Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/239

that of a quantity $Va$ of electricity placed at its centre, and at all points inside the sphere the potential will be simply increased by $V$.

The whole charge on the sphere due to an external system of influencing points $A_{1},A_{2}$, &c. is

$E=Va-e_{1}{\frac {a}{f_{1}}}-e_{2}{\frac {a}{f_{2}}}-etc.,$

(9)

from which either the charge $E$ or the potential $V$ may be calculated when the other is given.

When the electrified system is within the spherical surface the induced charge on the surface is equal and of opposite sign to the inducing charge, as we have before proved it to be for every closed surface, with respect to points within it.

160.] The energy due to the mutual action between an electrified point $e$, at a distance $f$ from the centre of the sphere greater than $a$ the radius, and the electrification of the spherical surface due to the influence of the electrified point and the charge of the sphere, is

$M=e\left({\frac {Va}{f}}-{\frac {ea}{f^{2}-a^{2}}}\right)={\frac {e}{f}}\left(E-{\frac {e^{2}a^{3}}{f\left(f^{2}-a^{2}\right)}}\right),$

(10)

where $V$ is the potential, and $E$ the charge of the sphere.

The repulsion between the electrified point and the sphere is therefore, by Art. 92,

${\begin{array}{ll}F&=ea\left({\frac {V}{f^{2}}}-{\frac {ef}{\left(f^{2}-a^{2}\right)^{2}}}\right)\\\\&={\frac {e}{f^{2}}}\left(E-e{\frac {a^{3}\left(2f^{2}-a^{2}\right)}{f\left(f^{2}-a^{2}\right)^{2}}}\right).\end{array}}$

(11)

Hence the force between the point and the sphere is always an attraction in the following cases–

(1) When the sphere is uninsulated.

(2) When the sphere has no charge.

(3) When the electrified point is very near the surface.

In order that the force may be repulsive, the potential of the sphere must be positive and greater than $e{\tfrac {f^{3}}{\left(f^{2}-a^{2}\right)^{2}}}$, and the charge of the sphere must be of the same sign as $e$ and greater than $e{\tfrac {a^{3}\left(2f^{2}-a^{2}\right)}{f\left(f^{2}-a^{2}\right)^{2}}}$.

At the point of equilibrium the equilibrium is unstable, the force