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,[1] 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 |
By this the magnetic energy per unit of volume becomes
| 12 g2 | ∂ξ | 2 | + | ∂η | 2 | + | ∂ζ | 2 | , | |||
| ∂t | ∂t | ∂t |
and since this must be the kinetic energy of the elastic medium, the density of the latter must be taken equal to g2, so that it must be the same in all substances.
It may further be asked what value we have to assign to the potential energy in the model, which must correspond to the electric energy in the electromagnetic field. Now, on account of (11) and (19), we can satisfy the equations (12) by putting Dx = gc (∂ζ/∂y − ∂η/∂z), &c., so that the electric energy (17) per unit of volume becomes
| 12 g2c2 | 1 | ∂ζ | − | ∂η | 2 + | 1 | ∂ξ | − | ∂ζ | 2 + | 1 | ∂η | − | ∂ξ | 2 . | |||
| ε1 | ∂y | ∂z | ε2 | ∂z | ∂x | ε3 | ∂x | ∂y |
This, therefore, must be the potential energy in the model.
It may be shown, indeed, that, if the aether has a uniform constant density, and is so constituted that in any system, whether homogeneous or not, its potential energy per unit of volume can be represented by an expression of the form
| 12 L | ∂ζ | − | ∂η | 2 + M | ∂ξ | − | ∂ζ | 2 + N | ∂η | − | ∂ξ | 2 , |
| ∂y | ∂z | ∂z | ∂x | ∂x | ∂y |
where L, M, N are coefficients depending on the physical properties
of the substance considered, the equations of motion will exactly
correspond to the equations of the electromagnetic field.
18. Theories of Neumann, Green, and MacCullagh.—A theory of light in which the elastic aether has a uniform density, and in which the vibrations are supposed to be parallel to the plane of polarization, was developed by Franz Ernst Neumann,[2] who gave the first deduction of the formulas for crystalline reflection. Like Fresnel, he was, however, obliged to introduce some illegitimate assumptions and simplifications. Here again Green indicated a more rigorous treatment.
By specializing the formula for the potential energy of an anisotropic body he arrives at an expression which, if some of his coefficients are made to vanish and if the medium is supposed to be incompressible, differs from (20) only by the additional terms
| 2 L | ∂ζ | ∂η | − | ∂η | ∂ζ | + M | ∂ξ | ∂ζ | − | ∂ζ | ∂ξ | + N | ∂η | ∂ξ | − | ∂ξ | ∂η | . | ||||||
| ∂y | ∂z | ∂y | ∂z | ∂z | ∂x | ∂z | ∂x | ∂x | ∂y | ∂x | ∂y |
If ξ, η, ζ vanish at infinite distance the integral of this expression
over all space is zero, when L, M, N are constants, and the same
will be true when these coefficients change from point to point,
provided we add to (21) certain terms containing the differential
coefficients of L, M, N, the physical meaning of these terms being
that, besides the ordinary elastic forces, there is some extraneous
force (called into play by the displacement) acting on all those
elements of volume where L, M, N are not constant. We may
conclude from this that all phenomena can be explained if we admit
the existence of this latter force, which, in the case of two contingent
bodies, reduces to a surface-action on their common boundary.
James MacCullagh[3] avoided this complication by simply assuming an expression of the form (20) for the potential energy. He thus established a theory that is perfectly consistent in itself, and may be said to have foreshadowed the electromagnetic theory as regards the form of the equations for transparent bodies. Lord Kelvin afterwards interpreted MacCullagh’s assumption by supposing the only action which is called forth by a displacement to consist in certain couples acting on the elements of volume and proportional to the components 12 {(∂ζ/∂y) − (∂η/∂z)}, &c., of their rotation from the natural position. He also showed[4] that this “rotational elasticity” can be produced by certain hidden rotations going on in the medium.
We cannot dwell here upon other models that have been proposed, and most of which are of rather limited applicability. A mechanism of a more general kind ought, of course, to be adapted to what is known of the molecular constitution of bodies, and to the highly probable assumption of the perfect permeability for the aether of all ponderable matter, an assumption by which it has been possible to escape from one of the objections raised by Newton (§ 4) (see Aether).
The possibility of a truly satisfactory model certainly cannot be denied. But it would, in all probability, be extremely complicated. For this reason many physicists rest content, as regards the free aether, with some such general form of the electromagnetic theory as has been sketched in § 16.
19. Optical Properties of Ponderable Bodies. Theory of Electrons.—If we want to form an adequate representation of optical phenomena in ponderable bodies, the conceptions of the molecular and atomistic theories naturally suggest themselves. Already, in the elastic theory, it had been imagined that certain material particles are set vibrating by incident waves of light. These particles had been supposed to be acted on by an elastic force by which they are drawn back towards their positions of equilibrium, so that they can perform free vibrations of their own, and by a resistance that can be represented by terms proportional to the velocity in the equations of motion, and may be physically understood if the vibrations are supposed to be converted in one way or another into a disorderly heat-motion. In this way it had been found possible to explain the phenomena of dispersion and (selective) absorption, and the connexion between them (anomalous dispersion).[5] These ideas have been also embodied into the electromagnetic theory. In its more recent development the extremely small, electrically charged particles, to which the name of “electrons” has been given, and which are supposed to exist in the interior of all bodies, are considered as forming the connecting links between aether and matter, and as determining by their arrangement and their motion all optical phenomena that are not confined to the free aether.[6]
It has thus become clear why the relations that had been established between optical and electrical properties have been found to hold only in some simple cases (§ 16). In fact it cannot be doubted that, for rapidly alternating electric fields, the formulae expressing the connexion between the motion of electricity and the electric force take a form that is less simple than the one previously admitted, and is to be determined in each case by
- ↑ “Reflection and Refraction,” Trans. Cambr. Phil. Soc. 7, p. 1 (1837); “Double Refraction,” ibid. p. 121 (1839).
- ↑ “Double Refraction,” Ann. d. Phys. u. Chem. 25 (1832), p. 418; “Crystalline Reflection,” Abhandl. Akad. Berlin (1835), p. 1.
- ↑ Trans. Irish Acad. 21, “Science,” p. 17 (1839).
- ↑ Math. and Phys. Papers (London, 1890), 3, p. 466.
- ↑ Helmholtz, Ann. d. Phys. u. Chem., 154 (1875), p. 582.
- ↑ H. A. Lorentz, Versuch einer Theorie der elektrischen u. optischen Erscheinungen in bewegten Körpern (1895) (Leipzig, 1906); J. Larmor, Aether and Matter (Cambridge, 1900).
