A History of the Theories of Aether and Electricity (1910) — part 12 of 29

The next advance in the subject was due to Fresnel,1[ who showed that in naturally active bodies the velocity of propa- gation of circularly polarized light is different according as the polarization is right-handed or left-handed. From this property the rotation of the plane of polarization of a plane- polarized ray may be immediately deduced ; for the plane- polarized ray may be resolved into two rays circularly polarized in opposite senses, and these advance in phase by different

• Mem. de 1'Institut, 1812, Part i, p. 218, sqq. ; Annales de Chim., ix (1818), p. 372; x (1819), p. 63. tCamb. Phil. Soc. Trans, i, p. 43.

J Mem. de 1'Inst. vii, p. 73. § Baltimore Lectures (ed. 1904), p. 31.

|| The term rotatory may be applied with propriety to the property discovered by Faraday, which will be discussed later.

H Annales de Chim. xxviii (1825), p. 147.

The Aether as an Elastic Solid. 1 75

amounts in passing through a given thickness of the substance-: at any stage they may be recompoundecl into a pkne-polarized ray, the azimuth of whose plane of polarization varies with the length of path traversed.

It is readily seen from this that a ray of light incident on a crystal of quartz will in general bifurcate into two refracted rays, each of which will be elliptically polarized, i.e. will be capable of resolution into two plane-polarized components which differ in phase by a definite amount. The directions of these refracted rays may be determined by Huygens' con- struction, provided the wave-surface is supposed to consist of a sphere and spheroid which do not touch.

The first attempt to frame a theory of naturally active bodies was made by MacCullagh in 1836.* Suppose a plane wave of light to be propagated within a crystal of quartz. Let (#, ?/, z) denote the coordinates of a vibrating molecule, when the axis of x is taken at right angles to the plane of the wave, and the axis of z at right angles to the axis of the crystal. Using Fand Zto denote the displacements parallel to the axes of y and z respectively at any time t, MacCullagh assumed that the differential equations which determine Y and

---

w "" ** w ^'w

where /* denotes a constant on which the natural rotatory property of the crystal depends. In order to avoid compli- cations arising from the ordinary crystalline properties of quartz, we shall suppose that the light is propagated parallel to the optic axis, so that we can take c, equal to c2.

Assuming first that the beam is circularly polarized, let it be represented by

(y f\

Y = A sin — (Ix - £), Z = ± A cos — (Ix - t),

• Trans. Royal Irish Acad., xvii. ; MacCullagh's Coll. Works, p. 63.

176 The Aether as an Elastic Solid.

*he ambiguous sign being determined according as the circular polarization. ^ ri^ht-handed or left-handed.

Substituting in ^e above differential equations, we have

or

Since I// denotes the velocity of propagation, it is evident that the reciprocals of the velocities of propagation of a right-handed and left-handed beam differ by the quantity

from which it is easily shown that the angle through which the plane of polarization of a plane-polarized beam rotates in unit length of path is

rV

If we neglect the variation of Ci with the period of the light, this expression satisfies Biot's law that the angle of rotation in unit length of path is proportional to the inverse square of the wave-length.

MacCullagh's investigation can be scarcely called a theory, for it amounts only to a reduction of the phenomena to empirical, though mathematical, laws ; but it was on this foundation that later workers built the theory which is now accepted.*

• The later developments of this theory will be discussed in a subsequent chapter ; hut mention may here he made of an attempt which was made in 1856 by Carl Neumann, then a very young man, to provide a rational basis for MacCullagh's equations. Neumann showed that the equations may be derived from the hypothesis that the relative displacement of one aethereal particle with respect to another acts on the latter according to the same law as an element of an electric current acts on a magnetic pole. Cf. the preface to C. Neumann's Die Drehung der Polarisationsebene des Lichtes, Halle, 1863.

The Aether as an Elastic Solid. 177

The great investigators who developed the theory of light after the death of Fresnel devoted considerable attention to the optical properties of metals. Their researches in this direction must now be reviewed.

