A rectangular wire-frame possesses a width d , mass m , resistance R , and a large length. A uniform magnetic field B is situated to the left of the frame. At t = 0 , a constant force F begins pushing the frame into the magnetic field. After a certain period, the frame will travel with a constant velocity until it fully enters the magnetic field. Determine this terminal velocity v₀ .

Step-by-step solution

When the rectangular frame enters the magnetic field with velocity v , an induced electromotive force (emf) is generated across its width d . The motional emf is given by: e = B d v The induced current in the frame is: i = e R = B d v R According to Lenz's law, the magnetic force F_m acting on the current-carrying wire opposes the motion of the frame. The magnitude of this force is: F_m = i d B = ( B d v R ) 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 The frame reaches