Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/307

The calculation of the effect on the magnetic pole would be complicated, but it is easy to see that it will consist of

(1) A dragging force, parallel to the direction of motion of the disk.

(2) A repulsive force acting from the disk.

(3) A force towards the axis of the disk.

When the pole is near the edge of the disk, the third of these forces may be overcome by the force towards the edge of the disk, indicated in Art. 667.

All these forces were observed by Arago, and described by him in the Annales de Chimie for 1826. See also Felici, in Tortolini's Annals, iv, p. 173 (1853), and v. p. 35; and E. Jochmann, in Crelle's Journal, lxiii, pp. 158 and 329; and Pogg. Ann. cxxii, p. 214 (1864). In the latter paper the equations necessary for determining the induction of the currents on themselves are given, but this part of the action is omitted in the subsequent calculation of results. The method of images given here was published in the Proceedings of the Royal Society for Feb. 15, 1872.

Spherical Current-Sheet.

670.] Let $\phi$ be the current-function at any point $Q$ of a spherical current-sheet, and let $P$ be the potential at a given point, due to a sheet of imaginary matter distributed over the sphere with surface-density $\phi$, it is required to find the magnetic potential and the vector-potential of the current-sheet in terms of $P$.
Fig. 39.

Let $a$ denote the radius of the sphere, $r$ the distance of the given point from the centre, and $p$ the reciprocal of the distance of the given point from the point $Q$ on the sphere at which the current-function is $\phi$.

The action of the current-sheet at any point not in its substance is identical with that of a magnetic shell whose strength at any point is numerically equal to the current-function.

The mutual potential of the magnetic shell and a unit pole placed at the point $P$ is, by Art. 410,

$\Omega =\iint \phi {\frac {dp}{da}}\,dS$.