Question

Voltage and current in an ac circuit are given by $V = 5\sin \left( {100\pi t - \dfrac{\pi }{6}} \right)$ and $I = 4\sin \left( {100\pi t + \dfrac{\pi }{6}} \right)$.
A. Voltage leads the current by $30^\circ $
B. Current leads the voltage by $60^\circ $
C. Voltage leads the current by $60^\circ $
D. Current and voltage are in phase

Answer 1

Hint: We know the ac voltage and current general equation with the help of the equation we can find ${\phi _1}$ and ${\phi _2}$. Then calculate the phase difference between current and voltage with the help of the given statement.

Complete step by step answer:
The circuit that is excited using an alternating source is called an AC circuit.We know the general equation of ac voltage and current is as follow,
$V = {V_0}\sin \left( {\omega t + {\phi _1}} \right) \ldots \ldots \left( 1 \right)$
$ \Rightarrow I = {I_0}\sin \left( {\omega t + {\phi _2}} \right) \ldots \ldots \left( 2 \right)$
As per the given problem the voltage and current in circuit are given by
$V = 5\sin \left( {100\pi t - \dfrac{\pi }{6}} \right)$
$ \Rightarrow I = 4\sin \left( {100\pi t + \dfrac{\pi }{6}} \right)$.
Comparing the given equation with the actual voltage and current we get,
${\phi _1} = - \dfrac{\pi }{6} \ldots \ldots \left( 3 \right)$
$ \Rightarrow {\phi _2} = \dfrac{\pi }{6} \ldots \ldots \left( 4 \right)$

According to phase difference formula for the voltage and current is represented as,
Phase difference = Phase angle for current + Phase angle for voltage
$\Delta \phi = {\phi _2} - {\phi _1}$
By putting equation and we get,
$\Delta \phi = \dfrac{\pi }{6} - \left( { - \dfrac{\pi }{6}} \right)$
$ \Rightarrow \Delta \phi = \dfrac{{2\pi }}{6}$
Simplifying numerator and denominator we get,
$ \Rightarrow \Delta \phi = \dfrac{\pi }{3}$
Converting $\dfrac{\pi }{3}$ into degree we get,
$\dfrac{\pi }{3} = 60^\circ $
Where, $\pi = 180^\circ $.
Hence the phase difference is positive then the current leads the voltage with a phase angle of $60^\circ $.

Therefore the correct option is B.

Note: This type of circuit is generally used for domestic and industrial purposes.In our solution we calculate the phase difference by subtraction the phase angle of voltage from phase angle of current. But if we subtract phase angle of current from phase angle of voltage then we get negative phase difference which also denotes that current is leading.