The most striking properties of metals are the power of brilliantly reflecting light at all angles of incidence, which is so well shown by the mirrors of reflecting telescopes, and the opacity, which causes a train of waves to be extinguished before it has proceeded many wave-lengths into a metallic medium. That these two attributes are connected appears probable from the fact that certain non- metallic bodies — e.g., aniline dyes — which strongly absorb the rays in certain parts of the spectrum, reflect those rays with almost metallic brilliance. A third quality in which metals differ from transparent bodies, and which, as we shall see, is again closely related to the other two, is in regard to the polarization of the light reflected from them. This was first noticed by Malus ; and in 1830 Sir David Brewster* showed that plane-polarized light incident on a metallic surface remains polarized in the same plane after reflexion if its polarization is either parallel or perpendicular to the plane of reflexion, but that in other cases the reflected light is polarized elliptically.

It was this discovery of Brews ter's which suggested to the mathematicians a theory of metallic reflexion. For, as we have seen, elliptic polarization is obtained when plane-polarized light is totally reflected at the surface of a transparent body ; and this analogy between the effects of total reflexion and metallic reflexion led to the surmise that the latter pheno- menon might be treated in the same way as Fresnel had treated the former, namely, by introducing imaginary quantities into the formulae of ordinary reflexion. On these principles mathe- matical formulae were devised by MacCullaghf and Cauchy^

*Phil. Trans., 1830.

• Proc. Roy. Irish Acad., i (1836), p. 2 ; ii (1843), p. 376 : Trans. Roy. Irish Acad., xviii (1837), p. 71 : MacCullagh's Coll. Works, pp. 58, 132, 230.

J Comptes Rendus, vii (1838), p. 953 ; riii (1839), pp. 553, 658, 961 ; xxvi (1848), p. 86.

N

178 The Aether as an Elastic Solid.

To explain their method, we shall suppose the incident light to be polarized in the plane of incidence. According to Fresnel's sine-law, the amplitude of the light (polarized in this way) reflected from a transparent body is to the amplitude of the incident light in the ratio

_ sin (i - r) sin (i + r)'

where i denotes the angle of incidence and r is determined from the equation

sin i = ft sin r.

MacCullagh and Cauchy assumed that these equations hold good also for reflexion at a metallic surface, provided the refractive index /* is replaced by a complex quantity

IJL = v(l — *v/ — 1) say,

where v and K are to be regarded as two constants characteristic of the metal. We have therefore

tan i - tan r (ju2 - sin2 i)% - cos i

jj — ,.-,.. - — • - - — -- "

tan i + tan r (fj2 - sin2 i)k + cos i If then we write

so that equations defining U and v are obtained by equating separately the real and the imaginary parts of this equation, we have

Ue^ ~ l - cos i J

TT v/ — 1

Ue v + cos and this may be written in the form

where

-=.2 U* + cos2^ - 2 U cos v cos i U" + cos2* + 2 U cos v cos i 2 U cos i sin v

tang =

U* - cos2*

The Aether as an Elastic Solid. 179

The quantities J and S are interpreted in the same way as

in Fresnel's theory of total reflexion : that is, we take J to mean the ratio of the intensities of the reflected and incident light, while 3 measures the change of phase experienced by the light in reflexion.

The case of light polarized at right angles to the plane of incidence may be treated in the same way.

When the incidence is perpendicular, U evidently reduces to v (1 + K2)*, and u reduces to - tan-1 K. For silver at perpen-

— 2

dicular incidence almost all the light is reflected, so J is nearly unity : this requires cos v to be small, and K to be very large. The extreme case in which K is indefinitely great but v indefinitely small, so that the quasi- index of refraction is a pure imaginary, is generally known as the case of ideal silver.

The physical significance of the two constants v and K was more or less distinctly indicated by Cauchy; in fact, as the difference between metals and transparent bodies depends on the constant K, it is evident that K must in some way measure the opacity of the substance. This will be more clearly seen if we inquire how the elastic-solid theory of light can be extended so as to provide a physical basis for the formulae of MacCullagh and Cauchy.* The sine-formula of Fresnel, which was the starting-point of our investigation of metallic reflexion, is a consequence of Green's elastic-solid theory : and the differences between Green's results and those which we have derived arise solely from the complex value which we have assumed for yu. We have therefore to modify Green's theory in such a way as to obtain a complex value for the index of refraction.

