Question

A line charge $l$ per unit length is pasted uniformly onto 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 $B$ exists as shown in figure. What is the angular velocity of the wheel when the field is suddenly switched off?

• A. $\frac{2\pi \lambda a^{2}B}{mR}$
• B. $\frac{\pi \lambda a^{2}B}{mR}$
• C. $\frac{3\pi \lambda a^{2}B}{mR}$
• D. $\frac{\pi \lambda a^{2}B}{2mR}$

Answer 1

Answer: (B)

$\tau =\frac{\Delta L}{\Delta t}=\frac{I\omega }{\Delta t}=\frac{m{R}^2\omega }{\Delta t}$
but $\tau =\lambda (2\pi R)ER=\lambda \left(\pi a^2\frac{\Delta B}{\Delta t}\right)$ $R=\frac{\lambda \pi a^{2}BR}{\Delta t}$
$\Rightarrow \frac{m{R}^2\omega }{\Delta t}=\frac{\lambda \pi a^{2}BR}{\Delta t}$
$\Rightarrow \omega =\frac{\lambda \pi a^{2}B}{mR}$