
813. Right and left-handed rays 403
814. In media which of themselves have the rotatory property the
velocity is different for right and left-handed configurations 403
815. In media acted on by magnetism the velocity is different for
opposite directions of rotation 404
816. The luminiferous disturbance, mathematically considered, is a
vector 404
817. Kinematic equations of circularly-polarized light 405
xxii CONTENTS.
Art. Page
818. Kinetic and potential energy of the medium 406
819. Condition of wave-propagation 406
820. The action of magnetism must depend on a real rotation about
the direction of the magnetic force as an axis 407
821. Statement of the results of the analysis of the phenomenon .. 407
822. Hypothesis of molecular vortices 408
823. Variation of the vortices according to Helmholtz's law .. .. 409
824. Variation of the kinetic energy in the disturbed medium .. 409 825.- Expression in terms of the current and the velocity .. .. 410
826. The kinetic energy in the case of plane waves 410
827. The equations of motion 411
828. Velocity of a circularly-polarized ray 411
829. The magnetic rotation 412
830. Researches of Verdet 413
831. Note on a mechanical theory of molecular vortices 415
CHAPTER XXII.
ELECTRIC THEOEY OF MAGNETISM.
832. Magnetism is a phenomenon of molecules 418
833. The phenomena of magnetic molecules may be imitated by
electric currents 419
834. Difference between the elementary theory of continuous magnets
and the theory of molecular currents 419
835. Simplicity of the electric theory 420
836. Theory of a current in a perfectly conducting circuit .. .. 420
837. Case in which the current is entirely due to induction .. .. 421
838. Weber's theory of diamagnetism 421
839. Magnecrystallic induction 422
840. Theory of a perfect conductor 422
841. A medium containing perfectly conducting spherical molecules 423
842. Mechanical action of magnetic force on the current which it
excites 423
843. Theory of a molecule with a primitive current 424
844. Modifications of Weber's theory 425
845. Consequences of the theory 425
CHAPTER XXIII.
THEORIES OF ACTION AT A DISTANCE.
846. Quantities which enter into Ampere's formula 426
847. Relative motion of two electric particles 426
CONTENTS. xxiii
Art. Page
848. Relative motion of four electric particles. Fechner's theory .. 427
849. Two new forms of Ampere's formula 428
850. Two different expressions for the force between two electric
particles in motion 428
851. These are due to Gauss and to Weber respectively 429
852. All forces must be consistent with the principle of the con
servation of energy 429
853. Weber's formula is consistent with this principle but that of
Gauss is not 429
854. Helmholtz's deductions from Weber's formula 430
855. Potential of two currents 431
856. Weber's theory of the induction of electric currents .. .. 431
857. Segregating force in a conductor 432
858. Case of moving conductors 433
859. The formula of Gauss leads to an erroneous result 434
860. That of Weber agrees with the phenomena 434
861. Letter of Gauss to Weber 435
862. Theory of Riemann 435
863. Theory of C. Neumann 435
864. Theory of Betti 436
865. Repugnance to the idea of a medium 437
866. The idea of a medium cannot be got rid of 437
ERRATA. VOL. II.
p. 11, 1.1, for r.
dV, d2 .lx
read W = m9-^— = — m, m,^- — — (-)• 2 2^'
„ equation (8), insert — before each side of this equation. p. 1 3, last line but one, dele — . p. 14, 1. 8, for XVII read XIV. p. 15, equation (5), for VpdS read Vpdxdydz. p. 16, 1. 4 from bottom, after equation (3) insert of Art. 389. p. 17, equation (14), for r read r5. p. 21, 1. 1, for 386 read 385.
„ 1. 7 from bottom for in read on. p. 28, last line but one, for 386 read 385.
dF dH _ <W d# p. 41, equation (10), for ^--^ ttffi ^-^'
p. 43, equation (14), put accents on #, ?/, z.
p. 50, equation (19), for — , &c. rmc? — , &c., inverting all the differ du x cL v
ential coefficients. p. 51, 1. 11, for 309 read 310. p. 61, 1. 16, for Y=Fsm0 read Z=Fsm6.
„ equation (10), for TT read 7i2. p. 62, equation (13), for § read f. p. 63, 1. 3, for pdr read pdv. p. 67, right-hand side of equation should be
4
p. 120, equation (1), for downwards read upwards.
„ equation (2), insert — before the right-hand member of each
equation.
p. 153, 1. 15, for =(3 read =/3'. p. 155, 1. 8, for A A read AP. p. 190, equation (11), for Fbq1 read Fb^. p. 192, 1. 22, for Tp read Tp. p. 193, after 1. 5 from bottom, insert, But they will be all satisfied pro
vided the n determinants formed by the coefficients having the
indices 1 ; 1, 2 ; 1, 2, 3, &c. ; 1, 2, 3, ..n are none of them
negative.
p. 197, 1. 22, for (x^ #15 &c.) read fax^&c. „ 1. 23, for (xlt 052, &c.) read (x-^x^)^ &c. p. 208, 1. 2 from bottom, for Ny£ read \Ny£.
p. 222, 1. 9 from bottom, for -^~ or % read -^ or -&
p. 235, equations (5), for - read ju j and in (6) for — read — •
p. 245, first number of last column in the table should be 1010. p. 258, 1. 14, for perpendicular to read along.
p. 265, 1. 2 after equation (9), for -~ read -=~«
ay ciy
ERRATA. VOL. II.
3 from bottom, for (-) read (-) -
p. ;281y equation (19), for n read %.
p. 282, 1. 8, for z2 read z*.
p. 289, equation (22), for 4a24 read 2af ; and for 4«'24 read 2 a'
p. 293, equation (17), dele — .
p. 300, 1. 7, for when read where.
„ 1. 17, insert — after =.
„ 1. 26, for Q* read ft.
p. 301, equation (4') for / read r\
„ equation (5), insert — after = .
p. 302, 1. 4 from bottom, for M= \ read M=—J-
„ 1. 3 from bottom, insert at the beginning M—
n the denominator of the last term should be c,
„ last line, before the first bracket, for c22 read c2. p. 303, 1. 1 1 from bottom, for ft' read ft', p. 306, 1. 14, for 277 read 4-77.
„ 1. 15, for >fAa read 2 V~Aa. 1. 19 should be
7 Tlf
„ lines 23 and 27, change the sign of --=— •
p. 316, equation (3), for =My- read my.
p. 317, 1. 7, for ~| read -3. p. 318, 1. 8 from bottom for 36 to 31 read ^36 to p. 320, 1. 9, for 627, read 672. „ last line, after = insert f.
p. 324, equation (14) should be - ~ (1 -H—y^)=~^ = constant.
TT y y
p. 325, 1. 5 from bottom, should be #=| ^-2 ^ (a^-a3).
p. 346, 1. 2, for 0 read 0^
p. 359, equation (2), /or ^^ read —Ex.
p. 365, equation (3), last term, dele y.
PART III.
MAGNETISM. CHAPTEK I.
ELEMENTARY THEORY OF MAGNETISM.
371.] CERTAIN bodies, as, for instance, the iron ore called load stone, the earth itself, and pieces of steel which have been sub jected to certain treatment, are found to possess the following properties, and are called Magnets. •
If, near any part of the earth's surface except the Magnetic Poles, a magnet be suspended so as to turn freely about a vertical axis, it will in general tend to set itself in a certain azimuth, and if disturbed from this position it will oscillate about if. An un- magnetized body has no such tendency, but is in equilibrium in all azimuths alike.
372.] It is found that the force which acts on the body tends to cause a certain line in the body, called the Axis of the Magnet, to become parallel to a certain line in space, called the Direction of the Magnetic Force.
Let us suppose the magnet suspended so as to be free to turn in all directions about a fixed point. To eliminate the action of its weight we may suppose this point to be its centre of gravity. Let it come to a position^of equilibrium. Mark two points on the magnet, and note their positions in space. Then let the magnet be placed in a new position of equilibrium, and note the positions in space of the two marked points on the magnet.
Since the axis of the magnet coincides with the direction of magnetic force in both positions, we have to find that line in the magnet which occupies the same position in space before and
VOL. II. B
2 ELEMENTARY THEORY OF MAGNETISM. [373-
after the motion. It appears, from the theory of the motion of >•;{ ^'bodies of invariable form, that such a line always exists, and that a motion equivalent to the actual motion might have taken place by simple rotation round this line.
To find the line, join the first and last positions of each of the marked points, and draw planes bisecting these lines at right angles. The intersection of these planes will be the line required, which indicates the direction of the axis of the magnet and the direction of the magnetic force in space.
The method just described is not convenient for the practical determination of these directions. We shall return to this subject when we treat of Magnetic Measurements.
The direction of the magnetic force is found to be different at different parts of the earth's surface. If the end of the axis of the magnet which points in a northerly direction be marked, it has been found that the direction in which it sets itself in general deviates from the true meridian to a considerable extent, and that the marked end points on the whole downwards in the northern fc hemisphere and upwards in the southern.
The azimuth of the direction of the magnetic force, measured from the true north in a westerly direction, is called the Variation, or the Magnetic Declination. The angle between the direction of the magnetic force and the horizontal plane is called the Magnetic Dip. These two angles determine the direction of the magnetic force, and, when the magnetic intensity is also known, the magnetic force is completely determined. The determination of the values of these three elements at different parts of the earth's surface, the discussion of the manner in which they vary according to the place and time of observation, and the investigation of the causes of the magnetic force and its variations, constitute the science of Terrestrial Magnetism.
373.] Let us now suppose that the axes of several magnets have been determined, and the end of each which points north marked. Then, if one of these be freely suspended and another brought near it, it is found that two marked ends repel each other, that a marked and an unmarked end attract each other, and that two unmarked ends repel each other.
If the magnets are in the form of long rods or wires, uniformly and longitudinally magnetized, see below, Art. 384, it is found that the greatest manifestation of force occurs when the end of one magnet is held near the end of the other, and that the
374-] LAW OF MAGNETIC FORCE. 3
phenomena can be accounted for by supposing- that like ends of the magnets repel each other, that unlike ends attract each other, and that the intermediate parts of the magnets have no sensible mutual action.
The ends of a long thin magnet are commonly called its Poles. In the case of an indefinitely thin magnet, uniformly magnetized throughout its length, the extremities act as centres of force, and the rest of the magnet appears devoid of magnetic action. In all actual magnets the magnetization deviates from uniformity, so that no single points can be taken as the poles. Coulomb, how ever, by using long thin rods magnetized with care, succeeded in establishing the law of force between two magnetic poles *.
The repulsion between two magnetic poles is in the straight line joining them, and is numerically equal to the product of the strengths of the poles divided by the square of the distance between them.
374.] This law, of course, assumes that the strength of each pole is measured in terms of a certain unit, the magnitude of which may be deduced from the terms of the law.
The unit-pole is a pole which points north, and is such that, when placed at unit distance from another unit-pole, it repels it with unit offeree, the unit of force being defined as in Art. 6. A pole which points south is reckoned negative.
If m1 and m2 are the strengths of two magnetic poles, I the distance between them, and / the force of repulsion, all expressed
numerically, then .
~
But if [m], [I/I and [F] be the concrete units of magnetic pole, length and force, then
whence it follows that
or [m] = \Il*T-lM*\.
The dimensions of the unit pole are therefore f as regards length, ( — 1) as regards time, and \ as regards mass. These dimensions are the same as those of the electrostatic unit of electricity, which is specified in exactly the same way in Arts. 41, 42.
* His experiments on magnetism with the Torsion Balance are contained in the Memoirs of the Academy of Paris, 1780-9, and in Biot's Traite de Physique, torn. iii.
4 ELEMENTARY THEORY OF MAGNETISM. [375-
375.] The accuracy of this law may be considered to have been established by the experiments of Coulomb with the Torsion Balance, and confirmed by the experiments of Gauss and Weber, and of all observers in magnetic observatories, who are every day making measurements of magnetic quantities, and who obtain results which would be inconsistent with each other if the law of force had been erroneously assumed. It derives additional support from its consistency with the laws of electromagnetic phenomena.
376.] The quantity which we have hitherto called the strength of a pole may also be called a quantity of ' Magnetism,' provided we attribute no properties to ' Magnetism ' except those observed in the poles of magnets.
Since the expression of the law of force between given quantities of 'Magnetism' has exactly the same mathematical form as the law of force between quantities of 'Electricity' of equal numerical value, much of the mathematical treatment of magnetism must be similar to that of electricity. There are, however, other properties of magnets which must be borne in mind, and which may throw some light on the electrical properties of bodies.
Relation between the Poles of a Magnet.
377.] The quantity of magnetism at one pole of a magnet is always equal and opposite to that at the other, or more generally thus : —
In every Magnet the total quantity of Magnetism (reckoned alge braically) is zero.
Hence in a field of force which is uniform and parallel throughout the space occupied by the magnet, the force acting on the marked end of the magnet is exactly equal, opposite and parallel to that on the unmarked end, so that the resultant of the forces is a statical couple, tending to place the axis of the magnet in a determinate direction, but not to move the magnet as a whole in any direction.
This may be easily proved by putting the magnet into a small vessel and floating it in water. The vessel will turn in a certain direction, so as to bring the axis of the magnet as near as possible to the direction of the earth's magnetic force, but there will be no motion of the vessel as a whole in any direction ; so that there can be no excess of the force towards the north over that towards the south, or the reverse. It may also be shewn from the fact that magnetizing a piece of steel does not alter its weight. It does alter the apparent position of its centre of gravity, causing it in these
380.] MAGNETIC 'MATTER/ 5
latitudes to shift along the axis towards the north. The centre of inertia, as determined by the phenomena of rotation, remains unaltered.
378.] If the middle of a long thin magnet be examined, it is found to possess no magnetic properties, but if the magnet be broken at that point, each of the pieces is found to have a magnetic pole at the place of fracture, and this new pole is exactly equal and opposite to the other pole belonging to that piece. It is impossible, either by magnetization, or by breaking magnets, or by any other means, to procure a magnet whose poles are un equal.
If we break the long thin magnet into a number of short pieces we shall obtain a series of short magnets, each of which has poles of nearly the same strength as those of the original long magnet. This multiplication of poles is not necessarily a creation of energy, for we must remember that after breaking the magnet we have to do work to separate the parts, in consequence of their attraction for one another.
379.] Let us now put all the pieces of the magnet together as at first. At each point of junction there will be two poles exactly equal and of opposite kinds, placed in contact, so that their united action on any other pole will be null. The magnet, thus rebuilt, has therefore the same properties as at first, namely two poles, one at each end, equal and opposite to each other, and the part between these poles exhibits no magnetic action.
Since, in this case, we know the long magnet to be made up of little short magnets, and since the phenomena are the same as in the case of the unbroken magnet, we may regard the magnet, even before being broken, as made up of small particles, each of which has two equal and opposite poles. If we suppose all magnets to be made up of such particles, it is evident that since the algebraical quantity of magnetism in each particle is zero, the quantity in the whole magnet will also be zero, or in other words, its poles will be of equal strength but of opposite kind.
Theory of Magnetic ''Matter?
380.] Since the form of the law of magnetic action is identical with that of electric action, the same reasons which can be given for attributing electric phenomena to the action of one ' flu id' or two ' fluids' can also be used in favour of the existence of a magnetic matter, or of two kinds of magnetic matter, fluid or
6 ELEMENTARY THEORY OF MAGNETISM. [380.
otherwise. In fact, a theory of magnetic matter, if used in a purely mathematical sense, cannot fail to explain the phenomena, provided new laws are freely introduced to account for the actual facts.
One of these new laws must be that the magnetic fluids cannot pass from one molecule or particle of the magnet to another, but that the process of magnetization consists in separating to a certain extent the two fluids within each particle, and causing the one fluid to be more concentrated at one end, and the other fluid to be more concentrated at the other end of the particle. This is the theory of Poisson.
A particle of a magnetizable body is, on this theory, analogous to a small insulated conductor without charge, which on the two- fluid theory contains indefinitely large but exactly equal quantities of the two electricities. When an electromotive force acts on the conductor, it separates the electricities, causing them to become manifest at opposite sides of the conductor. In a similar manner, according to this theory, the magnetizing force causes the two kinds of magnetism, which were originally in a neutralized state, to be separated, and to appear at opposite sides of the magnetized particle.
In certain substances, such as soft iron and those magnetic substances which cannot be permanently magnetized, this magnetic condition, like the electrification of the conductor, disappears when the inducing force is removed. In other substances, such as hard steel, the magnetic condition is produced with difficulty, and, when produced, remains after the removal of the inducing force.
This is expressed by saying that in the latter case there is a Coercive Force, tending to prevent alteration in the magnetization, which must be overcome before the power of a magnet can be either increased or diminished. In the case of the electrified body this would correspond to a kind of electric resistance, which, unlike the resistance observed in metals, would be equivalent to complete insulation for electromotive forces below a certain value.
