A rectangular wire-frame having a width d , mass m , resistance R , and a large length is positioned such that a uniform magnetic field B exists to its left. At t = 0 , a constant force F starts pushing the frame into the magnetic field. If v₀ is the constant terminal velocity the frame eventually attains, evaluate the expression for the velocity v of the frame at any time t before it fully enters the field.

Step-by-step solution

Let the velocity of the frame at time t be v . The induced motional emf in the frame is = B d v . The induced current is I = R = B d v R . The magnetic force opposing the motion of the frame is F_m = I d B = B^2 d^2 v R . The net force acting on the frame is F_ net = F - F_m = F - B^2 d^2 v R . Using Newton's second law, m dv dt = F - B^2 d^2 v R . When the frame attains the constant terminal velocity v₀ , the net force becomes zero, i.e., dv dt = 0 . F - B^2 d^2 v₀ R = 0 B^2 d^2 R = F v₀ . Substituting this back i