Take the plane of incidence as plane of xy, and the metallic surface as plane of yz. If the light is polarized in the plane of incidence, so that the light- vector is parallel to the axis of z, the incident light may be taken to be a function of the argument

ax + by + ct,

• This was done by Lord Rayleigh, Phil. Mag. xliii (1872), p. 321.

N 2

180 The Aether as an Elastic Solid.

where

a /p\l . b

• = - - COS I,

c \n c

/p*

• • I sin ^ ; \nj

here * denotes the angle of incidence, p the inertia of the aether,, and n its rigidity.

Let the reflected light be a function of the argument

OiX + by + ct,

where, in order to secure continuity at the boundary, b and c must have the same values as before. Since Green's formulae are to be still applicable, we must have

where sin i = ft sin r, but /j. has now a complex value. This- equation may be written in the form

n Let the complex value of /u,z be written

p

the real part being written pi/p in order to exhibit the analogy with Green's theory of transparent media : then we have

n n

But an equation of this kind must (as in Green's theory) represent the condition to be satisfied in order that the quantity

(a\x + by + ct) / - I t/

may satisfy the differential equation of motion of the aether ; from which we see that the equation of motion of the aether in the metallic medium is probably of the form

dzez A dez

This equation of motion differs from that of a Greenian

The Aether as an Elastic Solid. 181

elastic solid by reason of the occurrence of the term in dez/dt. But this is evidently a " viscous " term, representing something like a frictional dissipation of the energy of luminous vibra- tions : a dissipation which, in fact, occasions the opacity of the metal. Thus the term which expresses opacity in the equation of motion of the luminiferous medium appears as the origin of the peculiarities of metallic reflexion.* It is curious to notice how closely this accords with the idea of Huygens, that metals are characterized by the presence of soft particles which damp the vibrations of light.

There is, however, one great difficulty attending this explanation of metallic reflexion, which was first pointed out by Lord Rayleigh.f We have seen that for ideal silver ^ is real and negative : and therefore A must be zero and p± negative ; that is to say, the inertia of the luminiferous medium in the metal must be negative. This seems to destroy entirely the physical intelligibility of the theory as applied to the case of ideal silver.

The difficulty is a deep-seated one, and was not overcome for many years. The direction in which the true solution lies will suggest itself when we consider the resemblance which has already been noticed between metals and those substances which show "surface colour" — e.g. the aniline dyes. In the case of the latter substances, the light which is so copiously reflected from them lies within a restricted part of the spectrum ; and it therefore seems probable that the phenomenon is not to be attributed to the existence of dissipative terms, but that it belongs rather to the same class of effects as dispersion, and is to be referred to the same causes. In fact, dispersion means that the value of the refractive index of a substance with respect to any kind of light depends on the period of the light ; and we have only to suppose that the physical causes which operate in dispersion cause the refractive index

• It is easily seen that the amplitude is reduced by the factor e-™* when light travels one wave-length in the metal : K is generally called the coefficient of absorption. 1" Loc. cit.

182 The Aether as an Elastic Solid.

to become imaginary for certain kinds of light, in order to explain satisfactorily both the surface colours of the aniline dyes and the strong reflecting powers of the metals.

Dispersion was the subject of several memoirs by the founders of the elastic-solid theory. So early as 1830 Cauchy's attention was directed* to the possibility of constructing a mathematical theory of this phenomenon on the basis of Fresnel's " Hypothesis of Finite Impacts "f — i.e. the assumption that the radius of action of one particle of the luminiferous medium on its neighbours is so large as to be comparable with the wave-length of light. Cauchy supposed the medium to be formed, as in Navier's theory of elastic solids, of a system of point-centres of force : the force between two of these point-centres, m at (x, y, z), and //, at (x + A#, y + Ay, z + Az), may be denoted by m^/(r), where r denotes the distance between m and p. When this medium is disturbed by light- waves pro- pagated parallel to the z-axis, the displacement being parallel to the #>axis, the equation of motion of m is evidently

• p) -rT— ,

where £ denotes the displacement of m, (£ -i A If) the displace- ment of p, and (r + p) the new value of r. Substituting for p its value, and retaining only terms of the first degree in A?, this equation becomes

DT r dr

Now, by Taylor's theorem, since £ depends only on z, we have

Substituting, and remembering that summations which involve odd powers of Az must vanish when taken over all

• Bull, des Sc. Math, xiv (1830), p. 9 : " Sur la dispersion de la lumiere," . Exevcwe* de Math., 1836. t Cf. p. 132.

