$A$ series $LCR$ circuit has $R=200 \Omega$, $L=663 \text{ mH}$, and $C=26.5 \mu F$. The applied alternating voltage has an amplitude of $50 \text{ V}$ and a frequency of $60 \text{ Hz}$ such that $X_{L}=250 \Omega$ and $X_{C}=100 \Omega$. The peak current is: (in $\text{ A}$)

Two coils require $20 \ min$ and $60 \ min$ respectively to produce the same amount of heat energy when connected separately to the same source. If they are connected in parallel to the same source,the time required to produce the same amount of heat by the combination of coils will be . . . . . . $min$.

An inductance of $\frac{300}{\pi} \text{ mH}$,a capacitance of $\frac{1}{\pi} \text{ mF}$ and a resistance of $20 \ \Omega$ are connected in series with an a.c. source of $240 \text{ V}, 50 \text{ Hz}$. The phase angle of the circuit is

$A$ series $LCR$ circuit consists of an inductor $L$,a capacitor $C$,and a resistor $R$ connected across a source of emf $\varepsilon = \varepsilon_0 \sin \omega t$. When $\omega L = \frac{1}{\omega C}$,the current in the circuit is $I_0$. If the angular frequency of the source is changed to $\omega^{\prime}$,the current in the circuit becomes $\frac{I_0}{2}$. Then,the value of $\left|\omega^{\prime} L - \frac{1}{\omega^{\prime} C}\right|$ is

In a parallel $AC$ circuit with $R_1 = R_2 = R$ and $X_L = X_C = X$. An $AC$ source $V = V_0 \sin(\omega t)$ is connected across the circuit as shown in the figure. The $RMS$ value of the potential difference $V_A - V_B$ will be:

In the given circuit,the magnitudes of $V_{L}$ and $V_{C}$ are twice that of $V_{R}$. Given that $f=50\,Hz$ and $R=5\,\Omega$,the inductance of the coil is $\frac{1}{K\pi}\,mH$. The value of $K$ is: