§3.4.1–3 Multipole expansion

Christopher CrawfordPHY 311

2014-02-28

Outline• Review of boundary value problem

General 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])

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

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Polar waves – Legendre functions

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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

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

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Expansion of 2-pole potential• Electric dipole moment

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

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

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Monopole• Point-charge equivalent

of total charge in thedistribution

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Dipole• “center of charge”

of distribution– Significant when

total charge is zero

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Quadrupole

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