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

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

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

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

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

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

Inverting this system the charges become

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

and

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

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

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