Having a mass m and resistance r , a wire PQ can slide on smooth, horizontal parallel rails separated by a distance l . The rails have negligible resistance. A uniform magnetic field B is present in the rectangular region, and a resistance R connects the rails outside the field region. At t = 0 , the wire PQ is projected towards the right with a speed v₀ . Find the velocity v of the wire as a function of its displace

Step-by-step solution

When the wire moves with velocity v , the motional emf induced in it is given by: E = Bvl The total resistance of the circuit is R_ eq = R + r . The induced current is: I = E R_ eq = Bvl R + r The magnetic force acting on the wire opposes its motion (by Lenz's law) and is given by: F = -IlB = - ( Bvl R + r )lB = - B^2 l^2 v R + r Using Newton's second law, F = ma = mv dv dx : mv dv dx = - B^2 l^2 v R + r Canceling v from both sides: dv = - B^2 l^2 m(R + r) dx Integrating both sides with initial conditions v = v₀ at