Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/251

If, in the inverted system, the potential of the surface is unity, then the density at the point $P'$ is

$\sigma '={\frac {1}{4\pi \alpha '}}\left(1-\left({\frac {\beta }{p'}}\right)^{3}\right).$

If, in the original system, the density at $P$ is $\sigma$, then

${\frac {\sigma }{\sigma '}}={\frac {1}{r^{3}}},$

and the potential is ${\tfrac {1}{r}}$. By placing $O$ at a negative charge of electricity equal to unity, the potential will become zero over the surface, and the density at $P$ will be

$\sigma ={\frac {1}{4\pi }}{\frac {a^{2}-\alpha ^{2}}{\alpha r^{3}}}\left(1-{\frac {\beta ^{3}r^{3}}{\left(\beta ^{2}r^{2}+\left(b^{2}-\beta ^{2}\right)\left(p^{2}-\beta ^{2}\right)\right)^{\frac {3}{2}}}}\right)$

This gives the distribution of electricity on one of the spherical surfaces due to a charge placed at $O$. The distribution on the other spherical surface may be found by exchanging $a$ and $b$, $\alpha$ and $\beta$, and putting $q$ or $AQ$ instead of $p$.

To find the total charge induced on the conductor by the electrified point at $O$, let us examine the inverted system.

In the inverted system we have a charge $\alpha '$ at $A'$, and $\beta '$ at $B'$, and a negative charge ${\frac {\alpha '\beta '}{\sqrt {\alpha '^{2}+\beta '^{2}}}}$ at a point $C'$ in the line $A'B'$ such that

$AC:CB::\alpha '^{2}:\beta '^{2}$

If $AO'=a',\ OB'=b',\ OC'=c',$ we find

$c'^{2}={\frac {a'^{2}\beta '^{2}+b'^{2}\alpha '^{2}-\alpha '^{2}\beta '^{2}}{\alpha '^{2}+\beta '^{2}}}$

Inverting this system the charges become

${\frac {\alpha '}{a'}}={\frac {\alpha }{a}},\ {\frac {\beta '}{b'}}={\frac {\beta }{b}};$

and

$-{\frac {\alpha '\beta '}{\sqrt {\alpha {}^{2}+\beta {}^{2}}}}{\frac {1}{c'}}=-{\frac {\alpha \beta }{\sqrt {a{}^{2}\beta {}^{2}+b{}^{2}\alpha {}^{2}-\alpha {}^{2}\beta {}^{2}}}}.$

Hence the whole charge on the conductor due to a unit of negative electricity at $O$ is

${\frac {\alpha }{a}}+{\frac {\beta }{b}}-{\frac {\alpha \beta }{\sqrt {a{}^{2}\beta {}^{2}+b{}^{2}\alpha {}^{2}-\alpha {}^{2}\beta {}^{2}}}}.$