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Unformatted text preview: mple, the associated complex potential function is
F(z) = ∞1 + (∞2 - ∞1)
The complex conjugate of ∞ is therefore
§ = Im(F(z)) = (∞2 - ∞1)
For parallel plates, the complex potential F(z)= Az + B is just a linear function of z
(A, B real)
(B) Potential between two Coaxial Cylinders
Here we need to solve Laplace's equation in the xy-pane using polar coordinates:
Ô2∞ = ∞rr + ∞r + 2 ∞øø = 0
Since ∞ depends only on r by symmetry, we are reduced to
Ô2∞ = ∞rr + ∞r = 0
We can write this as
ln(∞r) = dr
Thus ln(∞r) = - ln(r) + K = ln(A) - ln(r), say
giving ∞ = A ln(r) + B
We can now solve for A and B by knowing the potentials on each of the two cylinders
and their radii.
For the associated complex potential, we use that fact that ln(r) is the real part of Ln(z),
F(z) = ALn(z) + B
29 with associated conjugate potential
§(z) = A Arg(z)
For a circular symmetry situation (potential independent of ø), the complex potential
F(z)= ALn(z) + B is just a linear function of Ln(z).
(C) Potential in an Angular Region
Here we have two plates at an angle å w...
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