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Question

A point charge Q is moving in a circular orbit of radius R in the x-y plane with an angular velocity ω. This can be considered as equivalent to a loop carrying a steady current Qω2π. A uniform magnetic field along the positive z-axis is now switched on, which increases at a constant rate from 0 to B in one second. Assume that the radius of the orbit remains constant. The applications of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant γ.

The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change


  1. γBQR22

  2. 2γBQR2

  3. γBQR22

  4. γBQR2


Solution

The correct option is A

γBQR22


ML=Q2mM=(Q2m)LMαL
where γ=Q2m
(Q2m)(Iω)=(Q2ω)(mR2ω)=QωR22
Induced electric field is opposite so,
ω=ωαtα=τI=(QE)RmR2=(Q)(BR2)mR2=QB2mωωQB2m=ωQB2mMf=QωR22=Q(ωQB2m)R22ΔM=MfMi=QBR24mm=γQBR22

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