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



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

the integral of which is

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

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

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

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