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 non-conducting spokes and is free to rotate without friction about its axis (Fig. 6.22). A uniform magnetic field extends over a circular region within the rim. It is given by,
`B=B_(0)k" "(ra le a,a lt R)`
=0 `" "` (otherwise)
What is the angular velocity of the wheel after the field is suddenly switched off?

Explanation

When magnetic field is switched off, there is a change in flux. This flux produces an induced electric field. The induced electric field lies along the periphery of the ring and exerts a torque. If E is the magnitude of electric field, then

`Exx2piR=(dB)/(dt)xxpia^(2)(because int vecE vec (dl)=(-dphi_(b))/(dt))`
`tau=qExxR=(lambdaxx2piR)ExxR=(dL)/(dt)`
`rArr" "(dL)/(dt)=lambdaxx2piR^(2)xx(dB)/(dt)xx(a^(2))/(2R)`
`"or "dL=lambdapiRa^(2)xxB(because dL=ldomega)`
`rArr" "omega=(lambdapia^(2)B)/(MR)`