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Concept
Quadrupôle électrostatique
Résumé
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.
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.
Source officielle
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