3.5 Two dimensional problems

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3.5 Two dimensional problems

• Cylindrical symmetry
• Conformal mapping

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Laplace operator in polar coordinates
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Example Two half pipes
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Conformal Mapping
Is there a simple solution?
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For two-dimensional problems complex analytical
function are a powerful tool of much elegance.
Maps (x,y) plane onto (u,v) plane.
For analytical functions the derivative exists.
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Analytical functions obey the Cauchy-Riemann
equations
which imply that g and h obey the Laplace
equation,
If g(x,y) fulfills the boundary condition it is
the potential.
If h(x,y) fulfills the boundary condition it is
the potential.
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g and h are conjugate. If gV then gconst gives
the equipotentials and hconst gives the field
lines, or vice versa.
If F(z) is analytical it defines a conformal
mapping.
A conformal transformation maps a rectangular
grid onto a curved grid, where the coordinate
lines remain perpendicular.
Example
w
z
Cartesian onto polar coordinates
Full plane
Polar onto Cartesian coordinates
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A corner of conductors
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Edge of a conducting plane
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Parallel Plate Capacitor
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