MULTIPOLE EXPANSION IN MAGNETOSTATICS

We derive the multipole expansions of the magnetostatic field and vect or potential of an arbitrary steady current density. A simplifying par ameterization of the (l + 1)th-order tensor of lth-order moments of th e current density in terms of an lth-order tensor b(i1...il) allows us to derive all orders in the multipole expansions using only Cartesian coordinates of tensors. We do not use a magnetic scalar potential or spherical harmonics. The field B(l)(r) of the lth-order magnetostatic multipole depends on only the 2l + 1 independent components of the sym metric traceless part b(i1...il)(s0) of b(i1...il) in exactly the same way as the field E(l)(r) of the lth-order electrostatic multipole dep ends on the l-th-order symmetric traceless tensor rho(i1...il)(s0) of multipole moments of the charge density. The vector potential that dep ends on only the symmetric traceless tensors b(i1...il)(s0) differs fr om the vector potential in the Coulomb gauge. Our derivation shows tha t the fact that only the symmetric traceless part of b(i1...il) contri butes to the magnetostatic field is a consequence of charge conservati on and gauge invariance. (C) 1998 American Association of Physics Teac hers.