A wire PQ of mass m and resistance r is capable of sliding on smooth, horizontal parallel rails separated by a distance l . The rails' resistance is negligible. A uniform magnetic field B exists in the rectangular region, and a resistance R connects the rails outside the field region. At t = 0 , the wire PQ is pushed towards the right with a speed v₀ . Determine the maximum distance the wire will move.

Step-by-step solution

When the wire moves with velocity v , the motional emf induced is given by: = B l v The total resistance of the circuit is R_ eq = R + r . The induced current in the circuit is: i = R_ eq = B l v R + r The magnetic force acting on the wire opposes its motion and is given by: F = -i l B = - ( B l v R + r ) l B = - B^2 l^2 v R + r Using Newton's second law, F = m a = m v dv dx : m v dv dx = - B^2 l^2 v R + r m dv dx = - B^2 l^2 R + r dv = - B^2 l^2 m (R + r) dx Integrating both sides from the initial state ( x = 0, v