In electromagnetism, 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 (donut-shaped) charge arrangements whereas polar (or magnetic) toroidal dipole (also called anapole) correspond to the field of a solenoid bent into a torus. A complex expression allows the current density J to be written as a sum of electric, magnetic, and toroidal moments using Cartesian or spherical differential operators. The lowest order toroidal term is the toroidal dipole. Its magnitude along direction i is given by 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. 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. The interaction of this term is most easily understood in the non-relativistic limit, where the Hamiltonian is where d, μ, and a are the electric, magnetic, and anapole moments, respectively, and σ is the vector of Pauli matrices. The nuclear toroidal moment of cesium was measured in 1997 by Wood et al..

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Related concepts (2)
Electric dipole moment
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The debye (D) is another unit of measurement used in atomic physics and chemistry. Theoretically, an electric dipole is defined by the first-order term of the multipole expansion; it consists of two equal and opposite charges that are infinitesimally close together, although real dipoles have separated charge.
Magnetic moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets), permanent magnets, elementary particles (such as electrons), composite particles (such as protons and neutrons), various molecules, and many astronomical objects (such as many planets, some moons, stars, etc).

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