$X P Q Y$ is a vertical smooth long loop having a total resistance $R$ where $P X$ is parallel to $Q

Question

$X P Q Y$ is a vertical smooth long loop having a total resistance $R$ where $P X$ is parallel to $Q Y$ and separation between them is $l$. A constant magnetic field $B$ perpendicular to the plane of the loop exists in the entire space. A rod $C D$ of length $L(L>l)$ and mass $m$ is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is $\_\_\_\_$ $\mathrm{m} / \mathrm{s} .(\mathrm{g}=$ acceleration due to gravity)

Professor PC Thomas Classes · Answer

As the rod falls, it cuts magnetic field lines, inducing an EMF $e = Bvl$. The induced current in the loop is $I = \frac{e}{R} = \frac{Bvl}{R}$. The magnetic field exerts an upward force on the length $l$ of the rod carrying the current: $F_m = IlB = \frac{B^2 v l^2}{R}$. The rod reaches terminal speed when the upward magnetic force equals the downward gravitational force, $F_m = mg$. Equating them, we get $\frac{B^2 v l^2}{R} = mg \Rightarrow v = \frac{mgR}{B^2 l^2}$.