An $a.c.$ source is applied to a series $LR$ circuit with $X_L = 3R$ and power factor is $X_1$. Now a capacitor with $X_c = R$ is added in series to the $LR$ circuit and the power factor is $X_2$. The ratio $X_1$ to $X_2$ is

In an $AC$ circuit,the potential differences across an inductance and a resistance joined in series are $16 \, V$ and $20 \, V$ respectively. The total potential difference of the source is .......... $V$.

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

If $L$ and $R$ are respectively the inductance and resistance,then the dimensions of $\frac{R}{L}$ will be

In an $LCR$ series circuit,an inductor $30 \, {mH}$ and a resistor $1 \, \Omega$ are connected to an $AC$ source of angular frequency $300 \, {rad/s}$. The value of capacitance for which the current leads the voltage by $45^{\circ}$ is $\frac{1}{x} \times 10^{-3} \, {F}$. Then the value of $x$ 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