Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/266

 The result of this investigation is that if $$f$$ is the diameter of the sphere, $$a$$ the chord of the radius of the bowl, and $$r$$ the chord of the distance of $$P$$ from the pole of the bowl, then the surface-density $$\sigma$$ on the inside of the bowl is

$\sigma=\frac{V}{2\pi^{2}f}\left\{ \sqrt{\frac{f^{2}-a^{2}}{a^{2}-r^{2}}}-\tan^{-1}\sqrt{\frac{f^{2}-a^{2}}{a^{2}-r^{2}}}\right\},$|undefined

and the surface-density on the outside of the bowl at the same point is

$\sigma+\frac{V}{2\pi f}.$

In the calculation of this result no operation is employed more abstruse than ordinary integration over part of a spherical surface. To complete the theory of the electrification of a spherical bowl we only require the geometry of the inversion of spherical surfaces.

181.] Let it be required to find the surface-density induced at any point of the bowl by a quantity $$q$$ of electricity placed at a point $$Q$$, not now in the spherical surface produced.

Invert the bowl with respect to $$Q$$, the radius of the sphere of inversion being $$R$$. The bowl $$S$$ will be inverted into its image $$S'$$, and the point $$P$$ will have $$P'$$ for its image. We have now to determine the density $$\sigma'$$ at $$P'$$ when the bowl $$S'$$ is maintained at potential $$V'$$, such that $$q=V'R$$, and is not influenced by any external force.

The density $$\sigma$$ at the point $$P$$ of the original bowl is then

$\sigma=-\frac{\sigma'R^{3}}{QP^{3}},$|undefined

this bowl being at potential zero, and influenced by a quantity $$q$$ of electricity placed at $$Q$$.

The result of this process is as follows:

Let the figure represent a section through the centre, $$O$$, of the sphere, the pole, $$C$$, of the bowl, and the influencing point $$Q$$. $$D$$ is a point which corresponds in the inverted figure to the unoccupied pole of the rim of the bowl, and may be found by the following construction.

Draw through $$Q$$ the chords $$EQE'$$ and $$FQF'$$, then if we suppose the radius of the sphere of inversion to be a mean proportional between the segments into which a chord is divided at $$Q$$, $$E'F'$$ will be the