An inductor \( 500\,\mathrm{mH} \), a capacitor \( 200\,\mu\mathrm{F} \) and a resistor \( 10\,\Omega \), are connected in series with a \( 150\,\mathrm{V} \) variable frequency a.c. source. The approximate frequency of a.c. source, at which power factor of the circuit is unity, is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 16\,\mathrm{Hz} \)
The power factor \( \cos\phi = R/Z \) is unity when \( Z = R \), that is when \( X_L = X_C \): the circuit is at resonance.
\( \displaystyle \omega_0 = \frac{1}{\sqrt{LC}}, \qquad f_0 = \frac{1}{2\pi\sqrt{LC}} \)
\( L = 500\,\mathrm{mH} = 0.5\,\mathrm{H} \)
\( C = 200\,\mu\mathrm{F} = 2\times10^{-4}\,\mathrm{F} \)
\( LC = 0.5\times2\times10^{-4} = 10^{-4}\,\mathrm{s^{2}} \Rightarrow \sqrt{LC} = 10^{-2}\,\mathrm{s} \)
\( \displaystyle f_0 = \frac{1}{2\times3.14\times10^{-2}}\,\mathrm{Hz} = \frac{100}{6.28}\,\mathrm{Hz} \approx 16\,\mathrm{Hz} \)
The resistance and the source voltage do not affect the resonant frequency.