Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/195



If $$\sigma_{1},\ \sigma_{2}$$ are the surface-densities on the opposed surfaces of a solid sphere of radius $$a$$, and a spherical hollow of radius $$b$$, then

If $$E_1$$ and $$E_2$$ be the whole charges of electricity on these surfaces,

The capacity of the enclosed sphere is therefore $$\frac{ab}{b-a}$$.

If the outer surface of the shell be also spherical and of radius $$c$$, then, if there are no other conductors in the neighbourhood, the charge on the outer surface is

Hence the whole charge on the inner sphere is

and that of the outer

If we put $$b=\infty$$, we have the case of a sphere in an infinite space. The electric capacity of such a sphere is $$a$$, or it is numerically equal to its radius.

The electric tension on the inner sphere per unit of area is

The resultant of this tension over a hemisphere is $$\pi a^{2}p=F$$ normal to the base of the hemisphere, and if this is balanced by a surface tension exerted across the circular boundary of the hemisphere, the tension on unit of length being $$T$$, we have

Hence

$$\begin{array}{l} F=\frac{b^{2}}{8}\frac{(A-B)^{2}}{(b-a)^{2}}=\frac{E_{1}^{2}}{8a^{2}},\\ \\T=\frac{b^{2}}{16\pi a}\frac{(A-B)^{2}}{(b-a)^{2}}.\end{array}$$