Velocity of the rod $v (t)$ ?

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Solution

Let at time $t$ velocity of rod be $v$ (towards right) and current in the circuit is $i$ (from $a$ to $b$ ). The magnetic force on it is $i l B$ (towards right). Writing the equation of motion of the rod.
$m \cdot \frac{d v}{d t} = i l B = (\frac{E_{0} - B l v}{R}) l B$
$\int_{0}^{v} \frac{d v}{\frac{E_{0} B l}{m R} - \frac{B^{2} l^{2}}{m R} v} = \int_{0}^{t} d t$
$∴ v = \frac{E_{0}}{B l} ⎛ ⎜ ⎜ ⎝ 1 - e^{\frac{B^{2} l^{2}}{m R} t} ⎞ ⎟ ⎟ ⎠$

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