Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/287

Since

${\begin{array}{ll}{\frac {d\phi }{dy'}}={\frac {d\psi }{dx'}},\\\\\int _{-\infty }^{+\infty }\rho \ dx'&=\int _{-\infty }^{+\infty }{\frac {1}{4\pi }}{\frac {1}{(R+y')}}{\frac {d\psi }{dx}}dx,\\\\&={\frac {1}{8}}{\frac {1}{R+y'}}\left({\frac {y'}{B}}-1\right).\end{array}}$

(35)

Integrating with respect to $y'$, we find

$\int _{0}^{B}\int _{-\infty }^{+\infty }\rho \ dx'dy'={\frac {1}{8}}-{\frac {1}{8}}{\frac {R+B}{B}}\log {\frac {R+B}{R}},$

(36)

$={\frac {1}{16}}{\frac {B}{R}}+{\frac {1}{48}}{\frac {B^{2}}{R^{2}}}-\mathrm {etc.}$

(37)

This is the total quantity of electricity which we must suppose distributed in space near the positive side of one of the cylindric plates per unit of circumference. Since it is only close to the edge of the plate that the density is sensible, we may suppose it all condensed on the surface of the plate without altering sensibly its action on the opposed plane surface, and in calculating the attraction between that surface and the cylindric surface we may suppose this electricity to belong to the cylindric surface.

The superficial charge on the positive surface of the plate per unit of length would have been $-{\frac {1}{8}}$, if there had been no curvature. Hence this charge must be multiplied by the factor $\left(1+{\frac {1}{2}}{\frac {B}{R}}\right)$ to get the total charge on the positive side.

In the case of a disk of radius $R$ placed midway between two infinite parallel plates at a distance $B$, we find for the capacity of the disk

${\frac {R^{2}}{B}}+2{\frac {\log _{e}2}{\pi }}R+{\frac {1}{2}}B$

(38)

Theory of Thomson's Guard-ring.

201.] In some of Sir W. Thomson's electrometers, a large plane surface is kept at one potential, and at a distance $a$ from this surface is placed a plane disk of radius $R$ surrounded by a large plane plate called a Guard-ring with a circular aperture of radius $R'$ concentric with the disk. This disk and plate are kept at potential zero.

The interval between the disk and the guard-plate may be regarded as a circular groove of infinite depth, and of breadth $R'-R$, which we denote by $B$.