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(B) In an $LC$ circuit,the charge on the capacitor at any time $t$ is given by $q(t) = q_0 \cos(\omega t)$,where $\omega = \frac{1}{\sqrt{LC}}$.The en

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$\frac{\pi}{4} \sqrt{LC}$

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(B) In an $LC$ circuit,the charge on the capacitor at any time $t$ is given by $q(t) = q_0 \cos(\omega t)$,where $\omega = \frac{1}{\sqrt{LC}}$.The energy stored in the electric field is $U_E = \frac{q^2}{2C}$ and the total energy is $U_{total} = \frac{q_0^2}{2C}$.We are given that the energy is shared equally between the electric and magnetic fields,so $U_E = \frac{1}{2} U_{total}$.$\frac{q^2}{2C} = \frac{1}{2} \left( \frac{q_0^2}{2C} \right) \implies q^2 = \frac{q_0^2}{2} \implies q = \frac{q_0}{\sqrt{2}}$.Substituting $q(t) = q_0 \cos(\omega t)$,we get $q_0 \cos(\omega t) = \frac{q_0}{\sqrt{2}}$,which means $\cos(\omega t) = \frac{1}{\sqrt{2}}$.Thus,$\omega t = \frac{\pi}{4}$.Substituting $\omega = \frac{1}{\sqrt{LC}}$,we get $t = \frac{\pi}{4\omega} = \frac{\pi}{4} \sqrt{LC}$.

$A$ fully charged capacitor $C$ with initial charge $q_0$ is connected to a coil of self-inductance $L$ at $t = 0$. The time at which the energy is stored equally between the electric and the magnetic fields is

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