An emf \( E = 200\sin 300t \) volt is applied across an inductor coil having a resistance of \( 1.0\,\Omega \). If the maximum current is found to be 10 A, then the value of inductance is approximately:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 67\,\mathrm{mH} \)
A coil with resistance \( R \) and inductance \( L \) in an ac circuit has an impedance \( Z = \sqrt{R^{2} + X_L^{2}} \), with \( X_L = \omega L \). Comparing \( E = 200\sin 300t \) with \( E = E_0\sin\omega t \) gives \( E_0 = 200\,\mathrm{V} \) and \( \omega = 300\,\mathrm{rad\,s^{-1}} \).
The maximum current is \( I_0 = E_0/Z \):
\( \displaystyle Z = \frac{E_0}{I_0} = \frac{200}{10} = 20\,\Omega \)
\( \displaystyle X_L = \sqrt{Z^{2} - R^{2}} = \sqrt{20^{2} - 1^{2}} = \sqrt{399} \approx 19.97\,\Omega \)
\( \displaystyle L = \frac{X_L}{\omega} = \frac{19.97}{300} \approx 0.0666\,\mathrm{H} \approx 67\,\mathrm{mH} \)