Question

An a.c. source is applied to a series LR circuit with \(X_L = 3R\) and power factor is \(X_1\). Now a capacitor with \(X_c = R\) is added in series to LR circuit and power factor is \(X_2\). The ratio \(X_1\) to \(X_2\) is

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

Hint:

Power factor is R over impedance; the net reactance changes when the capacitor is added.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The power factor of a series circuit is \(\cos\phi = \frac{R}{Z}\), where \(Z = \sqrt{R^2 + X^2}\) and \(X\) is the net reactance.

Step 2: Key Formula or Approach:
LR circuit: \(X = X_L = 3R\). LCR circuit: \(X = X_L - X_C = 3R - R = 2R\).

Step 3: Detailed Explanation:
\(X_1 = \frac{R}{\sqrt{R^2 + 9R^2}} = \frac{1}{\sqrt{10}}\).
\(X_2 = \frac{R}{\sqrt{R^2 + 4R^2}} = \frac{1}{\sqrt5}\).
\[ \frac{X_1}{X_2} = \frac{\sqrt5}{\sqrt{10}} = \frac{1}{\sqrt2} \]
So the ratio is \(1 : \sqrt2\).

Final Answer:
The ratio \(X_1 : X_2\) is \(1 : \sqrt{2}\), option (D). \[ \boxed{1:\sqrt{2}} \]