A line charge `lambda` per unit length is lodged uniformly onto the rim of a wheel of mass M and radius R. The wheel has light nonconducting spokes and is free to rotate without friction about its axis as per figure. A uniform magnetic field extends over a circular region within the rim. It is given by,
`B=-B_0 k ( r le a , a lt R)`
=0 (otherwise)
What is the angular velocity of the wheel after the field is suddenly switched off ?

Explanation

Magnetic flux linked with area `A=pia^2` of a circle, concentric with given rim of a wheel is,
`phi=AB cos0^@ ( because vecA||vecB)`
`therefore phi=(pia^2)B rArr - (dphi)/(dt)=pia^2 (-(dB)/(dt))`…(1)
Here as per the statement when magnetic field is suddenly switch off, above magnetic flux becomes zero. Because of this change in flux, emf is induced in the conducting rim of a wheel in a clockwise direction to increase the flux. This induced emf produces induced electric field given by,
`epsilon=oint vecE.vec(dl)`
`therefore epsilon=oint E dl cos 0^@`
`=E oint dl`
`therefore epsilon=E(2piR)`
`therefore -(dphi)/(dt) =E(2piR)`
`therefore pia^2 (-(dB)/(dt))=E(2piR)` [From equation (1)]
`therefore E=a^2/(2R)(-(dB)/(dt))`...(2)
Here induced electric field exerts electric force dF on charge `dq=lambda dl` (where `lambda` = linear charge = density on the rim of a wheel). Because of this tangential force, torque is exerted on dl length given by,
`dtau=(R)(dF)`
`therefore dtau=(R)(dq)E`
`therefore dtau=(R)( lambda dl) E`
`therefore int dtau=R lambdaEint dl`
`therefore tau=RlambdaE (2piR)=2piR^2lambdaE`
`therefore Ialpha=2piR^2 lambda xx a^2/(2R)(-(dB)/(dt))` [From equation (2)]
`therefore MR^2((domega)/(dt))=piRlambdaa^2 (-(dB)/(dt)) " " (because I=MR^2)`
`therefore domega=-(pilambdaa^2)/(MR)dB`
`therefore int_(omega_i)^(omega_f) domega=-(pilambdaa^2)/(MR) int_(B_i)^(B_f)dB`
`therefore (omega_f-omega_i)=-(pilambdaa^2)/(MR)(B_f-B_i)`
`therefore omega_f =-(pilambdaa^2)/(MR)(0-B_0)` (`because omega_i = 0, B_i=B_0` and `B_f=0` )
`therefore omega_f=(pilambdaa^2 B_0)/(MR)`
(In the direction , perpendicularly inside the plane of figure , so as to increase inward flux , according to Lenz.s law).