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