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



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

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

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