Maxwell's Equations

The memoir of 1864 contained an extension of the equations to the case of bodies in motion; the consideration of which naturally revives the question as to whether the aether is in any degree carried along with a body which moves through it.

Maxwell did not formulate any express doctrine on this subject.

But his custom was to treat matter as if it were merely a modification of the aether, distinguished only by altered values of such constants as the magnetic permeability and the specific inductive capacity; so that his theory may be said to involve the assumption that matter and aether move together.

In deriving the equations which are applicable to moving bodies, he made use of Faraday’s principle that the electromotive force induced in a body depends only on the relative motion of the body and the lines of magnetic force, whether one or the other is in motion absolutely.

From this principle it may be inferred that the equation which determines the electric force[34] in terms of the potentials, in the case of a body which is moving with velocity w, is

…

Maxwell thought that the scalar quantity ψ in this equation represented the electrostatic potential.

But the researches of other investigators[35] have indicated that it represents the sum of the electrostatic potential and the quantity (A.w).

The electromagnetic theory of light was moreover extended in this memoir so as to account for the optical properties of crystals. For this purpose Maxwell assumed that in crystals the values of the coefficients of electric and magnetic induction depend on direction, so that the equation

is replaced by

…

and similarly the equation

…

is replaced by

…

The other equations are the same as in isotropio media; so that the propagation of disturbance is readily seen to depend on the equation

…

If μ1, μ2, μ3 are supposed equal to each other, this equation is the same as the equation of motion of MacCullagh’s aether in crystalline media,[36] the magnetic force H corresponding to MacCullagh’s elastic displacement.

We may immediately infer that Maxwell’s electromagnetic equations yield a satisfactory theory of the propagation of light in crystals, provided it is assumed that the magnetic permeability is (for optical purposes) the same in all directions, and provided the plane of polarization is identified with the plane which contains the magnetic vector.

The direction of the ray is at right angles to the magnetic vector and the electric force, and that the wave-front is the plane of the magnetic vector and the electric displacement.[37]

After this Maxwell proceeded to investigate the propagation of light in metals.

The difference between metals and dielectrics, so far as electricity is concerned, is that the former are conductors; and it was therefore natural to seek the cause of the optical properties of metals in their ohmic conductivity.

This idea at once suggested a physical reason for the opacity of metals—namely, that within a metal the energy of the light vibrations is converted into Joulian heat in the same way as the energy of ordinary electric currents.

The equations of the electromagnetic field in the metal may be written

…

where k denotes the ohmic conductivity; whence it is seen that the electric force satisfies the equation

…

This is of the same form as the corresponding equation in the elastic-solid theory[38]; and, like it, furnishes a satisfactory general explanation of metallic reflexion.

It is correct in all details, so long as the period of the disturbance is not too short—i.e., so long as the light-waves considered belong to the extreme infra-red region of the spectrum;

But if we attempt to apply the theory to the case of ordinary light, we are confronted by the difficulty which Lord Rayleigh indicated in the elastic-solid theory.[39] and which attends all attempts to explain the peculiar properties of metals by inserting a viscous term in the equation.

The difficulty is that, in order to account for the properties of ideal silver, we must suppose the coefficient of Ë negative—that is, the dielectric constant of the metal must be negative, which would imply instability of electrical equilibrium in the metal.

The problem, as we have already remarked,[40] was solved only when its relation to the theory of dispersion was rightly understood.