Potential Field: Rotatable Potential Fields

There are three coefficients that describe a dipole magnetic field in the spherical harmonic expansion g₁₀, g₁₁, and h₁₁. 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. g₂₀ 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. g₂₁ has the dipole moment along z and separated in y; h₂₁ has the dipole moment along z and separated in x; h₂₂ has the dipole moment along y and separated in x; g₂₂ 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 g₃₀, g₃₁, h₃₁, g₃₂, h₃₂, g₃₃, and h₃₃. These correspond to octupoles created by separating two quadrupoles. The g₃₀ is a zonal moment created by separating two oppositely directed zonal quadrupoles along the z-azis (rotational). Two planar octupoles, g₃₁ and h₃₁, are constructed from linear quadrupoles; two planar types, g₃₃ and h₃₃, are constructed from planar quadrupoles and there are two cubic types, g₃₂ and h₃₂. 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.