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 $$B$$,

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

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;|undefined

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

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