In the $AC$ circuit shown,$E = E_0 \sin(\omega t + \phi)$ and $i = i_0 \sin(\omega t + \phi + \frac{\pi}{4})$. Then,the box contains:

When an inductor $L$ and a resistor $R$ in series are connected across a $15 \, V, 50 \, Hz$ a.c. supply, a current of $0.3 \, A$ flows in the circuit. The current differs in phase from the applied voltage by $(\frac{\pi}{3})^c$. The value of $R$ is:
$(\sin \frac{\pi}{6} = \cos \frac{\pi}{3} = \frac{1}{2}, \sin \frac{\pi}{3} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2})$

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.

The diagram shows a capacitor $C$ and a resistor $R$ connected in series to an $ac$ source. $V_1$ and $V_2$ are voltmeters and $A$ is an ammeter. Consider the following statements:
$I$. Readings in $A$ and $V_2$ are always in phase.
$II$. Reading in $V_1$ lags behind the reading in $V_2$ by $\frac{\pi}{2}$.
$III$. Readings in $A$ and $V_1$ are always in phase.
Which of these statements is/are correct?

An $AC$ source of angular frequency $\omega$ is connected across a resistor $R$ and a capacitor $C$ in series. The current flowing in the circuit is found to be $I$. Now,the frequency of the source is changed to $\frac{\omega}{3}$ (maintaining the same voltage),and the current in the circuit is found to be halved. What is the ratio of reactance to resistance at the original frequency?

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