If resistance is increased gradually in LCR circuit.

Options

A. quality factor increases
B. Resonance frequency gets increased
C. quality factor and resonance frequency remain same
D. band width increase

Answer

To solve the problem regarding the effects of gradually increasing resistance in an LCR circuit, we will analyze the relationships between resistance, quality factor, resonance frequency, and bandwidth step by step.

### Step-by-Step Solution:

1. **Understanding the LCR Circuit**:
An LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series or parallel. The behavior of this circuit is influenced by the values of L, C, and R.

**Hint**: Recall the basic components of an LCR circuit and their roles in determining circuit behavior.

2. **Resonance Frequency**:
The resonance frequency (ω₀) of an LCR circuit is given by the formula:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]
This indicates that the resonance frequency depends only on the inductance (L) and capacitance (C), and is independent of the resistance (R).

**Hint**: Focus on the formula for resonance frequency and note that it does not include R.

3. **Quality Factor (Q)**:
The quality factor (Q) is defined as:
\[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \]
This shows that the quality factor is inversely proportional to the resistance (R). Therefore, if R increases, Q will decrease.

**Hint**: Remember that a higher resistance leads to a lower quality factor, indicating a less sharp resonance peak.

4. **Bandwidth**:
The bandwidth (Δω) of the circuit can be defined as:
\[ \Delta \omega = \frac{R}{L} \]
This means that the bandwidth is directly proportional to the resistance (R). As R increases, the bandwidth also increases.

**Hint**: Consider how bandwidth relates to R and how it changes when R is increased.

5. **Analyzing the Options**:
- **Option 1**: Quality factor increases - **False** (Q decreases with increasing R).
- **Option 2**: Resonance frequency increases - **False** (ω₀ remains constant as it does not depend on R).
- **Option 3**: Quality factor and resonance frequency remain the same - **False** (Q decreases while ω₀ remains the same).
- **Option 4**: Bandwidth increases - **True** (Bandwidth increases with increasing R).

**Hint**: Evaluate each option based on the relationships derived from the formulas.

### Conclusion:
The correct answer is **Option 4**: Bandwidth increases as the resistance is gradually increased in the LCR circuit.