For which combination of working temperature, the efficiency of Carnot's engine is the least ?

Options

A. 100 K, 80 K
B. 40 K, 20 K
C. 80 K, 60 K
D. 60 K, 40 K

Answer

To determine the combination of working temperatures for which the efficiency of a Carnot engine is the least, we need to understand the formula for the efficiency of a Carnot engine, which is given by:

\[ \eta = 1 - \frac{T_2}{T_1} \]

where:
- \( \eta \) is the efficiency,
- \( T_1 \) is the absolute temperature of the hot reservoir,
- \( T_2 \) is the absolute temperature of the cold reservoir.

### Step-by-Step Solution:

1. **Understanding Efficiency**: The efficiency of a Carnot engine depends on the temperatures of the hot and cold reservoirs. The efficiency is maximized when the temperature difference between the two reservoirs is maximized.

2. **Efficiency Formula**: The efficiency can be expressed as:
\[ \eta = 1 - \frac{T_2}{T_1} \]
This means that to minimize efficiency, we need to maximize the ratio \( \frac{T_2}{T_1} \).

3. **Maximizing \( \frac{T_2}{T_1} \)**: To find the combination of temperatures that results in the least efficiency, we need to find the maximum value of \( \frac{T_2}{T_1} \).

4. **Evaluating Given Combinations**: We will evaluate the combinations of temperatures provided in the question (assuming they are given as options). For each combination, we will calculate \( \frac{T_2}{T_1} \).

- For example, if the options provided are:
- Option A: \( T_1 = 100 \, K, T_2 = 80 \, K \) → \( \frac{T_2}{T_1} = \frac{80}{100} = 0.8 \)
- Option B: \( T_1 = 40 \, K, T_2 = 20 \, K \) → \( \frac{T_2}{T_1} = \frac{20}{40} = 0.5 \)
- Option C: \( T_1 = 80 \, K, T_2 = 60 \, K \) → \( \frac{T_2}{T_1} = \frac{60}{80} = 0.75 \)
- Option D: \( T_1 = 60 \, K, T_2 = 40 \, K \) → \( \frac{T_2}{T_1} = \frac{40}{60} = 0.67 \)

5. **Finding Maximum \( \frac{T_2}{T_1} \)**: After calculating the ratios:
- Option A: \( 0.8 \)
- Option B: \( 0.5 \)
- Option C: \( 0.75 \)
- Option D: \( 0.67 \)

The maximum ratio \( \frac{T_2}{T_1} \) is \( 0.8 \) from Option A.

6. **Conclusion**: Since the efficiency is minimized when \( \frac{T_2}{T_1} \) is maximized, the combination of working temperatures that results in the least efficiency of the Carnot engine is:
- **Option A: \( T_1 = 100 \, K, T_2 = 80 \, K \)**.