Two ideal Carnot engines opeate 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 the temperature of the sink of first engine which isi 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?

Options

A. `(2T_(1)T_(2))/(T_(1) + T_(2))`
B. `(T_(1) + T_(2))/(2)`
C. 0
D. `sqrt(T_(1)T_(2))`

Answer

B

Explanation

Suppose `Q_(H)=` Heat supplied to first engine
`Q_(L)`= Heat extracted from first engine
`Q._(L)`= Heat extracted from `2^(nd)` engine
Work done by first engine= work done by second engine
`Q_(H)- Q_(L) = Q_(L)- Q._(L)`
`:. Q_(H)+ Q._(L) = 2Q_(L)`
but `Q_(L) prop T and Q._(L) prop T_(2) "also" Q_(H) prop T_(1)`
`:. T_(1) + T_(2) = 2T`
`:. T= (T_(1) +T_(2))/(2)`