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Q.
What is the value of inductance $ L $ for which the current is a maximum in a series $ LCR $ circuit with $ C = 10\, μF $ and $ ω = 1000^{−1}\, s $ ?
In resonance condition, maximum current flows in the circuit.
Current in $LCR$ series circuit,
$i=\frac{V}{\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}}}$
where $V$ is rms value of current, $R$ is resistance, $X_L$ is inductive reactance and $X_C$ is capacitive reactance.
For current to be maximum, denominator should be minimum which can be done, if
$X_{L}=X_{C}$
This happens in resonance state of the circuit ie,
$\omega L=\frac{1}{\omega C}$
or $L=\frac{1}{\omega^{2}C}\quad\ldots\left(i\right)$
Given, $\omega=1000\,s^{-1}$, $C=10\,\mu F=10\times10^{-6}\,F$
Hence, $L=\frac{1}{\left(1000\right)^{2}\times10\times10^{-6}}$
$=0.1\,H$
$=100\,mH$