A metallic ring of mass m and radius l (ring being horizontal) is
falling under gravity in a region having a magnetic field. If z is the
vertical direction, the z-component of magnetic field is Bz = Bo
(1+λ z). If R is the resistance of the ring and if the ring falls with a
velocity v, find the energy lost in the resistance. If the ring has
reached a constant velocity, use the conservation of energy to
determine v in terms of m, B, λ and acceleration due to gravity g.

Rate of change of magnetic flux:

dϕdt=ddtAB       =AdBdt      =AddtBo1+λz       =πL2Boλdzdt       =πL2BoλvBut, dϕdt=IRHence,IR=πL2BoλvI=πL2BoλRv

Energy lost per second:


I2R=πL2λ2Bo2v2R

Rate of change in PE:

Rate of change = mgdzdt
                      = mgv

​Equate rate of change in PE to energy lost per second:

mgv=πL2λBo2v2Rv=mgRπL2λBo2

 

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