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In-Depth Information
Figure 21.1.
Analyzing a steady temperature
system.
We have explained the mathematical basis for the importance of the Laplace equa-
tion, now we want to sketch solutions to some practical problems and show how they
also lead to this equation. In a sense, classical field theory of physics is a study of the
solutions to the equation. (Actually, we have only stated the two-dimensional Laplace
equation and we should include the three-dimensional version for the previous sen-
tence to hold.) The reader will have to look up in topics on physics certain physical
laws to which we refer in the discussion if they are unfamiliar with them.
Steady Temperature.
Assume the temperature of a body
X
is defined by a function
T(x,y). Basically, we are assuming that the temperature is the same on all planes par-
allel to the xy-plane, so that we have a two-dimensional problem. The rate at which
heat crosses a curve is proportional to the integral of the normal derivative of T along
the curve. Let us see what happens in a small rectangle of width Dx and height Dy.
See Figure 21.1. The rate of flow of heat to the right through the left edge of the rec-
tangle is approximately
Ky
T
x
∂
∂
-
D
,
where K is a thermal conductivity constant associated to the solid. Using the deriva-
tive of this function we can estimate the loss of heat as we pass from the left edge to
the right edge of the rectangle by
2
∂
∂
T
x
-
Ky
D
D
x
.
2
Similarly, we can estimate the loss of heat as we pass from the bottom edge to the top
edge of the rectangle by
2
∂
∂
T
y
-
Kx
D
D
y
.
2
Since we are assuming a steady state, the losses must sum to 0, and we get the
equation
