Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/304

of induced magnetization, till Faraday^[1] explained it by means of the electric currents induced in the disk on account of its motion through the field of magnetic force.

To determine the distribution of these induced currents, and their effect on the magnet, we might make use of the results already found for a conducting sheet at rest acted on by a moving magnet, availing ourselves of the method given in Art. 600 for treating the electromagnetic equations when referred to moving systems of axes. As this case, however, has a special importance, we shall treat it in a direct manner, beginning by assuming that the poles of the magnet are so far from the edge of the disk that the effect of the limitation of the conducting sheet may be neglected.

Making use of the same notation as in the preceding articles (656–667), we find for the components of the electromotive force parallel to $x$ and $y$ respectively,

$\sigma u={}$	$\gamma {\frac {dy}{dt}}-{\frac {d\psi }{dx}}$,
$\sigma v=-{}$	$\gamma {\frac {dx}{dt}}-{\frac {d\psi }{dy}}$,

(1)

where $\gamma$ is the resolved part of the magnetic force normal to the disk.

If we now express $u$ and $v$ in terms of $\phi$, the current-function,

and if the disk is rotating about the axis of $z$ with the angular velocity $\omega$,

Substituting these values in equations (1), we find

	$\sigma {\frac {d\phi }{dy}}=\gamma \omega x-{\frac {d\psi }{dx}}$,
$-{}$	$\sigma {\frac {d\phi }{dx}}=\gamma \omega y-{\frac {d\psi }{dy}}$.

(4)

(5)

Multiplying (4) by $x$ and (5) by $y$, and adding, we obtain

Multiplying (4) by $y$ and (5) by $-x$, and adding, we obtain

1. Exp. Res., 81.