Power transfer by a non-ideal cell to an external resistor can be same for two different values. Out of which on value `(K_(1))` is less than internal resistance of cell and another `(R_(2))` is greater than internal resistance of cell then:

Options

A. `R_(1)` will give less internal loss
B. `R_(1)` will give more battery life
C. `R_(2)` will give less efficiency
D. `R_(2)` will give more battery life

Explanation

To solve the problem regarding power transfer by a non-ideal cell to an external resistor, we need to analyze the relationship between the external resistance (R) and the internal resistance (r) of the cell. Let's denote the internal resistance of the cell as \( r \) and the electromotive force (emf) of the cell as \( e \).

### Step-by-Step Solution:

1. **Understanding the Current Flow**:
The current \( I \) flowing through the circuit can be expressed as:
\[ I = \frac{e}{r + R} \]
where \( R \) is the external resistance.

2. **Power Loss in the Internal Resistance**:
The power loss (or internal loss) in the internal resistance of the cell can be calculated using:
\[ P_{\text{internal}} = I^2 \cdot r \]
Substituting the expression for \( I \):
\[ P_{\text{internal}} = \left(\frac{e}{r + R}\right)^2 \cdot r \]

3. **Analyzing the Two Cases**:
- **Case 1**: \( R_1 < r \) (External resistance is less than internal resistance)
- **Case 2**: \( R_2 > r \) (External resistance is greater than internal resistance)

4. **Comparing Power Loss**:
- For \( R_1 < r \):
- The current \( I \) will be relatively high since \( R_1 \) is small compared to \( r \).
- This results in a higher internal loss because \( P_{\text{internal}} \) is proportional to \( I^2 \).
- For \( R_2 > r \):
- The current \( I \) will be lower since \( R_2 \) is larger compared to \( r \).
- This results in a lower internal loss.

5. **Battery Life Consideration**:
The battery life is inversely proportional to the current:
\[ \text{Battery Life} \propto \frac{1}{I} \]
Therefore, a lower current (as in the case of \( R_2 \)) will lead to a longer battery life.

6. **Conclusion**:
- \( R_1 \) will result in more internal loss and shorter battery life.
- \( R_2 \) will result in less internal loss and longer battery life.

### Final Answer:
The correct statement is that \( R_2 \) will give more battery life.