The Multipole Expansion

Stand far enough from any lump of charge and you stop being able to see its details. A complicated arrangement of positive and negative charge, viewed from across the room, produces a field that depends on only a few numbers: the total charge, then how lopsided it is, then how stretched. The multipole expansion is the statement of exactly which few numbers, in order of how quickly they fade with distance.

Expanding the distance

The potential of a distribution $\rho ({\mathbf{r}}^{\prime })$ is an integral of $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$. For a field point far outside the charge, $r\gg r^{\prime }$, that denominator expands in Legendre polynomials of the angle $\gamma$ between $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$:

$$\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}=\frac{1}{r}\sum _{\ell =0}^{\infty }{\left(\frac{r^{\prime }}{r}\right)}^{\ell }{P}_{\ell }(\cos \gamma ).$$

Each power of $r^{\prime }/r$ is smaller than the last, so the series is ordered by how fast each term dies off with distance. Feeding it into the potential gives

$$V(\mathbf{r})=\frac{1}{4\pi {\epsilon }_0}\left[\frac{Q}{r}+\frac{\mathbf{p}\cdot \hat{\mathbf{r}}}{r^2}+\cdots \right].$$

The terms have names

The first term is the monopole: the total charge $Q$, its field falling as $1/r^2$ like a point charge sitting at the origin. The second is the dipole,

$$\mathbf{p}=\int {\mathbf{r}}^{\prime }\rho ({\mathbf{r}}^{\prime })d^3{r}^{\prime },$$

with a field that falls faster, as $1/r^3$. Then the quadrupole at $1/r^4$, and so on. A distribution with no net charge still has a dipole field; one with no dipole moment still has a quadrupole. You only ever see the leading term that does not vanish, because it dominates everything below it far away.

Why it is useful

The point of the expansion is not really the far field of a known charge. It is the reverse: a neutral molecule, seen from outside, acts to leading order as a dipole, and that single vector $\mathbf{p}$ captures how it interacts with fields and with other molecules.

The energy of a dipole in an external field is $U=-\mathbf{p}\cdot \mathbf{E}$, so it wants to line up with the field, and the torque $\mathbf{p}\times \mathbf{E}$ is what turns it. This is the whole basis of how dielectrics respond and how a microwave oven heats water: the field grabs the dipoles and the details of the molecule never enter.

So the multipole expansion is a way of forgetting, on purpose. It keeps precisely the few numbers that survive being seen from far off, and it tells you the order in which to keep them.