Question

A non conductive insulating ring of mass $m$ and radius $R$, having charge $Q$ uniformly distributed over it's circumference is hanging by a insulated thread with the help of a small smooth ring (not rigidly fixed with bigger ring). A time varying magnetic field $B=B_0\sin \omega t$ is switched at $t=0$ and ring is released at the same time. The average magnetic moment of Ring in time interval $0$ is:

• A. $\frac{B_0{q}^2{R}^2}{2\pi m}$
• B. $\frac{B_0{q}^2{R}^2}{4\pi m}$
• C. $\frac{B_0{q}^2{R}^2}{\pi m}$
• D. zero

Answer 1

Answer: (A)

$\frac{B_0{q}^2{R}^2}{2\pi m}$

From electromagnetic Induction,

$\frac{E}{R^{\prime }}=\frac{-d}{dt}BA$

$\therefore \frac{E}{R^{\prime }}=\frac{-d}{dt}{B}_{o}A\sin wt$

$\therefore \frac{E}{R^{\prime }}=-B_{0}wA\cos wt=1$

Magnetic moment $\mu =iA$

At $t=0,\frac{E}{R^{\prime }}=B_{o}wA$

Note that

$I=\frac{B_o{q}^2}{4{\pi }^{2}m}$

$\therefore \mu =iA=2\pi R^2\times \frac{B_o{q}^2}{4{\pi }^{2}m}$

$\therefore \mu =\frac{B_o{q}^2{R}^2}{2\pi m}$