Multipole Expansion: Understanding Electric & Magnetic Fields

[leehufford]

Hello,

I was hoping someone could help make the concept of electric multipole/ magnetic multipole expansions clearer. I think my most fundamental question is:

Are dipole, quadrupole and up fields just a shortcut to using the superposition principle on a charge distribution in space or do they yield different fields altogether, and if so what do they represent? I understand that electric monopoles are just single charges and that magnetic monopoles don't exit, so the most basic magnetic field must be a dipole, but the whole concept of a multipole expansion just isn't sinking in. Thanks in advance,

Lee

 

[Khashishi]

It's just a mathematical decomposition, similar to a Fourier series decomposition of a periodic wave. You have different "harmonics", but it's all part of the same field.

 

[leehufford]

I'm still a little confused. So do all electric fields have all the terms of the expansion? Wouldn't a single point charge only have a monopole term? I guess I don't understand the use of "harmonics" in this context. Thanks for the reply,

Lee

 

[blue_leaf77]

do all electric fields have all the terms of the expansion?

The common thing to be expanded in multipoles are the potential energy, I have never known people expanding electric field from a source into multipoles, perhaps because electric field is a vector quantity so expanding it would involve nasty equations.

Wouldn't a single point charge only have a monopole term?

Yes a single point source doesn't have higher terms unless the monopole.
The main purpose of expanding into multipoles is to make the expression for potential easier, this is because at sufficiently large distances, a considerable contributions from the expansion terms might only end at certain order, neglecting orders beyond that would only lead to very small error.

 

[leehufford]

The common thing to be expanded in multipoles are the potential energy, I have never known people expanding electric field from a source into multipoles, perhaps because electric field is a vector quantity so expanding it would involve nasty equations.

Yes a single point source doesn't have higher terms unless the monopole.
The main purpose of expanding into multipoles is to make the expression for potential easier, this is because at sufficiently large distances, a considerable contributions from the expansion terms might only end at certain order, neglecting orders beyond that would only lead to very small error.

So does this mean that a an electric dipole would have a dipole field and a monopole field? Is the type of field just literally dependent on the number of charges? Why (physically) do the terms change in significance at larger distances? (It's mathematically obvious). I just don't see where these field components are coming from. Thanks for the reply.

 

[blue_leaf77]

I have never known people expanding electric field from a source into multipoles

Please forget what I said above, I just remembered that electric field can also be expressed in term of multipoles.

Again as Khashishi has said, multipole expansion is just a mathematical tool to study complicated charge distribution. Imagine you have a bulk of material whose charge density varies from place to place. You can expand the electric potential around such body, but what do monopole, dipole, quadrupole terms, and so on physically mean? It has no physical meaning I think, as you can't associate a physical entity to each of the expansion term.

 

[Khashishi]

A point charge can have dipole and higher moments, if the charge is not located at the origin.

 

[blue_leaf77]

A point charge can have dipole and higher moments, if the charge is not located at the origin.

I guess that's right.
And actually the OP can see from this that multipole expansion doesn't really have physical meaning. If it had, then the physics of the system being expanded is not coordinate invariant, which is not allowed. That's why again multipole terms should be understood as only mathematical objects, similar to Taylor expansion.