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

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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 from each other, maintained respectively at potentials and .

It is manifest that in this case the potential will be a function of the distance from the plane , and will be the same for all points of any parallel plane between and , 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 , , and when , ,

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, . Hence the distribution of electricity on the first plane has a surface-density , where

On the other surface, where the potential is , the surface- density will be equal and opposite to , and