In an `AC` circuit, the power factor

Options

A. is unity when the circuit contains an ideal resistance only
B. is unity when the circuit contains an ideal inductance only
C. is zero when the circuit contains an ideal resistance only
D. is zero when the circuit contains an ideal inductance only

Explanation

To solve the question regarding the power factor in an AC circuit, we need to analyze the power factor formula and the conditions under which it takes specific values. The power factor (PF) is defined as:

\[ \text{Power Factor (PF)} = \cos \phi = \frac{R}{\sqrt{R^2 + (X_L - X_C)^2}} \]

Where:
- \( R \) is the resistance,
- \( X_L \) is the inductive reactance,
- \( X_C \) is the capacitive reactance.

### Step-by-Step Solution:

1. **Understanding Power Factor**:
- The power factor is a measure of how effectively electrical power is being converted into useful work output. It is the cosine of the phase angle \( \phi \) between the voltage and current in an AC circuit.

2. **Case 1: Ideal Resistance Only**:
- If the circuit contains only an ideal resistance, then \( X_L = 0 \) and \( X_C = 0 \).
- The power factor becomes:
\[ \cos \phi = \frac{R}{\sqrt{R^2 + 0}} = \frac{R}{R} = 1 \]
- This means the power factor is unity (1) when the circuit contains only an ideal resistance. **(Correct)**

3. **Case 2: Ideal Inductance Only**:
- If the circuit contains only an ideal inductance, then \( R = 0 \) and \( X_C = 0 \).
- The power factor becomes:
\[ \cos \phi = \frac{0}{\sqrt{0 + X_L^2}} = 0 \]
- This means the power factor is zero when the circuit contains only an ideal inductance. **(Incorrect)**

4. **Case 3: Ideal Resistance Only (Revisited)**:
- As previously calculated, if the circuit contains only an ideal resistance, the power factor is 1. **(Incorrect)**

5. **Case 4: Ideal Inductance Only (Revisited)**:
- As previously calculated, if the circuit contains only an ideal inductance, the power factor is 0. **(Correct)**

### Conclusion:
- The correct statements regarding the power factor in the given options are:
- The power factor is unity (1) when the circuit contains an ideal resistance only.
- The power factor is zero (0) when the circuit contains an ideal inductance only.