The power factor of an $AC$ circuit containing peak current $2 \ A$ and peak voltage $1 \ V$ is $1/2$. The angle between voltage and current is: (in $^{\circ}$)

The current and voltage functions in an $AC$ circuit are:
$I = 100 \sin(100t) \, mA$
$V = 100 \sin(100t + \frac{\pi}{3}) \, V$
The power dissipated in the circuit is ...... $W$.

$A$ lamp consumes only $50 \%$ of maximum power in an $AC$ circuit. What is the phase difference between the applied voltage and the circuit current?

In an $AC$ circuit $E = 200 \sin(50t) \text{ V}$ and $I = 100 \sin(50t + \frac{\pi}{3}) \text{ mA}$. The power dissipated in the circuit is (Given: $\sin 30^{\circ} = \cos 60^{\circ} = 0.5$,$\sin 60^{\circ} = \cos 30^{\circ} = \frac{\sqrt{3}}{2}$) (in $watt$)

$(a)$ 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.

$A$ choke coil is connected to an $AC$ source of frequency $\frac{50}{\pi} \, Hz$. If the resistance of the coil is $1 \, \Omega$ and the inductance is $10 \, mH$,what would be the value of the power factor?