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 can use the formula for the efficiency of a Carnot engine:

\[ \eta = 1 - \frac{T_L}{T_H} \]

where:
- \( \eta \) is the efficiency,
- \( T_L \) is the lower temperature (in Kelvin),
- \( T_H \) is the higher temperature (in Kelvin).

We will evaluate the efficiency for each of the given options.

### Step 1: Calculate Efficiency for Each Option

**Option A:**
\( T_L = 80 \, \text{K}, T_H = 100 \, \text{K} \)
\[ \eta_A = 1 - \frac{80}{100} = 1 - 0.8 = 0.2 \]

**Option B:**
\( T_L = 20 \, \text{K}, T_H = 40 \, \text{K} \)
\[ \eta_B = 1 - \frac{20}{40} = 1 - 0.5 = 0.5 \]

**Option C:**
\( T_L = 60 \, \text{K}, T_H = 80 \, \text{K} \)
\[ \eta_C = 1 - \frac{60}{80} = 1 - 0.75 = 0.25 \]

**Option D:**
\( T_L = 40 \, \text{K}, T_H = 60 \, \text{K} \)
\[ \eta_D = 1 - \frac{40}{60} = 1 - \frac{2}{3} = 0.33 \]

### Step 2: Compare Efficiencies

Now, we compare the efficiencies calculated for each option:
- \( \eta_A = 0.2 \)
- \( \eta_B = 0.5 \)
- \( \eta_C = 0.25 \)
- \( \eta_D = 0.33 \)

### Step 3: Identify the Least Efficiency

From the calculated efficiencies, we can see that the least efficiency is:
\[ \eta_A = 0.2 \]

### Conclusion

Thus, the combination of working temperatures for which the efficiency of the Carnot engine is the least is **Option A**.

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