In a series $LCR$ circuit,the voltages across the inductance and the capacitance are not

$A$ capacitor of reactance $4 \sqrt{3} \Omega$ and a resistor of resistance $4 \Omega$ are connected in series with an $AC$ source of peak value $8 \sqrt{2} \text{ V}$. The power dissipation in the circuit is . . . . . $\text{W}$.

When a $DC$ voltage of $100 \, V$ is applied to an inductor, a $DC$ current of $5 \, A$ flows through it. When an $AC$ voltage of $200 \, V$ peak value is connected to the inductor, its inductive reactance is found to be $20\sqrt{3} \, \Omega$. The power dissipated in the circuit is . . . . . . $W$.

When an inductance $L$ and resistor $R$ are connected in series to $25 \ V$, $50 \ Hz$ supply, a current of $0.5 \ A$ flows in the circuit. The current lags behind in phase from applied voltage by $(\frac{\pi}{3})$ radian. The value of $R$ is $(\cos 60^{\circ} = \frac{1}{2})$ (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

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