Two long, smooth, horizontal conducting rails are fixed parallel to each other a distance l apart. A light, perfectly conducting rod of the same length l can slide on the rails without friction. The rails are connected by a resistor R at one end, forming a closed rectangular circuit. A uniform magnetic field B is directed vertically downward over the entire region. Initially the system is at rest. The rod is given an impulse so that it starts sliding to the right with speed v0 at t=0.
(i) Obtain an expression for the emf induced in the circuit at any instant when the rod has speed v.
(ii) Using the appropriate magnetic force on the charges in the rod, write the differential equation of motion for v(t) and hence show that v(t)=v0e−RmB2l2t, where m is the mass of the rod.
(iii) Derive an expression for the instantaneous power dissipated as heat in the resistor.
(iv) Prove that, at every instant, the rate at which the kinetic energy of the rod decreases is exactly equal to the electrical power found in part (iii), thereby confirming conservation of energy.