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# An LC circuit contains a 20 mH inductor and a 50 omega F capacitor with an initial charge of 10 mC.

Q7.12  An $LC$ circuit contains a $20mH$ inductor and a $50\mu F$ capacitor withan initial charge of $10mC$. The resistance of the circuit is negligible. Let the instant the circuit is closed be $t=0$.

( c)     At what time is the energy stored

(i) completely electrical (i.e., stored in the capacitor)?

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at any instant, the charge on the capacitor is:

$Q=Q_0cos(w_{natural}t)=Q_0cos(2\pi f_{natural}t)=Q_0cos\left ( \frac{2\pi t}{T} \right )$

Where time period :

$T=\frac{1}{f_{natural}}=\frac{1}{159}=6.28ms$

Now, when the total energy is purely electrical, we can say that

$Q=Q_0$

$Q_0=Q_0cos(\frac{2\pi}{T})$

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

this is possible when

$t=0,\frac{T}{2},T,\frac{3T}{2}....$

Hence Total energy will be purely electrical(stored in a capacitor) at

$t=0,\frac{T}{2},T,\frac{3T}{2}....$.

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