For the series $LCR$ circuit,$R = \frac{X_L}{2} = 2 X_C$. The impedance of the circuit and the phase difference between $V$ and $I$ will be

$A$ resistance of $40\, \Omega$ and an inductance of $95.5\, \text{mH}$ are connected in series in a $50\, \text{Hz}$ $AC$ circuit. The impedance of this combination is very nearly.....$\Omega$.

An inductance coil has a resistance of $80 \Omega$. When an $AC$ signal of frequency $480 \text{ Hz}$ is applied to the coil,the voltage leads the current by $45^{\circ}$. The inductance of the coil in henry is $\left[\sin 45^{\circ}=\cos 45^{\circ}=1 / \sqrt{2}\right]$

$A$ $220\, V$,$50\, Hz$ $ac$ source is connected to an inductance of $0.2\, H$ and a resistance of $20\, \Omega$ in series. What is the current in the circuit (in $, A$)?

When a $DC$ voltage of $100 \, V$ is applied to an inductor, a $DC$ current of $5 \, A$ flows through it. When an $AC$ voltage of $200 \, V$ peak value is connected to the inductor, its inductive reactance is found to be $20\sqrt{3} \, \Omega$. The power dissipated in the circuit is . . . . . . $W$.

In a circuit,a metal filament lamp is connected in series with a capacitor of capacitance $C \mu F$ across a $200 V, 50 Hz$ supply. The power consumed by the lamp is $500 W$ while the voltage drop across it is $100 V$. Assume that there is no inductive load in the circuit. Take $rms$ values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $\varphi$.
Assume,$\pi \sqrt{3} \approx 5$.
$(1)$ The value of $C$ is . . . . . .
$(2)$ The value of $\varphi$ is
Give the answers of the questions $(1)$ and $(2)$: