Goldhammer

In 1890 and the years immediately following appeared several memoirs relating to the fundamental equations of electro-magnetic theory.

Hertz, after presenting[70] the general content of Maxwell’s theory for bodies at rest, proceeded[71] to extend the equations to the case in which material bodies are in motion in the field.

In a really comprehensive and correct theory, as Hertz remarked, a distinction should be drawn between the quantities which specify the state of the aether at every point, and those which specify the state of the ponderable matter entangled with it.

This anticipation has been fulfilled by later investigators.

But Hertz considered that the time was not ripe for such a complete theory, and preferred, like Maxwell, to assume that the state of the compound system—matter plus aether—can be specified in the same way when the matter moves as when it is at rest; or, as Hertz himself expressed it, that “the aether contained within ponderable bodies moves with them.”

Maxwell’s own hypothesis with regard to moving systems[72] amounted merely to a modification in the equation

…

which represents the law that the electromotive force in a closed circuit is measured by the rate of decrease in the number of lines of magnetic induction which pass through the circuit.

This law is true whether the circuit is at rest or in motion; but in the latter case, the E in the equation must be taken to be the electromotive force in a stationary circuit whose position momentarily coincides with that of the moving circuit; and since an electromotive force [w. B] is generated in matter by its motion with velocity w in a magnetic field B, we see that E is connected with the electromotive force E′ in the moving ponderable body by the equation

…

so that the equation of electromagnetic induction in the moving body is

…

Maxwell made no change in the other electromagnetic equations, which therefore retained the customary forms

…

Hertz, however, impressed by the duality of electric and magnetic phenomena, modified the last of these equations by assuming that a magnetic force 4π [D.w] is generated in a dielectric which moves with velocity w in an electric field; such a force would be the magnetic analogue of the electromotive force of induction. A term involving curl [D.w] is then introduced into the last equation.

The theory of Hertz resembles in many respects that of Heaviside,[73] who likewise insisted much on the duplex nature of the electromagnetic field, and was in consequence disposed to accept the term involving curl [D.w] in the equations of moving media.

Heaviside recognized more clearly than his predecessors the distinction between the force E′, which determines the flux D, and the force E, whose curl represents the electric current.

In conformity with his principle of duality, he made a similar distinction between the magnetic force H′, which determines the flux B, and the force H, whose curl represents the “magnetic current”.

This distinction, as Heaviside showed, is of importance when the system is acted on by “impressed forces,” such as voltaic electromotive forces, or permanent magnetization; these latter must be included in E′ and K′, since they help to give rise to the fluxes D and B.

But they must not be included in E and H since their curls are not electric or magnetic currents; so that in general We have

…

where e and h denote the impressed forces.

…

Developing the theory by the aid of these conceptions, Heaviside was led to make a further modification.

An impressed force is best defined in terms of the energy which it communicates to the system; thus, if e be an impressed electric force, the energy communicated to unit volume of the electromagnetic system in unit time is e × the electric current.

In order that this equation may be true, it is necessary to regard the electric current in a moving medium as composed of the conduction-current, displacement-current, convectioncurrent, and also of the term curl [D.w], whose presence in the equation we have already noticed. This may be called the current of dielectric convection.

Thus the total current is

…

where ρw denotes the conduction-current; and the equation connecting current with magnetic force is

…

where h0, denotes the impressed magnetic forces other than that induced by motion of the medium.

…

We must now consider the advances which were effected during the period following the publication of Maxwell’s Treatise in some of the special problems of electricity and optics.

We have seen[74] that Maxwell accounted for the rotation of the plane of polarization of light in a medium subjected to a magnetic field K by adding to the kinetic energy of the aether, which is represented by

…

ρė2, a term

…

σ(ė. curl ∂e/∂θ, where σ is a magneto-optic constant characteristic of the substance through which the light is transmitted, and ∂/∂θ stands for Kx∂/∂x + Ky∂/∂y + Kz∂/∂z.

