Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/438

 

satisfy the equations, one positive and the other negative, the positive value being numerically greater than the negative.

818.] We may obtain the equations of motion from a consideration of the potential and kinetic energies of the medium. The potential energy, $V$, of the system depends on its configuration, that is, on the relative position of its parts. In so far as it depends on the disturbance due to circularly-polarized light, it must be a function of $r$, the amplitude, and $q$, the coefficient of torsion, only. It may be different for positive and negative values of $q$ of equal numerical value, and it probably is so in the case of media which of themselves rotate the plane of polarization.

The kinetic energy, $T$, of the system is a homogeneous function of the second degree of the velocities of the system, the coefficients of the different terms being functions of the coordinates.

819.] Let us consider the dynamical condition that the ray may be of constant intensity, that is, that $r$ may be constant.

Lagrange's equation for the force in $r$ becomes

${\frac {d}{dt}}{\frac {dT}{d{\dot {r}}}}-{\frac {dT}{dr}}+{\frac {dV}{dr}}=0$.

(5)

Since $r$ is constant, the first term vanishes. We have therefore the equation

$-{\frac {dT}{dr}}+{\frac {dV}{dr}}=0$,

(6)

in which $q$ is supposed to be given, and we are to determine the value of the angular velocity ${\dot {\theta }}$, which we may denote by its actual value, $n$.

The kinetic energy, $T$, contains one term involving $n^{2}$; other terms may contain products of $n$ with other velocities, and the rest of the terms are independent of $n$. The potential energy, $V$, is entirely independent of $n$. The equation is therefore of the form

$An^{2}+Bn+C=0$.

(7)

This being a quadratic equation, gives two values of $n$. It appears from experiment that both values are real, that one is positive and the other negative, and that the positive value is numerically the greater. Hence, if $A$ is positive, both $B$ and $C$ are negative, for, if $n_{1}$ and $n_{2}$ are the roots of the equation,

$A(n_{1}+n_{2})+B=0$.

(8)

The coefficient, $B$, therefore, is not zero, at least when magnetic force acts on the medium. We have therefore to consider the expression $Bn$, which is the part of the kinetic energy involving the first power of $n$, the angular velocity of the disturbance.