Charge `Q` is uniformaly distributed on a thin insulating ring of mass `m` which is initially at rest. To what angular velocity will the ring be accelerated when a magnetic field `B`, perpendicular to the plane of the ring, is switched on ?

Options

A. (a) `(QB)/(2m)`
B. (b) `(3QB)/(2m)`
C. ( c) `(QB)/(m)`
D. (d) `(QB)/(4m)`

Answer

A

Explanation

(a) In accordance with Faraday's law of electromagnetic induction, the changing magnetic field induces an electric field in the ring. Let us imagine the ring to be diveded into differential elements of length `ds` and denote the tangential component of the induced electric field by `E_(t)`. The change on element `ds` of the ring is `dQ = Q(ds)/(2pir)`, where `r` is the radius of teh ring . The force extered on it is `dF_(t) = dQE_(t)`, and the resultant torque is `dtau = rdF_(t)`.
Thus, the torque experienced by the ring is
`tay = int dtau = int rQ(ds)/(2pir) E_(t) = (Q)/(2pi) int E_(t) ds` ltbr. The induced electromotive force along the ring is directly proportional to the rate of change in the magnetic flux, we have
`int E_(t)ds = (dPhi)/(dt) = pir^(2)(dB)/(dt)`
As a result of the torque, the ring, which has a moment of inertia `I = mr^(2)`, starts to spin with angular accelaration `alpha`. During a time interval `dt` its angular velocity changes by
`domega = alphadt = (tau)/(I)dt = (Q)/(2pi)(pir^(2)(dB)/(dt))(1)/(mr^(2))dt = (Q)/(2m)dB`
Since the magnetic field strength increasing from zero to `B`, the final angular velocity of the ring will be
`omega = (QB//2m)`
`rarr` The final angular velocity does not depend on the radius of the ring, the time over which the magnetic flux changes, or even on how the magnetic flux increases with time.
`rarr` In our calculation we ignored the magnetic field produced by the ritating ring.
`rarr` Except in the case of a cylindrical sysmmetric unitform field, it is not possible to find the actual value of the induced electric field within the ring because the geometric structure of the magnetic field is unknown an we do not know the position of the ring in the magnetic fiels. We can determine the total induced electromotive force, but not the eldtric field itself.