Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/123

In the present case we have for any point on the normal $v$

${\frac {dV}{dx}}={\frac {dV_{1}}{dx}}+v{\frac {d^{2}V_{1}}{dx^{2}}}+\And \!\!\!{\text{c.}}$;

(28)

also, if the element of surface is $dS$, that of the volume of the element of the stratum may be written $dS\,dv$; and if $X$ is the whole force on a stratum of thickness $v$,

$X=-\iiint \left({\frac {dV_{1}}{dx}}+v{\frac {d^{2}V_{1}}{dx^{2}}}+etc.\right)\rho '\,dS\,dv$.

(29)

Integrating with respect to $v$, we find

$x=-\iint \rho '\,dS\left(v{\frac {dV_{1}}{dx}}+{\frac {v^{2}}{2}}{\frac {d^{2}V_{1}}{dx^{2}}}+\And \!\!\!{\text{c.}}\right)$

(30)

or, since

${\frac {dV_{2}}{dx}}={\frac {dV_{1}}{dx}}+v{\frac {d^{2}V_{1}}{dx^{2}}}+\And \!\!\!{\text{c.}}$;

(31)

$X=-\iint {1 \over 2}\,\rho '\,v\,dS\left({\frac {dV_{1}}{dx}}+{\frac {dV_{2}}{dx}}\right)+\And \!\!\!{\text{c.}}$;

(32)

When $v$ is diminished and $\rho '$ increased without limit, the product $\rho 'v$ remaining always constant and equal to $\sigma$, the expression for the force in the direction of $x$ on the electricity $\sigma \,dS$ on the element of surface $dS$ is

$X=-\sigma \,dS\,{1 \over 2}\left({\frac {dV_{1}}{dx}}+{\frac {dV_{2}}{dx}}\right)$;

(33)

that is, the force acting on the electrified element $\sigma \,dS$ in any given direction is the arithmetic mean of the forces acting on equal quantities of electricity placed one just inside the surface and the other just outside the surface close to the actual position of the element, and therefore the resultant mechanical force on the electrified element is equal to the resultant of the forces which would act on two portions of electricity, each equal to half that on the element, and placed one on each side of the surface and infinitely near to it.

80.] When a conductor is in electrical equilibrium, the whole of the electricity is on the surface.

We have already shewn that throughout the substance of the conductor the potential $V$ is constant. Hence $\nabla ^{2}V$ is zero, and therefore by Poisson's equation, $\rho$ is zero throughout the substance of the conductor, and there can be no electricity in the interior of the conductor.

Hence a superficial distribution of electricity is the only possible one in the case of conductors in equilibrium. A distribution throughout the mass can only exist in equilibrium when the body is a non-conductor.