Q 11-11-148JEE MainJEE Main 2019 (10 Jan, Shift 1)Medium
Three Carnot engines operate in series between a heat source at a temperature $T_1$ and a heat sink at temperature $T_4$. The first engine works between $T_1$ and $T_2$, the second between $T_2$ and $T_3$, and the third between $T_3$ and $T_4$, with $T_1 > T_2 > T_3 > T_4$. The three engines are equally efficient if:
Answer: (C) $T_2 = \left(T_1^2T_4\right)^{1/3};\ T_3 = \left(T_1T_4^2\right)^{1/3}$
Equal efficiencies $1 - \dfrac{T_2}{T_1} = 1 - \dfrac{T_3}{T_2} = 1 - \dfrac{T_4}{T_3}$ mean
$$\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_4}{T_3} = k,\qquad k^3 = \frac{T_4}{T_1}$$
$$T_2 = kT_1 = T_1^{2/3}T_4^{1/3} = \left(T_1^2T_4\right)^{1/3},\qquad T_3 = k^2T_1 = \left(T_1T_4^2\right)^{1/3}$$
Solution by Sreeraj P, M.Sc Physics