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: Conceptual Foundation: This question evaluates comprehension of fundamental series LCR circuit properties, including the quality factor (Q-factor), damping phenomena, and resonance characteristics.
Step 2: Exhaustive Analysis:

(A) In a purely inductive circuit (R = 0), the Quality factor is infinite. The quality factor for a series LCR circuit is mathematically defined as \( Q = \frac{\omega_0 L}{R} \), where \( \omega_0 \) represents the resonant frequency. When the resistance R approaches zero (indicating an ideal, lossless circuit), the Q-factor tends towards infinity (\( Q \to \infty \) as \( R \to 0 \)). This assertion is accurate.

(B) Resistance 'R' is solely responsible for the damping of oscillations. Within a basic LCR circuit, the resistor is the component responsible for energy dissipation (converted to heat), which leads to the decay or "damping" of oscillations. Inductors and capacitors, in an ideal scenario, store and release energy without any loss. Consequently, resistance is identified as the cause of damping. This assertion is accurate.

(C) The discharge of a capacitor is not oscillatory in character. In an LC or LCR circuit, energy stored in the capacitor is released through the inductor, which subsequently stores this energy in its magnetic field. This energy is then transferred back to the capacitor. This reciprocal energy transfer between the capacitor and inductor inherently produces oscillations. In the absence of resistance (R=0), these oscillations are undamped. With a positive resistance (R>0), the oscillations are damped. The declaration that the discharge is "not oscillatory" is incorrect (unless considering the specific case of overdamping, as oscillatory behavior is the general characteristic).

(D) The Q-factor is a measure of resonance sharpness in a driven oscillator. This statement directly reflects the fundamental definition of the Q-factor. A high Q-factor signifies a sharp, narrow resonance peak, indicating that the circuit is highly selective for frequencies in close proximity to its resonant frequency. Conversely, a low Q-factor results in a broad resonance curve. This assertion is accurate.
Step 3: Conclusion: Statements (A), (B), and (D) are confirmed as true, whereas statement (C) is determined to be false. Therefore, the correct option is (A).