A series combination of a wire of resistance of \( 2\,\Omega \), a coil of inductance \( 0.01\,\mathrm{H} \), and a capacitor is connected across a \( 200\,\mathrm{V} \), \( 50\,\mathrm{Hz} \) supply. The capacitance of the capacitor so that the circuit resonates will be approximately:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 10\times10^{-4}\,\mathrm{F} \)
A series LCR circuit is at resonance when \( X_L = X_C \), i.e. \( \omega L = 1/\omega C \). The resistance and the supply voltage do not affect the resonance condition.
\( \displaystyle C = \frac{1}{\omega^{2}L} \)
\( \omega = 2\pi f = 2\pi\times50\,\mathrm{Hz} = 100\pi\,\mathrm{rad\,s^{-1}} \)
\( \displaystyle C = \frac{1}{(100\pi\,\mathrm{rad\,s^{-1}})^{2}\times0.01\,\mathrm{H}} = \frac{1}{9.87\times10^{4}\times0.01}\,\mathrm{F} \)
\( C \approx 1.01\times10^{-3}\,\mathrm{F} \approx 10\times10^{-4}\,\mathrm{F} \)