This theory of magnetism, like the corresponding theory of electricity, is evidently too large for the facts, and requires to be restricted by artificial conditions. For it not only gives no reason why one body may not differ from another on account of having more of both fluids, but it enables us to say what would be the properties of a body containing an excess of one magnetic fluid. It is true that a reason is given why such a body cannot exist,
381.] MAGNETIC POLARIZATION. 7
but this reason is only introduced as an after-thought to explain this particular fact. It does not grow out of the theory.
381.] We must therefore seek for a mode of expression which shall not be capable of expressing too much, and which shall leave room for the introduction of new ideas as these are developed from new facts. This, I think, we shall obtain if we begin by saying that the particles of a magnet are Polarized.
Meaning of the term ' Polarization?
When a particle of a body possesses properties related to a certain line or direction in the body, and when the body, retaining these properties, is turned so that this direction is reversed, then if as regards other bodies these properties of the particle are reversed, the particle, in reference to these properties, is said to be polarized, and the properties are said to constitute a particular kind of polarization.
Thus we may say that the rotation of a body about an axis constitutes a kind of polarization, because if, while the rotation continues, the direction of the axis is turned end for end, the body will be rotating in the opposite direction as regards space.
A conducting particle through which there is a current of elec tricity may be said to be polarized, because if it were turned round, and if the current continued to flow in the same direction as regards the particle, its direction in space would be reversed.
In short, if any mathematical or physical quantity is of the nature of a vector, as defined in Art. 11, then any body or particle to which this directed quantity or vector belongs may be said to be Polarized *9 because it has opposite properties in the two opposite directions or poles of the directed quantity.
The poles of the earth, for example, have reference to its rotation, and have accordingly different names.
* The word Polarization has been used in a sense not consistent with this in Optics, where a ray of light is said to be polarized when it has properties relating to its sides, which are identical on opposite sides of the ray. This kind of polarization refers to another kind of Directed Quantity, which may be called a Dipolar Quantity, in opposition to the former kind, which may be called Unipolar.
When a dipolar quantity is turned end for end it remains the same as before. Tensions and Pressures in solid bodies, Extensions, Compressions and Distortions and most of the optical, electrical, and magnetic properties of crystallized bodies are dipolar quantities.
The property produced by magnetism in transparent bodies of twisting the plane of polarization of the incident light, is, like magnetism itself, a unipolar property. The rotatory property referred to in Art. 303 is also unipolar.
8 ELEMENTARY THEORY OF MAGNETISM. [382.
Meaning of the term ' Magnetic Polarization.''
382.] In speaking of the state of the particles of a magnet as magnetic polarization, we imply that each of the smallest parts into which a magnet may be divided has certain properties related to a definite direction through the particle, called its Axis of Magnetization, and that the properties related to one end of this axis are opposite to the properties related to the other end.
The properties which we attribute to the particle are of the same kind as those which we observe in the complete magnet, and in assuming that the particles possess these properties, we only assert what we can prove by breaking the magnet up into small pieces, for each of these is found to be a magnet.
Properties of a Magnetized Particle.
383.] Let the element dxdydz be a particle of a magnet, and let us assume that its magnetic properties are those of a magnet the strength of whose positive pole is mt and whose length is ds. Then if P is any point in space distant r from the positive pole and / from the negative pole, the magnetic potential at P will be
— due to the positive pole, and -- -^ due to the negative pole, or
If ds, the distance between the poles, is very small, we may put
/— r = dscos e, (2)
where e is the angle between the vector drawn from the magnet to P and the axis of the magnet, or
, N cose. (3)
Magnetic Moment.
384.] The product of the length of a* uniformly and longitud inally magnetized bar magnet into the strength of its positive pole is called its Magnetic Moment.
Intensity of Magnetization.
The intensity of magnetization of a magnetic particle is the ratio of its magnetic moment to its volume. We shall denote it by /.
The magnetization at any point of a magnet may be defined by its intensity and its direction. Its direction may be defined by its direction-cosines A, /u,, v.
385.] COMPONENTS OF MAGNETIZATION. 9
Components of Magnetization.
The magnetization at a point of a magnet (being a vector or directed quantity) may be expressed in terms of its three com ponents referred to the axes of coordinates. Calling these A, B, C,
A = I\, B = Iy., C=Iv,
and the numerical value of I is given by the equation (4)
ja = A*+B* + C2. (5)
385.] If the portion of the magnet which we consider is the
differential element of volume dxdydz, and if / denotes the intensity
of magnetization of this element, its magnetic moment is Idxdydz.
Substituting this for mds in equation (3), and remembering that
rcose = \(£-x)+iL(ri—y) + v(C—z), (6)
where £, 77, f are the coordinates of the extremity of the vector r drawn from the point (#, y, z), we find for the potential at the point (£, 77, () due to the magnetized element at (a?, y, z\
W= {A(£-x) + B(ri-y)+C({-z)}±;dxdydz. (7)
To obtain the potential at the point (£. r], f) due to a magnet of finite dimensions, we must find the integral of this expression for every element of volume included within the space occupied by the magnet, or
(8) Integrating by parts, this becomes
dc
where the double integration in the first three terms refers to the surface of the magnet, and the triple integration in the fourth to the space within it.
If I, m, n denote the direction-cosines of the normal drawn outwards from the element of surface dS, we may write, as in Art. 21 j the sum of the first three terms,
where the integration is to be extended over the whole surface of the magnet.
10 ELEMENTARY THEORY OF MAGNETISM. [386.
If we now introduce two new symbols a and p} defined by the equations <r =
(dA dB dC^
p: ~^ + ^ + ^;j
the expression for the potential may be written
386.] This expression is identical with that for the electric potential due to a body on the surface of which there is an elec trification whose surface-density is o-, while throughout its substance there is a bodily electrification whose volume-density is p. Hence, if we assume cr and p to be the surface- and volume-densities of the distribution of an imaginary substance, which we have called t magnetic matter,' the potential due to this imaginary distribution will be identical with that due to the actual magnetization of every element of the magnet.
The surface-density v is the resolved part of the intensity of magnetization 7 in the direction of the normal to the surface drawn outwards, and the volume-density p is the ' convergence' (see Art. 25) of the magnetization at a given point in the magnet.
This method of representing the action of a magnet as due to a distribution of f magnetic matter ' is very convenient, but we must always remember that it is only an artificial method of representing the action of a system of polarized particles.
On the Action of one Magnetic Molecule o 387.] If, as in the chapter on Spherical Harmonics, Art. 129,
we make d , d d d
~TL = ^ T~ + m ~j — \- n r "> W
dh dx dy dz
where I, m, n are the direction-cosines of the axis It, then the potential due to a magnetic molecule at the origin, whose axis is parallel to klt and whose magnetic moment is mlt is
y _ d ml ml (
'** ~5*77~"HAi'
where A.L is the cosine of the angle between h± and r.
Again, if a second magnetic molecule whose moment is m2, and whose axis is parallel to hz, is placed at the extremity of the radius vector r, the potential energy due to the action of the one magnet on the other is
387.] FORCE BETWEEN TWO MAGNETIZED PARTICLES. 11
(3)
(4)
where /u12 is the cosine of the angle which the axes make with each other, and Xls A2 are the cosines of the angles which they make with r.
Let us next determine the moment of the couple with which the first magnet tends to turn the second round its centre.
Let us suppose the second magnet turned through an angle d(f) in a plane perpendicular to a third axis &3, then the work done
against the magnetic forces will be -^ — dti, and the moment of the
a(f>
forces on the magnet in this plane will be
dW ml m2 ,dyl2 d\2^
~~d^ = ~^~\d$~ Al3^'
The actual moment acting on the second magnet may therefore be considered as the resultant of two couples, of which the first acts in a plane parallel to the axes of both magnets, and tends to increase the angle between them with a force whose moment is
while the second couple acts in the plane passing through r and the axis of the second magnet, and tends to diminish the angle between these directions with a force
3 m* m9
>~^cos(r/h)siu(r/^, (7)
where (f^), (?'^2); (^1^2) denote the angles between the lines r,
To determine the force acting on the second magnet in a direction parallel to a line 7/3, we have to calculate dW d* ,K
(9)
(10)
If we suppose the actual force compounded of three forces, R, H^ and H2, in the directions of r, ^ and ^2 respectively, then the force in the direction of ^3 is
(11)
12 ELEMENTARY THEORY OF MAGNETISM. [388.
Since the direction of h% is arbitrary, we must have
3 tYl-i tlfli\ ~\
_/L ^^ .— — vMl2 "~~ 1 2/5
(12)
The force 72 is a repulsion, tending to increase r ; H^ and ZT2 act on the second magnet in the directions of the axes of the first and second magnet respectively.
This analysis of the forces acting between two small magnets was first given in terms of the Quaternion Analysis by Professor Tait in the Quarterly Math. Journ. for Jan. 1860. See also his work on Quaternions, Art. 414.
Particular Positions.
388.] (1) If Aj and A2 are each equal to 1, that is, if the axes of the magnets are in one straight line and in the same direction, fj.12 = 1, and the force between the magnets is a repulsion
p. TT , TT Qm1m2 . .
Jic-f jczi-f/ZgTs -- 4 -- (13)
The negative sign indicates that the force is an attraction.
(2) If A: and A2 are zero, and /*12 unity, the axes of the magnets are parallel to each other and perpendicular to /, and the force is a repulsion 3m1m2
In neither of these cases is there any couple.
(3) If A! = 1 and A2 = 0, then /u12 = 0. (15)
The force on the second magnet will be - — *— 2 in the direction of its axis, and the couple will be — ^— 2 t tending to turn it parallel to the first magnet. This is equivalent to a single force - ^ 2
acting parallel to the direction of the axis of the second magnet, and cutting r at a point two-thirds of its length from m2.
Fig. 1. Thus in the figure (1) two magnets are made to float on water,
388.]
FORCE BETWEEN TWO SMALL MAGNETS.
13
being in the direction of the axis of m1 , but having- its own axis at right angles to that of ml. If two points, A, B, rigidly connected with % and m2 respectively, are connected by means of a string T, the system will be in equilibrium,, provided T cuts the line m1m2 at right angles at a point one-third of the distance from ml to m2 .
(4) If we allow the second magnet to turn freely about its centre till it comes to a position of stable equilibrium, ?Fwill then be a minimum as regards k2 , and therefore the resolved part of the force due to m2, taken in the direction of ^15 will be a maximum. Hence, if we wish to produce the greatest possible magnetic force at a given point in a given direction by means of magnets, the positions of whose centres are given, then, in order to determine the proper directions of the axes of these magnets to produce this effect, we have only to place a magnet in the given direction at the given point, and to observe the direction of stable equilibrium of the axis of a second magnet when its centre is placed at each of the other given points. The magnets must then be placed with their axes in the directions indicated by that of the second magnet.
Of course, in performing this experi ment we must take account of terrestrial magnetism, if it exists.
Let the second magnet be in a posi tion of stable equilibrium as regards its direction, then since the couple acting on it vanishes, the axis of the second magnet must be in the same plane with that of the first. Hence
(M2) = (V)+M2), (16)
and the couple being
Fig. 2.
m
(sin (h-^ /t>2) — 3 cos (h-^ r) sin (r h2)),
(17)
we find when this is zero
tan (^ r) = 2 tan (r 7*2) ,
(18)
or tan^Wg-B = 2 ta,nRm2ff2. (19)
When this position has been taken up by the second magnet the
dV
value of W becomes
where h2 is in the direction of the line of force due to ml at
14 ELEMENTARY THEORY OF MAGNETISM. [389.
Hence W
,-.V;
T ~1
* (20)
Hence the second magnet will tend to move towards places of greater resultant force.
The force on the second magnet may be decomposed into a force R, which in this case is always attractive towards the first magnet, and a force ffl parallel to the axis of the first magnet, where
H L = 3^ ** _ . (21)
^ 73 Ax2 + 1
In Fig. XVII, at the end of this volume, the lines of force and equipotential surfaces in two dimensions are drawn. The magnets which produce them are supposed to be two long cylindrical rods the sections of which are represented by the circular blank spaces, and these rods are magnetized transversely in the direction of the arrows.
Jf we remember that there is a tension along the lines of force, it is easy to see that each magnet will tend to turn in the direction of the motion of the hands of a watch.
That on the right hand will also, as a whole, tend to move towards the top, and that on the left hand towards the bottom of the page.
On the Potential Energy of a Magnet placed in a Magnetic Field.
389.] Let V be the magnetic potential due to any system of magnets acting on the magnet under consideration. We shall call V the potential of the external magnetic force.
If a small magnet whose strength is m, and whose length is ds, be placed so that its positive pole is at a point where the potential is T3 and its negative pole at a point where the potential is F', the potential energy of this magnet will be mCF—P'), or, if ds is measured from the negative pole to the positive,
dV - ,1X
m-f-ds. (1)
as
If / is the intensity of the magnetization, and A, p, v its direc tion-cosines, we may write,
mds =
dV dV dV dV and -7- = A-y--f-ju-^ — |- v^-> ds dx dy dz
and, finally, if A, B, C are the components of magnetization, A=\I, B=pl, C=vl,
390.] POTENTIAL ENERGY OP A MAGNET. 15
so that the expression (1) for the potential energy of the element
of the magnet becomes
To obtain the potential energy of a magnet of finite size, we must integrate this expression for every element of the magnet. We thus obtain
W = fff(A df + Bll^ + Cd-f) dxdydz (3)
J J J ^ dx dy dz '
as the value of the potential energy of the magnet with respect to the magnetic field in which it is placed.
The potential energy is here expressed in terms of the components of magnetization and of those of the magnetic force arising from external causes.
By integration by parts we may express it in terms of the distribution of magnetic matter and of magnetic potential
~ + -- + -dxdydzy (4)
where /, m, n are the direction-cosines of the normal at the element of surface dS. If we substitute in this equation the expressions for the surface- and volume-density of magnetic matter as given in Art. 386, the expression becomes
pdS. (5)
We may write equation (3) in the form
+ Cy}dxdydz, (6)
where a, ft and y are the components of the external magnetic force.
On the Magnetic Moment and Axis of a Magnet.
390.] If throughout the whole space occupied by the magnet the external magnetic force is uniform in direction and magnitude, the components a, /3, y will be constant quantities, and if we write
IJJAdxdydz=lK, jjJBdxdydz=mK, [((cdxdydz = nKt (7)
the integrations being extended over the whole substance of the magnet, the value of ^may be written
y). (8)
16 ELEMENTAEY THEORY OF MAGNETISM.
In this expression I, m, n are the direction-cosines of the axis of
the magnet, and K is the magnetic moment of the magnet. If
e is the angle which the axis of the magnet makes with the
direction of the magnetic force «£), the value of W may be written
JF = -K$cos€. (9)
If the magnet is suspended so as to be free to turn about a vertical axis, as in the case of an ordinary compass needle, let the azimuth of the axis of the magnet be $, and let it be inclined 0 to the horizontal plane. Let the force of terrestrial magnetism be in a direction whose azimuth is 5 and dip £, then
a = «$p cos £ cos bj (3 = «£j cos £ sin 8, y = «£) sin f; (10)
I = cos 0 cos <£, m = cos 0 sin <£, n — sin 0 ; (11)
whence W— — KQ (cos £ cos 6 cos ($ — 8) + sin ( sin e). (12)
The moment of the force tending to increase $ by turning the magnet round a vertical axis is
_ ^L=_K cos Ccos<9 sin (<J>-5). (13)
On the Expansion of the Potential of a Magnet in Solid Harmonics.
391.] Let V be the potential due to a unit pole placed at the point (£, T?, f). The value of F" at the point #, y, z is
r= {(f-*)2+(>/-,?o2 +(<r-*)Ti (i)
This expression may be expanded in terms of spherical harmonics, with their centre at the origin. We have then
(2)
when FQ = - , r being the distance of (f, 77, f ) from the origin, (3)
(4)
_
2~ 2r5
fee.
To determine the value of the potential energy when the magnet is placed in the field of force expressed by this potential, we have to integrate the expression for W in equation (3) with respect to x, y and z, considering £, 77, (" and r as constants.
If we consider only the terms introduced by F~0, Ft and V2 the result will depend on the following volume-integrals,
392.] EXPANSION OF THE POTENTIAL DUE TO A MAGNET. 17 lK = jjJAdxdydz, mK = fjfsdxdydz, nK =JJJ Cdxdydz; (6)
L=jjJAxdxdydz> M = jjj Bydxdydz, N =jjJCzdxdydz', (7)
P = (B* + Cy)dxdydz, Q =
R = ^y + Bnyndydz- (8)
We thus find for the value of the potential energy of the magnet placed in presence of the unit pole at the point (^17, Q, _
r5
This expression may also be regarded as the potential energy of the unit pole in presence of the magnet, or more simply as the potential at the point £ , 17, f due to the magnet.
