Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/242

Fig. 9

If $A$ and $B$ are two points, $A'$ and $B'$ their images, $O$ being the centre of inversion, and $R$ the radius of the sphere of inversion,

$OA\cdot OA'=R^{2}=OB\cdot OB'$

Hence the triangles $OAB$, $OB'A'$ are similar, and

$AB:A'B'::OA:OB'::OA\cdot OB:R^{2}.$

If a quantity of electricity $e$ be placed at $A$, its potential at $B$ will be

$V={\frac {e}{AB}}$

If $e'$ be placed at $A'$ its potential at $B'$ will be

$V'={\frac {e'}{A'B'}}.$

In the theory of electrical images

$e:e'::OA:R::R:OA'.$

Hence

$V:V'::R:OB,\,$

(17),

or the potential at $B$ due to the electricity at $A$ is to the potential at the image of $B$ due to the electrical image of $A$ as $R$ is to $OB$.

Since this ratio depends only on $OB$ and not on $OA$, the potential at $B$ due to any system of electrified bodies is to that at $B'$ due to the image of the system as $R$ is to $OB$.

If $r$ be the distance of any point $A$ from the centre, and $r'$ that of its image $A'$, and if $e$ be the electrification of $A$, and $e'$ that of $A'$, also if $L,S,K$ be linear, superficial, and solid elements at $A$, and $L',S',K'$ their images at $A'$, and $\lambda ,\sigma ,\rho ,\lambda ',\sigma ',\rho ',$ the corresponding line-surface and volume-densities of electricity at the two points, $V$ the potential at $A$ due to the original system, and $V'$ the potential at $A'$ due to the inverse system, then

$\left.{\begin{array}{c}{\frac {r'}{r}}={\frac {L'}{L}}={\frac {R^{2}}{r^{2}}}={\frac {r'^{2}}{r^{2}}},\ {\frac {S'}{S}}={\frac {R^{4}}{r^{4}}}={\frac {r'^{4}}{R^{4}}},\ {\frac {K'}{K}}={\frac {R^{6}}{r^{6}}}={\frac {r'^{6}}{R^{6}}};\\\\{\frac {e'}{e}}={\frac {R}{r}}={\frac {r'}{R}},\ {\frac {\lambda '}{\lambda }}={\frac {r}{R}}={\frac {R}{r'}},\\\\{\frac {\sigma '}{\sigma }}={\frac {r^{3}}{R^{3}}}={\frac {R^{3}}{r'^{3}}},\ {\frac {\rho '}{\rho }}={\frac {r^{5}}{R^{5}}}={\frac {R^{5}}{r'^{5}}}\\\\{\frac {V'}{V}}={\frac {r}{R}}={\frac {R}{r'}}.\end{array}}\right\}$

(18)[1]

If in the original system a certain surface is that of a conductor,

1. See Thomson and Tait's Natural Philosophy, § 515.