Page:ComstockInertia.djvu/8

This page has been validated.

8

Dr. D. F. Comstock on the

Since

[E, c u r l E]_{x} = - {[E, \frac{1}{V} \frac{\partial H}{\partial t}, H]}_{x}

,

the last two terms in the bracket of (9) become together

- \frac{1}{V^{2}} \frac{\partial}{\partial t} [E H]_{x}

,

which is minus the time rate of the density of momentum at the point. The time rate, however, refers to a point fixed in space, and to change to a point moving with the system we make use of the usual expression and write

- \frac{\partial}{\partial t} m_{x} = - \frac{\partial^{'}}{\partial t} m_{x} + v_{1} (l \frac{\partial m_{x}}{\partial x} + m \frac{\partial m_{x}}{\partial y} + n \frac{\partial m_{x}}{\partial z})

,

(10)

where $m_{x}$ is the $x$ -component of the density of momentum and ( $l, m, n$ ) are, as formerly, the direction cosines of the constant velocity ( $v_{1}$ ). The operator $\frac{\partial^{'}}{\partial t}$ now refers to the rate of change at a point moving with the system.

Substituting (10) for the two last terms in (9) and noticing that

\frac{\partial m_{x}}{\partial t}

may be written

\frac{\partial}{\partial x} \frac{\partial^{'}}{\partial t} \int_{R}^{P} m_{x} d x

,

where the integration is to be taken from $R$ (meaning merely from a point outside the system where $m_{x}$ is zero) up to the point $P$ in question, account being taken of any discontinuity at the bounding surface, we have in place of (9)

\frac{\partial X_{x}}{\partial x} + \frac{\partial X_{y}}{\partial y} + \frac{\partial X_{z}}{\partial z} = - ρ ℑ_{x}

$= \frac{\partial}{\partial x} {- \frac{1}{8 π} (E_{x}^{2} - E_{y}^{2} - E_{z}^{2}) - \frac{1}{8 π} (H_{x}^{2} - H_{y}^{2} - H_{z}^{2}) - v_{1} l m_{x} + \frac{\partial^{'}}{\partial t} \int_{R}^{P} m_{x} d x}$

$+ \frac{\partial}{\partial y} {- \frac{1}{4 π} (E_{x} E_{y} + H_{x} H_{y}) - v_{1} m m_{x} + \frac{\partial^{'}}{\partial t} \int_{R}^{P} m_{x} d y}$

+ \frac{\partial}{\partial z} {- \frac{1}{4 π} (E_{x} E_{z} + H_{x} H_{z}) - v_{1} n m_{x} + \frac{\partial^{'}}{\partial t} \int_{R}^{P} m_{x} d z}

(11)

This equation gives us what we were seeking, namely, the values of $X_{x}, X_{y}$ , and $X_{z}$ in terms of the electric and magnetic forces and the density of the momentum.