Question

In an AC circuit, the instantaneous $emf$ and current are given by
$\begin{align}
  & e=100\sin 30t \\
 & i=20\sin \left( 30t-\dfrac{\pi }{4} \right) \\
\end{align}$
In one cycle of AC, the average power consumed by the circuit and the wattless current are, respectively:
$A)50,10$
$B)\dfrac{1000}{\sqrt{2}},10$
$C)\dfrac{50}{\sqrt{2}},0$
$D)50,0$

Answer 1

Hint: In an AC circuit, the average power is calculated by using $rms$values of $emf$ and current. Wattless current is the amount of current flowing in an AC circuit, when the average power used in the AC circuit is equal to zero. It can be determined using $rms$ value of current.
Formula used:
$1){{P}_{avg}}={{e}_{rms}}{{I}_{rms}}\cos \phi $
where
 ${{P}_{avg}}$ is the average power consumed by an AC circuit
${{e}_{rms}}$ is the average value of emf in the AC circuit
${{I}_{rms}}$ is the average value of current in the AC circuit
$\phi $ is the phase constant
$2)I={{I}_{rms}}\sin \phi $
where
$I$ is the wattless current
${{I}_{rms}}$ is the average value of current in an AC circuit
$\phi $ is the phase constant

Complete step-by-step solution:
We are provided with the values of instantaneous $emf$ and current as follows:
$\begin{align}
  & e=100\sin 30t \\
 & i=20\sin \left( 30t-\dfrac{\pi }{4} \right) \\
\end{align}$
It can be understood that both these equations are given in the form of wave equations because we are dealing with alternating current and voltage.
From these wave equations, it is clear that the maximum values of $emf$ and current in the given AC circuit is $100$ and $20$, respectively. Let us represent them as follows.
${{e}_{\max }}=100V$
${{I}_{\max }}=20A$
where
${{e}_{\max }}$ is the maximum value of $emf$ in the given AC circuit
${{I}_{\max }}$ is the maximum value of current in the given AC circuit
To calculate the average power consumed by the AC circuit, we have to take $rms$ values of $emf$ and current.
We know that
${{e}_{rms}}=\dfrac{{{e}_{\max }}}{\sqrt{2}}$
and
${{I}_{rms}}=\dfrac{{{I}_{\max }}}{\sqrt{2}}$
Let this set of equations be denoted as X.
Substituting the values of ${{e}_{\max }}$ and ${{I}_{\max }}$ in the above set of equations, we have
${{e}_{rms}}=\dfrac{{{e}_{\max }}}{\sqrt{2}}=\dfrac{100}{\sqrt{2}}$
and
 ${{I}_{rms}}=\dfrac{{{I}_{\max }}}{\sqrt{2}}=\dfrac{20}{\sqrt{2}}$
Let this set of equations be represented by A.
Now, the average power consumed by an AC circuit is given by
${{P}_{avg}}={{e}_{rms}}{{I}_{rms}}\cos \phi $
where
${{P}_{avg}}$ is the average power consumed by an AC circuit
${{e}_{rms}}$ is the average value of emf in the AC circuit
${{I}_{rms}}$ is the average value of current in the AC circuit
$\phi $ is the phase constant
Let this be equation B.
Looking at the wave equations provided in the question, it is clear that phase constant $(\phi )$ is equal to $\dfrac{\pi }{4}$.
Substituting this value as well as the values from the set of equations given in A, in equation B, we have
${{P}_{avg}}={{e}_{rms}}{{I}_{rms}}\cos \phi =\left( \dfrac{100}{\sqrt{2}} \right)\left( \dfrac{20}{\sqrt{2}} \right)\cos \dfrac{\pi }{4}=1000\cos \dfrac{\pi }{4}=\dfrac{1000}{\sqrt{2}}$
Therefore, the average power consumed by the AC circuit is given by
${{P}_{avg}}=\dfrac{1000}{\sqrt{2}}$
Let this be equation C.
Now, let us understand what is meant by the wattless current. As the name suggests, it is the amount of current when there is no watt or no power. Wattless current is defined as the current in an AC circuit when the average power consumed by the AC circuit is equal to zero.
Mathematically, it is represented as:
$I={{I}_{rms}}\sin \phi $
where
$I$ is the wattless current
${{I}_{rms}}$ is the average value of current in an AC circuit
$\phi $ is the phase constant
Substituting the values of ${{I}_{rms}}$ and $\phi $ in the above equation, we have
$I={{I}_{rms}}\sin \phi =\dfrac{20}{\sqrt{2}}\sin \dfrac{\pi }{4}=\dfrac{20}{\sqrt{2}\times \sqrt{2}}=10$
Therefore, the amount of wattless current in the given AC circuit is given by
$I=10$
Let this be equation D.
From equation C and equation D, it is clear that the average power consumed and the wattless current in one cycle of the given AC circuit are $\dfrac{1000}{\sqrt{2}}$ and $10$, respectively.
Hence, the correct option to be marked is B.

Note: Students need to understand that $rms$ value of $emf$ as well as the current is taken in order to determine the average power consumed by the AC circuit. Root mean square $(rms)$ of an alternating voltage or current is defined as the DC value of alternating voltage or current, which would produce the same average power output. The formula for taking $rms$ value can easily be remembered. Students can take a look on the set of equations denoted by X, to go through the formulas.