For the series $LCR$ circuit connected with a $220 \ V$, $50 \ Hz$ a.c. source as shown in the figure, the power factor is $\frac{\alpha}{10}$. The value of $\alpha$ is . . . . . .

$A$ series $LCR$ circuit is connected to an a.c. source of $230 \ V, 50 \ Hz$. The circuit contains a resistance of $80 \ \Omega$,an inductor having inductive reactance $70 \ \Omega$,and a capacitor of capacitive reactance $130 \ \Omega$. The power factor of the circuit is $x$. The value of $x$ is

An e.m.f. $e=E_0 \cos \omega t$ is applied to a circuit containing $L, C$ and $R$ in series where $X_L=3 R$ and $X_C=R$. The average power dissipated in the circuit is

An $AC$ voltage $V=20 \sin 200 \pi t$ is applied to a series $LCR$ circuit which drives a current $I=10 \sin (200 \pi t + \frac{\pi}{3})$. The average power dissipated is: (in $W$)

In an $A.C.$ circuit,the $e.m.f.$ $(\varepsilon)$ and the current $(i)$ at any instant are given respectively by $E = E_o \sin \omega t$ and $i = I_o \sin(\omega t - \phi)$. The average power in the circuit over one cycle of $A.C.$ is:

The instantaneous values of current and potential difference in an alternating circuit are $i = \sin \omega t$ and $E = 100 \cos \omega t$ respectively. The $r.m.s.$ value of the wattless current (in $A$) in the circuit is: