Quadrupole

A quadrupole or quadrapole 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 quadrupole moment tensor Q is a rank-two tensor (3x3 matrix) and is traceless (i.e. ${\displaystyle Q_{xx}+Q_{yy}+Q_{zz}=0}$). The quadrupole moment tensor has thus 9 components, but because of the symmetry and zero-trace property, only 5 of these are independent.

For a discrete system of point charges (or masses in the case of a gravitational quadrupole), each with charge ${\displaystyle q_{l}}$ (or mass ${\displaystyle m_{l}}$) and position ${\displaystyle {\vec {r_{l}}}=(r_{xl},r_{yl},r_{zl})}$ relative to the coordinate system origin, the components of the Q matrix are defined by:

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