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A rod of mass $m$, resistance $R$ and length $L$ is sitting at rest on frictionless rails in a magnetic field as shown. At $t = 0$, the switch $S$ is closed and a voltage $V$ applied across the rails. Show all work while deriving the following results, clearly indicating the physical law used and reasoning process. Neatness and clarity count.

a) What is the net voltage across the resistance $R$ as a function of $\vert\vec{v}\vert$?

b) What is the current $I$ in the loop as a function of $\vec{v}$?

c) What is the force $\vec{F}$ on the rod as a function of $\vec{v}$?

d) What is the terminal velocity of the rod as $t \to \infty$?

10 points of extra Credit: Solve the first order, linear, ordinary, inhomogeneous differential equation and find the velocity of the rod $\vec{v}(t)$ as a function of time. Draw a qualitatively correct curve showing this function and show how it corresponds to your answer to d).