A line charge with linear charge density `lambda` is wound around an insulating disc of mass `M` and radius `R`, which is then suspended horizontally as shown in Fig. 3.90, so that it is free to rotate. In the central region, of radius `a`, there is a uniform magnetic field `B_(0)`, pointing up. Now the magnetic field is switched off, which causes the disc to rotate.
Find the angular speed with which the disc starts rotating.

Explanation

The induced electric field `E` due to the changing magnetic field is given by (from Faraday's law)
`rarr` `E. 2pir = -pia^(2)(dB)/(dt)` `rarr` `E = (-a^(2))/(2R)(dB)/(dt)`
Hence, induced electric field is tangential to the disc as shown in Fig. 3.90 and its magnitude is `E = (a^(2))/(2R)(dB)/(dt)`.
This electric field causes the disc to rotate. Now torque on the disc is `tau = (lambdapuR)ER = pilambd a^(2)R(dB)/(dt)`.
Angular impulse:
`L = int taudt = intpilambdaa`^(2)R(dB)/(dt)dt = pi` lambda `a^(2)Rint dB = pi alpha a^(2)RB_(0)`
Now `L = Iomega`
`rarr` `pi` lambda `a^(2)RB_(0) = (MR^(2))/(2)omega` `rarr` `omega = (2pi lambda a^(2)B_(0))/(MR)`