§3.4. 4 Multipole fields

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

2. Outline • Review of general multipole expansionInternal / externalmultipoles – HW6Relation to general solution in spherical coordinatesRevisit external boundary conditions at r=0, ∞Are there multipoles for other coordinate systems? • Lowest order multipolesMonopole– point charge (l=0, scalar)Dipole– center of charge (l=1, vector)– spherical dipole: boundary value problemQuadrupole– moment of inertia (l=2, tensor [matrix])– opposing dipoles: example calculationOctupole– 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 E0at 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