3.5 Two dimensional problems Cylindrical symmetry Conformal mapping

Laplace operator in polar coordinates

Example: Two half pipes

Conformal Mapping Is there a simple solution?

For two-dimensional problems complex analytical function are a powerful tool of much elegance. x iy Maps (x,y) plane onto (u,v) plane. For analytical functions the derivative exists. Examples:

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.

g and h are conjugate. If g=V then g=const gives the equipotentials and h=const 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. Cartesian onto polar coordinates: Example Polar onto Cartesian coordinates: Full plane z w

A corner of conductors

Edge of a conducting plane equipotentials field lines

Parallel Plate Capacitor