The diagram shows a circuit having a coil of resistance `R=10Omega` and inductance L connected to a conducting rod PQ which can slide on a perfectly conducting circular ring of radius 10 cm with its centre at 'P' Assume that friction & gravity are absent and a constant uniform magnetic field of 5 T exists as shown in figure. At `t=0` the circuit is switched on and simultaneously a time varying external torque is applied on the rod so that it rotates about P with a constant angular velocity 40 rad//s Find magnitude of this torque (in mu Nm) when current reaches half of its maximum value. Neglect the self inductance of the loop formed by the circuit.

Answer

`1250`

Explanation

Induced EMF `=1/2Bomegal^(2)`
At any time t
`L(di)/(dt)+iR=(Bomegal^(2))/2`
Solving for I, we get
`i=(Bomegal^(2))/(2R)[1-e^(-Rt//L]`
Torque about the hinge P is
`tau= underset0oversetlint idx B.x =1/2 iBl^(2)=(Bl^(2))/2 (Bomegal^(2))/(2R)[1-e^(-Rt//L]=(B^(2)omegal^(4))/(4R)[1-e^(-Rt//L)]`
Max. value occur at `t=infty` and half of this is equal to `i_(1)=(B^(2)omegal^(2))/(4R) when 1-e ^(-Rt//L)=1/2`
:. Torque at this instant `=(B^(2)omegal^(4))/(8R)=1.25`