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A thin piece of thermal conductor of constant thermal conductivity insulated on the lateral sides connects two reservoirs which are maintained at temperatures $T_{1}$ and $T_{2}$ as shown. Assuming that the system is in steady state, which of the following plots best represents the dependence of the rate of change of entropy of the ratio of temperatures $\mathrm{T}_{1} / \mathrm{T}_{2}$
Options:
- A
- B
- C
- D
Solution:
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The correct answer is:
$\begin{aligned} \mathrm{ds} &=\frac{-\mathrm{Qdt}}{\mathrm{T}_{1}}+\frac{\mathrm{Qdt}}{\mathrm{T}_{2}} \\ \frac{\mathrm{ds}}{\mathrm{dt}} &=-\mathrm{Q}\left[\frac{1}{\mathrm{~T}_{1}}-\frac{1}{\mathrm{~T}_{2}}\right] \\ &=\frac{-\left(\mathrm{T}_{1}-\mathrm{T}_{2}\right)}{\mathrm{R}}\left[\frac{\mathrm{T}_{2}-\mathrm{T}_{1}}{\mathrm{~T}_{1} \mathrm{~T}_{2}}\right] \\ &=+\left(\frac{\mathrm{T}_{1}^{2}-\mathrm{T}_{2}^{2}}{\mathrm{~T}_{1} \mathrm{~T}_{2}}\right) \frac{1}{\mathrm{R}} \\ \frac{\mathrm{ds}}{\mathrm{dt}} &=\left(\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}-\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}\right) \frac{1}{\mathrm{R}} \\ &=\left(\mathrm{x}-\frac{1}{\mathrm{x}}\right) \frac{1}{\mathrm{R}} \end{aligned}$
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