A V-shaped conducting rail has a constant resistance R connected across its vertex. A resistanceless conducting bar moves away from the vertex at a constant velocity v . A uniform magnetic field B exists perpendicular to the plane of the rails. If the external mechanical force required to maintain the constant velocity of the bar is proportional to t^n , then the value of n is ______.

Step-by-step solution

Let the angle of the V-shaped rail be 2 . As the bar moves with a constant velocity v , its distance from the vertex at time t is y = vt . The active length of the conducting bar inside the magnetic field is l = 2y = 2vt . The induced motional EMF in the bar is: E = B l v = B (2vt ) v = (2Bv^2 ) t The induced current in the loop is: I = E R = 2Bv^2 R t The magnetic force acting on the current-carrying bar opposes its motion and is given by: F_m = I l B = ( 2Bv^2 R t ) (2vt ) B = 4B^2v^3 ^2 R t^2 To maintain a const