An electric lamp connected in series with a capacitor and an a.c. source is glowing with certain brightness. On reducing the frequency of the source, the brightness of the lamp

In an $RL$ circuit,the resistance is $\pi \sqrt{3} \,\Omega$. The phase difference between the current and the voltage is $30^\circ$. The $ac$ frequency is $50 \,Hz$. The inductance is . . . . . . $Henry$.

An inductor of inductance $L$ and a resistor of resistance $R$ are joined in series and connected to a source of frequency $\omega$. The power dissipated in the circuit is

Which one of the following curves represents the variation of impedance $(Z)$ with frequency $(f)$ in a series $L-C-R$ circuit,when connected to an $AC$ source?

When an inductor $L$ and a resistor $R$ in series are connected across a $12 \, V, 50 \, Hz$ supply, a current of $0.5 \, A$ flows in the circuit. The current differs in phase from the applied voltage by $\frac{\pi}{3}$ radian. Then the value of $R$ is (in $\Omega$)

$A$ 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