Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/386

through the walls of an element of volume, is applicable to electric currents, and is perhaps preferable in point of form to that which we have given, but as it may be found in any treatise on Hydrodynamics we need not repeat it here.

Quantity of Electricity which passes through a given Surface.

296.] Let $\Gamma$ be the resultant current at any point of the surface. Let $dS$ be an element of the surface, and let $\epsilon$ be the angle between $\Gamma$ and the normal to the surface, then the total current through the surface will be

$\iint \Gamma \cos \epsilon \;ds,$

the integration being extended over the surface.

As in Art. 21, we may transform this integral into the form

$\iint \Gamma \cos \epsilon \;ds=\iiint \left({\frac {du}{dz}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}\right)dx\;dy\;dz$

(19)

in the case of any closed surface, the limits of the triple integration being those included by the surface. This is the expression for the total efflux from the closed surface. Since in all cases of steady currents this must be zero whatever the limits of the integration, the quantity under the integral sign must vanish, and we obtain in this way the equation of continuity (17).