Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/240

being an attraction when the bodies are nearer and a repulsion when they are farther off.

When the electrified point is within the spherical surface the force on the electrified point is always away from the centre of the sphere, and is equal to

${\frac {e^{2}af}{\left(a^{2}-f^{2}\right)^{2}}}.$

The surface-density at the point of the sphere nearest to the electrified point where it lies outside the sphere is

${\begin{array}{ll}\sigma _{1}&={\frac {1}{4\pi a^{2}}}\left\{Va-e{\frac {a(f+a)}{(f-a)^{2}}}\right\}\\\\&={\frac {1}{4\pi a^{2}}}\left\{E-e{\frac {a^{2}(3f-a)}{f(f-a)^{2}}}\right\}.\end{array}}$

(12)

The surface-density at the point of the sphere farthest from the electrified point is

${\begin{array}{ll}\sigma _{2}&={\frac {1}{4\pi a^{2}}}\left\{Va-e{\frac {a(f-a)}{(f+a)^{2}}}\right\}\\\\&={\frac {1}{4\pi a^{2}}}\left\{E+e{\frac {a^{2}(3f+a)}{f(f+a)^{2}}}\right\}.\end{array}}$

(13)

When $E$, the charge of the sphere, lies between

$e{\frac {a^{2}(3f-a)}{f(f-a)^{2}}}$ and $-e{\frac {a^{2}(3f+a)}{f(f+a)^{2}}}$

the electrification will be negative next the electrified point and positive on the opposite side. There will be a circular line of division between the positively and the negatively electrified parts of the surface, and this line will be a line of equilibrium.

If

$E=ea\left({\frac {1}{\sqrt {f^{2}-a^{2}}}}-{\frac {1}{f}}\right),$

(14)

the equipotential surface which cuts the sphere in the line of equilibrium is a sphere whose centre is the electrified point and whose radius is ${\sqrt {f^{2}-a^{2}}}$.

The lines of force and equipotential surfaces belonging to a case of this kind are given in Figure IV at the end of this volume.

Images in an Infinite Plane Conducting Surface.

161.] If the two electrified points $A$ and $B$ in Art. 156 are electrified with equal charges of electricity of opposite signs, the surface of zero potential will be the plane, every point of which is equidistant from $A$ and $B$.