Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/218

Within the sphere the equipotential surfaces are equidistant planes, and the lines of force are straight lines parallel to the axis, their distances from the axis being as the square roots of the natural numbers. The lines outside the sphere may be taken as a representation of those which would be due to the earth’s magnetism if it were distributed according to the most simple type.

144.] It appears from equation (52), by making $i=0$, that if $V$ satisfies Laplace’s equation throughout the space occupied by a sphere of radius $a$, then the integral

$\iint V\ dS=4\pi a^{2}V_{0},$

(69)

where the integral is taken over the surface of the sphere, $dS$ being an element of that surface, and $V_{0}$ is the value of $V$ at the centre of the sphere. This theorem may be thus expressed.

The value of the potential at the centre of a sphere is the mean value of the potential for all points of its surface, provided the potential be due to an electrified system, no part of which is within the sphere.

It follows from this that if $V$ satisfies Laplace’s equation through out a certain continuous region of space, and if, throughout a finite portion, however small, of that space, $V$ is constant, it will be constant throughout the whole continuous region.

If not, let the space throughout which the potential has a constant value $C$ be separated by a surface $S$ from the rest of the region in which its values differ from $C$, then it will always be possible to find a finite portion of space touching $S$ and out side of it in which $V$ is either everywhere greater or everywhere less than $C$.

Now describe a sphere with its centre within $S$, and with part of its surface outside $S$, but in a region throughout which the value of $V$ is every where greater or everywhere less than $C$.

Then the mean value of the potential over the surface of the sphere will be greater than its value at the centre in the first case and less in the second, and therefore Laplace’s equation cannot be satisfied throughout the space occupied by the sphere, contrary to our hypothesis. It follows from this that if $V=C$ throughout any portion of a connected region, $V=C$ throughout the whole of the region which can be reached in any way by a body of finite size without passing through electrified matter. (We suppose the body to be of finite size because a region in which $V$ is constant may be separated from another region in which it is