Page:ComstockInertia.djvu/5

For the total density of energy-flow ($f_{x},f_{y},f_{z}$) we must of course add to the above the components of the Poynting vector. Writing as usual $X,Y,Z$ and $\alpha ,\beta ,\gamma$ for the electric and magnetic force intensities and calling ($V$) the velocity of light, we have

${\begin{cases}f{}_{x}={\frac {V}{4\pi }}(\gamma Y-\beta Z)-(v_{x}X_{x}+v_{y}Y_{x}+v_{z}Z_{x}),\\f{}_{y}={\frac {V}{4\pi }}(\alpha Z-\gamma X)-(v_{x}X_{y}+v_{y}Y_{y}+v_{z}Z_{y}),\\f{}_{z}={\frac {V}{4\pi }}(\beta X-\alpha Y)-(v_{x}X_{z}+v_{y}Y_{z}+v_{z}Z_{z}),\end{cases}}$

(2)

These equations give the density of the total energy-flow through any purely electrical system, in which the ordinary electrical laws hold universally.

6. Consider an isolated electrical system moving as a whole through space with the constant velocity ($v_{1}$). A constant velocity will be possible if the system retains on the average the same internal structure. The total average rate of transfer of energy corresponding to the movement of such a system is evidently ($v_{1}\cdot W$), where $W$ is the total contained energy. Another expression for the same thing is to be obtained by integrating throughout the system the components along ($v_{1})$ of $(f_{x},f_{y},f_{z}$) given in equations (2). In order that the velocity ($v_{1}$) may appear explicitly, however, it is necessary that the velocity ($v$), which was used in equations (2), be written as the sum of ($v_{1}$) and another velocity ($v_{2}$). Then ($v_{2}$) is the velocity with respect to axes moving with the system.

If $l,m,n$ are the direction cosines of the constant velocity ($v_{1}$), we have for the total energy-flow ($F$) in the direction of ($v_{1}$),

$v_{1}=F=\int (lf_{x}+mf_{y}+nf_{z})d\tau \,$ $={\frac {lV}{4\pi }}\int (\gamma Y-\beta Z)d\tau -lv_{1}\int (lX_{x}+mY_{x}+nZ_{x})d\tau -l\int (v_{2x}X_{x}+v_{2y}Y_{x}+v_{2z}Z_{x})d\tau$ $+{\frac {mV}{4\pi }}\int (\alpha Z-\gamma X)d\tau -mv_{1}\int (lX_{y}+mY_{y}+nZ_{y})d\tau -m\int (v_{2x}X_{y}+v_{2y}Y_{y}+v_{2z}Z_{y})d\tau$ $+{\frac {nV}{4\pi }}\int (\beta X-\alpha Y)d\tau -nv_{1}\int (lX_{z}+mY_{z}+nZ_{z})d\tau -n\int (v_{2x}X_{z}+v_{2y}Y_{z}+v_{2z}Z_{z})d\tau$

(3)