GATE 2023 CE (CE2) – Question 31
The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of 100 °C and the other three sides are kept at a temperature of 0 °C. Ignoring the effect of discontinuities in the boundary conditions at the corners, the steady-state temperature at the center of the plate is obtained as T₀ °C. Due to symmetry, the steady-state temperature at the center will be same (T₀ °C), when any one side of the square is kept at a temperature of 100 °C and the remaining three sides are kept at a temperature of 0 °C. Using the principle of superposition, the value of T₀ is ______ (rounded off to two decimal places).
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Correct answer: 24.88 to 25.12
Explanation
The Laplace equation is linear, so solutions can be **superposed**.
**Step 1: decompose.** Let $T_1$ be the solution with side AB at 100 °C and the other three at 0 °C (centre value $T_0$). By symmetry (rotating the square by 90°), the solutions with BC, CD or DA at 100 °C and the rest at 0 °C also have the centre value $T_0$.
**Step 2: add all four.** The sum of the four solutions has boundary values $100+0+0+0=100$ °C on every side. The steady solution of the Laplace equation with all sides at 100 °C is simply a uniform 100 °C.
**Step 3: centre value.** At the centre the sum is $4T_0$, and it equals 100 °C:
$$4T_0=100\;\Rightarrow\;T_0=\mathbf{25.00\ ^\circ C}.$$