An LCR series circuit with L = 100 mH, C = \( 100\,\mu\mathrm{F} \), R = \( 10\,\Omega \) is connected to an AC source of emf 220 V. The resonant frequency of the circuit will be nearly
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 50\,\mathrm{Hz} \)
At resonance the inductive and capacitive reactances are equal, \( \omega L = 1/(\omega C) \). The resonant frequency depends only on \( L \) and \( C \); the resistance and the source emf do not affect it.
\( \displaystyle f_0 = \frac{1}{2\pi\sqrt{LC}} \)
\( L = 100\,\mathrm{mH} = 0.1\,\mathrm{H} \)
\( C = 100\,\mu\mathrm{F} = 10^{-4}\,\mathrm{F} \)
\( \displaystyle \sqrt{LC} = \sqrt{0.1 \times 10^{-4}} = \sqrt{10^{-5}} = 3.16\times10^{-3}\,\mathrm{s} \)
\( \displaystyle f_0 = \frac{1}{2\pi \times 3.16\times10^{-3}} = \frac{1}{1.99\times10^{-2}} \approx 50\,\mathrm{Hz} \)