Question

Which of following is true in a LCR circuit?
(A) In purely inductive circuit (R = 0), Quality factor is infinite.
(B) Resistance 'R' is alone responsible for damping of oscillations.
(C) Discharge of capacitor is not oscillatory in character.
(D) Q-factor is measure of sharpness of resonance in case of a driven oscillator.
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only
• B. (A), (B) and (C) only
• C. (A), (C) and (D) only
• D. (B), (C) and (D) only

Hint:

Remember the physical meaning of the Q-factor. It's a measure of the "quality" of an oscillator. \( Q = 2\pi \times \frac{\text{Energy stored per cycle}}{\text{Energy dissipated per cycle}} \). For R=0, energy dissipated is zero, so Q is infinite. High Q means low damping and sharp resonance.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question assesses the understanding of key characteristics of a series LCR circuit, including the quality factor (Q-factor), damping, and resonance.
Step 2: Detailed Explanation:
Let's analyze each statement:

(A) In purely inductive circuit (R = 0), Quality factor is infinite. The quality factor for a series LCR circuit is given by \( Q = \frac{\omega_0 L}{R} \), where \( \omega_0 \) is the resonant frequency. If the resistance R is zero (an ideal, lossless circuit), the Q-factor becomes infinite (\( Q \to \infty \) as \( R \to 0 \)). This statement is true.

(B) Resistance 'R' is alone responsible for damping of oscillations. In a simple LCR circuit, the resistor is the component that dissipates energy (as heat), causing the oscillations to die down or be "damped". Inductors and capacitors ideally store and release energy without loss. Therefore, resistance is the cause of damping. This statement is true.

(C) Discharge of capacitor is not oscillatory in character. In an LC or LCR circuit, the energy stored in the capacitor discharges through the inductor, which then stores the energy in its magnetic field. This energy is then transferred back to the capacitor. This exchange of energy between the capacitor and inductor results in oscillations. If R=0, the oscillations are undamped. If R>0, they are damped oscillations. The statement that the discharge is "not oscillatory" is false (except for the overdamped case, but oscillatory behavior is the general characteristic).

(D) Q-factor is measure of sharpness of resonance in case of a driven oscillator. This is a primary definition of the Q-factor. A high Q-factor implies a sharp, narrow resonance peak, meaning the circuit is highly selective to frequencies near its resonant frequency. A low Q-factor means a broad resonance curve. This statement is true.
Step 3: Final Answer:
Statements (A), (B), and (D) are true, while statement (C) is false. Therefore, the correct option is (A).