$A$ bar of mass $m$,length $d$,and resistance $R$ slides without friction in a horizontal plane,moving on parallel rails as shown in the figure. $A$ battery that maintains a constant emf $\varepsilon$ is connected between the rails,and a constant magnetic field $\vec{B}$ is directed perpendicularly to the plane of the page. Assuming the bar starts from rest,find the speed at time $t$.

• A
  $v = \frac{\varepsilon}{Bd}\left(1 - e^{\frac{B^2dt}{mR}}\right)$
• B
  $v = \frac{\varepsilon}{Bd}\left(1 - e^{\frac{B^2d^2t}{2mR}}\right)$
• C
  $v = \frac{\varepsilon}{Bd}\left(1 - e^{-\frac{B^2d^2t}{mR}}\right)$
• D
  $v = \frac{\varepsilon}{Bd}\left(1 - e^{\frac{Bdt}{mR}}\right)$