A conducting wire ab of length l, resistance r and mass m starts sliding at t = 0 down a smooth, vertical, thick pair of connected rails as shown in . A uniform magnetic field B exists in the space in a diraction perpendicular to the plane of the rails. (a) Write the induced emf in the loop at an instant t when the speed of the wire is v. (b) what would be the magnitude and direction of the induced current in the wire? (c) Find the downward acceleration of the wire at this instant. (d) After sufficient time, the wire starts moving with a constant velocity. Find this velocity `v_m. (e) Find the velocity of the wire as a function of time. (f) Find the displacement of the wire as a functong of time. (g) Show that the rate of heat developed inte wire is equal to the rate at which the gravitational potential energy is decreased after steady state is reached.

Explanation

When velocity of rod is `v`
induced emf `e=Bvl`
(b) Induiced current `I=(e)/(R)=(Bvl)/(R)`
Flux is increasing, induced current in the loop is clockwise.
(c) Magnetic force on the wire
`F_(m)=BIL=(B^(2)vl^(2))/(R)` , upward
`mg-F_(m)=ma`
`a=g-(F_(m))/(m)=g-(B^(2)l^(2)v)/(mR)`
(d) When `v=v_(m)` , `a=0`
`g-(B^(2)l^(2)v_(m))/(mR)=0 implies v_(m)=(mgR)/(B^(2)l^(2))`
(e) In steady state, the wire falls with constant velocity `v_(m)`
Rate of heat developed `P_(1)=I^(2)R=(B^(2)v_(m)^(2)l^(2)//R)=mgv_(m)`
Rate at which gravitational potenital enregy decreases
`p_(2)=(d)/(dt)(mgh)=mg ((dh)/dt)=mgv_(m)`
`P_(1)=P_(2)`