A metal rod `OA` of mass `m` amd length `r` is kept rotating with a constant angular speed in a vertical plane about a horizontal axis at the end `O`. The free end `A` is arranged to slide without friction along a fixed conducting circular ring in the same plane as that of rotation. A uniform and constant magnetic induction is applied perpendicular and into the plane of rotation as shows in Fig. An inductor `L` and an exernal resistance `R` are connected through a switch `S` between point `O` and point `C` on the ring to from an eletrical circuit. Neglect the resistance of the ring and rod. Initially, the switch is open.

a. What is the induced emf across the terminals of the switch ?
b. Switch `S` is closed at time `t = 0`.
i. Obtain an expression for the current as a funtion of time.
ii. In the steady state, obtain the time dependence of the torque required to maintain the constant angular speed, given that rod `OA` was along the positive X-axis at `t = 0`.

Answer

(a) `-(1)/(2)Br^(2)omega` (b) (i) `(Br^(2))/(2R) (1-e^((Rt)/(L)))rB (ii) (B^(2)r^(4)omega)/(4R)(I-e^(-RT//L))`