Question

A rod of length $l$ having uniformly distributed charge q is rotated about one end with constant frequency f its magnetic moment.

• A. $\pi fq{I}^2$
• B. $\frac{\pi fq{l}^2}{3}$
• C. $\frac{2\pi fq{l}^2}{3}$
• D. $2\pi fq{I}^2$

Answer 1

The correct option is B $\frac{\pi fq{l}^2}{3}$

Charge density $\lambda =\frac{q}{l}$

Consider small element $dx$ at a distance $x$ from free end=$dq=\lambda dx$

Current on rotations, $di=fdq$

$f\lambda dx$

magnetic moment $dM=di\times area$

$⟹f\lambda dx.\pi x^2$

$⟹$ Total magnetic moment $M=\int dM$

Limits of integration $0$ to $l$, $L=$length

$={\int }_0^{L}f\lambda dx.\pi x^2=f\lambda \pi {\int }_0^L{x}^{2}dx=f\lambda \pi \frac{L^3}{3}\lambda l=q\Rightarrow M=\frac{f\pi q{L}^2}{3}$