On ike Centre of a Magnet and its Primary and Secondary Axes.
392.] This expression may be simplified by altering the directions of the coordinates and the position of the origin. In the first place, we shall make the direction of the axis of x parallel to the axis of the magnet. This is equivalent to making
l—\^ m = 0, n — 0. (10)
If we change the origin of coordinates to the point (#', y', /), the directions of the axes remaining unchanged, the volume-integrals IK, mK and nK will remain unchanged, but the others will be altered as follows :
L'=L-lKx', M'=M-mKy', Nf = N-nKz/-f (11)
P'=P—K(mz'+ny), Q'=Q- K(nx' + lz'\ R' — R— K(ly' + mx'}.
If we now make the direction of the axis of x parallel to the axis of the magnet, and put
, Zl-M-N , R , Q , .
x'= —^ > y = Tr> z = -^> (13)
2A A A
then for the new axes M and N have their values unchanged, and the value of 1! becomes \ (M+N). P remains unchanged, and Q and R vanish. We may therefore write the potential thus,
VOL. II.
18 ELEMENTARY THEOEY OF MAGNETISM. \_392-
We have thus found a point, fixed with respect to the magnet, such that the second term of the potential assumes the most simple form when this point is taken as origin of coordinates. This point we therefore define as the centre of the magnet, and the axis drawn through it in the direction formerly defined as the direction of the magnetic axis may be defined as the principal axis of the magnet.
We may simplify the result still more by turning the axes of y
and z round that of x through half the angle whose tangent is
p -=£ — — . This will cause P to become zero, and the final form
of the potential may be written
Kt ttf-
3 2
This is the simplest form of the first two terms of the potential of a magnet. When the axes of y and z are thus placed they may be called the Secondary axes of the magnet.
We may also determine the centre of a magnet by finding the position of the origin of coordinates, for which the surface-integral of the square of the second term of the potential, extended over a sphere of unit radius, is a minimum.
The quantity which is to be made a minimum is, by Art. 141, 4 (Z2 + Mz + N*-MN-NL-LM] + 3 (P2 + Q2 +^2). (16)
The changes in the values of this quantity due to a change of position of the origin may be deduced from equations (11) and (12). Hence the conditions of a minimum are
21(2 L—M— N)+3nQ+3mR = 0, 2m(2M-N-L)+3lR+3nP = 0, (17)
2n (2N—Z—M)+3mP+3lQ = 0. If we assume I = I, m = 0, n = Q, these conditions become
2L-M—N=0, q = 0, R=0, (18)
which are the conditions made use of in the previous invest igation.
This investigation may be compared with that by which the potential of a system of gravitating matter is expanded. In the latter case, the most convenient point to assume as the origin is the centre of gravity of the system, and the most convenient axes are the principal axes of inertia through that point.
In the case of the magnet, the point corresponding to the centre of gravity is at an infinite distance in the direction of the axis,
394'] CONVENTION RESPECTING SIGNS. 19
and the point which we call the centre of the magnet is a point having- different properties from those of the centre of gravity. The quantities If, M, N correspond to the moments of inertia, and P, Q, R to the products of inertia of a material body, except that Z, M and N are not necessarily positive quantities.
When the centre of the magnet is taken as the origin, the spherical harmonic of the second order is of the sectorial form, having its axis coinciding with that of the magnet, and this is true of no other point.
When the magnet is symmetrical on all sides of this axis, as in the case of a figure of revolution, the term involving the harmonic of the second order disappears entirely.
393.] At all parts of the earth's surface, except some parts of the Polar regions, one end of a magnet points towards the north, or at least in a northerly direction, and the other in a southerly direction. In speaking of the ends of a magnet we shall adopt the popular method of calling the end which points to the north the north end of the magnet. When, however, we speak in the language of the theory of magnetic fluids we shall use the words Boreal and Austral. Boreal magnetism is an imaginary kind of matter supposed to be most abundant in the northern, parts of the earth, and Austral magnetism is the imaginary magnetic matter which prevails in the southern regions of the earth. The magnetism of the north end of a magnet is Austral, and that of the south end is Boreal. When therefore we speak of the north and south ends of a magnet we do not compare the magnet with the earth as the great magnet, but merely express the position which the magnet endeavours to take up when free to move. When, on the other hand, we wish to compare the distribution of ima ginary magnetic fluid in the magnet with that in the earth we shall use the more grandiloquent words Boreal and Austral magnetism.
394.] In speaking of a field of magnetic force we shall use the phrase Magnetic North to indicate the direction in which the north end of a compass needle would point if placed in the field of force.
In speaking of a line of magnetic force we shall always suppose it to be traced from magnetic south to magnetic north, and shall call this direction positive. In the same way the direction of magnetization of a magnet is indicated by a line drawn from the south end of the magnet towards the north end, and the end of the magnet which points north is reckoned the positive end.
20 ELEMENTARY THEORY OF MAGNETISM. \_394--
We shall consider Austral magnetism, that is, the magnetism of that end of a magnet which points north, as positive. If we denote its numerical value by m> then the magnetic potential
and the positive direction of a line of force is that in which V diminishes.
CHAPTER II.
MAGNETIC FORCE AND MAGNETIC INDUCTION.
395.] WE have already (Art. 386) determined the magnetic potential at a given point due to a magnet, the magnetization of which is given at every point of its substance, and we have shewn that the mathematical result may be expressed either in terms of the actual magnetization of every element of the magnet, or in terms of an imaginary distribution of ' magnetic matter,' partly condensed on the surface of the magnet and partly diffused through out its substance.
The magnetic potential, as thus denned, is found by the same mathematical process, whether the given point is outside the magnet or within it. The force exerted on a unit magnetic pole placed at any point outside the magnet is deduced from the potential by the same process of differentiation as in the corresponding electrical problem. If the components of this force are a, /3, y,
dV dV dV m
a= — > /3 = j-j y— j-- (1)
dx dy dz
To determine by experiment the magnetic force at a point within the magnet we must begin by removing part of the magnetized substance, so as to form a cavity within which we are to place the magnetic pole. The force acting on the pole will depend, in general, in the form of this cavity, and on the inclination of the walls of the cavity to the direction of magnetization. Hence it is necessary, in order to avoid ambiguity in speaking of the magnetic force within a magnet, to specify the form and position of the cavity within which the force is to be measured. It is manifest that when the form and position of the cavity is specified, the point within it at which the magnetic pole is placed must be regarded as
22 MAGNETIC FORCE AND MAGNETIC INDUCTION. [396.
no longer within the substance of the magnet, and therefore the ordinary methods of determining the force become at once applicable.
396.] Let us now consider a portion of a magnet in which the direction and intensity of the magnetization are uniform. Within this portion let a cavity be hollowed out in the form of a cylinder, the axis of which is parallel to the direction of magnetization, and let a magnetic pole of unit strength be placed at the middle point of the axis.
Since the generating lines of this cylinder are in the direction of magnetization, there will be no superficial distribution of mag netism on the curved surface, and since the circular ends of the cylinder are perpendicular to the direction of magnetization, there will be a uniform superficial distribution, of which the surface- density is /for the negative end, and —/for the positive end.
Let the length of the axis of the cylinder be 2 b, and its radius a. Then the force arising from this superficial distribution on a magnetic pole placed at the middle point of the axis is that due to the attraction of the disk on the positive side, and the repulsion of the disk on the negative side. These two forces are equal and in the same direction, and their sum is
---!=. (2)
From this expression it appears that the force depends, not on the absolute dimensions of the cavity, but on the ratio of the length to the diameter of the cylinder. Hence, however small we make the cavity, the force arising from the surface distribution on its walls will remain, in general, finite.
397.] We have hitherto supposed the magnetization to be uniform and in the same direction throughout the whole of the portion of the magnet from which the cylinder is hollowed out. Wlien the magnetization is not thus restricted, there will in general be a distribution of imaginary magnetic matter through the substance of the magnet. The cutting out of the cylinder will remove part of this distribution, but since in similar solid figures the forces at corresponding points are proportional to the linear dimensions of the figures, the alteration of the force on the magnetic pole due to the volume-density of magnetic matter will diminish indefinitely as the size of the cavity is diminished, while the effect due to the surface-density on the walls of the cavity remains, in general, finite.
If, therefore, we assume the dimensions of the cylinder so small
399-1 MAGNETIC FORCE IN A CAVITY. 23
that the magnetization of the part removed may be regarded as everywhere parallel to the axis of the cylinder, and of constant magnitude I, the force on a magnetic pole placed at the middle point of the axis of the cylindrical hollow will be compounded of two forces. The first of these is that due to the distribution of magnetic matter on the outer surface of the magnet, and throughout its interior, exclusive of the portion hollowed out. The components of this force are a, /3 and y, derived from the potential by equations (1). The second is the force 72, acting along the axis of the cylinder in the direction of magnetization. The value of this force depends on the ratio of the length to the diameter of the cylindric cavity.
398.] Case I. Let this ratio be very great, or let the diameter of the cylinder be small compared with its length. Expanding the
expression for R in terms of j- , it becomes
a quantity which vanishes when the ratio of b to a is made infinite. Hence, when the cavity is a very narrow cylinder with its axis parallel to the direction of magnetization, the magnetic force within the cavity is not affected by the surface distribution on the ends of the cylinder, and the components of this force are simply a, /3, y, where
dV dV dV ,,.
a = -- 7-, 0 = — -=-, y= — -—. (4)
dx dy dz
We shall define the force within a cavity of this form as the magnetic force within the magnet. Sir William Thomson has called this the Polar definition of magnetic force. When we have occasion to consider this force as a vector we shall denote it
*>7$.
399.] Case II. Let the length of the cylinder be very small
compared with its diameter, so that the cylinder becomes a thin disk. Expanding the expression for R in terms of - , it becomes
_ £+££-*..}, (5)
a 2 #3 3
the ultimate value of which, when the ratio of a to b is made infinite, is 4 TT J.
Hence, when the cavity is in the form of a thin disk, whose plane is normal to the direction of magnetization, a unit magnetic pole
24 MAGNETIC FORCE AND MAGNETIC INDUCTION. [400.
placed at the middle of the axis experiences a force 4 IT I in the direction of magnetization arising from the superficial magnetism on the circular surfaces of the disk *.
Since the components of J are A, B and (7, the components of this force are 4 -n A, 4 TT B and 4 TT C. This must be compounded with the force whose components are a, {3, y.
400.] Let the actual force on the unit pole be denoted by the vector 35, and its components by a, b and c, then a = a + 4 TT A,
0=/3 + 47T.£, (6)
C = y -f 4 TT C.
We shall define the force within a hollow disk, whose plane sides are normal to the direction of magnetization, as the Magnetic Induction within the magnet. Sir William Thomson has called this the Electromagnetic definition of magnetic force.
The three vectors, the magnetization 3, the magnetic force <!fj, and the magnetic induction S3 are connected by the vector equation
47:3. (7)
Line-Integral of Magnetic Force.
401.] Since the magnetic force, as denned in Art. 398, is that due to the distribution of free magnetism on the surface and through the interior of the magnet, and is not affected by the surface- magnetism of the cavity, it may be derived directly from the general expression for the potential of the magnet, and the line- integral of the magnetic force taken along any curve from the point A to the point B is
where VA and V^ denote the potentials at A and B respectively.
* On the force within cavities of other forms.
1. Any narrow crevasse. The force arising from the surface-magnetism is 47r/cos€ in the direction of the normal to the plane of the crevasse, where 6 is the angle between this normal and the direction of magnetization. When the crevasse is parallel to the direction of magnetization the force is the magnetic force £ ; when the crevasse is perpendicular to the direction of magnetization the force is the magnetic induction 93.
2. In an elongated cylinder, the axis of which makes an angle « with the direction of magnetization, the force arising from the surface-magnetism is 27r/sin e, perpendicular to the axis in the plane containing the axis and the direction of magnetization.
3. In a sphere the force arising from surface-magnetism is f IT I in the direction of magnetization.
402.] SURF ACE -INTEGRAL. 25
Surface-Integral of Magnetic Induction.
402.] The magnetic induction through the surface 8 is defined as the value of the integral
Q = ff%cos€dS, (9)
where 23 denotes the magnitude of the magnetic induction at the element of surface clS, and e the angle between the direction of the induction and the normal to the element of surface, and the integration is to be extended over the whole surface, which may be either closed or bounded by a closed curve.
If a, b, c denote the components of the magnetic induction, and /, m, n the direction-cosines of the normal, the surface-integral may be written
q = jj(la+mb+nG)d8. (10)
If we substitute for the components of the magnetic induction their values in terms of those of the magnetic force, and the magnetization as given in Art. 400, we find
Q = n(la + mp + ny)dS + 4 TT (lA + m£ + nC)dS. (11)
We shall now suppose that the surface over which the integration extends is a closed one, and we shall investigate the value of the two terms on the right-hand side of this equation.
Since the mathematical form of the relation between magnetic force and free magnetism is the same as that between electric force and free electricity, we may apply the result given in Art. 77 to the first term in the value of Q by substituting a, ft, y, the components of magnetic force, for X, Y, Z, the components of electric force in Art. 77, and M, the algebraic sum of the free magnetism within the closed surface, for e, the algebraic sum of the free electricity.
We thus obtain the equation
ny)48*x 4irM. (12)
Since every magnetic particle has two poles, which are equal in numerical magnitude but of opposite signs, the algebraic sum of the magnetism of the particle is zero. Hence, those particles which are entirely within the closed surface S can contribute nothing to the algebraic sum of the magnetism within S. The
26 MAGNETIC FORCE AND MAGNETIC INDUCTION. [403.
value of M must therefore depend only on those magnetic particles which are cut by the surface S.
Consider a small element of the magnet of length s and trans verse section kz, magnetized in the direction of its length, so that the strength of its poles is m. The moment of this small magnet will be ms, and the intensity of its magnetization, being the ratio of the magnetic moment to the volume, will be
/=£• (13)
Let this small magnet be cut by the surface S, so that the direction of magnetization makes an angle e' with the normal drawn outwards from the surface, then if dS denotes the area of the section, p = ds cos e/t ( 1 4)
The negative pole — m of this magnet lies within the surface S.
Hence, if we denote by dM the part of the free magnetism within S whic*h is contributed by this little magnet,
IS. (15)
To find M, the algebraic sum of the free magnetism within the closed surface S, we must integrate this expression over the closed
surface, so that
M=-
or writing A, .Z?, C for the components of magnetization, and I, m, n for the direction-cosines of the normal drawn outwards,
(16)
This gives us the value of the integral in the second term of equation (11). The value of Q in that equation may therefore be found in terms of equations (12) and (16),
Q = 47r3/-47rl/= 0, (17)
or, the surface-integral of the magnetic induction through any closed surface is zero.
403.] If we assume as the closed surface that of the differential element of volume dx dy dz, we obtain the equation
*! + *+* = 0. (18)
dx dy dz
This is the solenoidal condition which is always satisfied by the components of the magnetic induction.
405.] LINES OF MAGNETIC INDUCTION. 27
Since the distribution of magnetic induction is solenoidal, the induction through any surface bounded by a closed curve depends only on the form and position of the closed curve, and not on that of the surface itself.
404.] Surfaces at every point of which
la + mb + nc = 0 (19)
are called Surfaces of no induction, and the intersection of two such surfaces is called a Line of induction. The conditions that a curve, Sj may be a line of induction are
1 dx 1 dy \ dz , .
= 'L = . (20)
a ds I ds c ds
A system of lines of induction drawn through every point of a closed curve forms a tubular surface called a Tube of induction.
The induction across any section of such a tube is the same. If the induction is unity the tube is called a Unit tube of in duction.
All that Faraday * says about lines of magnetic force and mag netic sphondyloids is mathematically true, if understood of the lines and tubes of magnetic induction.
The magnetic force and the magnetic induction are identical outside the magnet, but within the substance of the magnet they must be carefully distinguished. In a straight uniformly mag netized bar the magnetic force due to the magnet itself is from the end which points north, which we call the positive pole, towards the south end or negative pole, both within the magnet and in the space without.
The magnetic induction, on the other hand, is from the positive pole to the negative outside the magnet, and from the negative pole to the positive within the magnet, so that the lines and tubes of induction are re-entering or cyclic figures.
The importance of the magnetic induction as a physical quantity will be more clearly seen when we study electromagnetic phe nomena. When the magnetic field is explored by a moving wire, as in Faraday's Exp. Res. 3076, it is the magnetic induction and not the magnetic force which is directly measured.
The Vector-Potential of Magnetic Induction.
405.] Since, as we have shewn in Art. 403, the magnetic in duction through a surface bounded by a closed curve depends on
* Exp. Res., series xxviii.
28 MAGNETIC FORCE AND MAGNETIC INDUCTION. [406.
the closed curve, and not on the form of the surface which is bounded by it, it must be possible to determine the induction through a closed curve by a process depending only on the nature of that curve, and not involving the construction of a surface forming a diaphragm of the curve.
