What is the physical meaning of the terms in the multipole expansion?

Question

I have a few questions on multipole expansions and I have read about the topic in many places but could not find an answer to my questions, so please be patient with me.

The electrostatic potential due to an arbitrary charge distribution $\rho(\mathbf{r}')$ at a given point $\mathbf{r}$ is given (up to a factor of $1/4\pi\epsilon_0$) by

$$ V(\mathbf{r})=\int_{V'}\frac{\rho(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} dV'$$

In case where $r\gg r'$, $V(\mathbf{r})$ can be multipole expanded to give

$$V(\mathbf{r})=V(\mathbf{r})_\text{mon}+V(\mathbf{r})_\text{dip}+V(\mathbf{r})_\text{quad}+\cdots$$

where

\begin{align} V(\mathbf{r})_\text{mon}& =\frac{1}{r}\int_{V~`}\rho(\mathbf{r}') dV',\\ V(\mathbf{r})_\text{dip}&=\frac{1}{r^2}\int_{V~`}\rho(\mathbf{r}') ~\hat{\mathbf{r}}\cdot\mathbf{r}'dV', \\ V(\mathbf{r})_\text{quad}&=\frac{1}{r^3}\int_{V~`}\rho(\mathbf{r}') ~\left(3(\hat{\mathbf{r}}\cdot\mathbf{r}')^2-r'^2\right)dV', \end{align} and so on.

Now here are my questions:

1. Is there an intuitive meaning of every one of these terms? For example, I can make sense of the monopole term in the following way: to the 1st approximation the charge distribution will look like a point charge sitting at the origin, which mathematically corresponds to what is called a monopole term, which is nothing but $Q/r$. Is this correct?

2. Now what is the meaning of the dipole term? I know that the word dipole comes from having 2 opposite charges, and the potential due to that configuration, if the charges are aligned along the $z$ axis symmetrically say, goes like $\frac{\cos\theta}{r^2}$. But from the multipole expansion there is a nonzero dipole term even, say, in the case of a single charge sitting at some distance from the origin. Why is it called a dipole term then? Is there a way to make sense of this term in the same way I made sense of the monopole term?

3. What is the intuitive meaning of the quadrupole term?

4. Is the multipole expansion an expansion in powers of $1/r$ only? or of $\cos\theta$ too?

5. Maybe this is not an independent question but I am wondering if there is something like a geometrical/pictorial meaning of every term in the multipole expansion.

Answer 1

For question 2: ("Why does a single charge away from the origin have a dipole term?")

Let's say you have a charge of +3 at point (5,6,7). Using the superposition principle, you can imagine that this is the superposition of two charge distributions

• Charge distribution A: A charge of +3 at point (0,0,0)

• Charge distribution B: A charge of -3 at point (0,0,0) and a charge of +3 at (5,6,7).

Obviously, when you add these together, you get the real charge distribution:

$$ (\text{real charge distribution}) = (\text{charge distribution A}) + (\text{charge distribution B}). $$

By the superposition principle: $$ (\text{Real }\mathbf E\text{ field}) = (\mathbf E\text{ field of charge distribution A}) + (\mathbf E\text{ field of charge distribution B}). $$

And, since the multipole expansion also obeys the superposition principle:

\begin{align} (\text{real monopole term}) & = (\text{monopole term of distribution A}) + (\text{monopole term of distribution B}),\\ (\text{real dipole term}) & = (\text{dipole term of distribution A}) + (\text{dipole term of distribution B}),\\ (\text{real quadrupole term}) & = (\text{quadrupole term of distribution A}) + (\text{quadrupole term of distribution B}), \end{align} and so on.

The field of charge distribution A is a pure monopole field, while the field of charge distribution B has no monopole term, only dipole, quadrupole, etc. Therefore, \begin{align} (\text{real monopole term}) & = (\text{monopole term of distribution A}), \\ (\text{real dipole term}) & = (\text{dipole term of distribution B}),\\ (\text{real quadrupole term}) & = (\text{quadrupole term of distribution B}), \end{align} and so on.

Even though it's unintuitive that the real charge distribution has a dipole component, it is not at all surprising that charge distribution B has a dipole component: It is two equal and opposite separated charges! And charge distribution B is exactly what you get after subtracting off the monopole component to look at the subleading terms of the expansion.

Answer 2

To understand the meaning of multipole expansion,firstly we need to ask ourselves about the evaluation of potential of a very random charge distribution.Recall that, if you forget about the multipole expansion you have no any simple device to know the potential of a random charge distribution, either you want potential near the distribution or very far away.

In our basic electrostatic courses we are usually taught about how to evaluate the potential of some symmetric charge distribution, for instance, potential for distributions which are spherically and cylindrically symmetric,and in most toughest cases the distribution would be of some parameter of polar angles or radial distances.

Now, the second thing which one always ignores (and that's why got stuck with the confusion about intuition) during the study of multipole expansions is that the dipole has nothing to do with a pair of positive and negative charge, or an octupole has nothing to do with a group of 4 positive and 4 negative charge. There charge distribution only take care of the variation of potential,

Answer 3

1. Is there an intuitive meaning of every one of these terms? For example, I can make sense of the monopole term in the following way: to the 1st approximation, the charge distribution will look like a point charge sitting at the origin, which mathematically corresponds to what is called a monopole term, which is nothing but $Q/r$. Is this correct?

First off, the nomenclature is rather unfortunate (goes as $2^n$, i.e $2^2=\text{quadrupole}$) and can be misleading. Mathematically, the monopole term is the zeroth order Legendre Polynomial ($L_0(x)=1$, $L_1(x)=x$, $L_2(x)=(3x^2-1)/2$) and so on (up to a normalization).

Physically speaking, the first term tells us something about the symmetry of the system depending on the location of the observer. Suppose the potential at a point A of the system of charges only has a monopole structure, this means that the charge distribution has full spatial invariance. More importantly, the first term tells you that if we want the total energy of the system, we need to only worry about the Potential at A because the monopole only couples to the electric Potential.

2. Now what is the meaning of the dipole term? I know that the word dipole comes from having 2 opposite charges, and the potential due to that configuration, if the charges are aligned along the z axis symmetrically say, goes like $\cos\theta/r^2$. But from the multipole expansion there is a nonzero dipole term even in the case of a single charge sitting at some distance from the origin say. Why is it called a dipole term then? Is there a way to make sense of this term in the same way I made sense of the monopole term?

The dipole term is the 1st order Legendre polynomial ($2^1=\text{dipole}$). It is possible to have higher order terms even when the net charge is zero. This means, the energy of the system depends on the interaction of the dipole moment with the Electric Field of the test charge. $\vec{d} \cdot\vec{E}$ couplings are studied in light-matter interactions. Another interesting point to note is that there is some kind of spatial symmetry breaking that emerges because dipole interactions can setup a preferred spatial axis (ex along the line joining the charges).

3. What is the intuitive meaning of the quadrupole term?

4. Is the multipole expansion an expansion in powers of $1/r$ only? or of $\cos\theta$ too?

5. Maybe this is not an independent question, but I am wondering if there is something like a geometrical/pictorial meaning of every term in the multipole expansion.

The quadrupole term is so named because it is the second order Legendre polynomial $2^2=4$. The quadrupole moment couples with the gradient of the electric field.

Anyway, this is my personal take on the subject. Mathematically, it is very interesting to ask why do we get Legendre polynomials out of this? It turns out that the Legendre polynomials can be generated by ortho-normalizing monomials (basis for this series expansion).