Explanation:
Current in \(LCR\) series circuit,
\(i = \frac{V}{{\sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} }}\)
where, \(V\) is rms value of voltage \(R\) 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, i.e., during the resonance of series \(LCR\) circuit
\({X_L} = {X_C}\quad {\rm{i}}{\rm{.e,}}\quad \omega L = \frac{1}{{\omega C}}\)
or \(L = \frac{1}{{{\omega ^2}C}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
Given,\(\omega = 1000{s^{ - 1}},\quad C = 10\mu F = 10 \times {10^{ - 6}}\,F\)
Hence,\(L = \frac{1}{{{{\left( {1000} \right)}^2} \times 10 \times {{10}^{ - 6}}}}\)
\( = 0.1\,H = 100\,mH\)