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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):.

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3. 3 Separation of Variables

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  1. 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.

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

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

  4. superposition Boundary condition (iii): Fourier sum Fourier coefficients

  5. Example:

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

  7. Set of functions is called

  8. Jean Bapitiste Joseph Fourier 21 March 1768 – 16 May 1830

  9. Example 3.4

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

  11. 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.

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

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

  14. The first Legendre polynomials

  15. Example 3.6

  16. Example 3.8

  17. Dipole: Multipole Expansion Approximate potential at large distance

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

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

  20. monopole dipole dipole moment The monopole and Dipole Terms

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

  22. 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.

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

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