Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/116

76 Let the force at a distance $$r$$ from a point at which a quantity $$e$$ of electricity is concentrated be $$R$$, where $$R$$ is some function of $$r$$. All central forces which are functions of the distance admit of a potential, let us write $$\tfrac {f(r)}{r}$$ for the potential function due to a unit of electricity at a distance $$r$$.

Let the radius of the spherical shell be $$a$$, and let the surface-density be $$\sigma$$. Let $$P$$ be any point within the shell at a distance $$p$$ from the centre. Take the radius through $$P$$ as the axis of spherical coordinates, and let $$r$$ be the distance from $$P$$ to an element $$dS$$ of the shell. Then the potential at $$P$$ is

and V must be constant for all values of $$p$$ less than $$a$$.

Multiplying both sides by $$p$$ and differentiating with respect to $$p$$,

Differentiating again with respect to $$p$$, , Since a and p are independent,

and the potential function is

The force at distance $$r$$ is got by differentiating this expression with respect to $$r$$, and changing the sign, so that

or the force is inversely as the square of the distance, and this therefore is the only law of force which satisfies the condition that the potential within a uniform spherical shell is constant. Now