Question

A line charge $\lambda$ per unit length is pasted uniformly on to the rim of a wheel of mass $m$ and radius $R$. The wheel has light non-conducting spokes and is free to rotate about a vertical axis as shown in figure. A uniform magnetic field extends over a radial region of radius $r$ given by $B=-B_0\hat{k}(r\le a;a<R)=0$ (otherwise). What is the angular velocity of the wheel when this field is suddenly switched.

• A. $\frac{-2B_0{\pi a}^{2}r}{mR}\hat{k}$
• B. $\frac{-B_0{\pi a}^{2}r}{3mR}\hat{k}$
• C. $\frac{B_0{\pi a}^2\lambda }{mR}\hat{k}$
• D. $\frac{{-B}_0{\pi a}^2\lambda }{mR}\hat{k}$
• E. None of these

Answer 1

The correct option is D $\frac{{-B}_0{\pi a}^2\lambda }{mR}\hat{k}$

Induced electric field ( in clockwise sence)
$E=\frac{a^2}{2R}\frac{dB}{dt}$
$I\omega =\frac{\tau }{d}t$
$\Rightarrow m{R}^2\omega =\int qERdt=\int \frac{q{a}^2}{2R}\frac{dB}{dt}Rdt=\frac{\lambda 2\pi R}{2R}{a}^{2}R\int dB$
$\Rightarrow \omega =\frac{B_0\pi a^2\lambda }{mR}$
$\Rightarrow \overrightarrow{\omega }=\frac{B_0\pi a^2\lambda }{mR}\hat{k}$(because clockwise sence)