§3.4.1–3 Multipole expansion Christopher Crawford PHY 416 2014-11-12

Review: Boundary value problems 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

Outline Multipole expansion Binomial series – expansion of functions 2-pole expansion – dipole field (first term) General multipole expansion Results from this week’s HW Calculation of multipoles Example: pure dipole spherical distribution of charge Lowest order multipoles Monopole – 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]) Spherical vs cylindrical vs Cartesian tensors l,m vs matrices, sextupole moment

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

Expansion of 2-pole potential Electric dipole moment

General multipole expansion Brute force method – see HW 8 for simpler approach

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