An inductor of \( 500\,\mathrm{mH} \) is in series with a resistance and a variable capacitor connected to a source of frequency \( 0.4\,\mathrm{kHz} \). The value of capacitance of the capacitor to get a maximum current will be
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 0.32\,\mu\mathrm{F} \)
The maximum current in a series LCR circuit occurs at resonance, when the inductive and capacitive reactances are equal, \( X_L = X_C \).
\( \displaystyle \omega L = \frac{1}{\omega C} \;\Rightarrow\; C = \frac{1}{\omega^{2}L} = \frac{1}{4\pi^{2}f^{2}L} \)
\( f = 0.4\,\mathrm{kHz} = 400\,\mathrm{Hz} \)
\( L = 500\,\mathrm{mH} = 0.5\,\mathrm{H} \)
\( \displaystyle C = \frac{1}{4\pi^{2} \times (400)^{2} \times 0.5} = \frac{1}{3.16\times10^{6}} = 3.2\times10^{-7}\,\mathrm{F} = 0.32\,\mu\mathrm{F} \)