Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductivity $2K$ while that in the middle has thermal conductivity $K$. The left end of the combination is maintained at temperature $3T$ and the right end at $T$. The rods are thermally insulated from outside. In steady state, temperature at the left junction is $T_1$ and that at the right junction is $T_2$. The ratio $T_1/T_2$ is
Answer: (C) $\dfrac{5}{3}$
Concept. Rods in series carry the same heat current $H = \dfrac{\Delta T}{R_{th}}$, with thermal resistance $R_{th} = \dfrac{l}{KA}$; resistances add.
Step 1. Resistances. Each outer rod ($2K$) has resistance $R$; the middle rod ($K$) has $2R$. Total $= 4R$.
Step 2. Heat current.
$$H = \frac{3T - T}{4R} = \frac{T}{2R}$$
Step 3. Junction temperatures. Each outer rod drops $HR = \dfrac{T}{2}$:
$$T_1 = 3T - \frac{T}{2} = \frac{5T}{2},\qquad T_2 = T + \frac{T}{2} = \frac{3T}{2}$$
Step 4. Ratio. $\dfrac{T_1}{T_2} = \dfrac53$.
Answer. $\dfrac53$ (Option C).
Solution by Sreeraj P, M.Sc Physics