A sinusoidal voltage of angular frequency 500 rad/s is applied to a series LCR circuit with \( R=5\,\Omega \), \( L=20\,\mathrm{mH} \) and \( C=400\,\mu\mathrm{F} \). The impedance of the circuit is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 5\sqrt{2}\,\Omega \)
In a series LCR circuit the impedance is \( Z=\sqrt{R^{2}+(X_L-X_C)^{2}} \), with the inductive reactance \( X_L=\omega L \) and the capacitive reactance \( X_C=1/\omega C \).
\( \begin{aligned} X_L &= \omega L \\ &= 500\,\mathrm{rad\,s^{-1}}\times20\times10^{-3}\,\mathrm{H} \\ &= 10\,\Omega \end{aligned} \)
\( \begin{aligned} X_C &= \frac{1}{\omega C} \\ &= \frac{1}{500\,\mathrm{rad\,s^{-1}}\times400\times10^{-6}\,\mathrm{F}} \\ &= 5\,\Omega \end{aligned} \)
\( \begin{aligned} Z &= \sqrt{R^{2}+(X_L-X_C)^{2}} \\ &= \sqrt{5^{2}+(10-5)^{2}}\,\Omega \\ &= \sqrt{50}\,\Omega \\ &= 5\sqrt{2}\,\Omega \end{aligned} \)