Multipole expansion in magnetostatics

We derive the multipole expansions of the magnetostatic field and vector potential of an arbitrary steady current density. A simplifying parametrization of the (l+1)th-order tensor of lth-order moments of the current density in terms of an lth-order tensor bi1…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 symmetric traceless part bi1…ils0 of bi1…il in exactly the same way as the field E(l)(r) of the lth-order electrostatic multipole depends on the lth-order symmetric traceless tensor ρi1…ils0 of multipole moments of the charge density. The vector potential that depends on only the symmetric traceless tensors bi1…ils0 differs from the vector potential in the Coulomb gauge. Our derivation shows that the fact that only the symmetric traceless part of bi1…il contributes to the magnetostatic field is a consequence of charge conservation and gauge invariance.