Three identical rods are joined as shown in the figure. The left and right ends are kept at \( 0^\circ C \) and \( 90^\circ C \) as shown in the figure. The temperature \( \theta \) at the junction of the rods is:
Show Hint
- Use heat balance at the junction when multiple rods are connected.
- If all rods are identical, the heat flux equation simplifies based on temperature differences.
- Remember, thermal equilibrium ensures that heat inflow equals heat outflow at steady state.
Step 1: Understanding Heat Conduction in Identical Rods The principle of thermal equilibrium states that heat flux through each rod should be equal at the junction. Since all rods are identical, they have the same thermal conductivity and length, so the heat flow rate through each rod is given by Fourier's Law: \[ Q = k A \frac{\Delta T}{L} \] Since \( k, A, \) and \( L \) are the same for all rods, we use the temperature differences for balancing heat flux.
Step 2: Applying Heat Balance at the Junction Let \( \theta \) be the temperature at the junction. The left rod has a temperature difference of \( \theta - 0 \), while the two right rods have a temperature difference of \( 90 - \theta \). At steady-state thermal equilibrium: \[ \text{Heat inflow} = \text{Heat outflow} \] \[ k A \frac{\theta - 0}{L} = 2 \times k A \frac{90 - \theta}{L} \] Canceling common terms: \[ \theta = 2 (90 - \theta) \] \[ \theta + 2\theta = 180 \] \[ 3\theta = 180 \] \[ \theta = 60^\circ C \]
Step 3: Verifying the Correct Option Comparing with the given options, the correct answer is: \[ \mathbf{60^\circ C} \]
