## Solution to 2008 Problem 36

 We use Faraday's Law in integral form\begin{align*}\int_{\text{closed loop}} \mathbf{E} \cdot d \mathbf{l} = -\frac{d}{dt}\left(\int_{\text{open surface}} \mathbf...We choose to traverse the loop in a clockwise direction. We also choose the reference direction of the current $I$ to be in the counter-clockwise direction. Then,\begin{align*}\int_{\text{closed loop}} \mathbf{E} \cdot d \mathbf{l} = V = \frac{Q}{C}\end{align*}where $V$ is the voltage difference between the upper left-hand corner of the circuit and the lower left-hand corner of the circuit and $Q$ is the charge on the upper plate of the capacitor. Also,\begin{align*}\frac{d}{dt}\left(\int_{\text{open surface}} \mathbf{B} \cdot d \mathbf{a}\right) = L \frac{dI}{dt} = L \frac{d...So, Faraday's Law is\begin{align*}-L \frac{d^2Q}{dt^2} = \frac{Q}{C}\end{align*}At $t = 0$, $Q = Q_0$ and $dQ/dt = 0$, so the solution to this equation is\begin{align*}Q(t) = Q_0 \cos \left(\frac{t}{\sqrt{LC}} \right)\end{align*}and\begin{align*}I(t) = \frac{dQ(t)}{dt} = -\frac{Q_0}{\sqrt{LC}} \sin \left(\frac{t}{\sqrt{LC}} \right)\end{align*}The energy stored in the inductor as a function of time is therefore\begin{align*}U = \frac{1}{2} L I^2 = \frac{Q_0^2}{2 C}\sin^2 \left(\frac{t}{\sqrt{LC}} \right)\end{align*}Thus, answer (A) is correct.