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(A) In a series $LCR$ circuit, maximum current flows at resonance, where the inductive reactance equals the capacitive reactance $(X_L = X_C)$.At reso

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$0.1$

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(A) In a series $LCR$ circuit, maximum current flows at resonance, where the inductive reactance equals the capacitive reactance $(X_L = X_C)$.At resonance, the potential difference across the inductor $(V_L)$ and the capacitor $(V_C)$ are equal in magnitude but opposite in phase, so $V_L = V_C = I_{max} X_L = I_{max} X_C$.The energy stored in the capacitor is $U_C = \frac{1}{2} C V_C^2$ and the energy stored in the inductor is $U_L = \frac{1}{2} L I_{max}^2$.Since $V_C = I_{max} X_C$, we have $U_C = \frac{1}{2} C (I_{max} X_C)^2 = \frac{1}{2} C I_{max}^2 (\frac{1}{\omega C})^2 = \frac{1}{2} I_{max}^2 (\frac{1}{\omega^2 C})$.Using $\omega^2 = \frac{1}{LC}$, we get $U_C = \frac{1}{2} I_{max}^2 (LC) = \frac{1}{2} L I_{max}^2$.Thus, $U_C = U_L$, which means the ratio $U_C : U_L = 1 : 1$. However, checking the provided options, there seems to be a discrepancy. Re-evaluating the question context: if the question implies a specific frequency or state, but standard resonance is assumed, the ratio is $1:1$. Given the options provided, if we calculate $X_L = \omega L$ and $X_C = \frac{1}{\omega C}$, at resonance $\omega = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{10^{-3} 	imes 2 	imes 10^{-6}}} = \frac{1}{\sqrt{2 	imes 10^{-9}}} \approx 22360 	ext{ rad/s}$. The ratio of energies $U_C/U_L = (\frac{1}{2} C V_C^2) / (\frac{1}{2} L I^2) = (\frac{1}{2} C (I X_C)^2) / (\frac{1}{2} L I^2) = (C X_C^2) / L = C (\frac{1}{\omega C})^2 / L = \frac{1}{\omega^2 L C}$. Since $\omega^2 = \frac{1}{LC}$, the ratio is $1$. None of the options match $1:1$. Assuming a typo in the question's premise, if the ratio was intended to be calculated differently, based on standard physics, the answer is $1:1$.

In a series $LCR$ circuit, $C = 2 \text{ } \mu\text{F}$, $L = 1 \text{ mH}$, and $R = 10 \text{ } \Omega$. What is the ratio of energies stored in the capacitor and inductor when maximum current flows through the circuit (in $: 1$)?

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