Quadrupole

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 - 3x3 matrix. There are several definitions, but it is normally stated in the traceless form (i.e. Q x x + Q y y + Q z z = 0 {displaystyle Q_{xx}+Q_{yy}+Q_{zz}=0}  ). The quadrupole moment tensor has thus 9 components, but because of transposition symmetry and zero-trace property, in this form only 5 of these are independent.The simplest example of an electric quadrupole consists of alternating positive and 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, regardless of the coordinate origin that has been chosen. But the quadrupole moment of the arrangement in the diagram cannot be reduced to zero, regardless of where we place the coordinate origin. The electric potential of an electric charge quadrupole is given by An extreme generalization ('point octopole') would be: Eight alternating point charges at the eight corners of a parallelepiped, e.g. of a cube with edge length a. The 'octopole moment' of this arrangement would correspond, in the 'octopole limit' lim a → 0 ; a 3 ⋅ Q → c o n s t . {displaystyle lim _{a o 0;,a^{3}cdot Q o { m {const.}}}}  , to a nonzero diagonal tensor of order three. Still higher multipoles, e.g. of order 2l, would be obtained by dipolar (quadrupolar, octopolar, ...) arrangements of point dipoles (quadrupoles, octopoles, ...), not point monopoles, of lower order, e.g. 2l-1.All known magnetic sources give dipole fields. However, it is possible to make a magnetic quadrupole by placing four identical bar magnets perpendicular to each other such that the north pole of one is next to the south of the other. Such a configuration cancels the dipole moment and gives a quadrupole moment, and its field will decrease at large distances faster than that of a dipole.The mass quadrupole is analogous to the electric charge quadrupole, where the charge density is simply replaced by the mass density and a negative sign is added because the masses are always positive and the force is attractive. The gravitational potential is then expressed as: