§3.4. 4 Multipole fields
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§3.4.4 Multipole fields
Christopher CrawfordPHY 311
2014-03-03
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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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