Question

In an electrical circuit ' \(R\) ', ' \(L\) ', ' \(C\) ' and an a.c. voltage source are all connected in series. When ' L ' is removed from the circuit, the phase difference between the voltage and the current in the circuit is \(\frac{\pi}{3}\). If instead ' \(C\) ' is removed from the circuit, the phase difference is again \(\frac{\pi}{3}\). The power factor of the circuit is \(\left(\tan \frac{\pi}{3}=\sqrt{3}\right)\)

• A. \(\frac{\sqrt{3}}{2}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{1}{\sqrt{2}}\)
• D. 1

Hint:

Removing L leaves an RC circuit and removing C leaves an RL circuit; equal phase angles mean X_L equals X_C.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Without L the circuit is RC, so \(\tan\phi=\frac{X_C}{R}\). Without C the circuit is RL, so \(\tan\phi=\frac{X_L}{R}\).

Step 2: Use the given data
Both phase angles are \(\frac{\pi}{3}\), so \(\frac{X_C}{R}=\frac{X_L}{R}=\tan\frac{\pi}{3}=\sqrt{3}\).

Step 3: Full circuit
Hence \(X_L=X_C\), the circuit is in resonance and the impedance is \(Z=R\).

Step 4: Power factor
\(\cos\phi=\frac{R}{Z}=1\), which is option (D).

Final Answer:
The power factor of the circuit is 1. \[ \boxed{\text{(D) }1} \]