Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/159

CHAPTER V.

MECHANICAL ACTION BETWEEN ELECTRIFIED BODIES.

103.] Let $V=C$ be any closed equipotential surface, $C$ being a particular value of a function $V$, the form of which we suppose known at every point of space. Let the value of $V$ on the outside of this surface be $V_{1}$, and on the inside $V_{2}$. Then, by Poisson’s equation

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

(1)

we can determine the density $\rho _{1}$ at every point on the outside, and the density $\rho _{2}$ at every point on the inside of the surface. We shall call the whole electrified system thus explored on the outside $E_{1}$, and that on the inside $E_{2}$. The actual value of $V$ arises from the combined action of both these systems.

Let $R$ be the total resultant force at any point arising from the action of $E_{1}$ and $E_{2}$, $R$ is everywhere normal to the equipotential surface passing through the point.

Now let us suppose that on the equipotential surface $V=C$ electricity is distributed so that at any point of the surface at which the resultant force due to $E_{1}$ and $E_{2}$ reckoned outwards is $R$, the surface-density is $\rho$, with the condition

$R=4\pi \sigma \ ;$

(2)

and let us call this superficial distribution the electrified surface $S$, then we can prove the following theorem relating to the action of this electrified surface.

If any equipotential surface belonging to a given electrified system be coated with electricity, so that at each point the surface-density $\sigma ={\frac {R}{4\pi }}$, where $R$ is the resultant force, due to the original electrical system, acting outwards from that point of the surface, then the potential due to the electrified surface at any point on