In series LCR circuit at resonance

Options

A. Power factor is zero
B. Power developed across the resistor is maximum
C. Power developed across the inductor is zero
D. Both (2) & (3)

Answer

To solve the question regarding the characteristics of a series LCR circuit at resonance, we will analyze each option provided in the question step by step.

### Step-by-Step Solution:

1. **Understanding Resonance in LCR Circuit:**
- In a series LCR circuit, resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC). This condition can be expressed as:
\[ XL = XC \]
- At resonance, the total impedance (Z) of the circuit is purely resistive, and can be expressed as:
\[ Z = R \]

2. **Power Factor at Resonance:**
- The power factor (PF) is defined as:
\[ PF = \cos(\phi) \]
- Where \(\phi\) is the phase angle between the voltage and current. At resonance, since \(XL = XC\), the phase angle \(\phi\) becomes 0:
\[ \phi = 0 \implies PF = \cos(0) = 1 \]
- Therefore, the statement that the power factor is 0 (Option A) is incorrect.

3. **Power Developed Across the Resistance:**
- The power developed across the resistance (P) in an AC circuit is given by:
\[ P = V_{rms} \cdot I_{rms} \cdot \cos(\phi) \]
- At resonance, since \(\phi = 0\), the power factor is 1, and thus the power developed across the resistance is maximum:
\[ P = V_{rms} \cdot I_{rms} \]
- Therefore, Option B is correct.

4. **Power Developed Across the Inductor:**
- The power developed across the inductor (P_L) can be calculated as:
\[ P_L = V_{rms} \cdot I_{rms} \cdot \sin(\phi) \]
- Since at resonance \(\phi = 0\), we have:
\[ P_L = V_{rms} \cdot I_{rms} \cdot \sin(0) = 0 \]
- Hence, Option C is also correct.

5. **Conclusion on Options:**
- Since Option A is incorrect, and Options B and C are correct, we can conclude that Option D, which states that both Options 2 and 3 are correct, is also correct.

### Final Answer:
- The correct options are B (power developed across the resistance is maximum) and C (power developed across the inductor is 0). Therefore, Option D is correct.