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12.3: Electrical Analogue

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  • Suppose that an alternating potential difference \( E=\hat{E}\sin\omega t\) is applied across an LCR circuit. We refer to Equation 11.6.3, and we see that the equation that governs the charge on the capacitor is

    \[ L\ddot{Q}+R\dot{Q}+\frac{Q}{C}=\hat{E}\sin\omega t. \label{12.3.1}\]

    We can differentiate both sides with respect to time, and divide by \( L\), and hence see that the current is given by

    \[ \ddot{I}+\frac{R}{L}\dot{I}+\frac{1}{LC}I=\frac{\hat{E}\omega}{L}\cos\omega t. \label{12.3.2}\]

    We can compare this directly with Equation 12.2.2, so that we have

    \[ \gamma = \frac{R}{L},\quad \omega_{0}^{2}=\frac{1}{\sqrt{LC}},\quad \hat{f}=\frac{\hat{E}\omega}{L}. \label{12.3.3}\]

    Then, by comparison with Equation 12.2.5, we see that I will lag behind \( E\) by \( \alpha\), where

    \[ \tan\alpha =\frac{\frac{R\omega}{L}}{\frac{1}{LC}-\omega^{2}}=\frac{R}{\frac{1}{C\omega}-L\omega}. \label{12.3.4}\]

    This is just what we obtain from the more familiar complex number approach to alternating current circuits.