Three rods made of the same material and having the same cross-section have been joined as shown in figure. Each rod is of same length. The left and right ends are kept at \(0^\circ C\) and \(90^\circ C\) respectively. The temperature of the junction will be
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At thermal equilibrium:
\[
\text{Heat entering}=\text{Heat leaving}
\]
Use electrical resistance analogy for thermal conduction problems.
Step 1: Understand the junction setup. Three identical rods meet at a junction. In the figure two rods connect the hot end at $90^\circ$C to the junction, and one rod connects the junction to the cold end at $0^\circ$C. We want the steady junction temperature $T$.
Step 2: Use the steady state rule. In steady state, the heat flowing into the junction equals the heat flowing out. No heat piles up. Step 3: Same thermal resistance. Each rod is the same material, length and cross section, so each has the same conducting factor $\tfrac{kA}{L}$. This factor will cancel later. Step 4: Write heat in and heat out. Two hot rods bring heat in, one cold rod carries it out. \[ 2\,\frac{kA}{L}(90 - T) = \frac{kA}{L}(T - 0) \] Step 5: Cancel and simplify. Cancel $\tfrac{kA}{L}$ from both sides. \[ 2(90 - T) = T \] \[ 180 - 2T = T \] Step 6: Solve for the junction temperature. \[ 180 = 3T \] \[ T = 60^\circ\text{C} \] \[ \boxed{60^\circ\text{C}} \]