The Aether as an Elastic Solid. 183

the point-centres within the sphere of influence of m, we obtain an equation of the form

fft d'K o^E d'Z

w = a&+Pw + y& + ---'

where a, )3, 7 . . . denote constants.

Each successive term on the right-hand side of this equation involves an additional factor (A^)2/X8 as compared with the pre- ceding term, where X denotes the wave-length of the light : so if the radii of influence of the point-centres were indefinitely small in comparison with the wave-length of the light, the equation would reduce to

8^_ cP£

ar- = "&*'

which is the ordinary equation of wave-propagation in one dimension in non-dispersive media. But if the medium is so coarse-grained that A is not large compared with the radii of influence, we must retain the higher derivates of £. Substi- tuting

in the differential equation with these higher derivates retained, we have

'2-irV

which shows that cb the velocity of the light in the medium, depends on the wave-length A ; as it should do in order to explain dispersion.

Dispersion is, then, according to the view of Fresnel and Cauchy, a consequence of the coarse-grainedness of the medium. Since the luminiferous medium was found to be dispersive only within material bodies, it seemed natural to suppose that in these bodies the aether is loaded by the molecules of matter, and that dispersion depends essentially on the ratio of the wave-length to the distance between adjacent material molecules.

184 The Aether as an Elastic Solid.

This theory, in one modification or another, held its ground until forty years later it was overthrown by the facts of anomalous dispersion.

The distinction between aether and ponderable matter was more definitely drawn in memoirs which were published independently in 1841-2 by F. E. Neumann* and Matthew O'Brien.f These authors supposed the ponderable particles to remain sensibly at rest while the aether surges round them, and is acted on by them with forces which are proportional to its displacement. ThusJ the equation of motion of the aether becomes

rP&

p •£ ~2 = - (k + ^n) grad div e - n curl curl e - Ce, ot

where C denotes a constant on which the phenomena of dis- persion depend. For polarized plane waves propagated parallel to the axis of x, this equation becomes

92e 92e „

fg^»5r*5

and substituting

e = e

where r denotes the period and V the velocity of the light, we have

G T,

772 - P 4^3 r '

an equation which expresses the dependence of the velocity on the period.

The attempt to represent the properties of the aether by those of an elastic solid lost some of its interest after the rise of the electromagnetic theory of light. But in 1867,

• Berlin Abhandlungen aus dem Jahre 1841, Zweiter Teil, p. 1 : Berlin, 1843. t Trans. Camb. Phil. Soc. vii (1842), p. 397. J O'Brien, loc. cit, §§ 15, 28.

The Aether as an Elastic Solid. 185

before the electromagnetic hypothesis had attracted much attention, an elastic-solid theory in many respects preferable to its predecessors was presented to the French Academy* by Joseph Boussinesq (b. 1842). Until this time, as we have seen, investigators had been divided into two parties, according as they attributed the optical properties of different bodies to variations in the inertia of the luminiferous medium, or to variations in its elastic properties. Boussinesq, taking up a position apart from both these schools, assumed that the aether is exactly the same in all material bodies as in interplanetary space, in regard both to inertia and to rigidity, and that the optical properties of matter are due to interaction between the aether and the material particles, as had been imagined more or less by Neumann and O'Brien. These material particles he supposed to be disseminated in the aether, in much the same way as dust-particles floating in the air.

