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Also the potential is undefined at the origin since

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Unformatted text preview: mple, the associated complex potential function is z F(z) = ∞1 + (∞2 - ∞1) h The complex conjugate of ∞ is therefore y § = Im(F(z)) = (∞2 - ∞1) h In short: 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: 1 1 Ô2∞ = ∞rr + ∞r + 2 ∞øø = 0 r r Since ∞ depends only on r by symmetry, we are reduced to 1 Ô2∞ = ∞rr + ∞r = 0 r We can write this as ∞rr 1 =r ∞ r d 1 or ln(∞r) = dr r Thus ln(∞r) = - ln(r) + K = ln(A) - ln(r), say A so ∞r = r 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), and so 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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