This may be done by finding a vector 21 related to 33, the magnetic induction, in such a way that the line-integral of SI, extended round the closed curve, is equal to the surface-integral of 33, extended over a surface bounded by the closed curve.
If, in Art. 24, we write F9 G, H for the components of SI, and a, b, c for the components of 33, we find for the relation between these components
dH dG dF dH dG dF
a= —
.j 7
dz dz ax ax ay
The vector SI, whose components are F, G, //, is called the vector- potential of magnetic induction. The vector-potential at a given point, due to a magnetized particle placed at the origin, is nume rically equal to the magnetic moment of the particle divided by the square of the radius vector and multiplied by the sine of the angle between the axis of magnetization and the radius vector, and the direction of the vector-potential is perpendicular to the plane of the axis of magnetization and the radius vector, and is such that to an eye looking in the positive direction along the axis of magnetization the vector-potential is drawn in the direction of rotation of the hands of a watch.
Hence, for a magnet of any form in which A^ B, C are the components of magnetization at the point xyz, the components of the vector-potential at the point f 77 £ are
(22)
where p is put, for conciseness, for the reciprocal of the distance between the points (f, 77, Q and (#, y, z), and the integrations are extended over the space occupied by the magnet.
406.] The scalar, or ordinary, potential of magnetic force, Art. 386, becomes when expressed in the same notation,
406.] VECTOR- POTENTIAL. 29
/v /y\ t-j /v\
Kemembering that ~ = — -~, and that the integral dx u/ £
has the value — 4 TT ( A) when the point (£, 77, f) is included within the limits of integration, and is zero when it is not so included, (A) being the value of A at the point (f, 77, (*), we find for the value of the ^-component of the magnetic induction,
dH _ dG_ dr] d£
f d^p dzp \ d'*p d2j) }
\dydr) dzdC' dx dr] dxd^S
7> r, ^ 7 7
-ri - ~ + B -/- -f- (7 7 \dxdydz d£jJJ ( dx dy d
The first term of this expression is evidently -- ^ , or a, the component of the magnetic force.
The quantity under the integral sign in the second term is zero for every element of volume except that in which the point (f, ry, £) is included. If the value of A at the point (f, r/, f) is (A), the value of the second term is 4 TT (A)9 where (A) is evidently zero at all points outside the magnet.
We may now write the value of the ^-component of the magnetic induction « = o+4w(^), (25)
an equation which is identical with the first of those given in Art. 400. The equations for b and c will also agree with those of Art. 400.
We have already seen that the magnetic force § is derived from the scalar magnetic potential V by the application of Hamilton's operator y , so that we may write, as in Art. 1 7,
£=-vF, (26)
and that this equation is true both without and within the magnet.
It appears from the present investigation that the magnetic induction S3 is derived from the vector-potential SI by the appli cation of the same operator, and that the result is true within the magnet as well as without it.
The application of this operator to a vector-function produces,
30 MAGNETIC FORCE AND MAGNETIC INDUCTION. [406.
in general, a scalar quantity as well as a vector. The scalar part, however, which we have called the convergence of the vector- function, vanishes when the vector-function satisfies the solenoidal
condition
dF dG dH
•Jl + -J~ + -7TF = °* df; dr] d£
By differentiating the expressions for F, G, If in equations (22), we find that this equation is satisfied by these quantities.
We may therefore write the relation between the magnetic induction and its vector-potential
23 = V %
which may be expressed in words by saying that the magnetic induction is the curl of its vector-potential. See Art. 25.
CHAPTER III
MAGNETIC SOLENOIDS AND SHELLS*.
On Particular Forms of Magnets.
407.] IF a long narrow filament of magnetic matter like a wire is magnetized everywhere in a longitudinal direction, then the product of any transverse section of the filament into the mean intensity of the magnetization across it is called the strength of the magnet at that section. If the filament were cut in two at the section without altering the magnetization, the two surfaces, when separated, would be found to have equal and opposite quan tities of superficial magnetization, each of which is numerically equal to the strength of the magnet at the section.
A filament of magnetic matter, so magnetized that its strength is the same at every section, at whatever part of its length the section be made, is called a Magnetic Solenoid.
If m is the strength of the solenoid, ds an element of its length, r the distance of that element from a given point, and e the angle which r makes with the axis of magnetization of the element, the potential at the given point due to the element is
m ds cos € m dr ..
— o = —s- ~r~ ds.
r2 r* ds
Integrating this expression with respect to s} so as to take into account all the elements of the solenoid, the potential is found
to be ,11^ V = m ( ) >
rl r2
T! being the distance of the positive end of the solenoid, and r^ that of the negative end from the point where V exists.
Hence the potential due to a solenoid, and consequently all its magnetic effects, depend only on its strength and the position of
* See Sir W. Thomson's 'Mathematical Theory of Magnetism,' Phil. Trans., 1850, or Reprint.
32 MAGNETIC SOLENOIDS AND SHELLS. [408.
its ends, and not at all on its form, whether straight or curved, between these points.
Hence the ends of a solenoid may be called in a strict sense its poles.
If a solenoid forms a closed curve the potential due to it is zero at every point, so that such a solenoid can exert no magnetic action, nor can its magnetization be discovered without breaking it at some point and separating the ends.
If a magnet can be divided into solenoids, all of which either form closed curves or have their extremities in the outer surface of the magnet, the magnetization is said to be solenoidal, and, since the action of the magnet depends entirely upon that of the ends of the solenoids, the distribution of imaginary magnetic matter will be entirely superficial.
Hence the condition of the magnetization being solenoidal is dA dB dC _ dx dy dz
where A, B, C are the components of the magnetization at any point of the magnet.
408.] A longitudinally magnetized filament, of which the strength varies at different parts of its length, may be conceived to be made up of a bundle of solenoids of different lengths, the sum of the strengths of all the solenoids which pass through a given section being the magnetic strength of the filament at that section. Hence any longitudinally magnetized filament may be called a Complex Solenoid.
If the strength of a complex solenoid at any section is m, then the potential due to its action is
ds where m is variable,
Cm dr
f% -
m\ mi /I
fll* 4* i 4*
/I /*> J I
l dm 7
ds
This shews that besides the action of the two ends, which may in this case be of different strengths, there is an action due to the distribution of imaginary magnetic matter along the filament with a linear density dm
/V. — - •"— j *
ds
Magnetic Shells. 409.] If a thin shell of magnetic matter is magnetized in a
SHELLS. 33
direction everywhere normal to its surface, the intensity of the magnetization at any place multiplied by the thickness of the sheet at that place is called the Strength of the magnetic shell at that place.
If the strength of a shell is everywhere equal, it is called a Simple magnetic shell; if it varies from point to point it may be conceived to be made up of a number of simple shells superposed and overlapping each other. It is therefore called a Complex magnetic shell.
Let dS be an element of the surface of the shell at Q, and 4> the strength of the shell, then the potential at any point, P, due to the element of the shell, is
d V = <J> — - dS cos €* r2
where e is the angle between the vector QP, or r and the normal drawn from the positive side of the shell.
But if du> is the solid angle subtended by dS at the point P
r2 da — dS cos e,
whence dF = <&da>,
and therefore in the case of a simple magnetic shell
or, the potential due to a magnetic shell at any point is the product of its strength into the solid angle subtended by its edge at the given point*.
410.] The same result may be obtained in a different way by supposing the magnetic shell placed in any field of magnetic force, and determining the potential energy due to the position of the shell.
If V is the potential at the element dS, then the energy due to this element is dy dy dy
* (^ -r- +m~j- + n ~r) <*** \ da dy dz'
or, the product of the strength of the shell into the part of the surface-integral of V due to the element dS of the shell.
Hence, integrating with respect to all such elements, the energy due to the position of the shell in the field is equal to the product of the strength of the shell and the surf ace -integral of the magnetic induction taken over the surface of the shell.
Since this surface-integral is the same for any two surfaces which
* This theorem is due to Gauss, General Theory of Terrestrial Magnetism, § 38. VOL. II. D
34 MAGNETIC SOLENOIDS AND SHELLS. [411-
have the same bounding- edge and do not include between them any centre of force, the action of the magnetic shell depends only on the form of its edge.
Now suppose the field of force to be that due to a magnetic pole of strength m. We have seen (Art. 76, Cor.) that the surface- integral over a surface bounded by a given edge is the product of the strength of the pole and the solid angle subtended by the edge at the pole. Hence the energy due to the mutual action of the pole and the shell is
and this (by Green's theorem. Art. 100) is equal to the product of the strength of the pole into the potential due to the shell at the pole. The potential due to the shell is therefore 4> co.
411.] If a magnetic pole m starts from a point on the negative surface of a magnetic shell, and travels along any path in space so as to come round the edge to a point close to where it started but on the positive side of the shell, the solid angle will vary continuously, and will increase by 4 TT during the process. The work done by the pole will be 4 TT 4> m, and the potential at any point on the positive side of the shell will exceed that at the neighbouring point on the negative side by 4 TT 4>.
If a magnetic shell forms a closed surface, the potential outside the shell is everywhere zero, and that in the space within is everywhere 4 TT 4>, being positive when the positive side of the shell is inward. Hence such a shell exerts no action on any magnet placed either outside or inside the shell.
412.] If a magnet can be divided into simple magnetic shells, either closed or having their edges on the surface of the magnet, the distribution of magnetism is called Lamellar. If <£ is the sum of the strengths of all the shells traversed by a point in passing from a given point to a point xy z by a line drawn within the magnet, then the conditions of lamellar magnetization are
,_<Z<I> d<}> d(f>
A = — =— , JD = -r— , L> = — T~ *
dx dy dz
The quantity, <J>, which thus completely determines the magnet ization at any point may be called the Potential of Magnetization. It must be carefully distinguished from the Magnetic Potential.
413.] A magnet which can be divided into complex magnetic shells is said to have a complex lamellar distribution of mag netism. The condition of such a distribution is that the lines of
415.] POTENTIAL DUE TO A LAMELLAE MAGNET. 35
magnetization must be such that a system of surfaces can be drawn cutting them at right angles. This condition is expressed by the well-known equation
Aff__<lB} ^A_<IC ^_<U ^dy dz> ^dz dx' ^dx dy '
Forms of the Potentials of Solenoidal and Lamellar Magnets. 414.] The general expression for the scalar potential of a magnet
where p denotes the potential at (#, y, z) due to a unit magnetic pole placed at f, TJ, £ or in other words, the reciprocal of the distance between (f, r;, Q, the point at which the potential is measured, and (#, y> z), the position of the element of the magnet to which it is due.
This quantity may be integrated by parts, as in Arts. 96, 386.
where I, m, n are the direction-cosines of the normal drawn out wards from dS, an element of the surface of the magnet.
When the magnet is solenoidal the expression under the integral sign in the second term is zero for every point within the magnet, so that the triple integral is zero, and the scalar potential at any point, whether outside or inside the magnet, is given by the surface- integral in the first term.
The scalar potential of a solenoidal magnet is therefore com pletely determined when the normal component of the magnet ization at every point of the surface is known, and it is independent of the form of the solenoids within the magnet.
415.] In the case of a lamellar magnet the magnetization is determined by c/>, the potential of magnetization, so that dcf) d<j> d$
•**• -— ~^ — j .£> = —7— , <-/ = — ; — •
ax ay dz
The expression for V may therefore be written
= fff, JJJ \
dp .
'
dx dx dy dy dz dz Integrating this expression by parts, we find
D 2
36 MAGNETIC SOLENOIDS AND SHELLS.
The second term is zero unless the point (f, r/, f) is included in the magnet, in which case it becomes 4 TT (<£) where (<£) is the value of <p at the point £, 77, f The surface-integral may be expressed in terms of rt the line drawn from (x, y, z] to (f, rj, f ), and 0 the angle which this line makes with the normal drawn outwards from dSt so that the potential may be written
where the second term is of course zero when the point (f, TJ, f) is not included in the substance of the magnet.
The potential, F, expressed by this equation, is continuous even at the surface of the magnet, where $ becomes suddenly zero, for if we write
fit =
and if £1L is the value of H at a point just within the surface, and 122 that at a point close to the first but outside the surface,
fla = ^ + 477^),
r2 = r,.
The quantity H is not continuous at the surface of the magnet.
The components of magnetic induction are related to 12 by the equations
d& da da
a= -- =— , 0= -- =-, c — -- -j- •
dx dy dz
416.] In the case of a lamellar distribution of magnetism we may also simplify the vector-potential of magnetic induction. Its ^-component may be written
By integration by parts we may put this in the form of the surface-integral
or F .
The other components of the vector-potential may be written down from these expressions by making the proper substitutions.
On Solid Angles. 417.] We have already proved that at any point P the potential
4 1 8.] SOLID ANGLES. 37
due to a magnetic shell is equal to the solid angle subtended by the edge of the shell multiplied by the strength of the shell. As we shall have occasion to refer to solid angles in the theory of electric currents, we shall now explain how they may be measured.
Definition. The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. This area is to be reckoned positive or negative according as it lies on the left or the right- hand of the path of the radius vector as seen from the given point.
Let (£, r], f) be the given point, and let (#, y, z) be a point on the closed curve. The coordinates- x, y, z are functions of s, the length of the curve reckoned from a given point. They are periodic functions of s, recurring whenever s is increased by the whole length of the closed curve.
We may calculate the solid angle o> directly from the definition thus. Using spherical coordinates with centre at (£, 77, Q, and putting
x — f = r sin0cos$, y— rj = r sin 0 sin^, z — C=rcos0, we find the area of any curve on the sphere by integrating
co = /(I— cos0) d$, or, using the rectangular coordinates,
the integration being extended round the curve s.
If the axis of z passes once through the closed curve the first term is 2 IT. If the axis of z does not pass through it this term is zero.
418.] This method of calculating a solid angle involves a choice of axes which is to some extent arbitrary, and it does not depend solely on the closed curve. Hence the following method, in which no surface is supposed to be constructed, may be stated for the sake of geometrical propriety.
As the radius vector from the given point traces out the closed curve, let a plane passing through the given point roll on the closed curve so as to be a tangent plane at each point of the curve in succession. Let a line of unit-length be drawn from the given point perpendicular to this plane. As the plane rolls round the
38 MAGNETIC SOLENOIDS AND SHELLS. [4 1 9.
closed curve the extremity of the perpendicular will trace a second closed curve. Let the length of the second closed curve be o-, then the solid angle subtended by the first closed curve is
00 = 27T — (7.
This follows from the well-known theorem that the area of a closed curve on a sphere of unit radius, together with the circum ference of the polar curve, is numerically equal to the circumference of a great circle of the sphere.
This construction is sometimes convenient for calculating the solid angle subtended by a rectilinear figure. For our own purpose, which is to form clear ideas of physical phenomena, the following method is to be preferred, as it employs no constructions which do not flow from the physical data of the problem.
419.] A closed curve s is given in space, and we have to find the solid angle subtended by s at a given point P.
If we consider the solid angle as the potential of a magnetic shell of unit strength whose edge coincides with the closed curve, we must define it as the work done by a unit magnetic pole against the magnetic force while it moves from an infinite distance to the point P. Hence, if cr is the path of the pole as it approaches the point P, the potential must be the result of a line-integration along this path. It must also be the result of a line-integration along the closed curve s. The proper form of the expression for the solid angle must therefore be that of a double integration with respect to the two curves s and a.
When P is at an infinite distance, the solid angle is evidently zero. As the point P approaches, the closed curve, as seen from the moving point, appears to open out, and the whole solid angle may be conceived to be generated by the apparent motion of the different elements of the closed curve as the moving point ap proaches.
As the point P moves from P to P' over the element do-, the element QQ' of the closed curve, which we denote by ds, will change its position relatively to P, and the line on the unit sphere corresponding to QQ' will sweep over an area on the spherical surface, which we may write
da = Udsdcr. (I)
To find FT let us suppose P fixed while the closed curve is moved parallel to itself through a distance da- equal to PPf but in the opposite direction. The relative motion of the point P will be the same as in the real case.
420.]
GENERATION OF A SOLID ANGLE.
39
During this motion the element QQ' will generate an area in the form of a parallelogram whose sides are parallel and equal to Q Q' and PP'. If we construct a pyramid on this parallelogram as base with its vertex at P, the solid angle of this pyramid will be the increment d& which we are in search of.
To determine the value of this solid angle, let 6 and tf be the angles which ds and dcr make with PQ respect ively, and let <£ be the angle between the planes of these two angles, then the area of the projection of the parallelogram ds .dcr on a. plane per pendicular to PQ or r will be
ds dcr sin Q sin 6' sin and since this is equal to r2 d<a, we find
Fig. 3.
Hence
du> = II ds dcr = -g sin Q sin 6' sin </> ds dcr. n = — - sin 6 sin 0' sin <>.
