Question 25: In a series L C R circuit, the inductance L is 10 mH , capacitance C is 1 \mu \mathrm{~F} and resistance R is 100 \Omega. The frequency at which resonance occurs is :
(1) 1.59 \mathrm{rad} / \mathrm{s}
(2) 1.59 kHz
(3) 15.9 \mathrm{rad} / \mathrm{s}
(4) 15.9 kHz
Answer: Option (2)
Explanation:
In a series LCR circuit, resonance occurs when the inductive reactance equals the capacitive reactance.
The resonant frequency is given by
f=\frac{1}{2\pi\sqrt{LC}}Given inductance,
L=10\,\mathrm{mH}=10\times10^{-3}\,\mathrm{H}=0.01\,\mathrm{H}Given capacitance,
C=1\,\mu\mathrm{F}=1\times10^{-6}\,\mathrm{F}Substituting the values,
\sqrt{LC}=\sqrt{0.01\times10^{-6}}=\sqrt{10^{-8}}=10^{-4} f=\frac{1}{2\pi\times10^{-4}} f\approx\frac{1}{6.28\times10^{-4}} f\approx1.59\times10^{3}\,\mathrm{Hz}=1.59\,\mathrm{kHz}Hence, the frequency at which resonance occurs is 1.59\,\mathrm{kHz}.
Therefore, the correct answer is Option (2).