If e denote the displacement at the point (x, y, z) in the aether, and e' the displacement of the ponderable particles at the same place, the equation of motion of the aether is

rfie ?P&'

P *jp = ~ (k + ^l) g11"1 div e + ^V2e - p, jp, (1)

where p and pl denote the densities of the aether and matter respectively, and k and n denote as usual the elastic constants of the aether. This differs from the ordinary Cauchy-Green equation only in the presence of the term pi&*'/dP, which represents the effect of the inertia of the matter. To this equation we must adjoin another expressing the connexion between the displacements of the matter and of the aether: if we assume that these are simply proportional to each other — say,

e' = Ae, (2)

• Journal de Math. (2) xiii (1868), pp. 313, 425 : cf. also Comptes Rendus, •cxvii (1893), pp. 80, 139, 193. Equations kindred to some of those of Boussinesq M-ere afterwards deduced by Karl Pearson, Proc. Lond. Math. Soc , xx (1889), p. 297, from the hypothesis that the strain-energy involves the velocities.

186 The Aether as an Elastic Solid.

where the constant A depends on the nature of the ponderable body — our equation becomes

32e

(p + P1A) ^ = - (k + Jw) grad div e + ^V2e, ot

which is essentially the same equation as is obtained in those older theories which suppose the inertia of the luminiferous medium to vary from one medium to another. So far there would seem to be nothing very new in Boussinesq's work. But when we proceed to consider crystal-optics, dispersion, and rotatory polarization, the advantage of his method becomes evident: he retains equation (1) as a formula universally true — at any rate for bodies at rest — while equation (2) is varied to suit the circumstances of the case. Thus dispersion can be explained if, instead of equation (2), we take the relation

e' = Ae - Z>V2e,

where D is a constant which measures the dispersive power of the substance : the rotation of the plane of polarization of sugar solutions can be explained if we suppose that in these bodies equation (2) is replaced by

e' = AQ + B curl e,

where B is a constant which measures the rotatory power ; and the optical properties of crystals can be explained if we suppose that for them equation (2) is to be replaced by the equations

ex' = Atfx, ey = Azeyt ez' = A3e,

When these values for the components of e' are substituted in equation (1), we evidently obtain the same formulae as were derived from the Stokes-Eankine-Eayleigh hypothesis of inertia different in different directions in a crystal; to which Boussinesq's theory of crystal-optics is practically equivalent.

The optical properties of bodies in motion may be accounted for by modifying equation (1), so that it takes the form

a a a ay ,

• • Wx— + Wy—- + W~ — C ,,

ct cv oy ozj

The Aether as an Elastic Solid. 187

where w denotes the velocity of the ponderable body. If the body is an ordinary isotropic one, and if we consider light propagated parallel to the axis of z, in a medium moving in that direction, the light- vector being parallel to the axis of x, the equation reduces to

d'ex d'ex id 9V

O — 7b ' — Q]A. I -f- IV — I 6r i

' O/2 ^W- • \ ^/ ^i/v /

C7t (j6 (7£ (72'/

substituting

«,-/(*- FO,

where V denotes the velocity of propagation of light in the medium estimated with reference to the fixed aether, we obtain

for V the value

/ n \k o\A

\p + pt p +

The absolute velocity of light is therefore increased by the amount piAw/(p + piA) owing to the motion of the medium ; and this may be written (/** - 1) wjfjc, where ju denotes the refractive index ; so that Boussinesq's theory leads to the same formula as had been given half a century previously by Fresnel.* It is Boussinesq's merit to have clearly asserted that all space, both within and without ponderable bodies, is occupied by one identical aether, the same everywhere both in inertia and elasticity; and that all aethereal processes are to be re- presented by two kinds of equations, of which one kind expresses the invariable equations of motion of the aether, while the other kind expresses the interaction between aether and matter. Many years afterwards these ideas were revived in connexion with the electromagnetic theory, in the modern forms of which they are indeed of fundamental importance.

• Cf. p. 115 sqq.

( 188 )

CHAPTEK VI.

FAKADAY.

