Quadrupole

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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.

A quadrupole is one of a sequence of configurations of electric charge 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.

1 Mathematical Definition
2 Electric multipoles
3 Magnetic multipoles
4 Gravitational multipoles
5 Distance dependence of multipole fields
6 References
7 See also
8 External links

[edit] Mathematical Definition

In the multipole expansion of a potential, V,

$V(\mathbf{x}) = {1\over4\pi\epsilon_0}\sum_{n=0}^{\infty}\left[|\mathbf{x}|^{-(n+1)}\int |\mathbf{y}|^n P_n(\cos\theta)\rho(\mathbf{y})d\mathbf{y}\right]$

where P_n is the degree-n Legendre polynomial, and θ is the angle between the vectors x and y, the n=2 term is the quadrupole.

[edit] Electric multipoles

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

In the theory of electricity there are two signs of electric charge. The net total charge is the monopole moment. If there are charges of two signs separated, then there is a dipole moment along the line connecting the charges. If there is no net charge, the lines of force resemble those of bar magnet, though they are electric, not magnetic. If there are charges of both signs, but separated in a more complicated way, an electric quadrupole, but not a monopole or dipole may be present, as depicted in the figure on the electric quadrupole. Configurations of electric charges of dipole and higher multipolar nature that change in time radiate electromagnetic radiation, whose character is named dipole, quadrupole, etc. according to a specific pattern generated by such sources. (There is no monopole radiation due to the conservation of electric charge.)

[edit] Magnetic multipoles

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 at the present time, 2006, be made in the laboratory or put to any known use. Thus, the magnetic sequence, though similar to the electric one, begins with the dipole case, which can consist of an ideal bar magnet. To make a 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 dipole or quadrupole moment will lead to the production of electromagnetic radiation.

[edit] Gravitational multipoles

The situation here is similar to the magnetic case; the difference is that mass is only positive, and so there is no dipole moment. There are gravitational monopoles; they are very commonly represented by ideal, stationary, spherically symmetric suns, planets, and so on. A gravitational quadrupole can be represented by two massive balls (say, lead) on opposite ends of a light rod, or, more simply, just as a long massive rod or a thin massive disk. A prolate (American football-shaped) or oblate spheroidal mass has a quadrupole moment. For example, the Earth is flattened at the poles, so it has a quadrupole moment. If a quadrupole (or higher order multipole) of mass rotates or oscillates (in vibration) it will emit gravitational radiation.

[edit] Distance dependence of multipole fields

The sequence monopole, dipole, quadrupole. can be extended to higher orders in a multipole expansion. A simple quadrupole is constructed by placing two opposing dipoles near each other, an octupole by adding another opposing quadrupole displaced from the one just made, and so on. The static fields of electric and magnetic multipoles fall off more and more rapidly as one moves away to an increasing radius r from the center. A monopole field (as for a single electric charge, or a single mass in Isaac Newton's law of universal gravitation, falls off with the inverse square law. A dipole field falls off as the cube of the distance, a quadrupole as the inverse fourth power, and so on.

When varying configurations of charges, currents, or masses are present, however, radiation is generally produced, and the radiation field falls off only as the first power of the distance, at large distances.