Consider a wire of mass m and length l that is free to slide on a pair of smooth vertical rails, as shown in the figure. A magnetic field B is present in the region, directed perpendicular to the plane of the rails. A capacitor of capacitance C connects the rails at their top end. Neglecting any electrical resistance, determine the acceleration of the wire.

Step-by-step solution

Let the velocity of the wire at any instant t be v downwards. The motional emf induced in the wire is given by e = Bvl . Since the electrical resistance of the circuit is negligible, the potential difference across the capacitor is equal to the induced emf, so V = Bvl . The charge on the capacitor at this instant is q = CV = CBvl . The current in the circuit is the rate of change of charge: i = dq dt = d dt (CBvl) = CBl dv dt = CBla where a = dv dt is the acceleration of the wire. The magnetic force acting on the c