Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/179

At a point of equilibrium $A_{1}$ is zero. If the first term that does not vanish is that in $r^{i}$, then

$V-V_{0}=A_{i}r^{i}\cos(i\theta +\alpha _{i})+$ terms in higher powers of r.

This gives $i$ sheets of the equipotential surface $V=V_{0}$, intersecting at angles each equal to ${\tfrac {\pi }{i}}$. This theorem was given by Rankine^[1].

It is only under certain conditions that a line of equilibrium can exist in free space, but there must be a line of equilibrium on the surface of a conductor whenever the electrification of the conductor is positive in one portion and negative in another.

In order that a conductor may be oppositely electrified in different portions of its surface, there must be in the field some places where the potential is higher than that of the body and others where it is lower. We must remember that at an infinite distance the potential is zero.

Let us begin with two conductors electrified positively to the same potential. There will be a point of equilibrium between the two bodies. Let the potential of the first body be gradually raised. The point of equilibrium will approach the other body, and as the process goes on it will coincide with a point on its surface. If the potential of the first body be now increased, the equipotential surface round the first body which has the same potential as the second body will cut the surface of the second body at right angles in a closed curve, which is a line of equilibrium.

Earnshaw’s Theorem.

116.] An electrified body placed in a field of electric force cannot be in stable equilibrium.

First, let us suppose the electricity of the moveable body ($A$), and also that of the system of surrounding bodies ($B$), to be fixed in those bodies.

Let $V$ be the potential at any point of the moveable body due to the action of the surrounding bodies ($B$), and let $e$ be the electricity on a small portion of the moveable body $A$ surrounding this point. Then the potential energy of $A$ with respect to $B$ will be

$M=\sum (Ve)$

where the summation is to be extended to every electrified portion of $A$.

1. ‚Summary of the Properties of certain Stream Lines‘, Phil. Mag., Oct. 1864. See also, Thomson and Tait’s Natural Philosophy, § 780 ; and Rankine and Stokes, in the Proc. R. S., 1867, p. 468 ; also W. R. Smith, Proc. R. S. Edin., 1869-70, p. 79.