# Can I replace a multipole expansion by a combination of separate dipoles?

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.

• 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! – Cham Jan 9 '19 at 18:07
• 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? – S. McGrew Jan 9 '19 at 21:00
• @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. – user3293204 Jan 10 '19 at 14:35