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§3.4. 1–3 Multipole expansion

§3.4. 1–3 Multipole expansion. Christopher Crawford PHY 311 2014-02-28. Outline. Review of boundary value problem General solution to Laplace equation Internal and external boundary conditions Orthogonal functions – extracting A n from f(x)

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§3.4. 1–3 Multipole expansion

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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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