A magnetic field B is confined to a region r a ≤ and

Question:

 A magnetic field B is confined to a region r 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 > a and mass m lies in the x-y 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 ∆t. Find the angular velocity ω of the ring after the field vanishes.

Solution:

The magnetic flux across the conducting ring reduces to zero from maximum when the magnetic field is reduced in ∆t.

Induced emf = E2πb

From Faraday’s law of emf,

The induced emf = rate of change of magnetic flux = Bπa2/∆t

Using the above equations, we get

E2πb = Bπa2/∆t

QE is the electric force experienced by the ring

Torque on the ring is Q.Ba2/2∆t

Change in angular momentum = torque × ∆t

The initial momentum is zero and the final momentum is Q.Ba2/2

ꞷ = Q.Ba2/2mb2

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