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(D) The total energy $U$ in an $LC$ circuit is constant and is given by $U = \frac{1}{2} \frac{Q_{0}^{2}}{C}$.When the energy stored in the inductor $

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$\frac{Q_{0}}{\sqrt{2}}$

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(D) The total energy $U$ in an $LC$ circuit is constant and is given by $U = \frac{1}{2} \frac{Q_{0}^{2}}{C}$.When the energy stored in the inductor $(U_L)$ and the capacitor $(U_C)$ are equal,each must be half of the total energy.So,$U_C = \frac{U}{2}$.Substituting the expressions for energy: $\frac{1}{2} \frac{q^2}{C} = \frac{1}{2} \left( \frac{1}{2} \frac{Q_{0}^{2}}{C} \right)$.Simplifying the equation: $\frac{q^2}{C} = \frac{Q_{0}^{2}}{2C}$.$q^2 = \frac{Q_{0}^{2}}{2}$.Taking the square root on both sides,we get $q = \frac{Q_{0}}{\sqrt{2}}$.

The charge of the capacitor in an $LC$ oscillatory circuit,when the energy associated with the inductor and capacitor are equal,is . . . . . . . ($Q_{0}$ is the initial charge on the capacitor.)

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