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