1. §5.3: Magnetic Multipole Expansion
Christopher Crawford
PHY 417

2. Outline Motivation – electric vs. magnetic dipole
Torque and force, dipole potential/field
Biot-Savart calculation of current loop
General derivation of axisymmetric multipoles
Boundary value problem – azimuthal currents New orthogonal function basis
Multipole expansion – electric vs. magnetic
Derivation of Helmholtz coil

3. Motivation – magnetic dipole
Original definition of B-field τ = m × B compare: τ = p × E
we need to relate this to Ampère’s law – next slide
Definition of magnetic multipole m=I a compare: p=q d
Dipole field – small current loop

4. Torque on magnetic dipole
Electric dipole
Remember pure spherical surface charge dipole
Torque
Potential
Magnetic dipole – Gilbert vs Ampere dipole
We will solve same BVP: spherical surface current dipole today

5. Electric vs. magnetic dipole force
Electric dipole
Magnetic dipole

6. Expansion of vector potential
From Griffiths

7. Dipole field From Griffiths – calculate the derivative
Coordinate-free notation – same tensor as in quadrupole!

8. General multipole expansion
Electric multipoles
No external monopole – just a constant potential offset
Magnetic multipoles
Derivatives P’(x) –> there is NO monopole at all

9. Review – setting up B.V.P.

10. A new orthogonal function basis
Derivative of Legendre polynomial – associated Legendre function