At t = 0 , a conducting wire ab of length l , resistance r , and mass m begins sliding down a smooth, vertical, thick pair of connected rails. A uniform magnetic field B is present in the space, directed perpendicular to the plane of the rails. After steady state is reached, what is the relationship between the rate of heat developed in the wire and the rate at which the gravitational potential energy decreases?

Step-by-step solution

Let the velocity of the wire at time t be v . The motional emf induced in the wire is given by E = B l v . The induced current in the circuit is i = E r = B l v r . The magnetic force acting on the wire opposes its downward motion and is given by F_m = i l B = B^2 l^2 v r . The equation of motion for the wire is: m dv dt = mg - F_m = mg - B^2 l^2 v r When the wire reaches its terminal velocity v_m , its acceleration becomes zero, so dv dt = 0 . mg = B^2 l^2 v_m r B^2 l^2 r = mg v_m Substituting this back into the e