Question

A thin circular disk of radius $R$ is uniformly charge with density $\sigma >0$ per unit area. the disk rotates about it axis with a uniform angular speed $\omega$. The magnetic moment of the disk is:-

• A. $4\pi R^4\sigma \omega$
• B. $2\pi R^4\sigma \omega$
• C. $\frac{\pi R^4}{2}\sigma \omega$
• D. $\frac{\pi R^4}{4}\sigma \omega$

Answer 1

The correct option is D $\frac{\pi R^4}{4}\sigma \omega$

Let the ring is made up of large number of elementary rings.

Charge on this ring

$dq=\sigma 2\pi xdx$

$dI=\frac{dq}{dt}$

$=2\pi \sigma xdxf(f=\frac{1}{dt})$

$=\sigma \omega xdx(\because \omega =2\pi f)$

$MagneticmomentdM=dI\times A$

$=\sigma \pi \omega x^{3}dx$

$M=\sigma \pi \omega \underset{0}{\overset{R}{\int }}{x}^{3}dx$

$=\frac{1}{4}\pi \sigma \omega R^4$