Two further verifications must here be mentioned. In the first place, though we shall speak almost exclusively of the propagation of light in transparent dielectrics, a few words may be said about the optical properties of conductors. The simplest assumption concerning the electric current C in a metallic body is expressed by the equation C = σE, where σ is the coefficient of conductivity. Combining this with his other formulae (we may say with (12) and (13)), Maxwell found that there must be an absorption of light, a result that can be readily understood since the motion of electricity in a conductor gives rise to a development of heat. But, though Maxwell accounted in this way for the fundamental fact that metals are opaque bodies, there remained a wide divergence between the values of the coefficient of absorption as directly measured and as calculated from the electrical conductivity; but in 1903 it was shown by E. Hagen and H. Rubens[22] that the agreement is very satisfactory in the case of the extreme infra-red rays.
In the second place, the electromagnetic theory requires that a surface struck by a beam of light shall experience a certain pressure. If the beam falls normally on a plane disk, the pressure is normal too; its total amount is given by c−1(i1 + i2 − i3), if i1, i2 and i3 are the quantities of energy that are carried forward per unit of time by the incident, the reflected, and the transmitted light. This result has been quantitatively verified by E. F. Nicholls and G. F. Hull.[23]
Maxwell’s predictions have been splendidly confirmed by the experiments of Heinrich Hertz[24] and others on electromagnetic waves; by diminishing the length of these to the utmost, some physicists have been able to reproduce with them all phenomena of reflection, refraction (single and double), interference, and polarization.[25] A table of the wave-lengths observed in the aether now has to contain, besides the numbers given in § 11, the lengths of the waves produced by electromagnetic apparatus and extending from the long waves used in wireless telegraphy down to about 0.6 cm.
17. Mechanical Models of the Electromagnetic Medium.—From the results already enumerated, a clear idea can be formed of the difficulties which were encountered in the older form of the wave-theory. Whereas, in Maxwell’s theory, longitudinal vibrations are excluded ab initio by the solenoidal distribution of the electric current, the elastic-solid theory had to take them into account, unless, as was often done, one made them disappear by supposing them to have a very great velocity of propagation, so that the aether was considered to be practically incompressible. Even on this assumption, however, much in Fresnel’s theory remained questionable. Thus George Green,[26] who was the first to apply the theory of elasticity in an unobjectionable manner, arrived on Fresnel’s assumption at a formula for the reflection coefficient An sensibly differing from (10).
In the theory of double refraction the difficulties are no less serious. As a general rule there are in an anisotropic elastic solid three possible directions of vibration (§ 6), at right angles to each other, for a given direction of the waves, but none of these lies in the wave-front. In order to make two of them do so and to find Fresnel’s form for the wave-surface, new hypotheses are required. On Fresnel’s assumption it is even necessary, as was observed by Green, to suppose that in the absence of all vibrations there is already a certain state of pressure in the medium.
If we adhere to Fresnel’s assumption, it is indeed scarcely possible to construct an elastic model of the electromagnetic medium. It may be done, however, if the velocities of the particles in the model are taken to represent the magnetic force H, which, of course, implies that the vibrations of the particles are parallel to the plane of polarization, and that the magnetic energy is represented by the kinetic energy in the model. Considering further that, in the case of two bodies connected with each other, there is continuity of H in the electromagnetic system, and continuity of the velocity of the particles in the model, it becomes clear that the representation of H by that velocity must be on the same scale in all substances, so that, if ξ, η, ζ are the displacements of a particle and g a universal constant, we may write
| Hx = g | ∂ξ | , Hy = g | ∂η | , Hz = g | ∂ζ | . |
| ∂t | ∂t | ∂t |
(19)
By this the magnetic energy per unit of volume becomes
| ½ g2 { ( | ∂ξ | ) | 2 | + ( | ∂η | ) | 2 | + ( | ∂ζ | ) | 2 | }, |
| ∂t | ∂t | ∂t |