3.5 Two dimensional problems

1. 3.5 Two dimensional problems • Cylindrical symmetry • Conformal mapping

2. Laplace operator in polar coordinates

3. Example: Two half pipes

4. Conformal Mapping Is there a simple solution?

5. iy x Examples: 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.

6. 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.

7. 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. Example w z Cartesian onto polar coordinates: Full plane Polar onto Cartesian coordinates:

8. A corner of conductors

9. equipotentials field lines Edge of a conducting plane

10. Parallel Plate Capacitor