Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/311

may be applied to current-sheets by substituting for the body supposed to be uniformly magnetized in the direction of $z$ with intensity $I$, a current-sheet having the form of its surface, and for which the current-function is

The currents in the sheet will be in planes parallel to that of $xy$, and the strength of the current round a slice of thickness $dz$ will be $I\,dz$. The magnetic potential due to this current-sheet at any point outside it will be

At any point inside the sheet it will be

The components of the vector-potential are

These results can be applied to several cases occurring in practice.

675.] (1) A plane electric circuit of any form.

Let $V$ be the potential due to a plane sheet of any form of which the surface-density is unity, then, if for this sheet we substitute either a magnetic shell of strength $I$ or an electric current of strength $I$ round its boundary, the values of $\Omega$ and of $F$, $G$, $H$ will be those given above.

(2) For a solid sphere of radius $a$,

		$V={\frac {4\pi }{3}}{\frac {a^{3}}{r}}$ when $r$ is greater than $a$,	(5)
	and⁠	$V={\frac {2\pi }{3}}(3a^{2}-r^{2})$ when $r$ is less than $a$.	(6)

Hence, if such a sphere is magnetized parallel to $z$ with intensity $I$, the magnetic potential will be

		$\Omega ={\frac {4\pi }{3}}I{\frac {a^{3}}{r^{3}}}z$ outside the sphere,	(7)
	and⁠	$\Omega ={\frac {4\pi }{3}}Iz$ inside the sphere.	(8)

If, instead of being magnetized, the sphere is coiled with wire in equidistant circles, the total strength of current between two small circles whose planes are at unit distance being $I$, then outside the sphere the value of $\Omega$ is as before, but within the sphere

This is the case already discussed in Art. 672.