Explain the motional $emf$ by the Lorentz force acting on the free charge carriers of a conductor.

(N/A) Consider any arbitrary charge $q$ in the conductor $PQ$ shown in the figure. When the rod moves with speed $v$,the charge will also be moving with speed $v$ in the magnetic field $B$.
The Lorentz force on this charge is $F = qvB$ in magnitude,and its direction is towards $Q$ (determined by the right-hand rule). All charges experience the same force in magnitude and direction,irrespective of their position in the rod $PQ$.
The work done in moving the charge from $P$ to $Q$ is $W = F \cdot l = (qvB)l = qvBl$.
Since $emf$ $(\varepsilon)$ is the work done per unit charge,
$\varepsilon = \frac{W}{q} = \frac{qvBl}{q} = Blv$.
This equation gives the $emf$ induced across the rod $PQ$ due to its motion in a magnetic field.