Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/88

CHAPTER V.

PARTICULAR PROBLEMS IN MAGNETIC INDUCTION.

A Hollow Spherical Shell.

431.] The first example of the complete solution of a problem in magnetic induction was that given by Poisson for the case of a hollow spherical shell acted on by any magnetic forces whatever.

For simplicity we shall suppose the origin of the magnetic forces to be in the space outside the shell.

If V denotes the potential due to the external magnetic system, we may expand V in a series of solid harmonics of the form

$V=C_{0}S_{0}+C_{1}S_{1}r+{\And }\!{c.}+C_{i}S_{i}r^{i}.$

(1)

wherer is the distance from the centre of the shell, S_i is a surface harmonic of order i, and C_i is a coefficient.

This series will be convergent provided r is less than the distance of the nearest magnet of the system which produces this potential. Hence, for the hollow spherical shell and the space within it, this expansion is convergent.

Let the external radius of the shell be a₂ and the inner radius a₁ and let the potential due to its induced magnetism be Ω. The form of the function Ω will in general be different in the hollow space, in the substance of the shell, and in the space beyond. If we expand these functions in harmonic series, then, confining our attention to those terms which involve the surface harmonic S_i, we shall find that if Ω₁ is that which corresponds to the hollow space within the shell, the expansion of Ω₁ must be in positive harmonics of the form A₁S_irⁱ, because the potential must not become infinite within the sphere whose radius is aa₁.

In the substance of the shell, where r₁ lies between a₁ and a₂, the series may contain both positive and negative powers of r, of the form

$A_{2}S_{i}r^{i}+B_{2}S_{i}r^{-(i+1)}.\,$

Outside the shell, where r is greater than a₂, since the series