Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/66

have the same bounding edge and do not include between them any centre of force, the action of the magnetic shell depends only on the form of its edge.

Now suppose the field of force to be that due to a magnetic pole of strength m. We have seen (Art. 76, Cor.) that the surface-integral over a surface bounded by a given edge is the product of the strength of the pole and the solid angle subtended by the edge at the pole. Hence the energy due to the mutual action of the pole and the shell is

$\Phi m\omega ,\,$

and this (by Green's theorem, Art. 100) is equal to the product of the strength of the pole into the potential due to the shell at the pole. The potential due to the shell is therefore Φω.

411.] If a magnetic pole m starts from a point on the negative surface of a magnetic shell, and travels along any path in space so as to come round the edge to a point close to where it started but on the positive side of the shell, the solid angle will vary continuously, and will increase by 4π during the process. The work done by the pole will be 4πΦm, and the potential at any point on the positive side of the shell will exceed that at the neighbouring point on the negative side by 4πΦ.

If a magnetic shell forms a closed surface, the potential outside the shell is everywhere zero, and that in the space within is everywhere 4πΦ, being positive when the positive side of the shell is inward. Hence such a shell exerts no action on any magnet placed either outside or inside the shell.

412.] If a magnet can be divided into simple magnetic shells, either closed or having their edges on the surface of the magnet, the distribution of magnetism is called Lamellar. If φ is the sum of the strengths of all the shells traversed by a point in passing from a given point to a point x y z by a line drawn within the magnet, then the conditions of lamellar magnetization are

$A={\frac {d\phi }{dx}},\quad B={\frac {d\phi }{dy}},\quad C={\frac {d\phi }{dz}}.$

The quantity, φ, which thus completely determines the magnetization at any point may be called the Potential of Magnetization. It must be carefully distinguished from the Magnetic Potential.

413.] A magnet which can be divided into complex magnetic shells is said to have a complex lamellar distribution of magnetism. The condition of such a distribution is that the lines of