TOWARDS the end of the year 1812, Davy received a letter in which the writer, a bookbinder's journeyman named Michael Faraday, expressed a desire to escape from trade, and obtain employment in a scientific laboratory. With the letter was enclosed a neatly written copy of notes which the young man — he was twenty-one years of age — had made of Davy's own public lectures. The great chemist replied courteously, and arranged an interview ; at which he learnt that his correspon- dent had educated himself by reading the volumes which came into his hands for binding. "There were two," Faraday wrote later, "that especially helped me, the 'Encyclopaedia Britannica,' from which I gained my first notions of electricity, and Mrs. Marcet's ' Conversations on Chemistry/ which gave me my foundation in that science." Already, before his applica- tion to Davy, he had performed a number of chemical experiments, and had made for himself a voltaic pile, with which he had decomposed several compound bodies.

At Davy's recommendation Faraday was in the following spring appointed to a post in the laboratory of the Koyal Institution, which had been established at the close of the eighteenth century under the auspices of Count Rumford ; and here he remained for the whole of his active life, first as assistant, then as director of the laboratory, and from 1833 onwards as the occupant of a chair of chemistry which was founded for his benefit.

For many years Faraday was directly under Davy's influence, and was occupied chiefly in chemical investigations. But in 1821, when the new field of inquiry opened by Oersted's

Faraday. 189

discovery was attracting attention, he wrote an Historical Sketch of Electro- Magnetism* as a preparation for which he carefully repeated the experiments described by the writers he was reviewing ; and this seems to have been the beginning of the researches to which his fame is chiefly due.

The memoir which stands first in the published volumes of Faraday's electrical workf was communicated to the Royal Society on November 24th, 1831. The investigation was inspired, as he tells us, by the hope of discovering analogies between the behaviour of electricity as observed in motion in currents, and the behaviour of electricity at rest on conductors. Static electricity was known to possess the power of " induction " — i.e., of causing an opposite electrical state on bodies in its neighbourhood ; was it not possible that electric currents might show a similar property ? The idea at first was that if in any circuit a current were made to flow, any adjacent circuit would be traversed by an induced current, which would persist exactly as long as the inducing current. Faraday found that this was not the case ; a current was indeed induced, but it lasted only for an instant, being in fact perceived only when the primary current was started or stopped. It depended, as he soon convinced himself, not on the mere existence of the inducing current, but on its variation.

Faraday now set himself to determine the laws of induction of currents, and for this purpose devised a new way of repre- senting the state of a magnetic field. Philosophers had been long accustomed? to illustrate magnetic power by strewing iron filings on a sheet of paper, and observing the curves in which they dispose themselves when a magnet is brought underneath.

•Published in Annals of Philosophy, ii (1821), pp. 195, 274; iii (1822), p. 107.

t Experimental Researches in Electricity, by Michael Faraday : 3 vols.

• The practice goes back at least as far as Niccolo Cabeo ; indeed the curves traced by Petrus Peregrinus on his globular lodestone (cf . p. 8) were projections of lines of force. Among eighteenth-century writers La Hire mentions the use of iron filings, Mem. de 1'Acad., 1717. Faraday had referred to them in his electro- magnetic paper of 1821, Exp. Res. ii, p. 127.

190 Faraday.

These curves suggested to Faraday* the idea of lines of magnetic force, or curves whose direction at every point coincides with the direction of the magnetic intensity at that point; the curves in which the iron filings arrange themselves on the paper resemble these curves so far as is possible subject to the condition of not leaving the plane of the paper.

With these lines of magnetic force Faraday conceived all space to be filled. Every line of force is a closed curve, which in some part of its course passes through the magnet to which it belongs, f Hence if any small closed curve be taken in space, the lines of force which intersect this curve must form a tubular surface returning into itself ; such a surface is called a tiibe of force. From a tube of force we may derive information not only regarding the direction of the magnetic intensity, but also regarding its magnitude; for the product of this magnitude} and the cross-section of any tube is constant along the entire length of the tube.§ On the basis of this result, Faraday conceived the idea of partitioning all space into compartments by tubes, each tube being such that this product has the same definite value. For simplicity, each of these tubes may be called a " unit line of force " ; the strength of the field is then indicated by the separation or concentration of the unit lines of force,! I so that the number of them which intersect a unit area placed at right angles to their direction

#They were first defined in Exp. Res., § 114 : "By magnetic curves, I mean the lines of magnetic forces, however modified hy the juxtaposition of poles, which could be depicted by iron filings ; or those to which a very small magnetic needle would form a tangent."

t Exp. Res. iii, p. 405.

