Question #43876

A pair of long, parallel, perfectly conducting rails are fixed on a horizontal rough plane (the $xy$-plane), separated by a distance $L$. The rails are parallel to the $y$-axis and are connected at $y=0$ by a wire of zero resistance. A uniform metal rod of mass $m$, length $L$, and resistance $R$ is placed perpendicularly across the rails. The coefficient of kinetic friction between the rod and the rails is $\mu$. A uniform magnetic field $\vec{B} = B_0 \hat{j} - B_0 \hat{k}$ exists in the entire region, where $\hat{j}$ is the unit vector along the positive $y$-axis and $\hat{k}$ is the unit vector vertically upwards. The acceleration due to gravity is $g$ in the $-\hat{k}$ direction. At $t=0$, the rod is located at $y > 0$ and is given an initial velocity $v_0 \hat{j}$. Assuming the rod remains in contact with the rails at all times, which of the following expressions gives the time $T$ taken by the rod to come to rest?