A partial differential equation
\(\frac{∂^2T}{∂x^2}+\frac{∂^2T}{∂y^2}=0\)
is defined for the two-dimensional field T: T(x, y), inside a planar square domain of size 2 m x 2 m. Three boundary edges of the square domain are maintained at value T = 50, whereas the fourth boundary edge is maintained T = 100.
The value of T at the center of the domain is
\(\frac{∂^2T}{∂x^2}+\frac{∂^2T}{∂y^2}=0\)
is defined for the two-dimensional field T: T(x, y), inside a planar square domain of size 2 m x 2 m. Three boundary edges of the square domain are maintained at value T = 50, whereas the fourth boundary edge is maintained T = 100.
The value of T at the center of the domain is
- 50.0
- 62.5
- 75.0
- 87.5
The Correct Option is B
Solution and Explanation
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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).
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