Q: 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 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 time ∆t. Find the angular velocity ω of the ring after the field vanishes.

Sol: Induced emf in the ring $\displaystyle e = \frac{\Delta \phi}{\Delta t}$

$\displaystyle e = \frac{B (\pi a^2)}{\Delta t}$

Applying V = E × d ; Where E = Electric field

e = E × 2 π b

$\displaystyle E \times 2 \pi b = \frac{B (\pi a^2)}{\Delta t}$

Torque acting on the ring = Force × Perpendicular distance

$\displaystyle \tau = Q E \times 2 b $

$\displaystyle \tau = 2 Q b ( \frac{B (\pi a^2)}{\Delta t (2 \pi b)} ) $

$\displaystyle \tau = \frac{Q B a^2}{2 \Delta t} $

Change Angular Momentum = Torque × Δ t

$\displaystyle L_f – L_i = \frac{Q B a^2}{2 \Delta t} \times \Delta t$

$\displaystyle L_f – 0 = \frac{Q B a^2}{2} $

$\displaystyle m b^2 \times \omega = \frac{Q B a^2}{2}$

$\displaystyle \omega = \frac{Q B a^2}{2 m b^2}$