Q7.12 (d)    An $LC$ circuit contains a $20mH$ inductor and a $50\mu F$ capacitor with an initial charge of $10mC$. The resistance of the circuit is negligible. Let the instant the circuit is closed be $t=0$.At what times is the total energy shared equally between the inductor and the capacitor?

The energy will be shared equally when the energy in the capacitor is half of the maximum energy it can store.i.e.

$\frac{Q^2}{2C}=\frac{1}{2}\frac{Q_0^2}{2C}$

From Here, we got

$Q=\frac{Q_0}{\sqrt{2}}$

So Now, we know the charge on the capacitor, we can calculate time for which

$\frac{Q_0}{\sqrt{2}}=Q_0cos\left ( \frac{2\pi t}{T} \right )$

$\frac{1}{\sqrt{2}}=cos\left ( \frac{2\pi t}{T} \right )$

From here,

$t=\frac{T}{8},\frac{3T}{8},\frac{5T}{8}..$

Hence for these times, the total energy will be shared equally between capacitor and inductor.

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