In LCR circuit during resonance

Options

A. power factor is zero
B. power factor is one
C. power developed acrtoss resistance is zero
D. power developed across capacitance is zero

Answer

To solve the problem regarding the LCR circuit during resonance, we will analyze the conditions and properties of the circuit 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. The circuit is powered by an alternating current (AC) source.

**Hint**: Remember that in an LCR circuit, the inductive reactance (XL) and capacitive reactance (XC) play crucial roles in determining the circuit's behavior.

2. **Resonance Condition**:
- At resonance, the inductive reactance (XL) equals the capacitive reactance (XC). This can be expressed as:
\[ XL = XC \]

**Hint**: Recall that resonance occurs when the frequency of the AC source matches the natural frequency of the circuit.

3. **Impedance at Resonance**:
- The total impedance (Z) of the circuit at resonance is minimized and is equal to the resistance (R) only:
\[ Z = R \]

**Hint**: Impedance is a measure of how much the circuit resists the flow of current. At resonance, the circuit behaves like a purely resistive circuit.

4. **Power Factor Calculation**:
- The power factor (cos φ) can be defined as:
\[ \text{Power Factor} = \cos φ = \frac{R}{Z} \]
- Since at resonance \( Z = R \), we have:
\[ \cos φ = \frac{R}{R} = 1 \]

**Hint**: The power factor indicates how effectively the circuit converts electrical power into useful work. A power factor of 1 means all the power is being used effectively.

5. **Power Developed Across Components**:
- At resonance, the power developed across the resistance is maximum, while the power developed across the capacitance is zero because the reactive power (due to the capacitor and inductor) cancels out.

**Hint**: Remember that in a purely resistive circuit, all the power is dissipated as heat in the resistor.

6. **Conclusion**:
- Therefore, the correct statements regarding the LCR circuit during resonance are:
- The power factor is 1.
- The power developed across the resistance is maximum.
- The power developed across the capacitance is zero.

**Final Answer**: The correct option is that the power factor is 1.