§3.4. 1–3 Multipole expansion Christopher Crawford PHY

Outline Review of boundary value problem General solution to Laplace equation Internal and external boundary conditions Orthogonal functions – extracting A n from f(x) Multipole expansion Binomial series – expansion of functions 2-pole expansion – dipole field (first term) General multipole expansion 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]) 2

Review: separation of variables k 2 = curvature of wave –> 0 [Laplacian] 3

Polar waves – Legendre functions 4

General solutions to Laplace eq’n 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 5

Expansion of functions Closely related to functions as vectors (basis functions) 6

Expansion of 2-pole potential Electric dipole moment 7

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

Example: integration of multipole Pure spherical dipole distribution – will use in Chapter 4, 6 9

Monopole Point-charge equivalent of total charge in the distribution 10

Dipole “center of charge” of distribution – Significant when total charge is zero 11

Quadrupole 12