Magnetic multipoles in spherical harmonic Does someone know explain me how to identify the multipoles magnetic terms of the multipolar expansion (Dipole, quadrupole, etc) in spherical harmonics?
 A: Monopole, dipole, quadrupole, octupole, hexadecapole, and so on ($2^\ell$-pole) moments are generally multiplied by spherical harmonics $Y_{\ell m}$ with $\ell=0,1,2,3,4,\dots$.
The magnetic field at infinity scales like that of the monopole, $1/r^2$, with the extra factor of $1/ r^\ell$ for each of the multipole moments described in the previous paragraph.
All the powers of two make it sounds more complex than it is. We get powers of two because it's the easiest way to cancel all the previous moments and leave the $\ell=\ell$ moment as the first nonzero one. Why?
The dipole is a positive and negative charge separated, two opposite multipoles. A quadrupole is a pair of two opposite dipoles, so the dipole charges cancels as well, and there are $2\times 2 = 4$ monopoles in this mode of the quadrupole (it's like a $2\times 2$ matrix with enties $((+1,-1),(-1,+1))$. An octupole may be modeled as two opposite quadrupoles shifted in the third direction so the quadrupole charge cancels as well and there are $2\times 4=8$ monopoles in the model, and so on.
But this arrangement of $2^\ell$ monopoles isn't the only way how we can get (electric or magnetic) charge distributions with the required moment.
