In an $LR$-circuit,the inductive reactance is equal to the resistance $R$ of the circuit. An e.m.f. $E = E_0 \cos(\omega t)$ is applied to the circuit. The power consumed in the circuit is:

An $AC$ source of angular frequency $\omega$ is connected across a resistor $R$ and a capacitor $C$ in series. The current registered is $I$. If the frequency of the source is changed to $\omega/3$ (maintaining the same voltage),the current in the circuit is found to be halved. Calculate the ratio of reactance to resistance at the original frequency $\omega$.

In an $RL$ circuit,the reactance of the coil is $\sqrt{3}$ times the resistance. The phase difference between the voltage and the current is:

$A$ $10\, V, 60\, W$ 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.

An a.c. source of angular frequency $\omega$ is connected across a resistor $R$ and a capacitor $C$ in series. The current is observed as $I$. Now the frequency of the source is changed to $\omega/4$,(keeping the voltage unchanged) the current is found to be $I/3$. The ratio of resistance to reactance at frequency $\omega$ is

With an alternating voltage source of frequency $f$,an inductor $L$,a capacitor $C$,and a resistor $R$ are connected in series. The voltage leads the current by $45^{\circ}$. The value of $L$ is $(\tan 45^{\circ} = 1)$.