Question

A metal rod of resistance $R$ is fixed along a diameter of a conducting ring of radius $r$. There is a magnetic field of magnitude $B$ perpendicular to the plane of the loop. The ring spins with angular velocity $\omega$ about its axis. The centre of the ring is joined to its rim by external wire $W$. The ring and $W$ have no resistance. The current in $W$ is

• A. $\frac{B r^{2} \omega}{R}$
• B. $\frac{B r^{2} \omega}{2 R}$
• C. $\frac{2 B r^{2} \omega}{R}$
• D. zero

Answer 1

Answer: (C)

$ \varepsilon=\frac{B r^{2} \cdot \omega}{2}$
Each half of metal rod has a resistance $\frac{R}{2} .$ Both halves are in parallel, so
$R_{e q}=\frac{R}{4}$
$\Rightarrow I=\frac{\varepsilon}{R_{e q}}=\frac{B r^{2} \omega}{2\left(\frac{R}{4}\right)}=\frac{2 B r^{2} \omega}{R}$