(2) (3)
420.] We may express the angles 6, 6', and $ in terms of and its differential coefficients with respect to s and o-, for
cos0= -=-,
/»/
cos<9'= •-=-, dcr
and sin 6 sin 6' cos cp = r
dsdcr
(4)
We thus find the following value for D2,
(5)
A third expression for II in terms of rectangular coordinates may be deduced from the consideration that the volume of the pyramid whose solid angle is d& and whose axis is r is J r* do) = J r* FT ds dcr.
But the volume of this pyramid may also be expressed in terms of the projections of r, ds, and dcr on the axis of #, y and zt as a determinant formed by these nine projections, of which we must take the third part. We thus find as the value of n,
n = -^
-=— > -^— > -=—
|
c— *i |
T\—y> |
<* -. l— *> |
|
-7— > dcr |
drj -j— > dcr |
T«' |
|
dx Ts* |
d_y_ 7 ^ ds |
dz ~ds" |
(6)
40 MAGNETIC SOLENOIDS AND SHELLS. [421.
This expression gives the value of FT free from the ambiguity of sign introduced by equation (5).
421.] The value of o>, the solid angle subtended by the closed curve at the point P, may now be written
a) = ndsdv-i-WQ, (7)
where the integration with respect to s is to be extended completely round the closed curve, and that with respect to <r from A a fixed point on the curve to the point P. The constant <o0 is the value of the solid angle at the point A. It is zero if A is at an infinite distance from the closed curve.
The value of o> at any point P is independent of the form of the curve between A and P provided that it does not pass through the magnetic shell itself. If the shell be supposed infinitely thin, and if P and Pf are two points close together, but P on the positive and P' on the negative surface of the shell, then the curves AP and AP/ must lie on opposite sides of the edge of the shell, so that PAP' is a line which with the infinitely short line PP forms a closed circuit embracing the edge. The value of o> at P exceeds that at P' by 47T, that is, by the surface of a sphere of radius unity.
Hence, if a closed curve be drawn so as to pass once through the shell, or in other words, if it be linked once with the edge
of the shell, the value of the integral I lUdsdv extended round
both curves will be 47r.
This integral therefore, considered as depending only on the closed curve s and the arbitrary curve AP, is an instance of a _ function of multiple values, since, if we pass from A to P along different paths the integral will have different values according to the number of times which the curve AP is twined round the curve s.
If one form of the curve between A and P can be transformed into another by continuous motion without intersecting the curve s, the integral will have the same value for both curves, but if during the transformation it intersects the closed curve n times the values of the integral will differ by 47m.
If s and a- are any two closed curves in space, then, if they are not linked together, the integral extended once round both is zero.
If they are intertwined n times in the same direction, the value of the integral is 4iTn. It is possible, however, for two curves
422.] VECTOR- POTENTIAL OF A CLOSED CURVE. 41
to be intertwined alternately in opposite directions, so that they are inseparably linked together though the value of the integral is zero. See Fig. 4.
It was the discovery by Gauss of this very integral, expressing the work done on a magnetic pole while de scribing a closed curve in presence of a closed electric current, and indicating the geometrical connexion between the two closed curves, that led him to lament the small progress made in the Geometry of Position since the time of Leibnitz, Euler and Vandermonde. We have now, how- Flg> 4>
ever, some progress to report, chiefly due to Riemann, Helmholtz and Listing.
422.] Let us now investigate the result of integrating with respect to s round the closed curve.
One of the terms of FT in equation (7) is
f — x dri dz _ di) d A dz^ , .
r3 da- ds ~~ da d£ W ds' If we now write for brevity
^ f 1 dx 7 „ f 1 dy .. TT f 1 dz F — I - -r- ds, G = I - -f- ds, R—\- ~ ds, (9)
J r ds J r ds J r ds
the integrals being taken once round the closed curve s, this term of FT may be written
da- d£ds and the corresponding term of / n ds will be
da- d£ Collecting all the terms of n, we may now write
This quantity is evidently the rate of decrement of co, the magnetic potential, in passing along the curve a-, or in other words, it is the magnetic force in the direction of da:
By assuming da- successively in the direction of the axes of x, y and z, we obtain for the values of the components of the magnetic force
42 MAGNETIC SOLENOIDS AND SHELLS. [423-
do> _ dH dG
Ot — ~~~ 7 f. — ~~j ~" T"T~
dt, d-r] d£
d<* _ dF dH
dr] d£ d£
do> _ dG dF
y =~ JT> — ,7 /• ~j
(11)
The quantities F, G, H are the components of the vector-potential of the magnetic shell whose strength is unity, and whose edge is the curve s. They are not, like the scalar potential o>, functions having a series of values, but are perfectly determinate for every point in space.
The vector-potential at a point P due to a magnetic shell bounded by a closed curve may be found by the following geometrical construction :
Let a point Q travel round the closed curve with a velocity numerically equal to its distance from P, and let a second point R start from A and travel with a velocity the direction of which is always parallel to that of Q, but whose magnitude is unity. When Q has travelled once round the closed curve join AR, then the line AR represents in direction and in numerical magnitude the vector-potential due to the closed curve at P.
Potential Energy of a Magnetic Shell placed in a Magnetic Field.
423.] We have already shewn, in Art. 410, that the potential energy of a shell of strength <£ placed in a magnetic field whose potential is T9 is
rffidV d7 dY\ 70
x-tJJ ('is +*?+•*)** ^
where I, m, n are the direction-cosines of the normal to the shell drawn from the positive side, and the surface-integral is extended over the shell.
Now this surface-integral may be transformed into a line-integral by means of the vector-potential of the magnetic field, and we
-+cf+^,
where the integration is extended once round the closed curve s which forms the edge of the magnetic shell, the direction of ds being opposite to that of the hands of a watch when viewed from the positive side of the shell.
If we now suppose that the magnetic field is that due to a
423.] POTENTIAL OF TWO CLOSED CURVES. 43
second magnetic shell whose strength is <£', the values of F, G, H will be
where the integrations are extended once round the curve /, which forms the edge of this shell.
Substituting these values in the expression for M we find
, ff I fdx dx dy dy dz dz^ .
Jf = —$$'// - (-J- -j-' + ir j' + -j--,,)dsds', (15) ^ JJ r ^ds ds ds ds ds ds'
where the integration is extended once round s and once round /. This expression gives the potential energy due to the mutual action of the two shells, and is, as it ought to be, the same when s and / are interchanged. This expression with its sign reversed, when the strength of each shell is unity, is called the potential of the two closed curves s and /. It is a quantity of great importance in the theory of electric currents. If we write e for the angle between the directions of the elements ds and ds', the potential of s and / may be written
(16)
It is evidently a quantity of the dimension of a line.
CHAPTER IV.
INDUCED MAGNETIZATION.
424.] WE have hitherto considered the actual distribution of magnetization in a magnet as given explicitly among the data of the investigation. We have not made any assumption as to whether this magnetization is permanent or temporary, except in those parts of our reasoning in which we have supposed the magnet broken up into small portions, or small portions removed from the magnet in such a way as not to alter the magnetization of any part.
We have now to consider the magnetization of bodies with respect to the mode in which it may be produced and changed. A bar of iron held parallel to the direction of the earth's magnetic force is found to become magnetic, with its poles turned the op posite way from those of the earth, or the same way as those of a compass needle in stable equilibrium.
Any piece of soft iron placed in a magnetic field is found to exhibit magnetic properties. If it be placed in a part of the field where the magnetic force is great, as between the poles of a horse-shoe magnet, the magnetism of the iron becomes intense. If the iron is removed from the magnetic field, its magnetic properties are greatly weakened or disappear entirely. If the magnetic properties of the iron depend entirely on the magnetic force of the field in which it is placed, and vanish when it is removed from the field, it is called Soft iron. Iron which is soft in the magnetic sense is also soft in the literal sense. It is easy to bend it and give it a permanent set, and difficult to break it.
Iron which retains its magnetic properties when removed from the magnetic field is called Hard iron. Such iron does not take
425.] SOFT AND HARD STEEL. 45
up the magnetic state so readily as soft iron. The operation of hammering-, or any other kind of vibration, allows hard iron under the influence of magnetic force to assume the magnetic state more readily, and to part with it more readily when the magnetizing force is removed. Iron which is magnetically hard is also more stiff to bend and more apt to break.
The processes of hammering, rolling, wire-drawing, and sudden cooling tend to harden iron, and that of annealing tends to soften it.
The magnetic as well as the mechanical differences between steel of hard and soft temper are much greater than those between hard and soft iron. Soft steel is almost as easily magnetized and de magnetized as iron, while the hardest steel is the best material for magnets which we wish to be permanent.
Cast iron, though it contains more carbon than steel, is not so retentive of magnetization.
If a magnet could be constructed so that the distribution of its magnetization is not altered by any magnetic force brought to act upon it, it might be called a rigidly magnetized body. The only known body which fulfils this condition is a conducting circuit round which a constant electric current is made to flow.
Such a circuit exhibits magnetic properties, and may therefore be called an electromagnet, but these magnetic properties are not affected by the other magnetic forces in the field. We shall return to this subject in Part IV.
All actual magnets, whether made of hardened steel or of load stone, are found to be affected by any magnetic force which is brought to bear upon them.
It is convenient, for scientific purposes, to make a distinction between the permanent and the temporary magnetization, defining the permanent magnetization as that which exists independently of the magnetic force, and the temporary magnetization as that which depends on this force. We must observe, however, that this distinction is not founded on a knowledge of the intimate nature of magnetizable substances : it is only the expression of an hypothesis introduced for the sake of bringing calculation to bear on the phenomena. We shall return to the physical theory of magnetization in Chapter VI.
425.] At present we shall investigate the temporary magnet ization on the assumption that the magnetization of any particle of the substance depends solely on the magnetic force acting on
46 INDUCED MAGNETIZATION. [425.
that particle. This magnetic force may arise partly from external causes, and partly from the temporary magnetization of neigh bouring particles.
A body thus magnetized in virtue of the action of magnetic force, is said to be magnetized by induction, and the magnetization is said to be induced by the magnetizing force.
The magnetization induced by a given magnetizing force differs in different substances. It is greatest in the purest and softest iron, in which the ratio of the magnetization to the magnetic force may reach the value 32, or even 45 *.
Other substances, such as the metals nickel and cobalt, are capable of an inferior degree of magnetization, and all substances when subjected to a sufficiently strong magnetic force, are found to give indications of polarity.
When the magnetization is in the same direction as the magnetic force, as in iron, nickel, cobalt, &c., the substance is called Para magnetic, Ferromagnetic, or more simply Magnetic. When the induced magnetization is in the direction opposite to the magnetic force, as in bismuth, &c., the substance is said to be Diamagnetic.
In all these substances the ratio of the magnetization to the magnetic force which produces it is exceedingly small, being only about — 4 o (H) o Q m the case °f bismuth, which is the most highly diamagnetic substance known.
In crystallized, strained, and organized substances the direction of the magnetization does not always coincide with that of the magnetic force which produces it. The relation between the com ponents of magnetization, referred to axes fixed in the body, and those of the magnetic force, may be expressed by a system of three linear equations. Of the nine coefficients involved in these equa tions we shall shew that only six are independent. The phenomena of bodies of this kind are classed under the name of Magnecrystallic phenomena.
When placed in a field of magnetic force, crystals tend to set themselves so that the axis of greatest paramagnetic, or of least diamagnetic, induction is parallel to the lines of magnetic force. See Art. 435.
In soft iron, the direction of the magnetization coincides with that of the magnetic force at the point, and for small values of the magnetic force the magnetization is nearly proportional to it.
* Thaten, Nova Ada, Reg. Soc. Sc., Upsal., 1863.
427.] PROBLEM OF INDUCED MAGNETIZATION. 47
As the magnetic force increases, however, the magnetization in creases more slowly, and it would appear from experiments described in Chap. VI, that there is a limiting value of the magnetization, beyond which it cannot pass, whatever be the value of the magnetic force.
In the following outline of the theory of induced magnetism, we shall begin by supposing the magnetization proportional to the magnetic force, and in the same line with it.
Definition of the Coefficient of Induced Magnetization.
426.] Let $ be the magnetic force, defined as in Art. 398, at any point of the body, and let 3 be the magnetization at that point, then the ratio of 3 to § is called the Coefficient of Induced Magnetization.
Denoting this coefficient by K, the fundamental equation of induced magnetism is
The coefficient K is positive for iron and paramagnetic substances, and negative for bismuth and diamagnetic substances. It reaches the value 32 in iron, and it is said to be large in the case of nickel and cobalt, but in all other cases it is a very small quantity, not greater than 0.00001.
The force <£) arises partly from the action of magnets external to the body magnetized by induction, and partly from the induced magnetization of the body itself, Both parts satisfy the condition of having a potential.
427.] Let V be the potential due to magnetism external to the body, let X2 be that due to the induced magnetization, then if U is the actual potential due to both causes
u= r+a. (2)
Let the components of the magnetic force «£), resolved in the directions of x, y, z, be a, /3, y, and let those of the magnetization 3 be A, B, C, then by equation (1),
A = K a,
*=K/3, (3)
C = Ky.
Multiplying these equations by dx, dy, dz respectively, and adding, we find
Adx + Bdy+Cdz = K(
48 INDUCED MAGNETIZATION. [427.
But since a, (3 and y are derived from the potential U, we may write the second member —KdU.
Hence, if /c is constant throughout the substance, the first member must also be a complete differential of a function of #, y and z, which we shall call $, and the equation becomes
i A d(b „ d(b d(b
where A = -f- , B = ~- , C — — - . (5)
ax dy dz
The magnetization is therefore lamellar, as defined in Art. 412.
It was shewn in Art. 386 that if p is the volume-density of free magnetism,
(dA dB dC.
P- — (-J- +-J- + T-}' x## dy dz '
which becomes in virtue of equations (3),
/da d(3 dy\
\lx dy dz' But, by Art. 77,
da dj3 dy _
dx dy dz ~
Hence (l+47r*)p = 0,
whence p = 0 (6)
throughout the substance, and the magnetization is therefore sole- noidal as well as lamellar. See Art. 407.
There is therefore no free magnetism except on the bounding surface of the body. If v be the normal drawn inwards from the surface, the magnetic surface-density is
d^> (-^
a- = j-- (7)
dv
The potential II due to this magnetization at any point may therefore be found from the surface-integral
«-//=
dS. (8)
The value of £1 will be finite and continuous everywhere, and will satisfy Laplace's equation at every point both within and without the surface. If we distinguish by an accent the value of H outside the surface, and if v be the normal drawn outwards, we have at the surface
Of =0.1 (9)
428.] POISSON'S METHOD. 49
da da'
— + ^ = -4™, by Art. 78,
= 4*8.^). , ., -..: : .
dU = -47rKj;> bF(4)»
fdV d^ , = -47rK(^+^),by(2).
We may therefore write the surface-condition
Hence the determination of the magnetism induced in a homo geneous isotropic body, bounded by a surface S, and acted upon by external magnetic forces whose potential is V9 may be reduced to the following mathematical problem.
We must find two functions H and H' satisfying the following conditions :
Within the surface S9 XI must be finite and continuous, and must satisfy Laplace's equation.
Outside the surface S, Of must be finite and continuous, it must vanish at an infinite distance, and must satisfy Laplace's equation.
At every point of the surface itself, H = Of, and the derivatives of H, Of and V with respect to the normal must satisfy equation (10). _
This method of treating the problem of induced magnetism is due to Poisson. The quantity k which he uses in his memoirs is not the same as *, but is related to it as follows :
47TK(£-l)+3/&= 0. (11)
The coefficient K which we have here used was introduced by J. Neumann.
428.] The problem of induced magnetism may be treated in a different manner by introducing the quantity which we have called, with Faraday, the Magnetic Induction.
The relation between 23, the magnetic induction, «£j, the magnetic force, and 3> the magnetization, is expressed by the equation
53 = $ + 471 3. (12)
The equation which expresses the induced magnetization in terms of the magnetic force is
3 = K$. (13)
VOL. IT. E
50 INDUCED MAGNETIZATION. [428.
Hence, eliminating- 3, we find
$ = (1+47TK)£ (14)
as the relation between the magnetic induction and the magnetic force in substances whose magnetization is induced by magnetic force.
In the most general case K may be a function, not only of the position of the point in the substance, but of the direction of the vector «jp, but in the case which we are now considering K is a numerical quantity.
If we next write ^ = I + 4 -n K} (15)
we may define /x as the ratio of the magnetic induction to the magnetic force, and we may call this ratio the magnetic inductive capacity of the substance, thus distinguishing it from K, the co efficient of induced magnetization.
