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 *N *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} **r**_{i × }**μ**_{i / 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.