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To determine the temperature at the junction $ x $ of three identical metal rods, follow this analysis:

Assuming No Heat Loss: The rods are made of the same material and are identical, ensuring consistent heat flow throughout.

Thermal Resistance: Let the thermal resistance of each rod be $ R $. The temperature at the junction is $ T_x $.

Heat Flow Equilibrium: Since the system reaches a steady state, the heat flowing into the junction from each side should equal the heat flowing out:

$ H = H_1 + H_2 $

Where:

$ H $ is the heat flowing into the junction from both ends.

$ H_1 $ and $ H_2 $ are the heat flows from the sides of the junction.

Application: Set up the heat balance equation:

$ \frac{T_x - 0}{R} = \frac{90 - T_x}{R} + \frac{90 - T_x}{R} $

Simplifying: Substitute and solve:

$ T_x = \frac{180}{3} = 60^\circ \text{C} $

Therefore, the temperature at the junction $ x $ is $ 60^\circ \text{C} $. This calculation assumes perfect thermal conduction and no heat loss outside the system.

Heat And Thermodynamics

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