In a circuit \(L\), \(C\) and \(R\) are connected in series with an alternating voltage source of frequency \(f\). If the current lags the voltage by \(45^\circ\), then the value of \(L\) will be
\(\frac{2\pi f C R + 1}{4\pi^2 f^2 C}\)
\(\frac{2\pi f C R - 1}{C}\)
\(\frac{2\pi f C R + 1}{2\pi f C}\)
\(\frac{2\pi f C R + 1}{C}\)
Solution:
Since current lags voltage, \(tan(45^\circ) = \frac{X_L - X_C}{R} = 1 ⇒ X_L - X_C = R\). Substituting \(X_L = 2\pi f L\) and \(X_C = \frac{1}{2\pi f C}\), we get \(2\pi f L = R + \frac{1}{2\pi f C} = \frac{2\pi f C R + 1}{2\pi f C}\). Thus, \(L = \frac{2\pi f C R + 1}{4\pi^2 f^2 C}\).
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