Assertion (A) : Quality factor of a series LCR circuit is `Q = (1)/(R) sqrt((L)/(C))`
Reason (R) : As bandwidth decreases, Q increases in a resonant LCR circuit.

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is NOT the correct explanation of A.
C. A is true but R is false
D. A is false and R is true

Explanation

To analyze the given statements regarding the quality factor \( Q \) of a series LCR circuit, we will break down the assertions and the reasoning step by step.

### Step 1: Understanding the Quality Factor \( Q \)

The quality factor \( Q \) of a series LCR circuit is defined as the ratio of the resonant frequency to the bandwidth of the circuit. Mathematically, it can be expressed as:

\[ Q = \frac{f_0}{\Delta f} \]

where \( f_0 \) is the resonant frequency and \( \Delta f \) is the bandwidth.

### Step 2: Formula for Quality Factor

The quality factor can also be expressed in terms of circuit components:

\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \]

where \( R \) is the resistance, \( L \) is the inductance, and \( C \) is the capacitance of the circuit. This formula indicates that \( Q \) is inversely proportional to resistance \( R \) and directly proportional to the square root of the ratio of inductance to capacitance.

### Step 3: Analyzing the Bandwidth

The bandwidth \( \Delta f \) of a series LCR circuit can be defined as:

\[ \Delta f = \frac{R}{2\pi L} \]

This indicates that the bandwidth is directly proportional to the resistance \( R \) and inversely proportional to the inductance \( L \).

### Step 4: Relationship Between Bandwidth and Quality Factor

From the definitions, we can see that as the bandwidth \( \Delta f \) decreases (meaning the circuit is more selective), the quality factor \( Q \) increases. This is because:

\[ Q = \frac{f_0}{\Delta f} \implies \text{If } \Delta f \text{ decreases, then } Q \text{ increases.} \]

This confirms the reasoning in the second statement.

### Step 5: Conclusion about the Assertions

Both statements can be evaluated as follows:

- **Assertion (A)**: The formula for the quality factor \( Q = \frac{1}{R} \sqrt{\frac{L}{C}} \) is correct.
- **Reason (R)**: The statement that as bandwidth decreases, \( Q \) increases is also correct.

Since both statements are true and the reason correctly explains the assertion, we conclude that:

- **Assertion (A)** is true.
- **Reason (R)** is true and correctly explains (A).

### Final Answer

Both the assertion and the reason are true, and the reason correctly explains the assertion.

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