Christopher Crawford PHY 311 2014-03-03 §3.4.4 Multipole fields Christopher Crawford PHY 311 2014-03-03

Outline Review of general multipole expansion Internal / external multipoles – HW6 Relation to general solution in spherical coordinates Revisit external boundary conditions at r=0, ∞ Are there multipoles for other coordinate systems? Lowest order multipoles Monopole – point charge (l=0, scalar) Dipole – center of charge (l=1, vector) – spherical dipole: boundary value problem Quadrupole – moment of inertia (l=2, tensor [matrix]) – opposing dipoles: example calculation Octupole – eight points (l=3 [cubic matrix]) (Sextupole?) – six rods Tensors – Spherical vs. Cartesian

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

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=∞

Monopole Point-charge equivalent: – total charge of the distribution External monopole?

Dipole “center of charge” of distribution External dipole field? Significance when total charge q=0

Review: pure spherical dipole Multipole moments Boundary Value Problem (BVP)

Quadrupole

Example: four-pole Sum over point charges Sum over opposing dipoles

Sextupole vs. Octupole

Spherical vs. Cartesian tensors Matrices vs. angular momentum