Question

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 $100^\circ$C and the other three sides are kept at a temperature of $0^\circ$C. Ignoring the effect of discontinuities at the corners, the steady-state temperature at the center of the plate is obtained as $T_0^\circ$C. Due to symmetry, the steady-state temperature at the center will be the same ($T_0^\circ$C) when any one side of the square is kept at a temperature of $100^\circ$C and the remaining three sides are kept at $0^\circ$C. Using the principle of superposition, find $T_0$ (rounded off to two decimal places).

Hint:

For linear PDEs like Laplace's equation, use symmetry and superposition: sum "one-side-heated" solutions to convert to an "all-sides-heated" uniform case, then back-calculate.

Answer 1

Step 1: Define a base problem.
Let Problem~$P_1$ be the plate with side AB at $100^\circ$C and the other three sides at $0^\circ$C; let the center temperature be $T_0^\circ$C.

Step 2: Use symmetry to create four problems.
By symmetry, if instead we heat any single side (AB, BC, CD, or DA) to $100^\circ$C with the others at $0^\circ$C, the center temperature is the same $T_0^\circ$C for each case.

Step 3: Superposition.
Superimpose the four single-side-heated solutions. On the boundary, each side receives one contribution of $100^\circ$C and three of $0^\circ$C, so every side becomes $100^\circ$C.
Thus the superposed boundary condition is all four sides at $100^\circ$C.

Step 4: Center temperature for the superposed problem.
For Laplace's equation with all boundaries at $100^\circ$C, the steady-state solution is uniform: $T(x,y)\equiv 100^\circ$C throughout the plate. Hence the center temperature is $100^\circ$C.

Step 5: Relate to $T_0$.
Temperatures add under superposition, so the center temperature of the superposed problem equals $T_{\text{center}}=T_0+T_0+T_0+T_0=4T_0$.
Therefore, $4T_0 = 100 \Rightarrow T_0 = 25^\circ$C.
\[ \boxed{T_0 = 25.00^\circ\text{C}} \]