# Reconstructing Charge Distribution from Multipole Expansion

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?

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.
• 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? Sep 17 '19 at 16:58