Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/321

of the hemisphere so that its surface is a little more than a hemisphere, and meets the surface of the sphere at right angles. Then we have a case of which we have already obtained the exact solution. See Art. 168.

If $A$ and $B$ be the centres of the two spheres cutting each other at right angles, $DD'$ a diameter of the circle of intersection, and $C$ the centre of that circle, then if $V$ is the potential of a conductor whose outer surface coincides with that of the two spheres, the quantity of electricity on the exposed surface of the sphere $A$ is

${\frac {1}{2}}V(AD+BD+AC-CD-BC)$

and that on the exposed surface of the sphere $B$ is

${\frac {1}{2}}V(AD+BD+BC-CD-AC)$

the total charge being the sum of these, or

$V(AD+BD-CD)$

If $\alpha$ and $\beta$ are the radii of the spheres, then, when $\alpha$ is large compared with $\beta$, the charge on $B$ is to that on $A$ in the ratio of

${\frac {3}{4}}{\frac {\beta ^{2}}{\alpha ^{2}}}\left(1+{\frac {1}{3}}{\frac {\beta }{\alpha }}+{\frac {1}{6}}{\frac {\beta ^{2}}{\alpha ^{2}}}+\mathrm {etc} .\right)$

Now let $\sigma$ be the uniform surface-density on $A$ when $B$ is removed, then the charge on $A$ is

$4\pi \alpha ^{2}\sigma$

and therefore the charge on $B$ is

$3\pi \beta ^{2}\sigma \left(1+{\frac {1}{3}}{\frac {\beta }{\alpha }}+\mathrm {etc} .\right)$

or, when $B$ is very small compared with $\alpha$, the charge on the hemisphere $B$ is equal to three times that due to a surface-density $\sigma$ extending over an area equal to that of the circular base of the hemisphere.

It appears from Art. 175 that if a small sphere is made to touch an electrified body, and is then removed to a distance from it, the mean surface-density on the sphere is to the surface-density of the body at the point of contact as $\pi ^{2}$ is to 6, or as 1.645 to 1.

225.] The most convenient form for the proof plane is that of a circular disk. We shall therefore shew how the charge on a circular disk laid on an electrified surface is to be measured.

For this purpose we shall construct a value of the potential function so that one of the equipotential surfaces resembles a circular flattened protuberance whose general form is somewhat like that of a disk lying on a plane.