Question

The power factor of the circuit is $\frac{1}{\sqrt{2}}$. The capacitance of the circuit is equal to -

• A. $400\mu \text{F}$
• B. $300\mu \text{F}$
• C. $500\mu \text{F}$
• D. $200\mu \text{F}$

Answer 1

The correct option is C $500\mu \text{F}$


Here,

$X_L=\omega L=100\times 0.1=10\Omega$

$X_C=\frac{1}{\omega C}=\frac{1}{100C}$

Now,

$\cos \varphi =\frac{R}{Z}=\frac{R}{\sqrt{R^2+(X_L-X_C{)}^2}}$

$\Rightarrow \frac{1}{\sqrt{2}}=\frac{10}{\sqrt{{10}^2+(10-X_C{)}^2}}$

$\Rightarrow 10-X_C=\pm 10$

$\Rightarrow 10-X_C=-10[\text{As}{X}_C\ne 0]$

$\Rightarrow X_C=20\Omega$

$\Rightarrow \frac{1}{100C}=20\Omega$

$\Rightarrow C=5\times {10}^{-4}\text{F}=500\mu \text{F}$

Hence, option $\left(C\right)$ is the correct answer.