5 3 Magnetic Multipole Expansion Christopher Crawford PHY

§ 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 2

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 3

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 – Torque – Potential 4

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

Expansion of vector potential • From Griffiths 6

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

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

Review – setting up B. V. P. 9

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