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I'm just curious about whether it is possible to build an ideal hexadecapole (16-pole) from point charges in 3D.

My intuition tells me no, because the point charge of a (electric) monopole "bound" a 0-dimensional region (a point), the 2 charges of a dipole bound a 1-dimensional region (a line segment), and so on (see picture below for magnetic multipoles, though I'm actually asking about either magnetic OR electric multipoles). Then it seems like the highest dimensional object we could construct in 3D is a octupole.

enter image description here

But I didn't use any math (or really physics either) to make my conjecture. So is there a way to prove that I'm right or is there actually a way to construct an ideal hexadecapole in 3-space?

Emilio Pisanty
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Bob Dylan
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    Note also that the dipole, quadrupole and octupole you drew are not ideal unless they are infinitesimally small. For the finite versions, there will be corrections of higher order that only truly die at infinity. – Emilio Pisanty Oct 13 '14 at 19:58
  • @EmilioPisanty I didn't draw it, it was just the best picture I could find from a quick google search. And yeah, I'm talking about ideal point charges at the vertices of an n-square (electric multipoles) or ideal infinitely thin bar magnets forming the sides of an n-square (magnetic multipoles). – Bob Dylan Oct 13 '14 at 20:05
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    Again, just because your dipoles have ideal point particles doesn't mean that the result is an ideal dipole field. For that you need to take the charge separation to zero whilst sending each individual charge to infinity in a controlled way (i.e. keeping the relevant multipole moment constant). If the charges and separations are finite, then the fields won't be pure - the dipole, for example, will have octupolar, 32-polar, 128-polar, and so on, contributions. (The others are forbidden by the symmetry.) – Emilio Pisanty Oct 13 '14 at 20:13
  • Either way, your question as posed is well answered by my linked answer. If you have further questions you should modify this one. – Emilio Pisanty Oct 13 '14 at 20:14
  • @EmilioPisanty: Lovely answer, Emilio. Thank you! Can we not go even beyond that? Since one can't build anything "ideal" in physics, and higher order moments fall off faster than dipole moments, wouldn't any non-ideal dipole contribution completely overwhelm any higher order moment at long distances? – CuriousOne Oct 13 '14 at 21:38

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