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, $\mathrm { T } _ { 1 }$ and $\mathrm { T } _ { 2 }$. The temperature of the hot reservoir of the first engine is $\mathrm { T } _ { 1 }$ 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 $\mathrm { T } _ { 1 }$ and $\mathrm { T } _ { 2 }$, if both the engines perform equal amount of work? (1) $\mathrm { T } = \frac { 2 \mathrm {~T} _ { 1 } \mathrm {~T} _ { 2 } } { \mathrm {~T} _ { 1 } + \mathrm { T } _ { 2 } }$ (2) $\mathrm { T } = \frac { \mathrm { T } _ { 1 } + \mathrm { T } _ { 2 } } { 2 }$ (3) $T = \sqrt { T _ { 1 } T _ { 2 } }$ (4) $\mathrm { T } = 0$
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, $\mathrm { T } _ { 1 }$ and $\mathrm { T } _ { 2 }$. The temperature of the hot reservoir of the first engine is $\mathrm { T } _ { 1 }$ 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 $\mathrm { T } _ { 1 }$ and $\mathrm { T } _ { 2 }$, if both the engines perform equal amount of work?\\
(1) $\mathrm { T } = \frac { 2 \mathrm {~T} _ { 1 } \mathrm {~T} _ { 2 } } { \mathrm {~T} _ { 1 } + \mathrm { T } _ { 2 } }$\\
(2) $\mathrm { T } = \frac { \mathrm { T } _ { 1 } + \mathrm { T } _ { 2 } } { 2 }$\\
(3) $T = \sqrt { T _ { 1 } T _ { 2 } }$\\
(4) $\mathrm { T } = 0$