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A quadrupole is one of a sequence of configurations of — for example — 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.

The traceless quadrupole moment of a system of charges (or masses, for example) is defined as

for a discrete system with individual charges q_n, or

for a continuous system with charge density ρ(x).

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.

If each charge is the source of a "1 / r" 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 n is the unit vector in the direction of R. Here, k is a constant that depends on the type of field, and the units being used.

Electric Quadrupole. Four charges, 2 positive and 2 negative. The arrows depict the electric fields, and the lines the equipotential surfaces.

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 ^[1]

where ε₀ is the electric permittivity.

For an example, see the quadrupole magnet article.

Because the existence of magnetic monopoles has never been confirmed, they are often assumed not to exist; certainly they cannot currently (2007) be detected or made in the laboratory. The source of the magnetic field, then, is a moving current of electric charges. In this case, the simple electric charge in the formula above must be replaced by a vector representing the electric current, and we will get a vector of three quadrupoles. This vector could then be inserted into a formula for the vector potential of the magnetic field, similar to the formula for the scalar potential given above.

Schematic quadrupole magnet("four-pole") 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 flow in the coils of tubing wrapped around the poles.

To make a magnetic quadrupole we could 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; the result is a configuration like that in the figure above with North poles in place of the positive charges and South in place of negative;. Such a configuration would have no dipole moment, and its field will decrease at large distances faster than that of a dipole—see below. Again, a changing magnetic quadrupole moment will lead to the production of electromagnetic radiation.

The mass quadrupole is very analogous to the electric charge quadrupole, where the charge density is simply replaced by the mass density. The gravitational potential is then expressed as:

For example, because the Earth is rotating, it is oblate (flattened at the poles). This gives it a nonzero quadrupole moment. While the contribution to the Earth's gravitational field from this quadrupole is extremely important for artificial satellites close to Earth, it is less important for the Moon, because the term falls quickly.

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.^[2]

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

where M is the mass of each hole, and x_i is the unit vector in the x-direction. As the system orbits, the x-vector will rotate, which means that it will have a nonzero second time derivative. Thus, the system will radiate gravitational waves. Energy lost in this way was indirectly detected in the Hulse-Taylor binary.

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.

1. ^ Jackson, John David (1975). Classical Electrodynamics. John Wiley & Sons. ISBN 047143132X. 
2. ^ Thorne, Kip S. (April 1980). "Multipole Expansions of Gravitational Radiation". Reviews of Modern Physics 52 (2). 

Quadrupole Ion Trap

• Multipole expansion