If we write U for the total magnetic potential compounded of T7, the potential due to external causes, and 12 for that due to the induced magnetization, we may express a, b, c, the components of magnetic induction, and a, (3, y, the components of magnetic force, as follows : dU
~}
a = "0 = -M'
dU e = ™=-*&'j
The components #, d, c satisfy the solenoidal condition
£+!+£=«• (17>
Hence, the potential U must satisfy Laplace's equation
at every point where /ot is constant, that is, at every point within the homogeneous substance, or in empty space.
At the surface itself, if v is a normal drawn towards the magnetic substance, and v one drawn outwards, and if the symbols of quan tities outside the substance are distinguished by accents, the con dition of continuity of the magnetic induction is
dv , dv dv , dv ,, dv , dv a-j- +6-j- +0-=- +a'-j- +V -r- +<f -j- = 0; (19) dx dy dz dx dy dz
429.] FARADAY'S THEORY OF MAGNETIC INDUCTION. 51 or, by equations (16),
fjf, the coefficient of induction outside the magnet, will be unity unless the surrounding medium be magnetic or diamagnetic.
If we substitute for U its value in terms of V and H, and for fj> its value in terms of K, we obtain the same equation (10) as we arrived at by Poisson's method.
The problem of induced magnetism, when considered with respect to the relation between magnetic induction and magnetic force, corresponds exactly with the problem of the conduction of electric currents through heterogeneous media, as given in Art. 309.
The magnetic force is derived from the magnetic potential, pre cisely as the electric force is derived from the electric potential.
The magnetic induction is a quantity of the nature of a flux, and satisfies the same conditions of continuity as the electric current does.
In isotropic media the magnetic induction depends on the mag netic force in a manner which exactly corresponds with that in which the electric current depends on the electromotive force.
The specific magnetic inductive capacity in the one problem corre sponds to the specific conductivity in the other. Hence Thomson, in his Theory of Induced Magnetism (Reprint, 1872, p. 484), has called this quantity the permeability of the medium.
We are now prepared to consider the theory of induced magnetism from what I conceive to be Faraday's point of view.
When magnetic force acts on any medium, whether magnetic or diamagnetic, or neutral, it produces within it a phenomenon called Magnetic Induction.
Magnetic induction is a directed quantity of the nature of a flux, and it satisfies the same conditions of continuity as electric currents and other fluxes do.
In isotropic media the magnetic force and the magnetic induction are in the same direction, and the magnetic induction is the product of the magnetic force into a quantity called the coefficient of induction, which we have expressed by p.
In empty space the coefficient of induction is unity. In bodies capable of induced magnetization the coefficient of induction is 1 + 4 TT K = /x, where K is the quantity already defined as the co efficient of induced magnetization.
429.] Let p, [k be the values of p on opposite sides of a surface
E
52 INDUCED MAGNETIZATION. [4^9-
separating two media, then if F, V are the potentials in the two media, the magnetic forces towards the surface in the two media
dV , dV' are -7- and -3-7- • Av dv
The quantities of magnetic induction through the element of
dV dV
surface dS are u-^-dS and u? -^-j-dS in the two media respect- r dv dv
ively reckoned towards dS.
Since the total flux towards dS is zero, dV ,dV
But by the theory of the potential near a surface of density o-,
dV dV
+ 4.47r(r:r= o.
dv dv
Hence -7- (l — —A + 4 TT or = 0.
c?i> V ju, /
If K! is the ratio of the superficial magnetization to the normal force in the first medium whose coefficient is jot, we have
4 77 KI =
Hence K± will be positive or negative according as /ut is greater or less than //. If we put ju = 4 TT /c + 1 and p' '= 4 77 / + 1 ,
"47T/+1
In this expression K and K' are the coefficients of induced mag netization of the first and second medium deduced from experiments made in air, and KX is the coefficient of induced magnetization of the first medium when surrounded by the second medium.
If K is greater than K, then /q is negative, or the apparent magnetization of the first medium is in the opposite direction from the magnetizing force.
Thus, if a vessel containing a weak aqueous solution of a para magnetic salt of iron is suspended in a stronger solution of the same salt, and acted on by a magnet, the vessel moves as if it were magnetized in the opposite direction from that in which a magnet would set itself if suspended in the same place.
This may be explained by the hypothesis that the solution in the vessel is really magnetized in the same direction as the mag netic force, but that the solution which surrounds the vessel is magnetized more strongly in the same direction. Hence the vessel is like a weak magnet placed between two strong ones all mag-
43°-] POISSON'S THEORY OP MAGNETIC INDUCTION. 53
netized in the same direction, so that opposite poles are in contact. The north pole of the weak magnet points in the same direction as those of the strong- ones, but since it is in contact with the south pole of a stronger magnet, there is an excess of south magnetism in the neighbourhood of its north pole, which causes the small magnet to appear oppositely magnetized.
In some substances, however, the apparent magnetization is negative even when they are suspended in what is called a vacuum.
If we assume K = 0 for a vacuum, it will be negative for these substances. No substance, however, has been discovered for which
K has a negative value numerically greater than — — , and therefore for all known substances /x is positive.
Substances for which K is negative, and therefore p less than unity, are called Diamagnetic substances. Those for which K is positive, and ^ greater than unity, are called Paramagnetic, Ferro magnetic, or simply magnetic, substances.
We shall consider the physical theory of the diamagnetic and paramagnetic properties when we come to electromagnetism, Arts. 831-845.
430.] The mathematical theory of magnetic induction was first given by Poisson *. The physical hypothesis on which he founded his theory was that of two magnetic fluids, an hypothesis which has the same mathematical advantages and physical difficulties as the theory of two electric fluids. In order, however, to explain the fact that, though a piece of soft iron can be magnetized by induction, it cannot be charged with unequal quantities of the two kinds of magnetism, he supposes that the substance in general is a non-conductor of these fluids, and that only certain small portions of the substance contain the fluids under circumstances in which they are free to obey the forces which act on them. These small magnetic elements of the substance contain each pre cisely equal quantities of the two fluids, and within each element the fluids move with perfect freedom, but the fluids can never pass from one magnetic element to another.
The problem therefore is of the same kind as that relating to a number of small conductors of electricity disseminated through a dielectric insulating medium. The conductors may be of any form provided they are small and do not touch each other.
If they are elongated bodies all turned in the same general
* Memoires de I'lnstitut, 1824.
54 INDUCED MAGNETIZATION. [43O.
direction, or if they are crowded more in one direction than another, the medium, as Poisson himself shews, will not be isotropic. Poisson therefore, to avoid useless intricacy, examines the case in which each magnetic element is spherical, and the elements are dissem inated without regard to axes. He supposes that the whole volume of all the magnetic elements in unit of volume of the substance is k.
We have already considered in Art. 314 the electric conductivity of a medium in which small spheres of another medium are dis tributed.
If the conductivity of the medium is ^ , and that of the spheres ju2, we have found that the conductivity of the composite system is
2) P = f*l-j
Putting fa = 1 and /ot2 = oc, this becomes
_ 1 + 2/fc
This quantity ju is the electric conductivity of a medium con sisting of perfectly conducting spheres disseminated through a medium of conductivity unity, the aggregate volume of the spheres in unit of volume being k.
The symbol ^ also represents the coefficient of magnetic induction of a medium, consisting of spheres for which the permeability is infinite, disseminated through a medium for which it is unity.
The symbol k, which we shall call Poisson's Magnetic Coefficient, represents the ratio of the volume of the magnetic elements to the whole volume of the substance.
The symbol K is known as Neumann's Coefficient of Magnet ization by Induction. It is more convenient than Poisson's.
The symbol ^ we shall call the Coefficient of Magnetic Induction. Its advantage is that it facilitates the transformation of magnetic problems into problems relating to electricity and heat.
The relations of these three symbols are as follows :
47TK
3 * =
3*
477
If we put K = 32, the value given by Thalen's* experiments on * Recherches sur les Proprietes Magnetiques dufer, Nova Ada, Upsal, 1863.
430.] POISSON'S THEORY OF MAGNETIC INDUCTION. 55
soft iron, we find k = |f|-. This, according to Poisson's theory, is the ratio of the volume of the magnetic molecules to the whole volume of the iron. It is impossible to pack a space with equal spheres so that the ratio of their volume to the whole space shall be so nearly unity, and it is exceedingly improbable that so large a proportion of the volume of iron is occupied by solid molecules whatever be their form. This is one reason why we must abandon Poisson's hypothesis. Others will be stated in Chapter VI. Of course the value of Poisson's mathematical investigations remains unimpaired, as they do not rest on his hypothesis, but on the experimental fact of induced magnetization.
CHAPTER V.
PARTICULAR PROBLEMS IN MAGNETIC INDUCTION.
A Hollow Spherical Shell.
431.] THE first example of the complete solution of a problem in magnetic induction was that given by Poisson for the case of a hollow spherical shell acted on by any magnetic forces whatever.
For simplicity we shall suppose the origin of the magnetic forces to be in the space outside the shell.
If V denotes the potential due to the external magnetic system, we may expand V in a series of solid harmonics of the form
7= CQ80 + C1S1r + to. + CiSiiA, (1)
where r is the distance from the centre of the shell, #< is a surface harmonic of order i, and Ci is a coefficient.
This series will be convergent provided r is less than the distance of the nearest magnet of the system which produces this potential. Hence, for the hollow spherical shell and the space within it, this expansion is convergent.
Let the external radius of the shell be a2 and the inner radius alf and let the potential due to its induced magnetism be H. The form of the function H will in general be different in the hollow space, in the substance of the shell, and in the space beyond. If we expand these functions in harmonic series, then, confining our attention to those terms which involve the surface harmonic Si9 we shall find that if Q^ is that which corresponds to the hollow space within the shell, the expansion of Q^ must be in positive har monics of the form Al St r*, because the potential must not become infinite within the sphere whose radius is a^.
In the substance of the shell, where r± lies between aL and a2, the series may contain both positive and negative powers of /*, of the form
Outside the shell, where r is greater than a2, since the series
HOLLOW SPHERICAL SHELL. 57
must be convergent however great r may be, we must have only negative powers of /, of the form
The conditions which must be satisfied by the function 12, are : It must be (1) finite, and (2) continuous, and (3) must vanish at an infinite distance, and it must (4) everywhere satisfy Laplace's equation.
On account of (1) Bl = 0.
On account of (2) when r = a^
(4-4,H2i+1-52=0, (2)
and when r = «2,
(^2-J3)^2i+1 + ^2-^3 = 0. (3)
On account of (3) Az = 0, and the condition (4) is satisfied everywhere, since the functions are harmonic.
But, besides these, there are other conditions to be satisfied at the inner and outer surface in virtue of equation (10), Art. 427.
At the inner surface where r = alt
, d£l9 d&, dV ,..
<1+4*«>V-ifr+4"'* = <)'
and at the outer surface where r = a2,
d dV
,KN 0.
From these conditions we obtain the equations
iCia12i+l = <), (6)
«22»+1-(^+l)^2)+(^+l)^3+47r^^22i+1=0^ (7) and if we put
we find
/ /, 2» + l\
4 = -(4™)^ + l)(l-Q) }NtClt (9)
[I a 2t+l^-j
2^+l+477K(^+l)(l-(^) )J^Ci, (10)
(11) «12i+1)^Ci. (12)
These quantities being substituted in the harmonic expansions give the part of the potential due to the magnetization of the shell. The quantity Ni is always positive, since 1 -f 4 ir K can never be negative. Hence A1 is always negative, or in other words, the
58 MAGNETIC PEOBLEMS. [432.
action of the magnetized shell on a point within it is always op posed to that of the external magnetic force whether the shell he paramagnetic or diamagnetic. The actual value of the resultant potential within the shell is
or (l + 4wjc)(2i+ l^NiCtS.r. (13)
432.] When K is a large number, as it is in the case of soft iron, then, unless the shell is very thin, the magnetic force within it is hut a small fraction of the external force.
In this way Sir W. Thomson has rendered his marine galvano meter independent of external magnetic force hy enclosing it in a tube of soft iron.
433.] The case of greatest practical importance is that in which i = 1. In this case
(14)
9(l+47TK)+2(477K)2(l-0')
= -477*13+ 8w«(l — (^) )UViQ, !> (15)
L X dr> —I
£3= 4 7TK(3 + 8 7TK)(#23 — «13)^V1 Ci.
The magnetic force within the hollow shell is in this case uniform and equal to
9(1+477*)
If we wish to determine K by measuring the magnetic force within a hollow shell and comparing it with the external magnetic force, the best value of the thickness of the shell may be found from the equation
1_
-
2 (4 TT K)2
The magnetic forc"e inside the shell is then half of its value outside. Since, in the case of iron, K is a number between 20 and 30, the thickness of the shell ought to be about the hundredth part of its radius. This method is applicable only when the value of K is large. When it is very small the value of A^ becomes insensible, since it depends on the square of K.
434-1 SPHERICAL SHELL. 59
For a nearly solid sphere with a very small spherical hollow,
. 2(4ir«)«
1J
4 77 K
The whole of this investigation might have been deduced directly from that of conduction through a spherical shell, as given in Art. 312, by putting ^ = (1 -f 47TK)/£2 in the expressions there given, remembering that A^ and A2 in the problem of conduction are equi valent to C1 + A1 and C1 + A2 in the problem of magnetic induction.
434.] The corresponding solution in two dimensions is graphically represented in Fig. XV, at the end of this volume. The lines of induction, which at a distance from the centre of the figure are nearly horizontal, are represented as disturbed by a cylindric rod magnetized transversely and placed in its position of stable equi librium. The lines which cut this system at right angles represent the equipotential surfaces, one of which is a cylinder. The large dotted circle represents the section of a cylinder of a paramagnetic substance, and the dotted horizontal straight lines within it, which are continuous with the external lines of induction, represent the lines of induction within the substance. The dotted vertical lines represent the internal equipotential surfaces, and are continuous with the external system. It will be observed that the lines of induction are drawn nearer together within the substance, and the equipotential surfaces are separated farther apart by the paramag netic cylinder, which, in the language of Faraday, conducts the lines of induction better than the surrounding medium.
If we consider the system of vertical lines as lines of induction, and the horizontal system as equipotential surfaces, we have, in the first place, the case of a cylinder magnetized transversely and placed in the position of unstable equilibrium among the lines of force, which it causes to diverge. In the second place, considering the large dotted circle as the section of a diamagnetic cylinder, the dotted straight lines within it, together with the lines external to it, represent the effect of a diamagnetic substance in separating the lines of induction and drawing together the equipotential surfaces, such a substance being a worse conductor of magnetic induction than the surrounding medium.
60 MAGNETIC PROBLEMS. [435-
Case of a Sphere in which the Coefficients of Magnetization are Different in Different Directions.
435.] Let a, (B, y be the components of magnetic force, and A, £, C those of the magnetization at any point, then the most general linear relation between these quantities is given by the equations A = ^0+^3/3+ q2y, \
£ = q9a+r2p+fly, { (1)
C = p2a+q1h2 + 7-3 y, )
where the coefficients r,jo, q are the nine coefficients of magnet ization.
Let us now suppose that these are the conditions of magnet ization within a sphere of radius a, and that the magnetization at every point of the substance is uniform and in the same direction, having the components A, 13, C.
Let us also suppose that the external magnetizing force is also uniform and parallel to one direction, and has for its components X, Y, Z.
The value of V is therefore
and that of &' the potential of the magnetization outside the sphere is
(3)
The value of H, the potential of the magnetization within the sphere, is 4-n-
(4)
o
The actual potential within the sphere is V-\- £1, so that we shall have for the components of the magnetic force within the sphere a = X — ^TtA, \ 0 = 7-J.ir-B, (5)
y =Z-
Hence
+i*r1)^+ twftjjB + iir&
C = &J+ r2Y+frZ, (6)
+(1 +
Solving these equations, we find A = r/^+K
'' (7)
43^.] CRYSTALLINE SPHERE. 61
where I/ /•/ = r± + ^ TT ( rB rl — p2 q2 4 r-± r2 —
;-A^i)>
&c.,
where D is the determinant of the coefficients on the right side of equations (6), and D' that of the coefficients on the left.
The new system of coefficients _p' ', /_, / will be symmetrical only when the system p, q, r is symmetrical, that is, when the co efficients of the form p are equal to the corresponding ones of the form q.
436.] The moment of the couple tending to turn the sphere about the axis of x from y towards z is
f. n Y\\ (Q\
— Jr2 ))* \ /
If we make
X = 0, Y = Fcos 0, Y = Fsin 0,
this corresponds to a magnetic force F in the plane of yz, and inclined to y at an angle 0. If we now turn the sphere while this force remains constant the work done in turning the sphere will
T27T
be / LdQ in each complete revolution. But this is equal to
0
Hence, in order that the revolving sphere may not become an inexhaustible source of energy, j»1/= fa', and similarly j»./= q2 and
These conditions shew that in the original equations the coeffi cient of B in the third equation is equal to that of C in the second, and so on. Hence, the system of equations is symmetrical, and the equations become when referred to the principal axes of mag netization, TI
A = rr*"i '
C =
(11)
The moment of the couple tending to turn the sphere round the axis of x is
62 MAGNETIC PROBLEMS. [437-
In most cases the differences between the coefficients of magnet ization in different directions are very small, so that we may put
This is the force tending to turn a crystalline sphere about the axis of oo from y towards z. It always tends to place the axis of greatest magnetic coefficient (or least diamagnetic coefficient) parallel to the line of magnetic force.
