6.6 Multipole expansion

Multipole expansion is a technique for approximating the potential and field of a complex charge distribution at points far from the source. Instead of evaluating a difficult integral exactly, you expand it as a series of terms with increasing angular complexity (monopole, dipole, quadrupole, ...), each falling off as a higher power of $1/r$. In practice, only the first few nonvanishing terms matter at large distances, which makes calculations far more tractable.

This is one of the central tools in electromagnetism. It shows up everywhere: molecular interactions, radiation theory, gravitational physics, and the structure of solutions to Laplace's equation in spherical coordinates.

Multipole expansion overview

The core idea is to exploit the fact that $|\vec{r}| \gg |\vec{r}'|$ (the observation point is far from the source). You expand the factor $\frac{1}{|\vec{r} - \vec{r}'|}$ in powers of $r'/r$, then integrate term by term against the charge density. Each term in the resulting series corresponds to a multipole moment of increasing order:

• Monopole ($l = 0$): total charge, potential falls as $1/r$
• Dipole ($l = 1$): charge separation, potential falls as $1/r^2$
• Quadrupole ($l = 2$): next correction, potential falls as $1/r^3$
• And so on: the $l$-th term falls as $1/r^{l+1}$

The expansion converges rapidly when $r'/r \ll 1$, so at large distances the lowest nonvanishing term dominates.

Electric potential of charge distributions

Potential in terms of charge density

The exact potential at $\vec{r}$ due to a volume charge density $\rho(\vec{r}')$ is:

$V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} \, d^3r'$

Here $\vec{r}'$ runs over the source region and $\vec{r}$ is the field point. The multipole expansion amounts to expanding the Green's function $1/|\vec{r} - \vec{r}'|$ using the identity:

$\frac{1}{|\vec{r} - \vec{r}'|} = \sum_{l=0}^{\infty} \frac{r'^l}{r^{l+1}} \frac{4\pi}{2l+1} \sum_{m=-l}^{l} Y_{lm}^*(\theta', \phi') \, Y_{lm}(\theta, \phi)$

valid for $r > r'$. Substituting this into the integral and pulling out the field-point dependence gives the multipole series.

Convergence of multipole expansion

• The expansion converges when $r > r'_{\text{max}}$, i.e., outside a sphere enclosing the entire charge distribution.
• The small parameter is $r'/r$, so convergence is faster the farther you are from the source.
• Each successive term is suppressed by an additional factor of roughly $a/r$, where $a$ is the characteristic size of the distribution.

Monopole moment

Net electric charge

The monopole moment is simply the total charge:

$Q = \int \rho(\vec{r}') \, d^3r'$

This is the $l = 0$ term. Its contribution to the potential is:

$V_{\text{mono}}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$

At very large distances, this term dominates (assuming $Q \neq 0$). It's the same as the potential of a point charge $Q$ at the origin, which makes intuitive sense: from far away, the details of the charge distribution wash out.

Dipole moment

Electric dipole definition

The electric dipole moment is defined as:

$\vec{p} = \int \vec{r}' \, \rho(\vec{r}') \, d^3r'$

For a simple system of two point charges $+q$ and $-q$ separated by displacement $\vec{d}$, this reduces to $\vec{p} = q\vec{d}$. But the integral definition is the general one that works for any charge distribution.

The dipole moment depends on the choice of origin unless the total charge $Q = 0$. If $Q = 0$, $\vec{p}$ is origin-independent, which is why the dipole term becomes the leading term for neutral distributions.

Potential of electric dipole

The $l = 1$ contribution to the potential is:

$V_{\text{dip}}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{\vec{p} \cdot \hat{r}}{r^2}$

Note the $1/r^2$ falloff (compared to $1/r$ for the monopole). The angular dependence goes as $\cos\theta$ when $\vec{p}$ is along the $z$-axis, which means the potential is positive on one side and negative on the other.

Quadrupole moment

Quadrupole tensor

The quadrupole moment tensor is defined as:

$Q_{ij} = \int \rho(\vec{r}') \left(3x_i' x_j' - r'^2 \delta_{ij}\right) d^3r'$

This tensor is symmetric ($Q_{ij} = Q_{ji}$) and traceless ($\sum_i Q_{ii} = 0$). Because of these constraints, a symmetric $3 \times 3$ matrix has 6 independent components, and the traceless condition removes one more, leaving 5 independent components. This matches the $2l + 1 = 5$ components for $l = 2$.

The subtraction of the $r'^2 \delta_{ij}$ piece is what makes the tensor traceless. Without it, you'd have the raw second moment of the charge distribution, which mixes in monopole information.

Potential of electric quadrupole

The quadrupole contribution to the potential is:

$V_{\text{quad}}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{1}{2} \sum_{i,j} \frac{Q_{ij} \, \hat{r}_i \hat{r}_j}{r^3}$

This falls off as $1/r^3$. The quadrupole term becomes the leading term when both $Q = 0$ and $\vec{p} = \vec{0}$ (e.g., for a linear quadrupole arrangement of charges).

