$A$ $10\, V, 60\, W$ bulb is to be connected to a $100\, V, 50\, Hz$ $AC$ source. To operate the bulb at its rated power,an induction coil is connected in series with it. Calculate the required self-inductance of the coil.

In a $L-R$ circuit of $4 \text{ mH}$ inductance and $3 \text{ } \Omega$ resistance, e.m.f. $E = \cos(1000t) \text{ V}$ is applied. The amplitude of current is (in $\text{ A}$)

$A$ $20 V$ $AC$ is applied to a circuit consisting of a resistor and a coil with negligible resistance. If the voltage across the resistor is $12 V$,the voltage across the coil is (in $V$)

An induction coil has an impedance of $10 \Omega$. When an $AC$ signal of frequency $1000 \ Hz$ is applied to the coil,the voltage leads the current by $45^\circ$. The inductance of the coil is

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 $(2)$:

An emf $E = 6 \cos(6000t) \ V$ is applied to an $L-R$ circuit of inductance $L = 4 \ mH$ and resistance $R = 7 \ \Omega$. The amplitude of the current in the circuit is . . . . . . (in $A$)