substituting the values of a, b, c from (6) we obtain

395

(7),

These are the equations satisfied by the components of electric displacement.

475. We shall now confine our attention to isotropic media. In this case AB= C=U, where U-2kK; hence (6) becomes

Let f=S, g= Sμ, h = Sv, S = €21π/Vr. (lx+my+nz− Vt),

Accordingly if denote the component of the external magnetic force perpendicular to the wave-front, the equations of magnetic force become

Propagation of Light.

476. We are now prepared to consider the propagation of light in a magnetized medium.

Let us suppose that plane waves of light are incident upon the surface of separation of air and a magnetized medium. Let the axis of x be the normal, and be drawn into the first medium, and let the axis of z be perpendicular to the plane of incidence; also let the direction of magnetization be parallel to the axis of z. Then p, p, 0, and none of the quantities are functions of z; whence the equations of motion become

f=A'S, g=A"S, h= AS, S=2/VT.(lx+my − Vt) ̧

Substituting in (10), we obtain

2

A'mA, A"= lA.

Hence, if V1, V2 denote the two values of V corresponding to the upper and lower signs, we see that two waves are propagated with velocities V1, V.

It is important to notice, that the directions of the two refracted waves corresponding to an incident wave are in general different.

397

PROPAGATION OF LIGHT.

To see this, let the suffixes 1 and 2 refer to the two refracted waves, and let the incident wave be

h = €2π/VT. (lx+my − Vt),

then the displacements in one of the refracted waves will be

and the displacements in the other wave will be obtained by changing the suffix from 1 to 2, and changing the signs of f1, 91. Now, if r1, r, be the angles of refraction, m, = sin r1, m1 = sin r2 ; and, since the coefficient of y must be the same in all three waves, we must have

V

√ 1

=

=

sin i

sin ri

sin 72

which shows that r, is different from r

х

J3

Y

2

Let B., B, be the component displacements in the plane z = 0, then since l cos r1, it follows that

Similarly

The component displacements perpendicular to the wave-fronts are evidently zero; whence, in real quantities, the displacements in the two waves are

and, consequently, the two waves are circularly polarized in opposite directions.

477. The results of the last article will enable us to explain the rotation of the plane polarization, when light is propagated through a magnetic field parallel to the direction of the lines of magnetic force. In this case

whence putting k = 1, since the field is a transparent dielectric, we obtain from (11),

accordingly if the waves are travelling along the negative direction

We shall hereafter show, that the amplitudes are not quite equal to one another, but are of the form P+Q and P-Q respectively, where Q is a quantity which depends upon the magnetic force. Since the magnetic effect is very small in transparent dielectrics, we may as a first approximation neglect the difference between A, and A,, whence dropping the suffixes, the vibrations in question become

Whence if be the angle through which the plane of polarization is rotated, measured towards the right hand of an observer who is looking along the direction of propagation of the ray,

ROTATORY POLARIZATION.

399

Expanding and V1 in powers of p, and putting p = Ca, we

obtain

accordingly

2

Since the wave is travelling in the negative direction of the axis of x, it follows that, if T be the thickness of the medium traversed by the wave, x=-T; whence (12) becomes

which shows that the plane of polarization of the emergent light is rotated, and that the direction of rotation depends upon that of the magnetic force.

478. It appears from Faraday's experiments, that the direction of rotation is the same as that of the amperean current, which would produce the magnetic force. Now a is measured along the positive direction of the axis of x, whence the amperean current circulates from the right hand to the left hand of an observer who is looking along the direction of propagation; accordingly C must be negative for glass, whilst for a medium such as perchloride of iron must be positive.

From these results we draw the following conclusions.

(i) The magnitude of the rotation is directly proportional to the magnetic force, and also to the thickness of the medium traversed; and it is inversely proportional to the square of the period of the light. Hence the rotation is greater for violet light than for red light.

(ii) The direction of rotation is the same as that of the amperean current which would produce the magnetic force, for media for which Hall's constant is negative; and in the opposite direction for media for which Hall's constant is positive.

(iii) When the direction of propagation is perpendicular to that of the magnetic force, it follows from (8), that the magnetic terms are zero; hence the magnetic force produces no optical effect. These results are in accordance with experiment; subject to the limitation, that the effect of rotatory dispersion is only approximately expressed by the first statement.