Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/238

The action of this superficial distribution on any point separated from $A$ by the surface is equal to that of a quantity of electricity $-e$, or

${\frac {4\pi aC}{AD\cdot Ad}}$

concentrated at $A$.

Its action on any point on the same side of the surface with $A$ is equal to that of a quantity of electricity

${\frac {4\pi Ca^{2}}{f\ AD\cdot Ad}}$

concentrated at $B$ the image of $A$.

The whole quantity of electricity on the sphere is equal to the first of these quantities if $A$ is within the sphere, and to the second if $A$ is without the sphere.

These propositions were established by Sir W. Thomson in his original geometrical investigations with reference to the distribution of electricity on spherical conductors, to which the student ought to refer.

159.] If a system in which the distribution of electricity is known is placed in the neighbourhood of a conducting sphere of radius $a$, which is maintained at potential zero by connexion with the earth, then the electrifications due to the several parts of the system will be superposed.

Let $A_{1},A_{2}$, &c. be the electrified points of the system, $f_{1},f_{2}$ &c. their distances from the centre of the sphere, $e_{1},e_{2}$, &c. their charges, then the images $B_{1},B_{2}$, &c. of these points will be in the same radii as the points themselves, and at distances ${\tfrac {a^{2}}{f_{1}}},\ {\tfrac {a^{2}}{f_{2}}}$ &c. from the centre of the sphere, and their charges will be

$-e{\frac {a}{f_{1}}},\ -e{\frac {a}{f_{2}}}\ etc.$

The potential on the outside of the sphere due to the superficial electrification will be the same as that which would be produced by the system of images $B_{1},B_{2}$, &c. This system is therefore called the electrical image of the system $A_{1},A_{2}$, &c.

If the sphere instead of being at potential zero is at potential $V$, we must superpose a distribution of electricity on its outer surface having the uniform surface-density

$\sigma ={\frac {V}{4\pi a}}$

The effect of this at all points outside the sphere will be equal to