A metal rod OA of mann 'm' and length 'r' is kept rotating with a constantangular speed `omega` in a vertical plane about a horizontal axis at the end O. The free end A is arraged to slide without friction along fixed conduction circular ring in the same plane as that of rotation. A uniform and constant magnetic induction `vec(B)` is applied perpendicular and into the plane of rotation as shown in the figure below. An inductor L and an external resistance R are connected through a swithch S between the point O and a point C on the ring to form an electrical circuit. Neglect the resistance of the ring and the rod. Initially, the switch is open.

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

Answer

`(a) E=1/2 Bomegar^(2) (b) (i) I=(Bomegar^(2)[1-e^(-Rt//L)])/(2R)` (ii) tau=(mgr)/2 cos omegat+(omegaB^(2)r^(4))/(4R)(1-e^(-Rt//L))`

Explanation

`de=B dx v=B xomegadx`
`e=Bromegaint_(0)^(r ) dx=1/2 Bomegar^(2)`

(b) (i) `I=e/R(1-e^(-t//tau))=1/2 (Bomegar^(2))/R (1-e^(-Rt//L))`
(ii) `tau= mgcostheta r/2+IrB.r/2`

`=mg costhetar/2+(Ibr^(2))/2`
`=(mgr)/2 costheta+(Br^(2))/2[1/2(Bomegar^(2))/R(1-e^(-Rt//L))]`
`=(mgr)/2 cosomegat+(B^(2)r^(4)omega)/(4R)(1-e^(-Rt//L))`