Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/283

 The capacity of the middle plate is

Correction for the Thickness of the Plate.

Since the middle plate is generally of a thickness which cannot be neglected in comparison with the distance between the plates, we may obtain a better representation of the facts of the case by supposing the section of the intermediate plate to correspond with the curve $$\psi=\psi'$$.

The plate will be of nearly uniform thickness, $$\beta=2b\psi'$$, at a distance from the boundary, but will be rounded near the edge.

The position of the actual edge of the plate is found by putting $$y'=0$$, whence

The value of $$\phi$$ at this edge is 0, and at a point for which $$x'=a$$ it is

$\frac{a+b\log_{e}2}{b}$

Hence the quantity of electricity on the plate is the same as if a strip of breadth

had been added to the plate, the density being assumed to be every where the same as it is at a distance from the boundary.

Density near the Edge.

The surface-density at any point of the plate is

The quantity within brackets rapidly approaches unity as $$x'$$ increases, so that at a distance from the boundary equal to $$n$$ times the breadth of the strip $$\alpha$$, the actual density is greater than the normal density by about $$\tfrac{1}{2^{2n+1}}$$ of the normal density.

In like manner we may calculate the density on the infinite planes

When $$x'=0$$, the density is $$2^{-\tfrac{1}{2}}$$ of the normal density.