A magnetic field $B =B_o \sin(wt) \hat{k}$ covers a large region where a wire AB slides smoothly over two parallel conductors separated by a distance d, as shown in the fig. The wire are in the x-y plane. The wire AB (of length d) has resistance R and the parallel wires have negligible resistance. It AB is moving velocity $v$, what is the force needed to keep the wire at constant velocity ?

Question

$(a) F = \frac{B_o^2 d^2}{R^2} \begin{bmatrix} v \sin wt + wx \cos wt \end{bmatrix} \sin wt$
$(b) F = \frac{B_o^2 R^2}{d^2} \begin{bmatrix} v \sin wt + wx \cos wt \end{bmatrix} \sin wt$
$(c) F = \frac{B_o^2 R^2}{d^2} \begin{bmatrix} v \sin wt + wx \cos wt \end{bmatrix}$
$(d) F = \frac{B_o^2 d^2}{R^2} \begin{bmatrix} v \sin wt + wx \cos wt \end{bmatrix}$