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A toroidal moment is a type of an electromagnetic moment, much like the more familiar electric dipole moment and magnetic dipole moment. Whereas the idealized point-like electric dipole carrier can be thought of as a pair of opposite charges (q and -q) separated by a distance d in the limit where d goes to zero as the electric dipole moment |p|=qd remains constant; and the point-like magnetic dipole carrier can be thought of as a planar current loop of area A and current I again in the limit that A goes to zero while the magnetic dipole moment |μ|=IA remains constant; then the point-like toroidal moment carrier can be thought of as a toroidal inductor coil in a limit where its size goes to zero.

A toroidal inductor coil is a current-carrying wire wound around a torus and can be thought of as an arrangement of magnetic dipoles (of magnitude μ) around in a ring (of radius R) so that each magnetic moment points along the ring. The magnitude of the toroidal moment then is given by

|t| = N μ R / 2

and its direction is through the axis of the ring in a direction given by the right hand rule. More generally, for any arrangement of magnetic dipoles, the toroidal moment is given by

t = ∑i ri × μi / 2

Toroidal moments are interesting creatures for a couple of reasons. The first is that they don't create any field (This is actually the reason why one uses toroidal inductor coils are sometimes instead of cylindrical ones: since the current in the coil doesn't create any field outside the torus, there is no loss due to radiation and there is less of a sensitivity to external fields). Therefore, when one makes a multipole expansion of the electromagnetic theory, one usually throws away the toroidal moment terms since they don't contribute to the field.

However, sometimes it is necessary to include the toroidal moments because they do interact with the field. Like the electric dipole couples directly to the electric field (i.e. with a potential energy given by U = - p.E) and the magnetic dipole couples directly to the magnetic field (U  = - μ.H), the toroidal moment couples directly to the curl of the magnetic field:

U = - t.( × H)

So in this case, the existence of a fixed toroidal moments, or a density of toroidal moments, would add a relevant term to the enegy function governing your system, and the moment carriers would effect the dynamics of the field and vice versa.

The second reason toroidal moments are interesting has to do with symmetries. All three moments I have been discussing are described by vectors (as opposed to the electric quadrupole moment, e.g., which is characterized by a rank-2 tensor). However, they are vector of slightly different kinds, characterized by how their sense changes when space is inverted and when time is reversed. The electric dipole is what is known as a polar vector (a.k.a. normal vector), its sense is reversed under space inversion but not under time reversal. The magnetic dipole is an axial vector (a.k.a. pseudovector), its sense is reversed under time reversal but not space inversion. The toroidal moment is a momentum-like vector that switches sign both under space inversion and time reversal (actually, the curl of the magnetic field is also a vector of this kind and therefore it is not surprising that it couples to the toroidal moment).

The spontaneous appearance in a physical system of a bulk quantity with this last behavior vis-a-vis space inversion and time reversal is associated with the linear magnetoelectric effect. Since materials with a large magnetoelectic effect would be very useful technologically, there recently has been a spot of interest in finding and studying material that spontaneously order in a way that generates a non-zero toroidal moment density. In analogy with ferromagnetic materials (which spontaneously magnetize and generate a net magnetic dipole moment) and ferroelectric materials (which spontaneously polarize and generate a net electric dipole moment), these hypothetical materials have been termed ferrotoroidic.

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