Magnetic Vortices

In 1876, A. Bartoli[72] thought of a different way of inferring the necessity for light-pressure.

He showed that, when radiant energy is transported from a cold body to a hot one by means of a moving mirror, the second law of thermodynamics would be violated unless a pressure were exerted on the mirror by the light.

The thermodynamical ideas introduced into the subject by Bartoli have proved very fruitful.

If a hollow vessel be at a definite temperature, the aether within the vessel must be full of radiation crossing from one side to the other: and hence the aether, when in radiative equilibrium with matter at a given temperature, is the seat of a definite quantity of energy per unit volume.

If U denote this energy per unit volume, and P the light-pressure on unit area of a surface exposed to the radiation, we may apply[73] the equation of available energy[74]

…

Since, as we have seen,

…

this equation gives

…

Therefore, U must be proportional to T4.

From this, it may be inferred that the intensity of emission of radiant energy by a body at temperature T is proportional to the fourth power of the absolute temperature—a law which was first discovered experimentally by Stefan[75] in 1879.

In the year in which Maxwell’s treatise was published, Sir William Crookes[76] obtained experimental evidence of a pressure accompanying the incidence of light.

But this was soon found to be due to thermal effects; and the existence of a true light-pressure was not confirmed experimentally[77] until 1899. Since then the subject has been considerably developed, especially in regard to the part played by the pressure of radiation in cosmical physics.

Another matter which received attention in Maxwell’s Treatise was the influence of a magnetic field on the propagation of light in material substances.

The theory of magnetic vortices [78] had its origin in Thomson’s speculations on this phenomenon.

Maxwell in his memoir of 1861–2 had attempted by the help of that theory to arrive at some explanation of it.

The more complete investigation which is given in the Treatise is based on the same general assumptions, namely, that in a medium subjected to a magnetic field there exist concealed vortical motions, the axes of the vortices being in the direction of the lines of magnetic force.

That waves of light passing through the medium disturb the vortices, which thereupon react dynamically on the luminous motion, and so affect its velocity of propagation.

Maxwell supposed that the magnetic vortices are affected by the light-waves in the same way as vortex-filaments in a liquid would be affected by any other coexisting motion in the liquid.

The latter problem had been already discussed in Helmholtz’s great memoir on vortex-motion; adopting Helmholtz’s results, Maxwell assumed for the additional term introduced into the magnetic force by the displacement of the vortices the value ∂e/∂θ, where e denotes the displacement of the medium (i.e. the light vector).

The operator ∂/∂θ denotes Hx∂/∂x + Hy∂/∂y + Hz∂/∂z, H denoting the imposed magnetic field. Thus the luminous motion, by disturbing the vortices, gives rise to an electric current in the medium, proportional to curl ∂e/∂θ.

Maxwell further assumed that the current thus produced interacts dynamically with the luminous motion in such a manner that the kinetic energy of the medium contains a term proportional to the scalar product of ė and curl ∂e/∂θ. The total kinetic energy of the medium may therefore be written

…

where p denotes the density of the medium, and o denotes a constant which measures the capacity of the medium to rotate the plane of polarization of light in a magnetic field.

The equation of motion may now be derived as in the elastic-solid theories of light: it is

…

When the light is transmitted in the direction of the lines of force, and the axis of x is taken parallel to this direction, the equation reduces to

…

and these equations, as we have seen,[79] furnish an explanation of Faraday’s phenomenon.

The term

…

in the kinetic energy may by partial integration be transformed into a term

…

together with surface-terms; or, again, into

…

together with surface-terms. These different forms all yield the same equation of motion for the medium; but, owing to the differences in the surface-terms, they yield different conditions at the boundary of the medium, and consequently give rise to different theories of reflexion.

The assumptions involved in Maxwell’s treatment of the magnetic rotation of light were such as might scarcely be justified in themselves; but since the discussion as a whole proceeded from sound dynamical principles, and its conclusions were in harmony with experimental results, it was fitted to lead to tho more perfect explanations which were afterwards devised by his successors.

Maxwell died in 1879 before his 49th year. Much yet remained to be done both in this and in the other investigations with which his name is associated; and the energies of the next generation were largely spent in extending and refining that conception of electrical and optical phenomena whose origin is correctly indicated in its name of Maxwell’s Theory.