GATE CE 2023 Set 2 — Question 31
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Engineering Mathematics → Partial Differential Equations → 2D Laplace Equation
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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 °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 °C. Due to symmetry, the steady-state temperature at the center will be same ( °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 is _________________________ (rounded off to two decimal places).
Correct answer
24.9 to 25.1
Solution
The steady-state temperature in a square plate is governed by the Laplace equation . Let be the temperature at the center when side AB is at 100 °C and others are at 0 °C. By symmetry, . Let be the center temperatures when sides BC, CD, and DA are respectively at 100 °C and others at 0 °C. By symmetry, . By the principle of superposition, if all four sides are at 100 °C, the temperature at the center is . In a steady state where all boundaries are at a constant temperature , the temperature everywhere inside the domain must be . Thus, °C.
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