Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/119

by $4\pi$, the displacement through a closed surface, reckoned outwards, is equal to the electricity within the surface.

Corollary. It also follows that if the surface is not closed but is bounded by a given closed curve, the total induction through it is $\omega$, where $\omega$ is the solid angle subtended by the closed curve at $0$. This quantity, therefore, depends only on the closed curve, and not on the form of the surface of which it is the boundary.

On the Equations of Laplace and Poisson.

77.] Since the value of the total induction of a single centre of force through a closed surface depends only on whether the centre is within the surface or not, and does not depend on its position in any other way, if there are a number of such centres $e_{l}$, $e_{2}$ , &c. within the surface, and $e_{1}'$, $e_{2}'$, &c. without the surface, we shall have

$\iint R\,\cos \epsilon \,dS=\,4\pi e$;

where $e$ denotes the algebraical sum of the quantities of electricity at all the centres of force within the closed surface, that is, the total electricity within the surface, resinous electricity being reckoned negative.

If the electricity is so distributed within the surface that the density is nowhere infinite, we shall have by Art. 64,

$4\pi \,e=4\pi \iiint \rho \,dx\,dy\,dz$

and by Art. 75,

$\iint R\,\cos \epsilon \,dS=\iiint \left({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}\right)\,dx\,dy\,dz$.

If we take as the closed surface that of the element of volume $dx$ $dy$ $dz$, we shall have, by equating these expressions,

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=4\pi \,\rho$;

and if a potential $V$ exists, we find by Art. 71 ,

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}+4\pi \,\rho =0$

This equation, in the case in which the density is zero, is called Laplace's Equation. In its more general form it was first given by Poisson. It enables us, when we know the potential at every point, to determine the distribution of electricity.