A magnetic field B is confined to a region `rle a` and points out of the paper (the z-axis), r = 0 being the centre of the circular region. A charged ring (charge Q) of radius b, b `gt` a and mass m lies in the xy-plane with its centre at the origin. The ring is free to rotate and is at rest. The magnetic field is brought to zero in time `Deltat`. Find the angular velocity `omega` of the ring after the field vanishes.

Explanation

In `Deltat` time flux linked with ring becomes zero from its maximum value which induces emf in ring electric field is also induced in ring.
Induced emf = Electric field x `(2pib) [ because V=Ed]`
But induced emf `epsilon=(Deltaphi)/(Deltat)=(pia^2B-0)/(Deltat)=(Bpia^2)/(Deltat)`
`therefore (Bpia^2)/(Deltat)=F(2pib)`
`therefore E=(Ba^2)/(2bDeltat)`
Force produced on ring `F=QE=(QBa^2)/(2bDeltat)`
Torque produced on ring ,
`T= rF sin theta =(b Q Ba^2)/(2bDeltat)=(QBa^2)/(2Deltat)`
Now torque = Rate of change in angular momentum,
`tau=(DeltaL)/(Deltat)`
`=(mvb-0)/(Deltat)`
`(QBa^2)/(2Deltat)=(mb^2omega)/(Deltat)`
`therefore omega=(QBa^2)/(2mb^2)`