An intuitive explanation of multipole expansion

[member 743765]

I could imagine the multipole expansion of a point charge not at the origin intuitively only up to the dipole moment but not higher my thought goes as follows : imagine u have only a point charge + q at r0 this is equivalent to have also in addition to it +q and -q at the origin which result in a dipole between - q at the origin and +q at r0 in addition a point charge +q at the origin but how to imagine higher moments?

 

[Orodruin]

which result in a dipole between - q at the origin and +q at r0

Ah, but that is not a dipole centered in the origin!
(It is also not q true dipole as a true dipole is point-like)

 

[Ibix]

That's not quite correct, because the dipole moment is for a dipole centered at the origin - i.e. charges equidistant from the origin, not one at the origin and one off the origin.

The point about these expansions is that they sometimes make maths easier. For a function $f(x)$ you can write a Taylor expansion $a_0+a_1x+a_2x^2+\ldots$. That's sometimes useful, especially when you can show that $a_i$ is small for all $i$ larger than some value, because you've replaced a mathematically messy function with a simpler one and shown that the difference is not important. But if you can't terminate the series early you need all the infinite terms and it's no better.

The multipole expansion of a 3d field is a similar approach for 3d fields, only it doesn't break the field up into polynomials. You aren't really meant to think of the components as sourced from different charge distributions. Just like you don't try to understand the sources of Taylor expansion components, you just hope to be able to treat some of them as small and ignore them and your textbook mathematical tools handle the survivors better than the original field.

I would suspect that the multipole expansion of a static point charge away from the origin is just a "check you can do the maths" exercise.

 

[Orodruin]

The simplest way of seeing this is along the line of separation from the origin. The field is proportional to $1/(x-d) = x^{-1}/(1-d/x)$. The multipole expansion along this line is given by Taylor expanding for small $\xi = d/x$:
$$
\frac 1x (1+\xi + \xi^2 + \xi^3 +\ldots)
$$
All multipoles are non-zero.

 

[member 743765]

Guys what I'm saying is true that's why it's an infinite series for each moment u need a higher moment to make it centered at the origin

 

[Ibix]

Not quite - a multipole expansion is the sum of the fields from an infinite set of infinitesimal multipoles all centered on the origin. You're describing adding a finite sized dipole centered half way between the origin and the off-centre charge. You can do that, of course, but it's not what's meant by the multipole expansion of a field.

 

[member 743765]

Not quite - a multipole expansion is the sum of the fields from an infinite set of infinitesimal multipoles all centered on the origin. You're describing adding a finite sized dipole centered half way between the origin and the off-centre charge. You can do that, of course, but it's not what's meant by the multipole expansion of a field.

No im saying the monopole term to be at the origin u created a dipole between the origin and r0 now to make the dipole centered at the origin u need a quadrupole and so on

 

[member 743765]

And yes I saw the expansion even derive it for a point charge the dipole moment is of finite length

 

[Hardy]

The point about these expansions is that they sometimes make maths easier. For a function $f(x)$ you can write a Taylor expansion $a_0+a_1x+a_2x^2+\ldots$. That's sometimes useful, especially when you can show that $a_i$ is small for all $i$ larger than some value, because you've replaced a mathematically messy function with a simpler one and shown that the difference is not important. But if you can't terminate the series early you need all the infinite terms and it's no better.

Interesting comparison of multi-pole expansion to Taylor's series. In the course of learning, it was more aligned to Fourier (or Legendre or Bessel) Series ( call it 'FLB' :) ) expansion and the orthogonal nature of these expansions. I understand that multipole expansion is different than any of these and has distinct formula based on charge distribution definition. FLB had dependencies on boundary condition. Never saw that it was 'like' Taylor's series, but I concede, the comparison valid, albeit not directly from my academia training.

To be honest, with computing power the way it is now and days, you could probably find your answers more efficiently with modeling the charge distribution with a large amount of point charges. Perhaps one of these days, like with the affect of AI, all these mathematical theorectic tools will not be as much of a foundation of EM teaching. I don't know, but if so, its sad to see and hard to swallow :( .

 

[Orodruin]

I understand that multipole expansion is different than any of these and has distinct formula based on charge distribution definition. FLB had dependencies on boundary condition.

It is no different at all. It is expanding the angular part of the solution in spherical harmonics. The radial coefficients will fall off with a power of r.