Multipole Expansion

When calculating the potential of a localized charge distribution from a significant distance ($r \gg r'$), it is often inefficient to perform the full integration. Instead, we expand the potential into a series of terms ordered by powers of $1/r$. This series is called the Multipole Expansion.

1. Approximate Potentials at Large Distances

Consider a charge distribution $\rho(\vec{r}')$. The exact potential at point $\vec{r}$ is given by: $$ V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{1}{\rsep} \rho(\vec{r}') d\tau' $$

As derived in the deep-dive, we can expand $1/\rsep$ using Legendre Polynomials. The resulting potential is a sum:

$$ V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \sum_{n=0}^{\infty} \frac{1}{r^{n+1}} \int (r')^n P_n(\cos\alpha) \rho(\vec{r}') d\tau' $$

$$ V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \left[ \frac{1}{r} \int \rho d\tau' + \frac{1}{r^2} \int r' \cos\alpha \rho d\tau' + \dots \right] $$ $$ = \underbrace{\frac{1}{4\pi\epsilon_0} \frac{Q}{r}}_{\text{Monopole}} + \underbrace{\frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \hat{\mathbf{r}}}{r^2}}_{\text{Dipole}} + \underbrace{\dots}_{\text{Quadrupole}} $$

Each term in this sum corresponds to a different multipole moment. The expansion allows us to classify complex distributions by their dominant behavior at long distances.

2. The Multipole Hierarchy

Usually, we are only interested in the first few terms. As $n$ increases, the terms die off faster with distance.

Monopole (n=0) $V_{mon} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$

The dominant term if the object has a net total charge $Q$. At large distances, any charged object looks like a point charge.

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

If $Q=0$, this term dominates. It arises from charge separation. The dipole moment is defined as $\mathbf{p} = \int \vec{r}' \rho(\vec{r}') d\tau'$.

Quadrupole (n=2) $V_{quad} \propto 1/r^3$

If both $Q=0$ and $\mathbf{p}=0$, the quadrupole term dominates. This occurs in highly symmetric arrangements, like two opposing dipoles.

Origin Dependence

While the monopole moment $Q$ is independent of the choice of origin, higher moments (like $\mathbf{p}$) are generally coordinate-dependent. However, if the lower-order moment is zero, the next moment becomes independent of the origin. For example, if $Q=0$, the dipole moment $\mathbf{p}$ is an intrinsic property of the distribution.

3. The Electric Field of a Dipole

For a "pure" dipole at the origin, the electric field is calculated by taking the negative gradient of the potential in spherical coordinates:

$$ \mathbf{E}_{dip}(r, \theta) = \frac{p}{4\pi\epsilon_0 r^3} (2\cos\theta \hat{\mathbf{r}} + \sin\theta \hat{\boldsymbol{\theta}}) $$

Notice that the field drops off as $1/r^3$. This explains why dipole interactions are short-range compared to monopole interactions.

Deep Dive: Derivation of the Dipole Potential & Field

🎓 Exam Tip: On the Axis ($\theta = 0$) (Click to roll/unroll)

In most exam questions, you are asked to find the field along the dipole axis (where $\theta = 0$ or $\theta = \pi$).

Since $\cos(0) = 1$ and $\sin(0) = 0$, the formula simplifies to:

$$ \mathbf{E}_{axis} = \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3} \hat{\mathbf{r}} $$

4. Interaction Energy between Distant Dipoles

A crucial application of the multipole expansion is calculating the energy of interaction between two localized charge distributions. For two neutral molecules separated by a distance $r \gg d$, we treat them as dipoles.

Why do we need this?

Interaction energy is fundamental to understanding molecular physics and chemistry. It explains:

Intermolecular Forces: The origin of van der Waals forces.
Dipole-Dipole Alignment: Why polar molecules in a liquid or crystal tend to orient themselves in specific ways.
Material Properties: The dielectric behavior of gases and fluids.

The interaction energy $U$ is the potential energy of the second dipole $\mathbf{p}_2$ in the electric field $\mathbf{E}_1$ produced by the first dipole $\mathbf{p}_1$: $$ U = -\mathbf{p}_2 \cdot \mathbf{E}_1 $$

Substituting the expression for the dipole field, we obtain the general interaction energy:

$$ U = \frac{1}{4\pi\epsilon_0 r^3} [ \mathbf{p}_1 \cdot \mathbf{p}_2 - 3(\mathbf{p}_1 \cdot \hat{\mathbf{r}})(\mathbf{p}_2 \cdot \hat{\mathbf{r}}) ] $$

This formula shows that the interaction energy depends strongly on the orientation of the dipoles. Because it scales as $1/r^3$, dipole-dipole interactions are much weaker at long distances than monopole-monopole ($1/r$) interactions.