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: Impedances:
$Z_1 = \sqrt{10}\,R$ and $Z_2 = \sqrt5\,R$.

Step 2: Ratio of power factors:
$\frac{\cos\phi_1}{\cos\phi_2} = \frac{Z_2}{Z_1} = \frac{\sqrt5}{\sqrt{10}} = \frac{1}{\sqrt2}$.

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