Question

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

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

Answer 1

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

Let us consider the disc to be made up of a larger number of concentric elementary rings.

Consider one such ring of radius $x$ and thickness $dx$

Charge on this elementary ring,

$dq=\sigma \times 2\pi xdx=2\pi \sigma xdx$

Current associated with this elementary ring,

$dl=\frac{dq}{dt}=dq\times f=\sigma \omega xdx$

Here $f$ is the frequency and $\omega =2\pi f$

Magnetic moment of this elementary ring,

$dM=dl\pi x^2=\pi \sigma \omega x^{3}dx$

Therefore, magnetic moment of the entire disc,

$M={\int }_0^R=\pi \sigma \omega {\int }_0^R{x}^{3}dx=\frac{\pi R^4}{4}\sigma \omega$