1. §3.4.1–3 Multipole expansion
Christopher Crawford
PHY 416

2. 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

3. 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

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

5. Expansion of 2-pole potential
Electric dipole moment

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

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