Thus we can experiment with electrical waves of sensible length, and thereby check theoretical developments; and we can push on the correspondence in properties between such waves and the waves of light, which are of very minute length. And it hardly admits of doubt, that in the case of a vacuum, where the complication of ponderable molecules with their disturbing free periods does not come in, absolute continuity will be found to exist in the transition from the one class to the other. But in the case of ponderable media, the two classes of waves will be influenced by free molecular periods of wholly different orders; so that any minute numerical correspondence is perhaps not to be anticipated. 410. By means of his experiments, Hertz proved the interference, reflection and polarization of electromagnetic waves; and from certain calculations based upon the results of his experiments, he has shown that the velocity of electromagnetic waves is approximately the same as that of light. Trouton' has further proved experimentally, that if electromagnetic waves are incident at the polarizing angle upon a bad conductor, the waves are not reflected when the direction of magnetic force is perpendicular to the plane of incidence; but when the direction of the former is parallel to the latter, reflection takes place at all angles of incidence. This experiment confirms Fresnel's hypothesis, that the vibrations of polarized light are perpendicular to the plane of polarization; and that upon Maxwell's theory, the disturbance which gives rise to optical effects is represented by the electric displacement. A complete discussion of the experiments of Hertz, and of the various other theories on the connection between light and electricity, belongs rather to a treatise on electromagnetism than to one on light. The reader, who desires further information upon these matters, is recommended to consult the original memoirs, and also Poincaré's Électricité et Optique, Part II., in which a very full account of Hertz's experiments is given. 1 Nature, 22nd Aug. 1889; see also Fitzgerald, Proc. Roy. Inst. March 21st, 1890. INTENSITY OF LIGHT. 361 Intensity of Light. 411. The intensity of light is usually measured on the electromagnetic theory, by the average energy per unit of volume. In a doubly refracting medium, the electrostatic energy per unit of volume is by (21). } (Pf+Qg+ Rh) = 2πμ1 (Â2X2 + В2μ2 + C2v2) S2 The electrokinetic energy is by (25). 1 ̧μ2 (a2 + B2 + y2) = 2πμ1V2S2, 8п It therefore follows that in any medium, the electrostatic and electrokinetic energies are equal. The energy therefore consists of two parts, one of which is a constant term, and the other is a periodic term. The first term is the average energy per unit of volume; and consequently the intensity of light on the electromagnetic theory, is proportional to the product of the magnetic permeability of the medium, the square of the amplitude, and the square of the velocity of propagation in the direction in which the wave is travelling. Conditions to be satisfied at the Surface of Separation 412. The conditions of continuity of force require, that the electric and magnetic forces parallel to the surface of separation should be the same in both media. These conditions furnish four equations. As regards the conditions to be satisfied perpendicular to the surface of separation, Maxwell has shown, Vol. 1. § 83, that if P, P' be the normal components of the electromotive force at the surface of separation of two media, whose specific inductive capacities are K, K', then PK-P'K' = 0 ; whence the components of the electric displacement perpendicular to the surface of separation must be the same in both media. Again, if μ, u' be the magnetic permeabilities of the two media, and a, a' the normal components of the magnetic force, Maxwell has shown Vol. II. § 428, that μα - μα' = 0; whence the components of the magnetic induction perpendicular to the surface of separation must be the same in both media1. 413. These conditions furnish altogether six equations, but we shall presently show that they reduce to only four; inasmuch as it will be proved later on, that the condition, that the electric displacement perpendicular to the surface of separation should be continuous, is analytically equivalent to the condition, that the magnetic force parallel to the line of intersection of the wave-front with the surface of separation should be the same in both media; and that the condition, that the magnetic induction perpendicular to the surface of separation should be continuous, is analytically equivalent to the condition, that the electric force parallel to the line of intersection of the wave-front with the surface of separation should be the same in both media. 414. The equations of motion (29) of an isotropic medium are of the same form, as those furnished by the elastic solid theory when is absolutely zero; for since there is no accumulation of free electricity dƒ/dx+dg/dy+dh/dz is always zero. We may therefore explain a variety of optical phenomena relating to isotropic media, by means of the electromagnetic theory just as well as by the elastic solid theory. There is however one important distinction between the two theories, viz. that the supposition that = 0, requires that m∞, and accordingly md may be finite; and in studying Green's theory of the reflection and refraction of light, we saw that under these circumstances it was necessary to 1 The continuity of electric displacement and magnetic induction can be at once deduced from the condition, that these quantities both satisfy an equation of the same form as the equation of continuity of an incompressible fluid. This equation is likewise the condition, that the directions of these quantities should be parallel to the wave-front. REFLECTION AND REFRACTION. 363 introduce a pressural wave. Nothing of the kind occurs in the electromagnetic theory, and accordingly we are relieved from one of the difficulties of the elastic solid theory. We shall now proceed to consider the reflection and refraction of light at the common surface of two isotropic media. Reflection and Refraction1. 415. Instead of beginning with the case of an isotropic medium, we shall suppose that the reflecting surface consists of a plate of Iceland spar, which is cut perpendicularly to its axis; so that we can pass to the case of an isotropic medium by putting a = c. The wave surface consists of the sphere x2 + y2 + z2 = c2, and the planetary ellipsoid x2/c2+(y2 + z2)/a2 = 1. 416. Let A, A', A, be the amplitudes of the incident, reflected and refracted waves; and let us first suppose that the incident light is polarized in the plane of incidence, so that the refracted ray is the ordinary ray. The condition that the electric forces parallel to the plane of incidence should be continuous, gives The condition that the corresponding components of magnetic induction should be continuous, gives (AA) V cos i = A ̧c cos r... ... ..(31). But if I, I', I, be the square roots of the intensities, it follows from § 411, that I, A,c' (I + I') sin i = I1 sin r, (I — I') cos i = I1 cos r ; 1 J. J. Thomson, Phil. Mag. Ap. 1880; Lorentz, Schlömilch Zeitschrift XXII.; Fitzgerald, Phil. Trans. 1880; Lord Rayleigh, Phil. Mag. Aug. 1881 which are the same as Fresnel's formulæ. Since the light is refracted according to the ordinary law, these formulæ are true in the case of two isotropic media. 417. In the next place, let the light be polarized perpendicularly to the plane of incidence, so that the refracted ray is an extraordinary ray. The conditions that the electric displacement perpendicular to the reflecting surface should be continuous, gives The condition that the electric forces parallel to this surface should be continuous, gives also and V2 (A — A') cos i = A ̧c2 cos r ............ (34). Now if V, be the velocity of the extraordinary wave where p is the perpendicular from the point of incidence, on to the tangent plane to the ellipsoid, at the extremity of the extraordinary ray. Equations (33) and (34) accordingly become The formulæ are the same as those furnished by MacCullagh's theory; sce § 253. |