Question

Three metal rods of the same material and identical in all respects are joined as shown in the figure. The temperatures at the ends of these rods are maintained as indicated. Assuming no heat energy loss occurs through the curved surfaces of the rods, the temperature at the junction is 

• A. 20°C
• B. 45°C
• C. 60°C
• D. 30°C

Hint:

In thermal equilibrium, the rate of heat flow through each part of the system is equal, and this can be used to find the temperature at the junction.

Answer 1

The Correct Option is D

Given that the rods are identical in all respects and made of the same material, the heat conduction through each rod will be governed by the same thermal conductivity. The junction temperature will be determined by the principle of thermal equilibrium, which ensures that the rate of heat flow through each rod is equal. Let the heat flow through each rod be denoted by \( Q_1 \), \( Q_2 \), and \( Q_3 \) for the rods with temperatures at the ends \( 20^\circ C \), \( 30^\circ C \), and \( 60^\circ C \), respectively. The rate of heat flow through a rod is given by Fourier’s law of heat conduction: \[ Q = \frac{kA \Delta T}{L} \] where: - \( k \) is the thermal conductivity, - \( A \) is the cross-sectional area of the rod, - \( \Delta T \) is the temperature difference across the rod, - \( L \) is the length of the rod. Since all rods are identical, the thermal conductivities \( k \), areas \( A \), and lengths \( L \) are the same for all the rods. The heat flow is therefore directly proportional to the temperature difference across each rod. Thus, we have: \[ Q_1 \propto 20^\circ C - T \] \[ Q_2 \propto 30^\circ C - T \] \[ Q_3 \propto 60^\circ C - T \] At thermal equilibrium, the heat flow through each rod must be equal, so: \[ 20 - T = 30 - T = 60 - T \] Solving this system gives \( T = 30^\circ C \). Thus, the temperature at the junction is \( 30^\circ C \).