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I am reading Jackson, Classical Electrodynamics, and I have a question regarding the electrodynamic multipole expansion (with page numbers I refer to the 3rd edition). So on page 409, he gives in equation 9.9 the general formula for the expansion of the vector potential:

$$\lim_{kr \rightarrow \infty} \mathbf A (\mathbf x) = \frac{\mu_0}{4\pi} \frac{e^{ikr}}{r} \sum_n \frac{(-ik)^n}{n!} \int \mathbf j (\mathbf x' ) (\mathbf n \cdot \mathbf x')^n d^3 x'$$

In the following, he takes the orders $n=0$ and $n=1$ and constructs the vector potential with the "physical" multipole moments, i. e. in order $n=0$ the electric dipole moment and for $n=1$ the electric quadrupole tensor and the magnetic dipole moment. Then on p. 415 he says, that from order $n=2$ upwards, the labor becomes increasingly prohibitive to do this decomposition in physical moments again. Instead, he indicates, that a systematic expansion is much easier (what, of course, I understand).

So I'm wondering now, if this decomposition has ever been done for $n=2$ (to get the electric octupole and the magnetic quadrupole tensor) or even higher orders, and if so, if is there a paper where this sophisticated task has been written down?

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  • $\begingroup$ Even where referring to standard literature, it is very helpful for people reading this if you just post the equations you are talking about. $\endgroup$ – ACuriousMind Jul 18 '14 at 14:32
  • $\begingroup$ Ok, I have added the basic equation of the expansion I refer to. $\endgroup$ – Tornado Jul 18 '14 at 14:40
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    $\begingroup$ @TornadoXXL Protip: don't read Jackson $\endgroup$ – Yossarian Jul 18 '14 at 14:53
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    $\begingroup$ Why not? Jackson is the standard book of electrodynamics. $\endgroup$ – Tornado Jul 18 '14 at 15:32
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    $\begingroup$ @TornadoXXL because youtube.com/watch?v=mm-4PltMB2A $\endgroup$ – Yossarian Jul 18 '14 at 16:24

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