A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.
Mathematical definition
The quadrupole moment tensor Q is a rank-two tensor—3×3 matrix. There are several definitions, but it is normally stated in the traceless form (i.e. ). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and zero-trace property, in this form only five of these are independent.
For a discrete system of point charges or masses in the case of a gravitational quadrupole, each with charge , or mass , and position relative to the coordinate system origin, the components of the Q matrix are defined by:
The indices run over the Cartesian coordinates and is the Kronecker delta. This means that must be equal, up to sign, to distances from the point to mutually perpendicular hyperplanes for the Kronecker delta to equal 1.
In the non-traceless form, the quadrupole moment is sometimes stated as:
with this form seeing some usage in the literature regarding the fast multipole method. Conversion between these two forms can be easily achieved using a detracing operator.
For a continuous system with charge density, or mass density, , the components of Q are defined by integral over the Cartesian space r:
As with any multipole moment, if a lower-order moment, monopole or dipole in this case, is non-zero, then the value of the quadrupole moment depends on the choice of the coordinate origin. For example, a dipole of two opposite-sign, same-strength point charges, which has no monopole moment, can have a nonzero quadrupole moment if the origin is shifted away from the center of the configuration exactly between the two charges; or the quadrupole moment can be reduced to zero with the origin at the center. In contrast, if the monopole and dipole moments vanish, but the quadrupole moment does not, e.g. four same-strength charges, arranged in a square, with alternating signs, then the quadrupole moment is coordinate independent.
If each charge is the source of a " potential" field, like the electric or gravitational field, the contribution to the field's potential from the quadrupole moment is:
where R is a vector with origin in the system of charges and R̂ is the unit vector in the direction of R. That is to say, for are the Cartesian components of the unit vector pointing from the origin to the field point. Here, is a constant that depends on the type of field, and the units being used.
Electric quadrupole
A simple example of an electric quadrupole consists of alternating positive and negative charges, arranged on the corners of a square. The monopole moment (just the total charge) of this arrangement is zero. Similarly, the dipole moment is zero, regardless of the coordinate origin that has been chosen. But the quadrupole moment of the arrangement in the diagram cannot be reduced to zero, regardless of where we place the coordinate origin. The electric potential of an electric charge quadrupole is given by
where is the electric permittivity, and follows the definition above.
Alternatively, other sources include the factor of one half in the tensor itself, such that:
, and
which makes more explicit the connection to Legendre polynomials which result from the multipole expansion, namely here
Generalization: higher multipoles
An extreme generalization ("point octopole") would be: Eight alternating point charges at the eight corners of a parallelepiped, e.g., of a cube with edge length a. The "octopole moment" of this arrangement would correspond, in the "octopole limit" to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order , would be obtained by dipolar (quadrupolar, octopolar, ...) arrangements of point dipoles (quadrupoles, octopoles, ...), not point monopoles, of lower order, e.g., .
Magnetic quadrupole
All known magnetic sources give dipole fields. However, it is possible to make a magnetic quadrupole by placing four identical bar magnets perpendicular to each other such that the north pole of one is next to the south of the other. Such a configuration cancels the dipole moment and gives a quadrupole moment, and its field will decrease at large distances faster than that of a dipole.