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

Answer 1

Answer: (B)

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