3. 3 Separation of Variables

Cartesian coordinates Cylindrical coordinates Spherical coordinates 3. 3 Separation of Variables We seek a solution of the form 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):

superposition Boundary condition (iii): Fourier sum Fourier coefficients

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.

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

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

Angular equation Solutions Legendre polynomials Rodrigues formula Orthogonality The second solution can (usually) be excluded because it becomes infinite at q=0, p.

The first Legendre polynomials

Example 3.6

Example 3.8

Dipole: Multipole Expansion Approximate potential at large distance

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

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

monopole dipole dipole moment The monopole and Dipole Terms

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

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.

3. 3 Separation of Variables