Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductiv…

Question

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 T1 and that at the right junction is T2. The ratio T1/T2 is:

1. A. 3/2
2. B. 4/3
3. C. 5/3 (Correct answer)
4. D. 5/4

Correct Answer

Option C — 5/3

Detailed Solution & Explanation

The correct answer is $\dfrac{5}{3}$.

Key Points

• Thermal resistance $R=\dfrac{l}{KA}$. The two side rods have $R=\dfrac{l}{2KA}$ each and the middle rod $R=\dfrac{l}{KA}$, so $R_{eq}=\dfrac{2l}{KA}$.
• In steady state the heat current is the same through each rod. Across the first rod: $\dfrac{3T-T_1}{l/2KA}=\dfrac{3T-T}{R_{eq}}$, giving $T_1=\dfrac{5T}{2}$.
• Across the last rod: $\dfrac{T_2-T}{l/2KA}=\dfrac{2T}{R_{eq}}$, giving $T_2=\dfrac{3T}{2}$.
• $\dfrac{T_1}{T_2}=\dfrac{5T/2}{3T/2}=\dfrac{5}{3}$.