§5.3: Magnetic Multipole Expansion Christopher Crawford PHY 417 2015-02-06

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

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

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

Electric vs. magnetic dipole force Electric dipole Magnetic dipole

Expansion of vector potential From Griffiths

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

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

Review – setting up B.V.P.

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