3. 3 Separation of Variables We seek a solution of the form Cartesian coordinatesCylindrical coordinates Spherical coordinates Not always possible! Usually only for the appropriate symmetry.

Example 3.3 Special boundary conditions (constant potential on planes):

Special choice of the separation constants to be able to fulfill the boundary conditions. Boundary conditions (i, ii, iv):

Boundary condition (iii): Fourier sum Fourier coefficients superposition

Example:

Contributions of the first terms of the Fourier sum at x=0. a) n=1, b) n<6, c) n<11, d) n<101

Set of functions is called

Jean Bapitiste Joseph Fourier 21 March 1768 – 16 May 1830

Example 3.4

Example 3.5 An infinitely long metal pipe is grounded, but one end is maintained at a given potential.

Spherical Coordinates Use for problems with spherical symmetry. Laplace’s equation: Boundary conditions on the surface of a sphere, origin, and infinity. Solution as a product

Assume azimuthal symmetry Solution as a product Separation constant Radial equation Solution

Angular equation Solutions Legendre polynomials The second solution can (usually) be excluded because it becomes infinite at  Rodrigues formula Orthogonality

The first Legendre polynomials

Example 3.6

Example 3.8

Multipole Expansion Approximate potential at large distance Dipole:

Potential of a general charge distribution at large distance Warning! The integral depends on the direction of r.

Addition theorem for Legendre polynomials: Spherical harmonics: solutions for 3D separation Angular distribution at large distance

The monopole and Dipole Terms monopole dipole dipole moment

A quadrupole has no dipole moment. physical dipole “pure” dipole is the limit Dipole moments are vectors and add accordingly.

In general, multipole moments depend on the choice of the coordinate system. Has a dipole moment. If Q=0 the dipole moment does not depend on the coordinate system.

The electric field of a dipole along the z-axis.