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Question

(A) At $t=0$,the capacitor is uncharged and the flux through the inductor is $\phi$. Using the relation $\phi = L I$,the initial current in the circui

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current in the circuit is $I(t)=(\phi / L) \cos \omega_{0} t$

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(A) At $t=0$,the capacitor is uncharged and the flux through the inductor is $\phi$. Using the relation $\phi = L I$,the initial current in the circuit at $t=0$ is $I_{0} = \phi / L$. The circuit forms an $LC$ oscillator. The differential equation for the charge $q$ on the capacitor is $L \frac{d^{2} q}{d t^{2}} + \frac{q}{C} = 0$,which simplifies to $\frac{d^{2} q}{d t^{2}} + \omega_{0}^{2} q = 0$,where $\omega_{0} = 1 / \sqrt{LC}$. The general solution for the charge is $q(t) = A \sin(\omega_{0} t + \delta)$. Since $q(0) = 0$,we have $\delta = 0$,so $q(t) = A \sin(\omega_{0} t)$. The current is $I(t) = \frac{dq}{dt} = A \omega_{0} \cos(\omega_{0} t)$. At $t=0$,$I(0) = A \omega_{0} = I_{0} = \phi / L$. Therefore,$A = \phi / (L \omega_{0})$. Substituting $A$ back,the current is $I(t) = (\phi / L) \cos(\omega_{0} t)$. Thus,option $A$ is correct.

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