Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/306

and for the component of the magnetic force normal to the disk due to the currents,

If $\Omega _{2}$ is the magnetic potential due to external magnets, and if we write

the component of the magnetic force normal to the disk due to the magnets will be

We may now write equation (18), remembering that

$\gamma =\gamma _{1}+\gamma _{2}$

Integrating twice with respect to $z$, and writing $R$ for ${\frac {\sigma }{2\pi }}$,

If the values of $P$ and $Q$ are expressed in terms of $r$, $\theta$, and $\zeta$, where

equation (24) becomes, by integration with respect to $\zeta$,

669.] The form of this expression shews that the magnetic action of the currents in the disk is equivalent to that of a trail of images of the magnetic system in the form of a helix.

If the magnetic system consists of a single magnetic pole of strength unity, the helix will lie on the cylinder whose axis is that of the disk, and which passes through the magnetic pole. The helix will begin at the position of the optical image of the pole in the disk. The distance, parallel to the axis between consecutive coils of the helix, will be $2\pi {\frac {R}{\omega }}$. The magnetic effect of the trail will be the same as if this helix had been magnetized everywhere in the direction of a tangent to the cylinder perpendicular to its axis, with an intensity such that the magnetic moment of any small portion is numerically equal to the length of its projection on the disk.