What is the value of inductance \(L\) for which the current is a maximum in a series \(L C R\) circuit with \(C=10 \mu \mathrm{F}\) and \(\omega=1000 \mathrm{~s}^{-1}\) ?

Current in \(L C R\) 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 \(\quad L=\frac{1}{\omega^{2} C}\) ...(i)
\(\begin{aligned} \text { Given, } \omega &=1000 \mathrm{~s}^{-1}, C=10 \mu \mathrm{F} \\ &=10 \times 10^{-6} \mathrm{~F} \\ \text { Hence, } L &=\frac{1}{(1000)^{2} \times 10 \times 10^{-6}} \\ &=0.1 \mathrm{H} \\ &=100 \mathrm{mH} \end{aligned}\)