Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/192

CHAPTER VIII.

SIMPLE CASES OF ELECTRIFICATION.

Two Parallel Planes.

124.] We shall consider, in the first place, two parallel plane conducting surfaces of infinite extent, at a distance $c$ from each other, maintained respectively at potentials $A$ and $B$.

It is manifest that in this case the potential $V$ will be a function of the distance $z$ from the plane $A$, and will be the same for all points of any parallel plane between $A$ and $B$, except near the boundaries of the electrified surfaces, which by the supposition are at an infinitely great distance from the point considered.

Hence, Laplace’s equation becomes reduced to

${\frac {d^{2}V}{dz^{2}}}=0$

the integral of which is

$V=C_{1}+C_{2}z$;

and since when $z=0$, $V=A$, and when $z=c$, $V=B$,

$V=A+(B-A){\frac {z}{c}}$

For all points between the planes, the resultant electrical force is normal to the planes, and its magnitude is

$R={\frac {A-B}{c}}$.

In the substance of the conductors themselves, $R=0$. Hence the distribution of electricity on the first plane has a surface-density $\sigma$, where

$4\pi \sigma =R={\frac {A-B}{c}}$

On the other surface, where the potential is $B$, the surface- density $\sigma '$ will be equal and opposite to $\sigma$, and

$4\pi \sigma '=-R={\frac {B-A}{c}}$