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

The capacity of the middle plate is

${\frac {1}{2\pi }}{\frac {S'}{B}}={\frac {1}{2\pi }}\left\{{\frac {S}{B}}+L{\frac {1}{\pi }}\log _{e}2\right\}$

(16)

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

$x'=b\log \cos \psi '$

(17)

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

$\alpha '={\frac {B}{\pi }}\log _{e}\left(2\cos {\frac {\pi \beta }{2B}}\right)$

(18)

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

${\begin{array}{ll}{\frac {1}{4\pi }}{\frac {d\phi }{dx'}}&={\frac {1}{4\pi b}}{\frac {e^{\frac {x'}{b}}}{\sqrt {e^{\frac {2x'}{b}}-1}}}\\\\&={\frac {1}{4\pi b}}\left(1+{\frac {1}{2}}e^{-{\frac {2x'}{b}}}+{\frac {3}{8}}e^{-{\frac {4x'}{b}}}-\mathrm {etc} .\right)\end{array}}$

(19)

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

$={\frac {1}{4\pi b}}{\frac {e^{\frac {x}{b}}}{\sqrt {e^{\frac {2x'}{b}}+1}}}$

(20)

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