Page:ComstockInertia.djvu/17

Here ($w$) is the density of the total electromagnetic energy, $S_{x},S_{y},S_{z}$ are the components of the Poynting vector, $\Im _{x},\Im _{y},\Im _{z}$ are the components of the total electromagnetic force on unit charge, $(\rho$) is the density of electrification at the given point, and $v_{x},v_{y},v_{z}$ represent the velocity through space of this electrification. Thus

$W={\frac {1}{8\pi }}\{(X^{2}+Y^{2}+Z^{2})+(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\}$,

where $X,Y,Z$, and $\alpha ,\beta ,\gamma$, are the electric and magnetic force intensities respectively, and

$\Im _{x}=X+(v_{y}\gamma -v_{z}\beta )$,

$\Im _{y}=Y+(v_{z}\alpha -v_{x}\gamma )$,

$\Im _{z}=Z+(v_{x}\beta -v_{y}\alpha )$.

Equation (23) states merely that the rate of increase of energy in an elementary volume is equal to the activity of any foreign (i. e., non-electrical) forces which may act therein minus the outward flow of energy.

Now suppose we consider an electromagnetic system bounded by a rigid surface ($AB$), which moves uniformly through space with the velocity ($v_{1}$) along the axis of ($x$); and further suppose that the volume inside this closed surface is divided into two parts by the plane partition ($CD$) which is perpendicular to the x-axis and which, although fixed in the moving system, coincides at a given instant with the plane ($C'D'$) fixed in space. If this system be considered as isolated, then no disturbance passes through the bounding surface ($AB$).

In equation (23) the time derivative of the energy-density