Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/194

To express the charge in terms of the difference of potentials, we have

$E_{1}={\frac {1}{4\pi }}{\frac {S}{c}}(B-A)=q(B-A)$

The coefficient ${\tfrac {1}{4\pi }}{\tfrac {S}{c}}=q$ represents the charge due to a difference of potentials equal to unity. This coefficient is called the Capacity of the surface $S$, due to its position relatively to the opposite surface.

Let us now suppose that the medium between the two surfaces is no longer air but some other dielectric substance whose specific inductive capacity is $K$, then the charge due to a given difference of potentials will be $K$ times as great as when the dielectric is air, or

$E_{1}={\frac {KS}{4\pi c}}(B-A)$

The total energy will be

${\begin{array}{ll}Q&={\frac {KS}{8\pi c}}(B-A)^{2},\\\\&={\frac {2\pi }{KS}}E_{1}^{2}c.\end{array}}$

The force between the surfaces will be

${\begin{array}{ll}F=pS&={\frac {KS}{8\pi }}{\frac {(B-A)^{2}}{c^{2}}},\\\\&={\frac {2\pi }{KS}}E_{1}^{2}.\end{array}}$

Hence the force between two surfaces kept at given potentials varies directly as $K$, the specific capacity of the dielectric, but the force between two surfaces charged with given quantities of electricity varies inversely as $K$.

Two Concentric Spherical Surfaces.

125.] Let two concentric spherical surfaces of radii $a$ and $b$, of which $b$ is the greater, be maintained at potentials $A$ and $B$ respectively, then it is manifest that the potential $V$ is a function of $r$ the distance from the centre. In this case, Laplace’s equation becomes

${\frac {d^{2}V}{dr^{2}}}+{\frac {2}{r}}{\frac {dV}{dr}}=0.$

The integral of this is

$V=C_{1}+C_{2}r^{-1};$

and the condition that $V=A$ when $r=a$, and $V=B$ when $r=b$, gives for the space between the spherical surfaces,