An inductive coil has a resistance of $100 \Omega$. When an a.c. 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 series $LR$ circuit,$X_L=R$,the power factor is $P_1$. If a capacitor of capacitance $C$ with $X_C=X_L$ is added to the circuit,the power factor becomes $P_2$. The ratio of $P_1$ to $P_2$ will be

In an $L-C-R$ series circuit,if $V$,$V_R$,$V_L$,and $V_C$ are the voltages across the source,resistor,inductor,and capacitor respectively at any instant,choose the correct relation.

$A$ bulb and a capacitor are connected in series across an $AC$ supply. $A$ dielectric is then placed between the plates of the capacitor. The glow of the bulb:

$A$ charged capacitor discharges through a resistance $R$ with time constant $\tau$. The two are now placed in series across an $AC$ source of angular frequency $\omega = \frac{1}{\tau}$. The impedance of the circuit will be-

$A$ series combination of resistor $R$ and capacitor $C$ is connected to an $A$.$C$. source of angular frequency $\omega$. Keeping the voltage same,if the frequency is changed to $\frac{\omega}{3}$,the current becomes half of the original current. Then the ratio of capacitive reactance and resistance at the former frequency is