$A$ coil of $0.01 \, H$ inductance and $1 \, \Omega$ resistance is connected to a $200 \, V, 50 \, Hz$ $AC$ supply. Find the impedance of the circuit and the time lag between the maximum alternating voltage and current.

(A) Given: Inductance $L = 0.01 \, H$,Resistance $R = 1 \, \Omega$,Voltage $V = 200 \, V$,Frequency $f = 50 \, Hz$.
$1$. Calculate the inductive reactance $X_L$:
$X_L = 2 \pi f L = 2 \times 3.1416 \times 50 \times 0.01 = 3.1416 \, \Omega$.
$2$. Calculate the impedance $Z$:
$Z = \sqrt{R^2 + X_L^2} = \sqrt{1^2 + (3.1416)^2} = \sqrt{1 + 9.8696} = \sqrt{10.8696} \approx 3.297 \, \Omega$.
$3$. Calculate the phase angle $\phi$:
$\tan \phi = \frac{X_L}{R} = \frac{3.1416}{1} = 3.1416$.
$\phi = \tan^{-1}(3.1416) \approx 72.34^{\circ}$.
$4$. Calculate the time lag $\Delta t$:
$\Delta t = \frac{\phi}{\omega} = \frac{\phi}{2 \pi f} = \frac{72.34 \times \pi / 180}{2 \times \pi \times 50} = \frac{72.34}{36000} \approx 0.00201 \, s$ or $2.01 \, ms$.