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 $B_{z} = B_{0} (1+ \lambda z)$. 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, $\lambda$ and acceleration due to gravity g.

Here a relation is established between the current induced, velocity of free falling ring and power lost. Let the mass of ring be m and radius l;

$\phi _{m}=B_{z}(\pi l^{2})=B_{0}(1+\lambda z)(\pi l)$

Applying Faraday's law of EMI, we have emf induced given by

$\frac{d\phi }{dt}=$ rate of change of flux. Also, by Ohm's law

$B_{0}(\pi l^{2})\lambda \frac{dz}{dt}=IR$

We have

$I=\frac{\pi l^{2}B_{0}\lambda }{R}v$

Energy lost/second =

$I^{2}R=\frac{(\pi l^{2}\lambda )^{2}B_{0}^{2}v^{2}}{R}$

Rate of change of

$PE=mg\frac{dz}{dt}=mgv$ [ as kinetic energy is constant for v=constant]

According to the law of conservation of energy

Thus,

$mgv=\frac{(\pi l^{2}\lambda B_{0})^{2}v^{2}}{R}$ or

$v=\frac{mgR}{(\pi l^{2}\lambda B_{0})^{2}}$

This is the required expression of velocity.