A long rectangular conducting loop of width '$l$' mass '$m$' and resistance '$R$' is placed partly in a perpendicular magnetic field '$B$'. With what velocity should it be pushed downwards so that it may continue to fall without any acceleration? ($g$ = acceleration due to gravity)

When the loop falls with a velocity $v$, the motional emf induced in the top horizontal segment of the loop is given by:

$\varepsilon = Bvl$

The induced current in the loop is:

$I = \dfrac{\varepsilon}{R} = \dfrac{Bvl}{R}$

This current experiences an upward magnetic force due to the magnetic field $B$. The magnitude of this force is:

$F_m = IlB = \left(\dfrac{Bvl}{R}\right)lB = \dfrac{B^2l^2v}{R}$

For the loop to fall without any acceleration, the net force on it must be zero. Therefore, the upward magnetic force must balance the downward gravitational force:

$F_m = mg$

$\dfrac{B^2l^2v}{R} = mg$

Solving for $v$, we get:

$v = \dfrac{mgR}{B^2l^2}$

Answer: $\dfrac{mgR}{B^2 l^2}$