Toroidal moment

A toroidal moment is an independent term in the multipole expansion of electromagnetic fields besides magnetic and electric multipoles. In the electrostatic multipole expansion, all charge and current distributions can be expanded into a complete set of electric and magnetic multipole coefficients. However, additional terms arise in an electrodynamic multipole expansion. The coefficients of these terms are given by the toroidal multipole moments as well as time derivatives of the electric and magnetic multipole moments. While electric dipoles can be understood as separated charges and magnetic dipoles as circular currents, axial (or electric) toroidal dipoles describes toroidal charge arrangements whereas polar (or magnetic) toroidal dipole (also called anapole) correspond to the field of a solenoid bent into a torus.

Classical toroidal dipole moment[]

A complex expression allows the current density J to be written as a sum of electric, magnetic, and toroidal moments using Cartesian^[1] or spherical^[2] differential operators. The lowest order toroidal term is the toroidal dipole. Its magnitude along direction i is given by

${\displaystyle T_{i}={\frac {1}{10c}}\int [r_{i}(\mathbf {r} \cdot \mathbf {J} )-2r^{2}J_{i}]\mathrm {d} ^{3}x.}$

Since this term arises only in an expansion of the current density to second order, it generally vanishes in a long-wavelength approximation.

However, a recent study comes to the result that the toroidal multipole moments are not a separate multipole family, but rather higher order terms of the electric multipole moments.^[3]

Quantum toroidal dipole moment[]

In 1957, Yakov Zel'dovich found that because the weak interaction violates parity symmetry, a spin 1⁄2 Dirac particle must have a toroidal dipole moment, also known as an anapole moment, in addition to the usual electric and magnetic dipoles.^[4] The interaction of this term is most easily understood in the non-relativistic limit, where the Hamiltonian is

ℋ ∝ −