Question

An a.c. source is applied to a series LR circuit with $\text{X}_{\text{L}} = 3\text{R}$ and power factor is $\text{X}_1$. Now a capacitor with $\text{X}_{\text{c}} = \text{R}$ is added in series to LR circuit and the power factor is $\text{X}_2$. The ratio $\text{X}_1$ to $\text{X}_2$ is

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

Hint:

Power factor increases as reactance decreases (moving closer to resonance).

Answer 1

The Correct Option is C

Step 1: Concept
Power factor $\cos \phi = R/Z$.

Step 2: Meaning
Case 1: $Z_1 = \sqrt{R^2 + (3R)^2} = \sqrt{10}R$. $\cos \phi_1 = 1/\sqrt{10}$.
Case 2: $Z_2 = \sqrt{R^2 + (3R-R)^2} = \sqrt{R^2 + 4R^2} = \sqrt{5}R$. $\cos \phi_2 = 1/\sqrt{5}$.

Step 3: Analysis
Ratio $X_1/X_2 = \frac{1/\sqrt{10}}{1/\sqrt{5}} = \sqrt{\frac{5}{10}} = \frac{1}{\sqrt{2}}$.

Step 4: Conclusion
The ratio is $1 : \sqrt{2}$. Final Answer: (C)