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Q.

In the following electrical network at t < 0 (as given the figure), key is placed on (1) till the capacitor got fully charged. Key is placed on (2) at t = 0. Time when the energy in both the capacitor and the inductor will be same for the first time is

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a

$\frac{2 π}{3} \sqrt{L C}$

b

$\frac{π}{3} \sqrt{L C}$

c

$\frac{π}{4} \sqrt{L C}$

d

$\frac{3 π}{4} \sqrt{L C}$

answer is A.

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Detailed Solution

When energy on both is same, means energy on capacitor is half of its maximum energy.
$\frac{q^{2}}{2 C} = \frac{1}{2} \frac{Q^{2}}{2 C} \Rightarrow q = \frac{Q}{\sqrt{2}}$
$\Rightarrow Q \cos ω t = \frac{Q}{\sqrt{2}} \Rightarrow \cos ω t = \frac{1}{\sqrt{2}}$
$\Rightarrow ω t = \frac{π}{4} \Rightarrow t = \frac{π}{4 ω} = \frac{π}{4} \sqrt{L C}$

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