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_O (I + lambdaz)`. 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.

Explanation

Magnetic flux linked with ring,
`phi=AB costheta (because theta =0)`
`theta =Bzpil^2`
`phi=B_0(1+lambdaz)pil^2`
Now according to Faraday.s law induced emf,
`epsilon=(dphi)/(dt)=d/(dt)[B_0 (1+lambdaz)pil^2]`
`epsilon=B_0 pil^2 lambda (dz)/(dt)`
`epsilon=B_0 pil^2 lambdav`
Induced current `I=S/R`
`I=(B_0pil^2lambdav)/R`
Heat produced per unit time,
`H=I^2R=((Bpil^2lambdav)/R)^2R`
`H=(B_0^2pi^2l^4lambda^2v^2)/R`
This heat per unit time is produced at the cost of rate of change of potential energy.
Rate of change of P.E. = Heat produced per unit time
(`because` Here that K.E. remains constant)
`d/(dt)(mgz)=H`
`mg(dz)/(dt)=(B_0^2pi^2l^4lambda^2v^2)/R`
`therefore mgv=(B_0^2pi^2l^4lambda^2v^2)/R`
`therefore v=(mgR)/(B_0^2pi^2l^4lambda^2)`
which is required equation of terminal velocity .