§3.4.4 Multipole fields

Christopher CrawfordPHY 311

2014-03-03

Outline• Review of general multipole expansion

Internal / external multipoles – HW6Relation to general solution in spherical coordinatesRevisit external boundary conditions at r=0, ∞Are there multipoles for other coordinate systems?

• Lowest order multipolesMonopole – point charge (l=0, scalar)Dipole – center of charge (l=1, vector)– spherical dipole: boundary value problemQuadrupole – moment of inertia (l=2, tensor [matrix])– opposing dipoles: example calculationOctupole – eight points (l=3 [cubic matrix])(Sextupole?) – six rods

• Tensors – Spherical vs. Cartesian

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Review: general multipole expansion• Brute force method – see HW 6 for simpler approach

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General solution; boundary conditions• Multipoles Q(l)

int, Q(l)ext are essentially the coefficients Al, Bl

• Generalized external boundary conditions – multipoles

• Examples– point charge Q at r=0

– External field E0 at r=∞

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Monopole• Point-charge equivalent:

– total charge of the distribution

• External monopole?

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Dipole

• “center of charge” of distribution

• External dipole field?

• Significance when total charge q=0

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Review: pure spherical dipole• Multipole moments

• Boundary Value Problem (BVP)

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Quadrupole

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Example: four-pole• Sum over point charges

• Sum over opposing dipoles

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Sextupole vs. Octupole

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Spherical vs. Cartesian tensors• Matrices vs. angular momentum

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