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,
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 ?
Correct Option: , 2
(2) Let $Q_{H}=$ Heat taken by first engine
$Q_{L}=$ Heat rejected by first engine
$Q_{2}=$ Heat rejected by second engine
Work done by $1^{\text {st }}$ engine $=$ work done by $2^{\text {nd }}$ engine
$W=Q_{H}-Q_{L}=Q_{L}-Q_{2} \Rightarrow 2 Q_{L}=Q_{H}+Q_{2}$
$2=\frac{\theta_{H}}{\theta_{L}}+\frac{\theta_{2}}{\theta_{L}}$
Let $\mathrm{T}$ be the temperature of cold reservoir of first engine. Then in carnot engine.
$\frac{Q_{H}}{Q_{L}}=\frac{T_{1}}{T}$ and $\frac{Q_{L}}{Q_{2}}=\frac{T}{T_{2}}$
$\Rightarrow 2=\frac{T_{1}}{T}+\frac{T_{2}}{T}$ using (i)
$\Rightarrow 2 T=T_{1}+T_{2} \Rightarrow T=\frac{T_{1}+T_{2}}{2}$