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# A resistor R, inductor L, and capacitor C are connected in series to an AC source of rms voltage $\Delta V$ and variable frequency. If the operating frequency is twice the resonance frequency, find the energy delivered to the circuit during one period.

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The solution is the same as in the previous problem only without introducing the numerical values. The energy supplied in one period is determined from the power relation and the relations of the resonant and regular frequency:

\begin{align*} E&=PT\\ &=\dfrac{V_{\text{rms}}^{2}R\omega^{2}}{R^{2}\omega^{2}+L^{2}(\omega^{2}-\omega_{0}^{2})^{2}}\dfrac{2\pi}{\omega}\\ &=\dfrac{V_{\text{rms}}R\cdot4\omega_{0}^{2}}{R^{2}\cdot4\omega_{0}^{2}+L^{2}\cdot9\omega_{0}^{4}}\dfrac{2\pi}{2\omega_{0}}\\ &=\dfrac{4\pi V_{\text{rms}}^{2}R}{\omega_{0}(4R^{2}+9L^{2}\omega_{0}^{2})}\\ &=\dfrac{4\pi V_{\text{rms}}^{2}R}{\dfrac{1}{\sqrt{LC}}\bigg(4R^{2}+\dfrac{9L^{2}}{LC}\bigg)}\\ &=\boxed{\dfrac{4\pi V_{\text{rms}}^{2}RC\sqrt{LC}}{4R^{2}C+9L}} \end{align*}

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