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

 CHAPTER V. PARTICULAR PROBLEMS IN MAGNETIC INDUCTION. A Hollow Spherical Shell. 431.] 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 wherer is the distance from the centre of the shell, Si is a surface harmonic of order i, and Ci 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 Si, 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 A1Siri, 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

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