In a series `L-R` circuit, under which condition the power loss will be least for an a.c voltage source?

Options

A. high resistance and high inductance
B. high resistance and low inductance
C. low resistance and high inductance
D. low resistance and low inductance

Answer

To determine the condition under which the power loss will be least in a series L-R circuit connected to an AC voltage source, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Power Loss in AC Circuits**:
The power loss \( P \) in an AC circuit is given by the formula:
\[ P = V_{\text{rms}} \times I_{\text{rms}} \times \cos \phi \]
where \( V_{\text{rms}} \) is the root mean square voltage, \( I_{\text{rms}} \) is the root mean square current, and \( \cos \phi \) is the power factor.

2. **Identify the Power Factor**:
The power factor \( \cos \phi \) can be expressed in terms of resistance \( R \) and impedance \( Z \):
\[ \cos \phi = \frac{R}{Z} \]
where \( Z \) is the total impedance of the circuit.

3. **Analyze the Impedance**:
In a series L-R circuit, the impedance \( Z \) is given by:
\[ Z = \sqrt{R^2 + (X_L)^2} \]
where \( X_L \) is the inductive reactance, which is proportional to the inductance \( L \) and the frequency \( f \) of the AC source:
\[ X_L = 2\pi f L \]

4. **Determine Conditions for Minimum Power Loss**:
To minimize power loss \( P \), we need to maximize \( Z \) and minimize \( R \). This means:
- **Lower Resistance \( R \)**: A lower resistance will contribute less to power loss.
- **Higher Inductance \( L \)**: A higher inductance will increase the inductive reactance \( X_L \), thus increasing the impedance \( Z \).

5. **Conclusion**:
The condition for the least power loss in a series L-R circuit under an AC voltage source is when the resistance \( R \) is low and the inductance \( L \) is high.

### Final Answer:
The power loss will be least for an AC voltage source in a series L-R circuit when there is **lower resistance and higher inductance**.