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Unformatted text preview: in the two-dimensional case. Since T satisfies Laplace's equation, it is also called the heat
potential, and is, as usual, the real part of a complex potential
F(z) = T(x, y) + iﬂ(x, y)
The equipotentials T = const are called isotherms and the curves § = const are heat
flow lines. The conjugate derivative F'(z) gives the heat flow vector field, measured in
units of energy per unit time.
36 (A) Temperature between parallel plates
Going back to Topic 9 on around p. 28, we find that the complex potential is just linear:
F(z) = Az + B
where A and B can be found from the temperatures of the two plates and their distance
(B) Insulated Hot Wire
A Hot Wire (500º) of radius 1 mm. in the center inside a cool cylinder (60º) or radius
100 mm. on the outside: Again going back to Topic 9, we use
F(z) = ALn(z) + B
Looking at the real part:
500 = Aln(1) + B = B
60 = Aln(100) + B = Aln(100) + 500
So we get A ‡ -95.54 and so
F(z) = -95.54Ln(z) + 500
This gives the temperature as the real part:
T(x, y) = -95.54 ln r + 500,
where r is...
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