This theory was developed further in 1879 by FitzGerald,[75] who brought it into closer connexion with the electromagnetic theory of light by identifying the curl of the displacement e of the aethereal particles with the electric displacement; the derivate of e with respect to the time then corresponds to the magnetic force.

Being thus in possession of a definitely electromagnetic theory of the magnetic rotation of light, FitzGerald proceeded to extend it so as to take account of a closely related phenomenon.

In 1876 J. Kerr[76] had shown experimentally that when plane-polarized light is regularly reflected from either pole of an iron electromagnet, the reflected ray has a component polarized in a plane at right angles to the ordinary reflected ray.

Shortly after this discovery had been made known, FitzGerald[77] had proposed to explain it by means of the same term in the equations which accounts for the magnetic rotation of light in transparent bodies.

His argument was that if the incident plane-polarized ray be resolved into two rays circularly polarized in opposite senses, the refractive index will have different values for these two rays, and hence the intensities after reflexion will be different; so that on recompounding them, two plane-polarized rays will be obtained—one polarized in the plane of incidence, and the other polarized at right angles to it.

The analytical discussion of Kerr’s phenomenon, which was given by FitzGerald in his memoir of 1879, was based on these ideas; the most essential features of the phenomenon were explained, but the investigation was in some respects imperfect.[78]

A new and fruitful conception was introduced in 1879–1880, when H. A. Rowland[79] suggested a connexion between the magnetic rotation of light and the phenomenon which had been discovered by his pupil Hall.[80]

Hall’s effect may be regarded as a rotation of conduction-currents under the influence of a magnetic field; and if it be assumed that displacement-currents in dielectrics are rotated in the same way, the Faraday effect may evidently be explained.

Considering the matter from the analytical point of view, the Hall effect may be represented by the addition of a term k [K.S] to the electromotive force, where K denotes the impressed magnetic force, and S denotes the current: so Rowland assumed that in dielectrics there is an additional term in the electric force, proportional to [K.Ḋ], i.e. proportional to the rate of increase of [K.D].

The total electric force round a circuit is proportional to the rate of decrease of the total magnetic induction through the circuit: so the total magnetic induction through the circuit must contain a term proportional to the integral of [K.D] taken round the circuit: and therefore the magnetic induction at any point must contain a term proportional to [curl K.D]. We may therefore write

…

where σ denotes a constant. But if this be combined with the customary electromagnetic equations

…

and all the vectors except B be eliminated (K being treated as a constant), we obtain the equation

…

where ∂/∂θ stands for Kx∂/∂x + Ky∂/∂y + Kz∂/∂z (K20/0c + Ky0/0y + K20/02); and this is identical with the equation which Maxwell had given[81] for the motion of the aether in magnetized media.

It follows that the assumptions of Maxwell and of Rowland, different though they are physically, lead to the same analytical equations-at any rate so far as concerns propagation through a homogeneous medium.

The connexions of Hall’s phenomenon with the magnetic rotation of light, and with the reflexion of light from magnetized metals, were extensively studied [82] in the years following the publication of Rowland’s memoir: but it was not until the modern theory of electrons had been developed that a satisfactory representation of the molecular processes involved in magnetooptic phenomena was attained.

The allied phenomenon of rotary polarization in naturally active bodies was investigated in 1892 by Goldhammer.[83]

In the clastic-solid theory of Boussinesq [84], the rotation of the plane of polarization of saccharine solutions had been represented by substituting the equation

…

in place of the usual equation

…

Goldhammer now proposed to represent rotatory power in the electromagnetic theory by substituting the equation

…

in place of the customary equation

…

the constant k being a measure of the natural rotatory power of the substance concerned. The remaining equations are as usual,

…

Eliminating H and E, we have

…

For a plane wave which is propagated parallel to the axis of x, this equation reduces to

…

and, as MacCullagh had shown in 1836,[85] these equations are competent to represent the rotation of the plane of polarization.

In the closing years of the nineteenth century, the general theory of aether and electricity assumed a new form. But before discussing the memoirs in which the new conception was unfolded, we shall consider the progress which had been made since the middle of the century in the study of conduction in liquid and gaseous media.