If I want to be able to model a magnetic field flux density $\mathbf{B}$ from a magnetic source located at the origin at a position $\mathbf{r}$, it is my understanding that I can represent $\mathbf{B}(\mathbf{r})$ using a multipole expansion, given that I am not exactly at the location of the source, but there is a tradeoff between the accuracy of that representation, and either the order (number of poles) of the expansion, or the distance to the magnetic source $r = ||\mathbf{r}||$. In other words, if I am very close to the source and therefore $r$ is very small, I will need a large number of terms in my multipole expansion to get an accurate estimate of $\mathbf{B}(\mathbf{r})$.

My questions are the following:

  1. Can I, instead of performing a high order multipole expansion, model the field as a combination of lower-order multipoles?

  2. Is there some sort of equivalence between a single multipole expansion and a sum of different dipole expansions? I would guess that you should be able to represent any field distribution as the combination of many dipoles, as I believe that is what happens microscopically, but I may be wrong there.

  • $\begingroup$ I think that it is physically "obvious" that any static magnetic field (no time dependance, no radiation) could be expressed as the field of a distribution of elementary dipoles, judiciously placed in space. I may be wrong, since it isn't mathematically obvious! $\endgroup$ – Cham Jan 9 '19 at 18:07
  • $\begingroup$ Are you asking if an arbitrary static magnetic field can be represented as a field of dipoles distributed through space? Or are you asking if a static magnetic field can be represented as a bunch of dipoles all located at a source point? $\endgroup$ – S. McGrew Jan 9 '19 at 21:00
  • $\begingroup$ @S.McGrew I'm asking the former. If you could represent a static magnetic field which a bunch of dipoles distributed through space, perhaps provided that you are not concerned with the values exactly at the positions of the different dipoles. $\endgroup$ – user3293204 Jan 10 '19 at 14:35

I know it is possible to construct any charge distribution in the form of a field of dipoles, because a dielectric medium simply reduces the effective charge of a point charge, by forming a field of dipoles.

Therefore it is possible to represent any charge distribution- and any static electric field- with a field of dipoles.

The same argument can be applied to the magnetic field, because the time-independent equations for the electric and magnetic fields have essentially the same form. Sorry, I can't provide the math, but the argument seems sound.

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