Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/262

The charge of each image is measured by its distance from the origin and is always positive.

The total charge of the sphere $A$ is therefore

$E_{a}=\sum _{s=0}^{s=\infty }{\frac {1}{{\frac {1}{a}}+s\left({\frac {1}{a}}+{\frac {1}{b}}\right)}}-{\frac {ab}{a+b}}\sum _{s=1}^{s=\infty }{\frac {1}{s}}.$

Each of these series is infinite, but if we combine them in the form

$E_{a}=\sum _{s=1}^{s=\infty }{\frac {a^{2}b}{s(a+b)(s(a+b)-a)}}$

the series becomes converging.

In the same way we find for the charge of the sphere $B$,

${\begin{array}{ll}E_{b}&=\sum _{s=1}^{s=\infty }{\frac {ab}{s(a+b)-b}}-{\frac {ab}{a+b}}\sum _{s=-1}^{s=-\infty }{\frac {1}{s}},\\\\&=\sum _{s=1}^{s=\infty }{\frac {ab^{2}}{s(a+b)\{s(a+b)-b\}}}.\end{array}}$

The values of $E_{a}$ and $E_{b}$ are not, so far as I know, expressible in terms of known functions. Their difference, however, is easily expressed, for

${\begin{array}{ll}E_{a}-E_{b}&=\sum _{s=-\infty }^{s=\infty }{\frac {ab}{b+s(a+b)}},\\\\&={\frac {\pi ab}{a+b}}\cot {\frac {\pi b}{a+b}}.\end{array}}$

When the spheres are equal the charge of each for potential unity is

${\begin{array}{ll}E_{a}&=a\sum _{s=1}^{s=\infty }{\frac {1}{2s(2s-1)}}\\\\&=a\left(1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+etc.\right),\\\\&=a\log _{e}2=1.0986a.\end{array}}$

When the sphere $A$ is very small compared with the sphere $B$ the charge on $A$ is

$E_{a}={\frac {a^{2}}{b}}\sum _{s=1}^{s=\infty }{\frac {1}{s^{2}}}$ approximately;

or $E_{a}={\frac {\pi ^{2}}{6}}{\frac {a^{2}}{b}}.$

The charge on $B$ is nearly the same as if $A$ were removed, or

$E_{b}=b.$

The mean density on each sphere is found by dividing the charge by the surface. In this way we get