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) $|undefined

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.