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Two Carnot engines $\mathrm{A}$ and $\mathrm{B}$ 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 $\mathrm{B}$ receives the heat at temperature $\mathrm{T}$ and rejects to its sink at temperature $T_2$. For what value of $T$ the efficiencies of the two engines are equal:
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Verified Answer
The correct answer is:
$\sqrt{T_1 T_2}$
We know that efficiency of carnot engine $\eta=1-\frac{T_2}{T_1}$
Where $T_1=$ temperature of source $T_2=$ temp of $\operatorname{sink}$ For engine $A, n_{\mathrm{A}}=1-\frac{T}{T_1}$ an engine $B, n_{\mathrm{B}}=1-\frac{T_2}{T}$ From question
$\begin{array}{l}
n_A =n_B \\
1-\frac{T}{T_1} =1-\frac{T_2}{T} \\
\Rightarrow \frac{T}{T_1}=\frac{T_2}{T} =\mathrm{T}^2 \\
\Rightarrow \mathrm{T}^2 =\mathrm{T}_1 \mathrm{~T}_2 \\
\Rightarrow \mathrm{T} =\sqrt{T_1 T_2}
\end{array}$
Where $T_1=$ temperature of source $T_2=$ temp of $\operatorname{sink}$ For engine $A, n_{\mathrm{A}}=1-\frac{T}{T_1}$ an engine $B, n_{\mathrm{B}}=1-\frac{T_2}{T}$ From question
$\begin{array}{l}
n_A =n_B \\
1-\frac{T}{T_1} =1-\frac{T_2}{T} \\
\Rightarrow \frac{T}{T_1}=\frac{T_2}{T} =\mathrm{T}^2 \\
\Rightarrow \mathrm{T}^2 =\mathrm{T}_1 \mathrm{~T}_2 \\
\Rightarrow \mathrm{T} =\sqrt{T_1 T_2}
\end{array}$
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