Two Carnot engines A and B are operated in series. The engine A receives heat from the source at temperature `T_1` and rejects the heat to the sink at temperature T. The second engine B receives the heat at temperature T and rejects to its sink at temperature `T_(2)`. For what value of T the efficiencies of the two engines are equal?

Options

A. `T=(2T_(1)T_(2))/(T_(1)+T_(2))`
B. `T=sqrt(T_(1)T_(2))`
C. `T=(T_(1)+T_(2))/(2)`
D. `T=0`

Answer

To solve the problem of finding the temperature T at which the efficiencies of two Carnot engines A and B are equal, we can follow these steps:

### Step 1: Write the efficiency formula for Carnot engines
The efficiency (η) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_{\text{sink}}}{T_{\text{source}}} \]
For engine A, the sink temperature is T and the source temperature is \(T_1\). Therefore, the efficiency of engine A (\(\eta_1\)) can be written as:
\[ \eta_1 = 1 - \frac{T}{T_1} \]

### Step 2: Write the efficiency formula for engine B
For engine B, the sink temperature is \(T_2\) and the source temperature is T. Thus, the efficiency of engine B (\(\eta_2\)) can be expressed as:
\[ \eta_2 = 1 - \frac{T_2}{T} \]

### Step 3: Set the efficiencies equal to each other
According to the problem, we need to find the temperature T such that the efficiencies of both engines are equal:
\[ \eta_1 = \eta_2 \]
Substituting the expressions for \(\eta_1\) and \(\eta_2\):
\[ 1 - \frac{T}{T_1} = 1 - \frac{T_2}{T} \]

### Step 4: Simplify the equation
By canceling the 1s on both sides, we get:
\[ -\frac{T}{T_1} = -\frac{T_2}{T} \]
This simplifies to:
\[ \frac{T}{T_1} = \frac{T_2}{T} \]

### Step 5: Cross-multiply to solve for T
Cross-multiplying gives us:
\[ T^2 = T_1 \cdot T_2 \]

### Step 6: Take the square root
To find T, we take the square root of both sides:
\[ T = \sqrt{T_1 \cdot T_2} \]

### Final Answer
Thus, the value of T for which the efficiencies of the two engines are equal is:
\[ T = \sqrt{T_1 \cdot T_2} \]

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