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Let $\rho$ be a static, discrete or continuous charge distribution, and $\phi(\mathbf{r})$ the corresponding electric potential. We may expand $\phi$ in a multipole series, $$ \phi(\mathbf{r}) = \frac{1}{r}\sum_{n \geq 0} \int \text{d}^3r' \rho(\mathbf{r}')\left(\frac{r'}{r}\right)^nP_n(\cos\theta).$$ My question is, if given the multipole expansion, one can determine the source $\rho$.

Evidently, this is possible if given the multipole expansion of two equally and oppositely charged point sources a distance $a$ apart. From the dipole moment, one can deduce $a$, and thus reconstruct the charge distribution. But is this true in general?

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But is this true in general?

No.

For a simple counterexample, consider two uniform solid spheres of charge, of different radii but with identical total charge $Q$: they will have vanishing multipole moments for all $l\geq 1$, and identical monopole moment, but the distributions are different.

There are equivalent examples for all multipole moments ─ the multipole distribution is basically blind to the radial aspects of the distribution, given suitable rescaling of the charges ─ but they're all basically copies of that monopole example.

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  • $\begingroup$ Fair enough. But, if we cannot obtain $\rho$ exactly, it still seems one can recover information about the geometry of the charge distribution. Is there a clear mapping from the multipole expansion to the geometry of the source generating the field? $\endgroup$ – fruitegg Sep 17 at 16:58
  • $\begingroup$ Yes. The multipole moments. (In other words: the moments themselves capture all the information there is to get, from the external field, about the internal shape of the distribution. If they look too abstract and inscrutable, then that's just because you don't yet have enough practice interpreting them.) $\endgroup$ – Emilio Pisanty Sep 17 at 17:45

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