Question

Two similar metallic rods of the same length \( l \) and area of cross section \( A \) are joined and maintained at temperatures \( T_1 \) and \( T_2 \) (\( T_1>T_2 \)) at one of their ends as shown in the figure. If their thermal conductivities are \( K \) and \( \frac{K}{2} \) respectively. The temperature at the joining point in the steady state is: 

• A. \(\frac{T_1 + T_2}{2}\)
• B. \(\frac{2(T_1 - T_2)}{3}\)
• C. \(\frac{2T_1 + T_2}{3}\)
• D. \(\frac{T_1 - T_2}{2}\)
• E. \(\frac{3(T_1 - T_2)}{2}\)

Hint:

In heat transfer through series of conductors, the heat flow rate is constant, and the temperature at interfaces depends on the conductivities.

Answer 1

The Correct Option is C

In a steady state, the heat flow through each rod must be equal, hence: \[ \frac{K (T_1 - T)}{l} = \frac{\frac{K}{2} (T - T_2)}{l} \] Solving for \( T \), the temperature at the joining point: \[ K (T_1 - T) = \frac{K}{2} (T - T_2) \implies 2(T_1 - T) = T - T_2 \] \[ 2T_1 - 2T = T - T_2 \implies 3T = 2T_1 + T_2 \implies T = \frac{2T_1 + T_2}{3} \] This equation shows that the temperature \( T \) at the junction is a weighted average of \( T_1 \) and \( T_2 \), more influenced by \( T_1 \) due to the higher thermal conductivity of the first rod.