Question

As shown in the figure, if the value of $rms$ current is $2.2\text{A}$, the power factor of the box is

• A. $\frac{1}{\sqrt{2}}$
• B. $1$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\frac{1}{\sqrt{2}}$

Given:-
$i_{rms}=2.2\text{A};L=\frac{1}{\pi }\text{H};R=100\Omega$

$V_{rms}=220\text{V};\omega =100\pi \text{rad/s}$

Power factor is given by

$\cos \varphi =\frac{R}{Z}$

Now, $i_{rms}=\frac{V_{rms}}{Z}$

$\therefore Z=\frac{V_{rms}}{i_{rms}}=\frac{220}{2.2}=100\Omega$

Also, we know that
$Z^2=R^2+(X_L-X_C{)}^2$

${100}^2={100}^2+(X_L-X_C{)}^2$

$\Rightarrow (X_L-X_C{)}^2=0$,

$\Rightarrow X_L=X_C$

$\Rightarrow X_C=X_L=\omega L=100\pi \times \frac{1}{\pi }$

$\therefore X_C=100\Omega$

Now, Power factor for the box

$=\frac{R}{Z_{Box}}=\frac{R}{\sqrt{R^2+X_C^2}}$

$=\frac{100}{\sqrt{{100}^2+{100}^2}}=\frac{1}{\sqrt{2}}$

Hence, option $\left(\text{A}\right)$ is correct.