A conducting wire of length l , situated perpendicular to a...

When the conducting wire moves in the magnetic field, the free electrons experience a magnetic Lorentz force.

The magnitude of the magnetic force on an electron of charge $e$ is given by:
$F_m = evB$

This force causes electrons to accumulate at one end of the wire, creating an electric field $E$ along its length. The electric field exerts an electric force on the electrons in the opposite direction, with magnitude:
$F_e = eE$

In the steady state, the redistribution of electrons stops when the electric force balances the magnetic force:
$F_e = F_m$

Substituting the expressions for the forces, we get:
$eE = evB$

$E = vB$

Alternatively, the motional emf developed across the ends of the wire is $\varepsilon = Bvl$. The uniform electric field $E$ is related to the potential difference by $E = \dfrac{\varepsilon}{l}$.

Therefore, $E = \dfrac{Bvl}{l} = vB$.