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This module allows you to explore the magnetic field configurations of multipolar magnatic fields of different order and degree.

Dipole tilt angle and longitude

There are three coefficients that describe a dipole magnetic field in the spherical
harmonic expansion g_{10}, g_{11}, and h_{11}. These correspond
to the projection of the dipole moment along the z-axis (rotation), the x-axis,
and the y-axis (in the rotational equator).

There are five coefficients that describe a quadrupole magnetic field in a spherical
harmonic expansion. The five coefficients correspond to quadrupoles constructed
from antiparallel, separated dipole moments. g_{20} corresponds to a separation
along the rotation axis. This is called a zonal (rotationally symmetric) moment.
The other four are planar with separation of the dipoles perpendicular to the dipole
direction. g_{21} has the dipole moment along z and separated in y; h_{21}
has the dipole moment along z and separated in x; h_{22} has the dipole
moment along y and separated in x; g_{22} has the dipole moment along a
direction at 45° to the x- and y-axes and a separation along a line at 45°
to the x- and y-directions. That these five configurations provide all possible
quadrupole fields (separated dipole pairs) can be shown by expressing the dipoles
as separated monopoles and comparing.

That there are seven coefficients that describe an octupole magnetic field can be
shown by expressing the dipoles as separated monopoles and comparing the spherical
harmonic expansion g_{30}, g_{31}, h_{31}, g_{32},
h_{32}, g_{33}, and h_{33}. These correspond to octupoles
created by separating two quadrupoles. The g_{30} is a zonal moment created
by separating two oppositely directed zonal quadrupoles along the z-azis (rotational).
Two planar octupoles, g_{31} and h_{31}, are constructed from linear
quadrupoles; two planar types, g_{33} and h_{33}, are constructed
from planar quadrupoles and there are two cubic types, g_{32} and h_{32}.
That these seven octupole configurations provide all possible octupole fields (separated
quadrupole pairs) can be seen by expressing the moments as distributed monopoles
and comparing.