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Multipole Expansion in Physics: Theory and Applications

Explore the multipole expansion method in physics, including monopoles, dipoles, quadrupoles, and octupoles. Learn how to calculate multipoles and apply them in solving boundary value problems. Understand the general solutions to Laplace's equation in various coordinate systems.

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Multipole Expansion in Physics: Theory and Applications

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  1. §3.4.1–3 Multipole expansion Christopher Crawford PHY 311 2014-02-28

  2. Outline • Review of boundary value problemGeneral solution to Laplace equationInternal and external boundary conditionsOrthogonal functions – extracting An from f(x) • Multipole expansionBinomial series– expansion of functions2-pole expansion – dipole field (first term)General multipole expansion • Calculation of multipolesExample: pure dipole spherical distribution of charge • Lowest order multipolesMonopole– point charge (l=0, scalar)Dipole– two points (l=1, vector)Quadrupole– four points (l=2, tensor [matrix])Octupole– eight points (l=3, tensor [cubic matrix])

  3. Review: separation of variables • k2 = curvature of wave –> 0 [Laplacian]

  4. Polar waves – Legendre functions

  5. General solutions to Laplace eq’n • Cartesian coordinates – no general boundary conditions! • Cylindrical coordinates – azimuthal continuity • Spherical coordinates – azimuthal and polar continuity • Boundary conditions • Internal: 2 conditions across boundary • External: 1 condition (flux or potential) on boundary • Orthogonality – to extract components

  6. Expansion of functions • Closely related to functions as vectors (basis functions)

  7. Expansion of 2-pole potential • Electric dipole moment

  8. General multipole expansion • Brute force method – see HW 6 for simpler approach

  9. Example: integration of multipole • Pure spherical dipole distribution – will use in Chapter 4, 6

  10. Monopole • Point-charge equivalentof total charge in thedistribution

  11. Dipole • “center of charge” of distribution • Significant when total charge is zero

  12. Quadrupole

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