Multipole expansion in magnetostatics

Abstract

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.

Magnetometers for tracking metallic targets

“…The lth magnetic moment m i1...i l is a symmetric traceless tensor of lth order having 2l + 1 independent components. The magnetic field of the lth multipole may be expressed as (for derivation see [8])…”

Singular behaviour of the electromagnetic field (the static case revisited)

“…It appears that cumbersome calculations are involved for higher n if we apply the formulae for higher order derivatives of 1/r as in the previous section. However, we can employ an invariance property of the electrostatic field to the substitutions of all moments P (n) , for all n, by their corresponding symmetric and trace-free STF projections P (n) [9,10,11]. Retaining the notation p for the first order moment, this invariance stands for the invariance of the multipole expansion of the electrostatic field to the following substitutions:…”

“…From this last equation, it is seen that the multipole expansion of B(r) (r = 0) is invariant to the substitution M (3) → M (3) [9,10].…”

Multipole Analysis of Circular Cylindircal Magnetic Systems

“…Multipole theory in electromagnetism [120,121,122,123,124] has a long and fruitful history. In fact, Clerk Maxwell [125], used the same analysis to evaluate many of the integrals he obtained in solving various electromagnetic problems.…”