 Solution

For $B_0\ne 0$, Newton's second law givesDefine the exponential relaxation time $\tau =mR/(B_0^2{L}^2)$ and terminal velocity $v_{\infty }={\mathcal{E}}_0/(B_{0}L)$. Solving the first-order equation for $v=\dot{s}$ and then integrating givesBoth prescribed initial values hold. The current approaches zero as the bar approaches terminal velocity. If $B_0=0$, there is no magnetic force and instead $\dot{s}=v_0$, $s=s_0+v_{0}t$; the constant current is ${\mathcal{E}}_0/R$.