J Within the substance of magnetized bodies we must in this connexion under- stand the magnetic intensity to be that experienced in a crevice whose sides are perpendicular to the lines of magnetization : in other words, we must take it to be what since Maxwell's time has been called the magnetic induction.

§ Exp. Res., § 3073. This theorem was first proved by the French geometer Michel Chasles, in his memoir on the attraction of an ellipsoidal sheet, Journal de 1'Ecole Polyt. xv (1837), p. 266.

|| Ibid., § 3122. "The relative amount of force, or of lines of force, in a given space is indicated by their concentration or separation — i.e., by their number in that space."

Faraday. 191

at any point measures the intensity of the magnetic field at that point.

Faraday constantly thought in terms of lines of force. " I cannot refrain," he wrote, in 1851,* " from again expressing my conviction of the truthfulness of the representation, which the idea of lines of force affords in regard to magnetic action. All the points which are experimentally established in regard to that action — i.e. all that is not hypothetical — appear to be well and truly represented by it."f

Faraday found that a current is induced in a circuit either when the strength of an adjacent current is altered, or when a magnet is brought near to the circuit, or when the circuit itself is moved about in presence of another current or a magnet. He saw from the firstj that in all cases the induction depends on the relative motion of the circuit and the lines of magnetic force in its vicinity. The precise nature of this dependence was the subject of long-continued further experiments. In 1832 he found§ that the currents produced by induction under the same circumstances in different wires are proportional to the conducting powers of the wires — a result which showed that the induction consists in the production of a definite electromotive force, independent of the nature of the wire, and dependent only on the intersections of the wire and the magnetic curves. This electromotive force is produced whether the wire forms a closed circuit (so that a current flows) or is open (so that electric tension results).

All that now remained was to inquire in what way the electromotive force depends on the relative motion of the wire and the lines of force. The answer to this inquiry is, in

• Exp. Res., § 3174.

t Some of Faraday's most distinguished contemporaries were far from sharing this conviction. " I declare," wrote Sir George Airy in 1855, " that I can hardly imagine anyone who practically and numerically knows this agreement " between observation and the results of calculation based on action at a distance, "to hesitate au instant in the choice between this simple and precise action, on the one hand, and anything so vague and varying as lines of force, on the other hand." Cf. Bence Jones's Life of Faraday, ii, p. 353.

I Exp. Res., § 116. § Ibid., § 213.

192 Faraday.

Faraday's own words,* that "whether the wire moves directly or obliquely across the lines of force, in one direction or another, it sums up the amount of the forces represented by the lines it has crossed," so that " the quantity of electricity thrown into a current is directly as the number of curves intersected."t The induced electromotive force is, in fact, simply proportional to the number of the unit lines of magnetic force intersected by the wire per second.

This is the fundamental principle of the induction of currents. Faraday is undoubtedly entitled to the full honour of its discovery ; but for a right understanding of the progress of electrical theory at this period, it is necessary to remember that many years elapsed before all the conceptions involved in Faraday's principle became clear and familiar to his contem- poraries ; and that in the meantime the problem of formulating the laws of induced currents was approached with success from other points of view. There were indeed many obstacles to the direct appropriation of Faraday's work by the mathematical physicists of his own generation ; not being himself a mathe- matician, he was unable to address them in their own language ; and his favourite mode of representation by moving lines of force repelled analysts who had been trained in the school of Laplace and Poisson. Moreover, the idea of electromotive force itself, which had been applied to currents a few years previously in Ohm's memoir, was, as we have seen, still involved in obscurity and misapprehension.

A curious question which arose out of Faraday's theory was whether a bar-magnet which is rotated on its own axis carries its lines of magnetic force in rotation with it. Faraday himself believed that the lines of force do not rotate J: on this view a revolving magnet like the earth is to be regarded as moving through its own lines of force, so that it must become charged at the equator and poles with electricity of opposite signs ; and if a wire not partaking in the earth's rotation were to have sliding contact with the earth at a pole and at the

• Exp. Res., § 3082. t Ibid., § 3115. % Ibid., § 3090.

Faraday. 193