The corresponding case in two dimensions is represented in Fig. XVI.
If we suppose the upper side of the figure to be towards the north, the figure represents the lines of force and equipotential surfaces as disturbed by a transversely magnetized cylinder placed with the north side eastwards. The resultant force tends to turn the cylinder from east to north. The large dotted circle represents a section of a cylinder of a crystalline substance which has a larger coefficient of induction along an axis from north-east to south-west than along an axis from north-west to south-east. The dotted lines within the circle represent the lines of induction and the equipotential surfaces, which in this case are not at right angles to each other. The resultant force on the cylinder is evidently to turn it from east to north.
437.] The case of an ellipsoid placed in a field of uniform and parallel magnetic force has been solved in a very ingenious manner by Poisson.
If V is the potential at the point (as, y, z\ due to the gravitation
dV of a body of any form of uniform density p, then — -=- is the
potential of the magnetism of the same body if uniformly mag netized in the direction of x with the intensity I = p.
For the value of -- =— 8# at any point is the excess of the value clx
of V3 the potential of the body, above V, the value of the potential when the body is moved — §x in the direction of x.
If we supposed the body shifted through the distance — 8#, and its density changed from p to — p (that is to say, made of repulsive
dV instead of attractive matter,) then — y-8# would be the potential
due to the two bodies.
Now consider any elementary portion of the body containing a volume b v. Its quantity is pbv, and corresponding to it there is
437-] ELLIPSOID. 63
an element of the shifted body whose quantity is — pbv at a distance — 8#. The effect of these two elements is equivalent to that of a magnet of strength pbr and length 8#. The intensity of magnetization is found hy dividing the magnetic moment of an element by its volume. The result is p 8#.
dV Hence -=- 8# is the magnetic potential of the body magnetized
rl V
with the intensity p bx in the direction of x, and — is that of
ax
the body magnetized with intensity p.
This potential may be also considered in another light. The body was shifted through the distance — 8# and made of density —p. Throughout that part of space common to the body in its two positions the density is zero, for, as far as attraction is con cerned, the two equal and opposite densities annihilate each other. There remains therefore a shell of positive matter on one side and of negative matter on the other, and we may regard the resultant potential as due to these. The thickness of the shell at a point where the normal drawn outwards makes an angle e with the axis of a? is 8 a? cos e and its density is p. The surface-density is therefore
dV p bx cos 6, and, in the case in which the potential is — , the
surface-density is p cos e.
In this way we can find the magnetic potential of any body uniformly magnetized parallel to a given direction. Now if this uniform magnetization is due to magnetic induction, the mag netizing force at all points within the body must also be uniform and parallel.
This force consists of two parts, one due to external causes, and the other due to the magnetization of the body. If therefore the external magnetic force is uniform and parallel, the magnetic force due to the magnetization must also be uniform and parallel for all points within the body.
Hence, in order that this method may lead to a solution of the
clV
problem of magnetic induction, -=- must be a linear function of
doc
the coordinates x, y> z within the body, and therefore V must be a quadratic function of the coordinates.
Now the only cases with which we are acquainted in which V is a quadratic function of the coordinates within the body are those in which the body is bounded by a complete surface of the second degree, and the only case in which such a body is of finite dimen-
64 MAGNETIC PROBLEMS. [437-
sions is when it is an ellipsoid. We shall therefore apply the method to the case of an ellipsoid.
be the equation of the ellipsoid, and let 4>0 denote the definite integral
f
'0
Then if we make
dfr
the value of the potential within the ellipsoid will be
70 = - (L x2 + My* + Nz*} + const. (4)
2
If the ellipsoid is magnetized with uniform intensity / in a direction making angles whose cosines are I, m, n with the axes of #, y, z, so that the components of magnetization are
A = II, B = Im, C = In, the potential due to this magnetization within the ellipsoid will be
a = —I(Llx + Mmy + Nnz). (5)
If the external magnetizing force is Ǥ, and if its components are a, ft, y, its potential will be
r=Xx + Yy + Zz. (6)
The components of the actual magnetizing force at any point within the body are therefore
X-AL, Y-BM, Z-CN. (7)
The most general relations between the magnetization and the magnetizing force are given by three linear equations, involving nine coefficients. It is necessary, however, in order to fulfil the condition of the conservation of energy, that in the case of magnetic induction three of these should be equal respectively to other three, so that we should have
A = K,(X-AL) + Kfs(Y-BM) + K'2(Z-CN}, B = K\ (X-AL) + K2i(Y-BM) + K\(Z-CN], (8)
C = K'2(X-AL) + K\(Y-BM) + Kz(Z-CN}. From these equations we may determine J, B and C in terms of X, Y} Z, and this will give the most general solution of the problem.
The potential outside the ellipsoid will then be that due to the
* See Thomson and Tait's Natural Philosophy, § 522.
438.] ELLIPSOID. 65
magnetization of the ellipsoid together with that due to the external magnetic force.
438.] The only case of practical importance is that in which
K\ = K2 = K3 = 0. (9)
|
We have then If the ellipsoid flattened form, |
A — |
"i |
X 1 |
(10) and is of the planetary or : (ID |
|
7? |
K2 |
T ' JJ V |
||
|
C = has two b= c |
1+K2M~ K3 g l+K3N axes equal, a |
(12) l-e
M = N = 2 , (±-^ sin-'*- ™) . \ e* e2 ' J
If the ellipsoid is of the ovary or elongated form
a — b = A/1 — e*c; (13)
In the case of a sphere, when e = 0,
— .«. -^- ^ j
In the case of a very flattened planetoid L becomes in the limit equal to 4 TT, and M and JV become 7r2 - •
In the case of a very elongated ovoid L and M approximate to the value 2 TT, while N approximates to the form
a2,, 2c ,
and vanishes when e = 1 .
It appears from these results that —
(1) When K, the coefficient of magnetization, is very small, whether positive or negative, the induced magnetization is nearly equal to the magnetizing force multiplied by K, and is almost independent of the form of the body.
VOL. II. F
66 MAGNETIC PROBLEMS.
(2) When K is a large positive quantity, the magnetization depends principally on the form of the body,, and is almost independent of the precise value of /c, except in the case of a longitudinal force acting on an ovoid so elongated that NK is a small' quantity though K is large.
(3) If the value of K could be negative and equal to — we
should have an infinite value of the magnetization in the case of a magnetizing force acting normally to a flat plate or disk. The absurdity of this result confirms what we said in Art. 428.
Hence, experiments to determine the value of K may be made on bodies of any form provided K is very small, as it is in the case of all diamagnetic bodies, and all magnetic bodies except iron, nickel, and cobalt.
If, however, as in the case of iron, K is a large number, experi ments made on spheres or flattened figures are not suitable to determine K ; for instance, in the case of a sphere the ratio of the magnetization to the magnetizing force is as 1 to 4.22 if K = 30, as it is in some kinds of iron, and if K were infinite the ratio would be as 1 to 4.19, so that a very small error in the determination of the magnetization would introduce a very large one in the value of K.
But if we make use of a piece of iron in the form of a very elongated ovoid, then, as long as NK is of moderate value com pared with unity, we may deduce the value of K from a determination of the magnetization, and the smaller the value of JV the more accurate will be the value of K.
In fact, if NK be made small enough, a small error in the value of N itself will not introduce much error, so that we may use any elongated body, such as a wire or long rod, instead of an ovoid.
We must remember, however, that it is only when the product JV~/c is small compared with unity that this substitution is allowable. In fact the distribution of magnetism on a long cylinder with flat ends does not resemble that on a long ovoid, for the free mag netism is very much concentrated towards the ends of the cylinder, whereas it varies directly as the distance from the equator in the case of the ovoid.
The distributi6n of electricity on a cylinder, however, is really comparable with that on an ovoid, as we have already seen, Art. 152.
439-] CYLINDER. 67
These results also enable us to understand why the magnetic moment of a permanent magnet can be made so much greater when the magnet has an elongated form. If we were to magnetize a disk with intensity / in a direction normal to its surface, and then leave it to itself, the interior particles would experience a constant demagnetizing force equal to 4 TT I, and this, if not sufficient of itself to destroy part of the magnetization, would soon do so if aided by vibrations or changes of temperature.
If we were to magnetize a cylinder transversely the demagnet izing force would be only 2 TT I.
If the magnet were a sphere the demagnetizing force would be £*/.
In a disk magnetized transversely the demagnetizing force is
a
7T2 - 1) and in an elongated ovoid magnetized longitudinally it
0
a2 2c
is least of all, being 4 TT -^ 7 log --- G a
Hence an elongated magnet is less likely to lose its magnetism than a short thick one.
The moment of the force acting on an ellipsoid having different magnetic coefficients for the three axes which tends to turn it about the axis of #, is
Hence, if *2 and K3 are small, this force will depend principally on the crystalline quality of the body and not on its shape, pro vided its dimensions are not very unequal, but if K2 and *3 are considerable, as in the case of iron, the force will depend principally on the shape of the body, and it will turn so as to set its longer axis parallel to the lines of force.
If a sufficiently strong, yet uniform, field of magnetic force could be obtained, an elongated isotropic diamagnetic body would also set itself with its longest dimension parallel to the lines of magnetic force.
439.] The question of the distribution of the magnetization of an ellipsoid of revolution under the action of any magnetic forces has been investigated by J. Neumann*. Kirchhofff has extended the method to the case of a cylinder of infinite length acted on by any force.
* Crelle, bd. xxxvii (1848). t Crelle, bd. xlviii (1854).
F 2
68 MAGNETIC PROBLEMS. [439-
Green, in the 17th section of his Essay, has given an invest igation of the distribution of magnetism in a cylinder of finite length acted on by a uniform external force parallel to its axis. Though some of the steps of this investigation are not very rigorous, it is probable that the result represents roughly the actual magnetization in this most important case. It certainly expresses very fairly the transition from the case of a cylinder for which K is a large number to that in which it is very small, but it fails entirely in the case in which K is negative, as in diamagnetic substances.
Green finds that the linear density of free magnetism at a distance x from the middle of a cylinder whose radius is a and whose length is 2 I, is
px
ea +e
where p is a numerical quantity to be found from the equation
0.231863 — 2 \ogep + 2p = - -— The following are a few of the corresponding values of p and K.
K K
oo 0
336.4 0.01
62.02 0.02
48.416 0.03
29.475 0.04
20.185 0.05
14.794 0.06
11.802 0.07
9.137 0.08
7.517 0.09
6.319 0.10
0.1427 1.00
0.0002 10.00
0.0000 oo
negative imaginary.
When the length of the cylinder is great compared with its radius, the whole quantity of free magnetism on either side of the middle of the cylinder is, as it ought to be,
M= v2aKX.
Of this \p M is on the flat end of the cylinder, and the distance of the centre of gravity of the whole quantity M from the end
a
of the cylinder is - P
When K is very small p is large, and nearly the whole free magnetism is on the ends of the cylinder. As K increases p diminishes, and the free magnetism is spread over a greater distance
44O-] FORCE ON PARA- AND DIA-MAGNETIC BODIES. 69
from the ends. When K is infinite the free magnetism at any point of the cylinder is simply proportional to its distance from the middle point, the distribution being similar to that of free electricity on a conductor in a field of uniform force.
440.] In all substances except iron, nickel, and cobalt, the co efficient of magnetization is so small that the induced magnetization of the body produces only a very slight alteration of the forces in the magnetic field. We may therefore assume, as a first approx imation, that the actual magnetic force within the body is the same as if the body had not been there. The superficial magnetization
dV dV
of the body is therefore, as a first approximation, K -j- , where -=-
is the rate of increase of the magnetic potential due to the external magnet along a normal to the surface drawn inwards. If we now calculate the potential due to this superficial distribution, we may use it in proceeding to a second approximation.
To find the mechanical energy due to the distribution of mag netism on this first approximation we must find the surface-integral
taken over the whole surface of the body. Now we have shewn in Art. 100 that this is equal to the volume-integral
/*/*/* ~^r~T7 ^ j 77" 2
taken through the whole space occupied by the body, or, if R is the resultant magnetic force,
E = -
Now since the work done by the magnetic force on the body during a displacement 8# is Xbos where X is the mechanical force in the direction of SB, and since
/
= constant,
which shews that the force acting on the body is as if every part of it tended to move from places where R2 is less to places where it is greater with a force which on every unit of volume is
rf.JP K dx '
70 MAGNETIC PEOBLEMS.
If K is negative, as in diamagnetic bodies, this force is, as Faraday first shewed, from stronger to weaker parts of the magnetic field. Most of the actions observed in the case of diamagnetic bodies depend on this property.
Skip's Magnetism.
441.] Almost every part of magnetic science finds its use in navigation. The directive action of the earth's magnetism on the compass needle is the only method of ascertaining the ship's course when the sun and stars are hid. The declination of the needle from the true meridian seemed at first to be a hindrance to the appli cation of the compass to navigation, but after this difficulty had been overcome by the construction of magnetic charts it appeared likely that the declination itsylf would assist the mariner in de termining his ship's place.
The greatest difficulty in navigation had always been to ascertain the longitude ; but since the declination is different at different points on the same parallel of latitude, an observation of the de clination together with a knowledge of the latitude would enable the mariner to find his position on the magnetic chart.
But in recent times iron is so largely used in the construction of ships that it has become impossible to use the compass at all without taking into account the action of the ship, as a magnetic body, on the needle.
To determine the distribution of magnetism in a mass of iron of any form under the influence of the earth's magnetic force, even though not subjected to mechanical strain or other disturb ances, is, as we have seen, a very difficult problem.
In this case, however, the problem is simplified by the following considerations.
The compass is supposed to be placed with its centre at a fixed point of the ship, and so far from any iron that the magnetism of the needle does not induce any perceptible magnetism in the ship. The size of the compass needle is supposed so small that we may regard the magnetic force at any point of the needle as the same.
The iron of the ship is supposed to be of two kinds only.
(1) Hard iron, magnetized in a constant manner.
(2) Soft iron, the magnetization of which is induced by the earth or other magnets.
In strictness we must admit that the hardest iron is not only
SHIP'S MAGNETISM. 71
capable of induction but that it may lose part of its so-called permanent magnetization in various ways.
The softest iron is capable of retaining what is called residual magnetization. The actual properties of iron cannot be accurately represented by supposing it compounded of the hard iron and the soft iron above defined. But it has been found that when a ship is acted on only by the earth's magnetic force, and not subjected to any extraordinary stress of weather, the supposition that the magnetism of the ship is due partly to permanent magnetization and partly to induction leads to sufficiently accurate results when applied to the correction of the compass.
The equations on which the theory of the variation of the compass is founded were given by Poisson in the fifth volume of the Memoires de I'Institut, p. 533 (1824).
The only assumption relative to induced magnetism which is involved in these equations is, that if a magnetic force X due to external magnetism produces in the iron of the ship an induced magnetization, and if this induced magnetization exerts on the compass needle a disturbing force whose components are JT', Y'9 Z', then, if the external magnetic force is altered in a given ratio, the components of the disturbing force will be altered in the same ratio.
It is true that when the magnetic force acting on iron is very great the induced magnetization is no longer proportional to the external magnetic force, but this want of proportionality is quite insensible for magnetic forces of the magnitude of those due to the earth's action.
Hence, in practice we may assume that if a magnetic force whose value is unity produces through the intervention of the iron of the ship a disturbing force at the compass needle whose com ponents are a in the direction of #, d in that of y, and g in that of z, the components of the disturbing force due to a force X in the direction of x will be aX, dX, and gX.
If therefore we assume axes fixed in the ship, so that x is towards the ship's head, y to the starboard side, and z towards the keel, and if X, Y, Z represent the components of the earth's magnetic force in these directions, and X', Y', Z' the components of the combined magnetic force of the earth and ship on the compass needle, X' = X+aX+bY+c Z+P, )
Y' = Y+dX+eY+fZ+Q, (1)
72 MAGNETIC PROBLEMS. [44 1-
In these equations #, #, c, d, e,f, g, h, Jc are nine constant co efficients depending on the amount, the arrangement, and the capacity for induction of the soft iron of the ship.
P, Q, and E are constant quantities depending on the permanent magnetization of the ship.
It is evident that these equations are sufficiently general if magnetic induction is a linear function of magnetic force, for they are neither more nor less than the most general expression of a vector as a linear function of another vector.
It may also be shewn that they are not too general, for, by a proper arrangement of iron, any one of the coefficients may be made to vary independently of the others.
