A) \[\pi \sqrt{LC}\]
B) \[\frac{\pi }{4}\sqrt{LC}\]
C) \[2\pi \sqrt{LC}\]
D) \[\sqrt{LC}\]
Correct Answer: B
Solution :
[b] Charge on the capacitor at any time 'f', \[q={{q}_{0}}\cos \omega t\] ...(i) At time when energy stored equally in electric and magnetic field, at this time Energy of a capacitor \[=\frac{1}{2}\]Total energy \[\frac{1}{2}\frac{{{q}^{2}}}{C}=\frac{1}{2}\left( \frac{1}{2}\frac{q_{0}^{2}}{C} \right)\Rightarrow q=\frac{{{q}_{0}}}{\sqrt{2}}\] From equation (i) \[\frac{{{q}_{0}}}{\sqrt{2}}={{q}_{0}}\cos \omega t\] \[\cos \omega t=\frac{1}{\sqrt{2}}\Rightarrow \omega t={{\cos }^{-1}}\left( \frac{1}{\sqrt{2}} \right)=\frac{\pi }{4}\] \[t=\frac{\pi }{4\omega }=\frac{\pi }{4}\sqrt{LC}\left( \therefore \omega =\frac{1}{\sqrt{LC}} \right)\]
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