1. Christopher Crawford PHY 416 2014-11-17
§3.4.4 Multipole fields
Christopher Crawford
PHY 416

2. Outline
Review of general multipole expansion Internal / external multipoles – HW8 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]) Octupole – eight points (l=3 [cubic matrix]) (Sextupole?) – six rods
Tensors – Spherical vs. Cartesian

3. General multipole expansion
Results from HW 6

4. 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=∞

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

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

7. Dipole field
Internal and external

8. Quadrupole

9. Octupole

10. Spherical vs. Cartesian tensors
Matrices vs. angular momentum