Electromagnetic induction

Question

A pair of parallel horizontal conducting rails of negligible resistance shorted at one end is fixed on a table. The distance between the rails is L . A conducting massless rod of resistance R can slide on the rails frictionlessly. The rod is tied to a massless string which passes over a pulley fixed to the edge of the table. A mass m, tied to the other end of the string hangs vertically. A constant magnetic field B exists perpendicular to the table. If the system is released from rest, calculate the terminal velocity achieved by the rod.

Answer 1

If v is the instantaneous velocity of the rod at any time t, the induced emf will be ε = BvL.

∴ Induced current in rod

Due to this current, the rod in magnetic field B will experience a force

F = BiL opposite to motion ......(ii)

If vT is terminal velocity of rod, and T the tension in the string, y then free-body diagrams of rod and mass m are shown in Fig. 3.99.

So equation of motion of mass m is mg-T = ma .......(iii)

For terminal velocity, a = 0

mg−T = 0

T = mg ......(iv)

Equation of motion of rod is

T−BiL= 0 × a = 0 (as rod is massless)

T = BiL .......(v)

Using Eq. (iv), we get, mg = BiL