Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/296

shell as consisting of two surfaces parallel to the plane of $xy$ the first, whose equation is $z={\frac {1}{2}}c$, having the surface-density ${\frac {\phi }{c}}$, and the second, whose equation is $z=-{\frac {1}{2}}c$, having the surface-density $-{\frac {\phi }{c}}$.

The potentials due to these surfaces will be

${\frac {1}{c}}P_{\left(z-{\frac {c}{2}}\right)}$ and $-{\frac {1}{c}}P_{\left(z+{\frac {c}{2}}\right)}$.

respectively, where the suffixes indicate that $z-{\frac {c}{2}}$ is put for $z$ in the first expression, and $z+{\frac {c}{2}}$ for $z$ in the second. Expanding

these expressions by Taylor s Theorem, adding them, and then making $c$ infinitely small, we obtain for the magnetic potential due to the sheet at any point external to it,

657.] The quantity $P$ is symmetrical with respect to the plane of the sheet, and is therefore the same when $-z$ is substituted for $z$.

$\Omega$ magnetic potential, changes sign when $-z$ is put for $z$.

At the positive surface of the sheet

At the negative surface of the sheet

Within the sheet, if its magnetic effects arise from the magnetization of its substance, the magnetic potential varies continuously from $2\pi \phi$ at the positive surface to $-2\pi \phi$ at the negative surface.

If the sheet contains electric currents, the magnetic force within it does not satisfy the condition of having a potential. The magnetic force within the sheet is, however, perfectly determinate.

The normal component,

is the same on both sides of the sheet and throughout its substance.

If $\alpha$ and $\beta$ be the components of the magnetic force parallel to