The quadrupole moment has 9 components, but because of the rotational symmetry and trace property, only 5 of these are independent. As with all types of moments except the monopole, the value of the quadrupole moment depends on the choice of the coordinate origin. For example, the basic dipole can have a quadrupole moment if the origin is shifted away from the center of the two charges. However, the quadrupole moment of the basic dipole can also be reduced to zero with a particular choice of the origin.
The classic example of an electric quadrupole is shown in the picture. There are two positive and two 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, when the coordinate origin is at the center of the picture. The quadrupole moment of this arrangement, however, cannot be reduced to zero, regardless of where we place the coordinate origin. The electric potential of an electric charge quadrupole is given by
All known magnetic sources give dipole fields. However, to make a magnetic quadrupole it is possible to place two identical bar magnets parallel to each other such that the North pole of one is next to the South of the other and vice versa. 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.
Magnetic quadrupoles like the one depicted on the right are being used to focus particle beams in a particle accelerator. There are four steel pole tips, two opposing magnetic north poles and two opposing magnetic south poles. The steel is magnetized by a large electric current that flows in the coils of tubing wrapped around the poles.
Changing magnetic quadrupole moments give production of electromagnetic radiation.
The mass quadrupole moment is also important in General Relativity because, if it changes in time, it can produce gravitational radiation, similar to the electromagnetic radiation produced by change electric or magnetic quadrupoles. (In particular, the second time derivative must be nonzero.) The mass monopole represents the total mass-energy in a system, and does not change in time — thus it gives off no radiation. Similarly, the mass dipole represents the center of mass of a system, which also does not change in time — thus it also gives off no radiation. The mass quadrupole, however, can change in time, and is the lowest-order contribution to gravitational radiation.
The simplest and most important example of a radiating system is a pair of black holes with equal masses orbiting each other. If we place the coordinate origin right between the two black holes, and one black hole at unit distance along the x-axis, the system will have no dipole moment. Its quadrupole moment will simply be
Just as electric charge and current multipoles contribute to the electromagnetic field, mass and mass-current multipoles contribute to the gravitational field in General Relativity, because GR also includes "gravitomagnetic" effects. Changing mass-current multipoles can also give off gravitational radiation. However, contributions from the current multipoles will typically be much smaller than that of the mass quadrupole.