Quadrupole

A quadrupole is a descriptor used in physics to characterize a system whose external field is not fully described by a monopole or dipole moment, but whose next nonzero contribution is a quadrupole moment. It often arises in charge, current, or mass distributions that are arranged with zero net charge (or mass) and zero net dipole moment, yet possess a nonzero second-order moment. The concept is widely used in electromagnetism, gravitation, and quantum physics to describe how fields deviate from those produced by simpler configurations.

In electrostatics, the quadrupole moment is represented by a symmetric, traceless rank-2 tensor Qij, defined for a charge distribution ρ(r) as Qij = ∫ ρ(r) (3 xi xj − r^2 δij) d3r, where xi are Cartesian coordinates. The electrostatic potential at far distance r scales as 1/r^3, with a directional dependence given by n̂i n̂j, where n̂ is the unit vector along r. Consequently, the electric field from a quadrupole falls off roughly as 1/r^4. The quadrupole moment provides information about the shape of the charge distribution beyond its total charge and dipole moment.

In gravity, the quadrupole moment describes how a mass distribution departs from spherical symmetry and contributes to the external gravitational field; time variations of this moment are the leading source of gravitational radiation in general relativity. Magnetic and higher-order quadrupole moments arise from particular current configurations.

Applications include nuclear and atomic spectroscopy, where quadrupole moments affect energy levels and transitions, and quadrupole antennas and mass spectrometers, which exploit quadrupole fields for signal transmission and ion filtering.