Multipoles as a superposition of dipoles?

While navigating on Stack Exchange, I was suddenly wondering about a possibility that I never saw described in my textbooks on electrodynamics.

Suppose you have a distribution of electric charges in space with a given total charge $$Q$$, total dipolar moment $$\vec{p}$$ and quadrupole components $$Q_{ij}$$. Neglecting all higher multipoles, the total electric potential is the following: $$\begin{equation}\tag{1} \phi(x) = \frac{k Q}{r} + \frac{k}{r^3} \, \vec{p} \cdot \vec{r} + \frac{k}{2 r^5} \, Q_{ij} \, x_i \, x_j. \end{equation}$$ To simplify things, suppose that the total charge vanishes: $$Q = 0$$, and the distribution is discrete (the charges can be counted: $$n = 1, 2, 3, \dots, N$$). It's then possible to consider pairs of opposite charges as forming a set of simple dipoles, so by the principle of superposition of fields: $$\begin{equation}\tag{2} \phi(x) = \sum_{n=1}^N \frac{k}{||\vec{r} - \vec{r}_n ||^3} \, \vec{p}_n \cdot (\vec{r} - \vec{r}_n). \end{equation}$$ Of course, the global dipole moment in (1) is $$\begin{equation}\tag{3} \vec{p} = \sum_{n=1}^N \vec{p}_n. \end{equation}$$ Is there a simple closed form giving the quadrupole cartesian components $$Q_{ij}$$ in (1) as a function of the dipolar moments $$\vec{p}_n$$ and the dipoles position $$\vec{r}_n$$? Also, what constraints can we get on the $$\vec{p}_n$$ and $$\vec{r}_n$$ (and their total $$Q_{ij}$$) so that all other multipolar elements vanish, except $$Q_{ij}$$?

• Nov 11 '21 at 21:31