Title: 3.5 Two dimensional problems
13.5 Two dimensional problems
- Cylindrical symmetry
- Conformal mapping
2Laplace operator in polar coordinates
3Example Two half pipes
4 Conformal Mapping
Is there a simple solution?
5For 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.
6Analytical 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.
7g 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
8A corner of conductors
9Edge of a conducting plane
10Parallel Plate Capacitor
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