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

2. 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

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

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

8. Quadrupole

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

10. Sextupole vs. Octupole

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