Question

An inductor of inductan L, a capacitor of capacitan C and a resistor of resistan R are connected in series to an ac sour. The quality factor of the circuit is

• A. $\sqrt{\frac{L}{CR^2}}$
• B. $\sqrt{\frac{LR^2}{C}}$
• C. $\sqrt{\frac{LC}{R^2}}$
• D. $\sqrt{\frac{L^2C}{R}}$

Hint:

Quality Factor of RLC Circuit:

• $Q = \frac\omega_0 LR$ for series RLC
• $\omega_0 = \frac1\sqrtLC$
• Substituting: $Q = \sqrt\fracLCR^2$
• High $Q$ means low energy loss and sharp resonance.

Answer 1

The Correct Option is A

The quality factor $Q$ in a series RLC circuit is a measure of the sharpness of resonance and energy efficiency. It is given by: \[ Q = \frac{\omega_0 L}{R} = \frac{1}{R} \cdot \frac{L}{\sqrt{LC}} = \frac{1}{R} \sqrt{\frac{L}{C}} = \sqrt{\frac{L}{CR^2}} \] We derived this by substituting the resonance angular frequency $\omega_0 = \frac{1}{\sqrt{LC}}$ into the $Q$ formula. Therefore, the correct expression is $\sqrt{\frac{L}{CR^2}}$.

Answer 2

Step 1: Understand the circuit components
The circuit consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series with an AC source.

Step 2: Define the quality factor (Q)
The quality factor \( Q \) of a series RLC circuit is a measure of how underdamped the circuit is, and it indicates the sharpness of the resonance.
It is defined as:
\[ Q = \frac{\text{Reactance at resonance}}{\text{Resistance}} = \frac{X_L}{R} = \frac{X_C}{R} \]

Step 3: Recall reactances at resonance frequency
At resonance frequency \( \omega_0 = \frac{1}{\sqrt{LC}} \), the inductive reactance \( X_L \) and capacitive reactance \( X_C \) are equal in magnitude:
\[ X_L = \omega_0 L = \frac{L}{\sqrt{LC}} = \sqrt{\frac{L}{C}} \]

Step 4: Calculate quality factor
Using the reactance and resistance, the quality factor is:
\[ Q = \frac{X_L}{R} = \frac{\sqrt{\frac{L}{C}}}{R} = \sqrt{\frac{L}{C R^2}} \]

Step 5: Conclusion
Therefore, the quality factor of the series RLC circuit is:
\[ Q = \sqrt{\frac{L}{C R^2}} \]