Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/426

 

797.] The velocity of the propagation of the wave is therefore completely determined from the equation $U=0$, or

${\frac {l^{2}}{V^{2}-a^{2}}}+{\frac {m^{2}}{V^{2}-b^{2}}}+{\frac {n^{2}}{V^{2}-c^{2}}}=0$,

(8)

There are therefore two, and only two, values of $V^{2}$ Corresponding to a given direction of wave-front.

If $\lambda$, $\mu$, $\nu$ are the direction-cosines of the electric current whose components are $u$, $v$, $w$

$\lambda :\mu :\nu ::{\frac {1}{a^{2}}}F^{\prime \prime }:{\frac {1}{b^{2}}}G^{\prime \prime }:{\frac {1}{c^{2}}}H^{\prime \prime }$,

(9)

then

$l\lambda +m\mu +n\nu =0$;

(10)

or the current is in the plane of the wave-front, and its direction in the wave-front is determined by the equation

${\frac {l}{\lambda }}(b^{2}-c^{2})+{\frac {m}{\mu }}(c^{2}-a^{2})+{\frac {n}{\nu }}(a^{2}-b^{2})=0$.

(11)

These equations are identical with those given by Fresnel if we define the plane of polarization as a plane through the ray perpendicular to the plane of the electric disturbance.

According to this electromagnetic theory of double refraction the wave of normal disturbance, which constitutes one of the chief difficulties of the ordinary theory, does not exist, and no new assumption is required in order to account for the fact that a ray polarized in a principal plane of the crystal is refracted in the ordinary manner^[1].

 

Relation between Electric Conductivity and Opacity.

798.] If the medium, instead of being a perfect insulator, is a conductor whose conductivity per unit of volume is $C$, the disturbance will consist not only of electric displacements but of currents of conduction, in which electric energy is transformed into heat, so that the undulation is absorbed by the medium.

If the disturbance is expressed by a circular function, we may write

$F=e^{-pz}\cos(nt-qz)$,

(1)

for this will satisfy the equation

${\frac {d^{2}F}{dz^{2}}}=\mu K{\frac {d^{2}F}{dt^{2}}}+4\pi \mu C{\frac {dF}{dt}}$,

(2)

$q^{2}-p^{2}=\mu Kn^{2}$,

(3)

$2pq=4\pi \mu Cn$.

(4)

1. See Stokes' 'Report on Double Refraction'; Brit. Assoc. Reports, 1862, p. 255.