Higher order multipole moments

General multipole expansion

The full expansion in spherical coordinates is:

$V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \sum_{l=0}^{\infty} \frac{1}{r^{l+1}} \sum_{m=-l}^{l} \frac{4\pi}{2l+1} \, q_{lm} \, Y_{lm}(\theta, \phi)$

where the spherical multipole moments are:

$q_{lm} = \int \rho(\vec{r}') \, r'^l \, Y_{lm}^*(\theta', \phi') \, d^3r'$

The $Y_{lm}(\theta, \phi)$ are spherical harmonics, which form a complete orthonormal set on the unit sphere. Each $l$ value corresponds to a $2^l$-pole, and there are $2l + 1$ independent moments at each order.

Contribution of higher order terms

• The $l$-th term scales as $(a/r)^l \cdot (1/r)$, so at large $r$ higher-order terms are strongly suppressed.
• For most applications, truncating at $l = 2$ (quadrupole) is sufficient.
• Higher-order terms matter when you need high precision, when the lower moments vanish, or when the source has intricate angular structure.

Applications of multipole expansion

Far-field approximation

In the far-field region ($r \gg a$, where $a$ is the source size), the first nonvanishing multipole term gives an excellent approximation. This is the regime where multipole expansion is most useful.

For example, a neutral molecule with a dipole moment $\vec{p}$ produces a potential that looks like a pure dipole field at distances much larger than the molecular size. You don't need to know the detailed charge distribution; the single vector $\vec{p}$ captures the essential far-field behavior.

Electrostatic interactions

The interaction energy between two well-separated charge distributions can be written as a sum over products of their multipole moments. The leading terms are:

• Monopole-monopole: $\propto Q_1 Q_2 / R$ (Coulomb's law between net charges)
• Monopole-dipole: $\propto Q_1 p_2 / R^2$ (charge interacting with a dipole)
• Dipole-dipole: $\propto p_1 p_2 / R^3$ (two dipoles interacting)

where $R$ is the separation between the distributions. This hierarchy is central to understanding intermolecular forces (van der Waals interactions, for instance, involve dipole-dipole and higher terms).

Multipole expansion in spherical coordinates

Spherical harmonics

Spherical harmonics $Y_{lm}(\theta, \phi)$ are the natural basis functions for expanding angular dependence on a sphere. Key properties:

• Orthonormality: $\int Y_{lm}^* Y_{l'm'} \, d\Omega = \delta_{ll'}\delta_{mm'}$
• Completeness: any well-behaved function on the sphere can be expanded in $Y_{lm}$'s
• They are eigenfunctions of $\nabla^2$ in the angular part, which is why they appear naturally when solving Laplace's equation in spherical coordinates

The multipole expansion is really just the statement that solutions to Laplace's equation (valid outside the source) can be written as a sum of $r^{-(l+1)} Y_{lm}(\theta,\phi)$ terms, with coefficients determined by the source.

Multipole moments in spherical coordinates

The number of independent components at each order:

Multipole expansion of magnetic fields

Magnetic scalar potential

Outside of current sources (where $\vec{J} = 0$), you can define a magnetic scalar potential $\phi_m$ such that $\vec{B} = -\mu_0 \nabla \phi_m$. This potential admits a multipole expansion analogous to the electric case:

$\phi_m(\vec{r}) = \frac{1}{4\pi} \sum_{l=0}^{\infty} \frac{1}{r^{l+1}} \sum_{m=-l}^{l} M_{lm} \, Y_{lm}(\theta, \phi)$

Magnetic multipole moments

There is one critical difference from the electric case: magnetic monopoles do not exist (as far as we know). This means:

• The $l = 0$ (monopole) term is always zero: $\nabla \cdot \vec{B} = 0$ forbids it.
• The magnetic dipole ($l = 1$) is the leading term. For a current loop with area $A$ carrying current $I$, the magnetic dipole moment is $\vec{m} = I A \hat{n}$.
• Magnetic quadrupole and higher moments exist and are defined analogously to the electric case, but the absence of the monopole term means the dipole always dominates at large distances.

The far field of any localized current distribution looks like a magnetic dipole field, just as the far field of a neutral charge distribution looks like an electric dipole field.

Limitations of multipole expansion

Convergence radius

The expansion is valid only for $r > r'_{\text{max}}$, i.e., outside the smallest sphere centered at the origin that encloses all the charge. Inside this sphere, the series diverges. If you need the field inside or near the source, you must use the interior expansion (powers of $r^l$ instead of $r^{-(l+1)}$) or compute the integral directly.

Accuracy near charge distribution

As you approach the source, the ratio $a/r$ grows toward 1, and convergence slows dramatically. You need more and more terms to get an accurate answer. At some point, direct numerical integration of the Coulomb integral becomes more practical than summing a slowly converging multipole series.

A useful rule of thumb: if you're at a distance of only 2-3 times the source size, you may need terms up to $l \sim 5$ or higher for percent-level accuracy. At 10 times the source size, the dipole (or lowest nonvanishing) term alone is often sufficient.