Thus, a long thin rod of iron under the action of a longitudinal magnetic force acquires poles, the strength of each of which is numerically equal to the cross section of the rod multiplied by the magnetizing force and by the coefficient of induced magnet ization. A magnetic force transverse to the rod produces a much feebler magnetization, the effect of which is almost insensible at a distance of a few diameters.
If a long iron rod be placed fore and aft with one end at a distance x from the compass needle, measured towards the ship's head, then, if the section of the rod is A, and its coefficient of magnetization K, the strength of the pole will be A K X, and, if
A = — — , the force exerted by this pole on the compass needle
will be aX. The rod may be supposed so long that the effect of the other pole on the compass may be neglected.
We have thus obtained the means of giving any required value to the coefficient a.
If we place another rod of section B with one extremity at the same point, distant x from the compass toward the head of the vessel, and extending to starboard to such a distance that the distant pole produces no sensible effect on the compass, the dis turbing force due to this rod will be in the direction of x, and
B K.Y bx* equal to — x - , or if B = , the force will be b Y.
X2 K '
This rod therefore introduces the coefficient b.
A third rod extending downwards from the same point will introduce the coefficient <?.
The coefficients d, e,f may be produced by three rods extending to head, to starboard, and downward from a point to starboard of
44i.] SHIP'S MAGNETISM. 73
the compass, and g, h, k by three rods in parallel directions from a point below the compass.
Hence each of the nine coefficients can be separately varied by means of iron rods properly placed.
The quantities P, Q, R are simply the components of the force on the compass arising from the permanent magnetization of the ship together with that part of the induced magnetization which is due to the action of this permanent magnetization.
A complete discussion of the equations (1), and of the relation between the true magnetic course of the ship and the course as indicated by the compass, is given by Mr. Archibald Smith in the Admiralty Manual of the Deviation of the Compass.
A valuable graphic method of investigating the problem is there given. Taking a fixed point as origin, a line is drawn from this point representing in direction and magnitude the horizontal part of the actual magnetic force on the compass-needle. As the ship is swung round so as to bring her head into different azimuths in succession, the extremity of this line describes a curve, each point of which corresponds to a particular azimuth.
Such a curve, by means of which the direction and magnitude of the force on the compass is given in terms of the magnetic course of the ship, is called a Dygogram.
There are two varieties of the Dygogram. In the first, the curve is traced on a plane fixed in space as the ship turns round. In the second kind, the curve is traced on a plane fixed with respect to the ship.
The dygogram of the first kind is the Lima9on of Pascal, that of the second kind is an ellipse. For the construction and use of these curves, and for many theorems as interesting to the mathe matician as they are important to the navigator, the reader is referred to the Admiralty Manual of the Deviation of the Compass.
CHAPTER VI.
WEBER'S THEORY OF INDUCED MAGNETISM.
442.] WE have seen that Poisson supposes the magnetization of iron to consist in a separation of the magnetic fluids within each magnetic molecule. If we wish to avoid the assumption of the existence of magnetic fluids, we may state the same theory in another form, hy saying that each molecule of the iron, when the magnetizing force acts on it, becomes a magnet.
Weber's theory differs from this in assuming that the molecules of the iron are always magnets, even before the application of the magnetizing force, but that in ordinary iron the magnetic axes of the molecules are turned indifferently in every direction, so that the iron as a whole exhibits no magnetic properties.
When a magnetic force acts on the iron it tends to turn the axes of the molecules all in one direction, and so to cause the iron, as a whole, to become a magnet.
If the axes of all the molecules were set parallel to each other, the iron would exhibit the greatest intensity of magnetization of which it is capable. Hence Weber's theory implies the existence of a limiting intensity of magnetization, and the experimental evidence that such a limit exists is therefore necessary to the theory. Experiments shewing an approach to a limiting value of magnetization have been made by Joule * and by J. Miiller f.
The experiments of Beetz J on electrotype iron deposited under the action of magnetic force furnish the most complete evidence of this limit, —
A silver wire was varnished, and a very narrow line on the
* Annals of Electricity, iv. p. 131, 1839 ; Phil Mag. [4] ii. p. 316. t Pogg., Ann. Ixxix. p. 337, 1850. + Pogg. cxi. 1860.
443-] THE MOLECULES OF IRON ARE MAGNETS. 75
metal was laid bare by making1 a fine longitudinal scratch on the varnish. The wire was then immersed in a solution of a salt of iron, and placed in a magnetic field with the scratch in the direction of a line of magnetic force. By making the wire the cathode of an electric current through the solution, iron was deposited on the narrow exposed surface of the wire, molecule by molecule. The filament of iron thus formed was then examined magnetically. Its magnetic moment was found to be very great for so small a mass of iron, and when a powerful magnetizing force was made to act in the same direction the increase of temporary magnetization was found to be very small, and the permanent magnetization was not altered. A magnetizing force in the reverse direction at once reduced the filament to the condition of iron magnetized in the ordinary way.
Weber's theory, which supposes that in this case the magnetizing force placed the axis of each molecule in the same direction during the instant of its deposition, agrees very well with what is observed.
Beetz found that when the electrolysis is continued under the action of the magnetizing force the intensity of magnetization of the subsequently deposited iron diminishes. The axes of the molecules are probably deflected from the line of magnetizing force when they are being laid down side by side with the mole cules already deposited, so that an approximation to parallelism. can be obtained only in the case of a very thin filament of iron.
If, as Weber supposes, the molecules of iron are already magnets, any magnetic force sufficient to render their axes parallel as they are electrolytically deposited will be sufficient to produce the highest intensity of magnetization in the deposited filament.
If, on the other hand, the molecules of iron are not magnets, but are only capable of magnetization, the magnetization of the deposited filament will depend on the magnetizing force in the same way in which that of soft iron in general depends on it. The experiments of Beetz leave no room for the latter hy pothesis.
443.] We shall now assume, with Weber, that in every unit of volume of the iron there are n magnetic molecules, and that the magnetic moment of each is m. If the axes of all the molecules were placed parallel to one another, the magnetic moment of the unit of volume would be
M = n m,
76 WEBER'S THEORY OF INDUCED MAGNETISM. [443.
and this would be the greatest intensity of magnetization of which the iron is capable.
In the unmagnetized state of ordinary iron Weber supposes the axes of its molecules to be placed indifferently in all directions.
To express this, we may suppose a sphere to be described, and a radius drawn from the centre parallel to the direction of the axis of each of the n molecules. The distribution of the extremities of these radii will express that of the axes of the molecules. In the case of ordinary iron these n points are equally distributed over every part of the surface of the sphere, so that the number of molecules whose axes make an angle less than a with the axis
of x is n .
- (I - cos a),
and the number of molecules whose axes make angles with that of ^, between a and a-f da is therefore
n . j - sin a a a. 2t
This is the arrangement of the molecules in a piece of iron which has never been magnetized.
Let us now suppose that a magnetic force X is made to act on the iron in the direction of the axis of a?, and let us consider a molecule whose axis was originally inclined a to the axis of so.
If this molecule is perfectly free to turn, it will place itself with its axis parallel to the axis of a?, and if all the molecules did so, the very slightest magnetizing force would be found sufficient to develope the very highest degree of magnetization. This, how ever, is not the case.
The molecules do not turn with their axes parallel to a?, and this is either because each molecule is acted on by a force tending to preserve it in its original direction, or because an equivalent effect is produced by the mutual action of the entire system of molecules.
Weber adopts the former of these suppositions as the simplest, and supposes that each molecule, when deflected, tends to return to its original position with a force which is the same as that which a magnetic force D, acting in the original direction of its axis, would produce.
The position which the axis actually assumes is therefore in the direction of the resultant of X and D.
Let APB represent a section of a sphere whose radius represents, on a certain scale, the force D.
443-] DEFLEXION OF AXES OF MOLECULES. 77
Let the radius OP be parallel to the axis of a particular molecule in its original position.
Let SO represent on the same scale the magnetizing force X which is supposed to act from 8 towards 0. Then, if the molecule is acted on by the force X in the direction SO, and by a force D in a direction parallel to OP, the original direction of its axis, its axis will set itself in the direction SP, that of the resultant of X and D.
Since the axes of the molecules are originally in all directions, P may be at any point of the sphere indifferently. In Fig. 5, in which X is less than D, SP, the final position of the axis, may be in any direction whatever, but not indifferently, for more of the molecules will have their axes turned towards A than towards JS. In Fig. 6, in which X is greater than D, the axes of the molecules will be all confined within the cone STT' touching the sphere.
Fig. 5.
Hence there are two different cases according as X is less or greater than D.
Let a = AOP, the original inclination of the axis of a molecule
to the axis of x. 0 = ASP, the inclination of the axis when deflected by
the force X.
(3 = SPO, the angle of deflexion. SO = X, the magnetizing force.
OP = D, the force tending towards the original position. SP = R, the resultant of X and D.
m = magnetic moment of the molecule.
Then the moment of the statical couple due to X, tending to diminish the angle 0, is
mL = mX sin#,
and the moment of the couple due to D, tending to increase 6, is mL —
78 WEBER'S THEORY OF INDUCED MAGNETISM. [443.
Equating these values, and remembering that /3 = a — 0, we find
J)sina
tan0 = - -- (1)
X +D cos a
to determine the direction of the axis after deflexion.
We have next to find the intensity of magnetization produced in the mass by the force X, and for this purpose we must resolve the magnetic moment of every molecule in the direction of #, and add all these resolved parts.
The resolved part of the moment of a molecule in the direction of x is m cos 0.
The number of molecules whose original inclinations lay between
a and a -{-da is % .
-smaaa. 2
We have therefore to integrate
/= f* — cos 6 tin a da, (2)
JQ 2
'remembering that 0 is a function of a.
We may express both 9 and a in terms of JR, and the expression to be integrated becomes
' (3)
the general integral of which is
In the first case, that in which X is less than D, the limits of integration are R = D + X and R = D— X. In the second case, in which X is greater than D, the limits are R = X+ D and R = X-D.
When X is less than D, I = | ~X. (5)
2
When X is equal to D, I = -mn. (6)
3
1 712
When X is greater than D, I — mn(\ -- — ) ; (7)
* o J\. I
and when X becomes infinite / = mn. (8)
According to this form of the theory, which is that adopted
by Weber *, as the magnetizing force increases from 0 to D, the
* There is some mistake in the formula given by Weber (Trans. Acad. Sax. i. p. 572 (1852), or Pogg., Ann. Ixxxvii. p. 167 (1852)) as the result of this integration, the steps of which are not given by him. His formula is
444-] L1MIT OF MAGNETIZATION. 79
magnetization increases in the same proportion. When the mag netizing force attains the value D, the magnetization is two-thirds of its limiting value. When the magnetizing force is further increased, the magnetization, instead of increasing indefinitely, tends towards a finite limit.
D 2D 3D 4D
Fig. 7.
The law of magnetization is expressed in Fig. 7, where the mag netizing force is reckoned from 0 towards the right and the mag netization is expressed by the vertical ordinates. Weber's own experiments give results in satisfactory accordance with this law. It is probable, however, that the value of D is not the same for all the molecules of the same piece of iron, so that the transition from the straight line from 0 to E to the curve beyond E may not be so abrupt as is here represented.
444.] The theory in this form gives no account of the residual magnetization which is found to exist after the magnetizing force is removed. I have therefore thought it desirable to examine the results of making a further assumption relating to the conditions under which the position of equilibrium of a molecule may be permanently altered.
Let us suppose that the axis" of a magnetic molecule, if deflected through any angle /3 less than /30, will return to its original position when the deflecting force is removed, but that if the deflexion j3 exceeds ^0, then, when the deflecting force is removed, the axis will not return to its original position, but will be per manently deflected through an angle /3 — j30, which may be called the permanent set of the molecule.
This assumption with respect to the law of molecular deflexion is not to be regarded as founded on any exact knowledge of the intimate structure of bodies, but is adopted, in our ignorance of the true state of the case, as an assistance to the imagination in following out the speculation suggested by Weber.
Let L = Dsin /30, (9)
80 WEBER'S THEORY OF INDUCED MAGNETISM. [444.
then, if the moment of the couple acting on a molecule is less than ml/, there will be no permanent deflexion, but if it exceeds mL there will be a permanent change of the position of equilibrium.
To trace the results of this supposition, describe a sphere whose centre is 0 and radius OL = L.
As long as X is less than L everything will be the same as in the case already considered, but as soon as X exceeds L it will begin to produce a permanent deflexion of some of the molecules.
Let us take the case of Fig. 8, in which X is greater than L but less than D. Through S as vertex draw a double cone touching the sphere L. Let this cone meet the sphere D in P and Q. Then if the axis of a molecule in its original position lies between OA and OP, or between OB and OQ, it will be deflected through an angle less than /30, and will not be permanently deflected. But if
Fig. 8. Fig. 9.
the axis of the molecule lies originally between OP and OQ, then a couple whose moment is greater than L will act upon it and will deflect it into the position SP, and when the force X ceases to act it will not resume its original direction, but will be per manently set in the direction OP.
Let us put
L = Xsin00 when 0 = PSA or QSB,
then all those molecules whose axes, on the former hypotheses, would have values of 6 between 00 and TT — 00 will be made to have the value 00 during the action of the force X.
During the action of the force X, therefore, those molecules whose axes when deflected lie within either sheet of the double cone whose semivertical angle is 00 will be arranged as in the former case, but all those whose axes on the former theory would lie outside of these sheets will be permanently deflected, so that their axes will form a dense fringe round that sheet of the cone which lies towards A.
445-] MODIFIED THEORY. 81
As X increases, the number of molecules belonging to the cone about B continually diminishes, and when X becomes equal to D all the molecules have been wrenched out of their former positions of equilibrium, and have been forced into the fringe of the cone round A, so that when X becomes greater than D all the molecules form part of the cone round A or of its fringe.
When the force X is removed, then in the case in which X is less than L everything returns to its primitive state. When X is between L and D then there is a cone round A whose angle
AOP = 00 + /30,
and another cone round B whose angle BOQ = 00-/30.
Within these cones the axes of the molecules are distributed uniformly. But all the molecules, the original direction of whose axes lay outside of both these cones, have been wrenched from their primitive positions and form a fringe round the cone about A.
If X is greater than D, then the cone round B is completely dispersed, and all the molecules which formed it are converted into the fringe round A, and are inclined at the angle 00-f-/30.
445.] Treating this case in the same way as before, we find for the intensity of the temporary magnetization during the action of the force X, which is supposed to act on iron which has never before been magnetized,
When X is less than L, I = - M -_- •
3 J-f
When X is equal to It, I = - M -=j •
When X is between L and 2),
When X is equal to D,
'
When X is greater than D>
When X is infinite, I = M.
When X is less than L the magnetization follows the former law, and is proportional to the magnetizing force. As soon as X exceeds L the magnetization assumes a more rapid rate of increase
VOL. n. G
82 WEBER'S THEORY OF INDUCED MAGNETISM. [445.
on account of the molecules beginning to be transferred from the one cone to the other. This rapid increase, however, soon conies to an end as the number of molecules forming the negative cone diminishes, and at last the magnetization reaches the limiting value M.
If we were to assume that the values of L and of D are different for different molecules, we should obtain a result in which the different stages of magnetization are not so distinctly marked.
The residual magnetization, /', produced by the magnetizing force X, and observed after the force has been removed, is as follows :
When X is less than I/, No residual magnetization.
When X is between L and D,
When X is equal to D,
T2 2
When X is greater than D,
'-J
When X is infinite,
If we make
M = 1000, L = 3, .# = 5,
we find the following values of the temporary and the residual magnetization : —
Magnetizing Temporary Residual
Force. Magnetization. Magnetization.
x i r
000
1 133 0
2 267 0
3 400 0
4 729 280
5 837 410
6 864 485
7 882 537
8 897 574
oo 1000 810
446.] TEMPORARY AND RESIDUAL MAGNETIZATION. 83
These results are laid down in Fig. 10.
10
I 2 3 4 5 6 7 8 J
JHcufn.etizin.tp jforce
Fig. 10.
The curve of temporary magnetization is at first a straight line from X = 0 to X = L. It then rises more rapidly till X = 1), and as X increases it approaches its horizontal asymptote.
The curve of residual magnetization begins when X = _Z/, and approaches an asymptote at a distance = .8lJf.
It must be remembered that the residual magnetism thus found corresponds to the case in which, when the external force is removed, there is no demagnetizing force arising from the distribution of magnetism in the body itself. The calculations are therefore applicable only to very elongated bodies magnetized longitudinally. In the case of short, thick bodies the residual magnetism will be diminished by the reaction of the free magnetism in the same way as if an external reversed magnetizing force were made to act upon it.
446.] The scientific value of a theory of this kind, in which we make so many assumptions, and introduce so many adjustable constants, cannot be estimated merely by its numerical agreement with certain sets of experiments. If it has any value it is because it enables us to form a