• # question_answer Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, ${{T}_{1}}$and ${{T}_{2}}$. The temperature of the hot reservoir of the first engine is ${{T}_{1}}$ and the temperature of the cold reservoir of the second engine is${{T}_{2}}$. $T$ is temperature of the sink of first engine which is also the source for the second engine. How is $T$ related to ${{T}_{1}}$ and${{T}_{2}}$, if both the engines perform equal amount of work?  [JEE MAIN Held on 07-01-2020 Evening] A) $T=\frac{{{T}_{1}}+{{T}_{2}}}{2}$ B) $T=\sqrt{{{T}_{1}}{{T}_{2}}}$ C) $T=\frac{2{{T}_{1}}{{T}_{2}}}{{{T}_{1}}+{{T}_{2}}}$     D) $T=0$

 [a] Let${{Q}_{1}}$: Heat input to first engine ${{Q}_{C}}$: Heat rejected by first engine ${{Q}_{2}}$: Heat rejected by second engine ${{T}_{C}}$: Lower temperature of first engine $W={{Q}_{1}}{{Q}_{C}}={{Q}_{C}}{{Q}_{2}}$ $\Rightarrow \,\,\,\,2{{Q}_{C}}={{Q}_{1}}+{{Q}_{2}}$ $\Rightarrow 2{{T}_{C}}={{T}_{1}}+{{T}_{2}}\Rightarrow {{T}_{C}}={{T}_{1}}+{{T}_{2}}$