In a previous question, I noted that if you have a charge distribution with nonzero charge, then it is possible to choose an origin (at the centre of charge) which makes its dipole moment vanish, and which therefore takes out the subleading order in the multipole expansion of the resulting electrostatic potential $$ \Phi(\mathbf{r})=\frac{Q}{|\mathbf{r}-\mathbf{r}_c|}+O\left(\frac{1}{|\mathbf{r}-\mathbf{r}_c|^{3}}\right). $$ Following on that standard observation, I asked for a result for the next layer up on the multipole expansion: given a neutral system, with a nonzero dipole moment, is it possible to choose an origin which will completely kill the coefficients on the subleading multipole, i.e. which will make all the quadrupole moments vanish.
The answer to that, as it turns out, is much more restrictive than the scalar case: this is possible if and only if the quadrupole is cylindrically symmetric (i.e. if and only if it has two equal eigenvalues) and the dipole moment lies along that axis of symmetry.
I now want to ratchet up the position on the multipole ladder, and ask about the next nontrivial step:
- Given a charge distribution with zero total charge and dipole moment, is it possible to choose an origin such that the octupole moments will vanish completely? What restrictions need to be imposed on the quadrupole and octupole moments for this to be possible?
There is also the obvious question of what happens for some arbitrary multipole $\ell$ at some finite point up the ladder, but given how hard the quadrupole case was, I'd like to keep it simple: kind of like in this other question along similar lines, I feel that there isn't yet a useful pattern that can be generalized to arbitrary multipoles, particularly since the passage from quadrupoles to higher-rank tensor takes out the eigenvectors as a tool of analysis, and I don't really know of good equivalent tools for higher-rank tensors. So I'd like to keep things focused for now and keep the general case for later.
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