In an $AC$ circuit,the current is $i = 5 \sin(100t - \frac{\pi}{2}) \ A$ and the voltage is $e = 200 \sin(100t) \ V$. The power consumption in the circuit is (given $\cos 90^{\circ} = 0$): (in $W$)

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 . . . . . .

From which of the following options can the power factor of an $AC$ circuit be zero?

What is power factor and voltage-current relationship in an $AC$ circuit?

$(a)$ For circuits used for transporting electric power, a low power factor implies large power loss in transmission. Explain.
$(b)$ Power factor can often be improved by the use of a capacitor of appropriate capacitance in the circuit. Explain.

$A$ group of electric lamps having a total power rating of $1000 \text{ W}$ is supplied by an $AC$ voltage $E = 200 \sin(310t + 60^\circ)$. Then the $rms$ value of the circuit current is