Question

The power factor of the ac circuit given below is $\frac{1}{\sqrt{2}}$. The capacitance of the capacitor in the circuit is equal to

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

Answer 1

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

Power factor $\cos \varphi =\frac{R}{Z}=\frac{R}{\sqrt{R^2+(X_L-X_C{)}^2}}$
Squaring both sides. ${\left(\frac{1}{\sqrt{2}}\right)}^2=\left[\frac{R}{\sqrt{R^2+(X_L-X_C{)}^2}}\right]$
$R^2+(X_L-X_C{)}^2=2R^2$
$X_L-X_C=\pm R$
$L\omega -\frac{1}{\omega C}=\pm 10$

$(0.1\times 100)-\frac{1}{100C}=\pm 10$
$\frac{1}{100C}-10=10$ $[X_c>X_L]$
$\frac{1}{100C}=20$
$C=\frac{1}{2000}=0.5\times {10}^{-3}$
$C=500\times {10}^{-6}\text{F}$
$C=500\mu \text{F}$