So my intuition of the multipole expansion is that we have a certain charge distribution, and the ME tells us that we can approximate that distribution as a superposition of monopoles, dipoles, etc. Physically we could have a certain dipole for instance, then adding a quadropole to this takes us a step closer to the original charge distribution. Repeat this an infinite number of times, adding higher order terms like octopoles, 32-poles etc and we would end up with the exact charge distribution. Correct me if I am wrong, but would this be the intuition behind ME?
Multipole Expansion – Intuitive Guide to Charge and Dipole Moments
chargedipoledipole-momentmultipole-expansion
Related Solutions
First, $\vec{r}^\prime$ is a vector that goes from the origin to the source of charge. If the source is a volumetric distribution, one must sum all contributions of charge, that's why one integrates over all the volume, say $\mathcal{V}$; the (correct) expression for the potential should be $$V( \vec{r}) = \frac{1}{4 \pi \epsilon _{0}} \int_\mathcal{V} \frac{\rho (\vec{r}^\prime)}{ℛ} d\mathcal{V}^\prime$$ so that all dependence of $V$ remains on $\vec{r}$. Then, $r^\prime$ is just the magnitude $|\vec{r}^\prime|$, being the distance from the origin to the source of charge.
Second, usually, the series expansion of a function $f(x)$ about some point $x_0$ is useful because if you want to know the value of $f$ near $x_0$, you may just take some few terms of the expansion; it is as seeing the plot of $f$ with a magnifying glass. You should remember this from your first calculus courses, it is done a lot in physics. Here the expansion about $\epsilon=0$ will be useful since $\epsilon\to0$ implies $r\to\infty$ (just really big, if you will). The (correct) expression $$V(\vec{r}) = \frac{1}{4 \pi \epsilon _{0}} \sum ^{\infty}_{n=0}\frac{1}{r^{n+1}} \int(r')^n\,P_{n}(\cos \theta^\prime)\,\rho( \vec{r}')\,d\mathcal{V}'$$ is just another way of writing the series expansion in terms of $r$, $r^\prime$ and $\theta^\prime$, where $P_n$ are the Legendre polynomials (Griffiths defines them there, ain't he?). This expression is useful, as it means, explicitly, that $$V(\vec{r})=\frac{1}{4\pi\epsilon_0}\left[\frac{1}{r}\int\rho(\vec{r}')\,d\mathcal{V}'+\frac{1}{r^2}\int{r'}\cos\theta'\,\rho(\vec{r}')\,d\mathcal{V}'+\frac{1}{r^3}\left(\cdots\right)+\ldots\right]$$ so that if you want to evaluate the potential for points far from the source (big $r$), then you may just neglect higher order terms in $r$ and just take the $1/r$ (monopole) term; and so on if you're considering a better approximation, you may take the $1/r^2$ (dipole) term, etc... That's the real usefulness of the series expansion; in a lot of situations evaluating $V( \vec{r}) = \frac{1}{4 \pi \epsilon _{0}} \int \frac{\rho (\vec{r}')}{ℛ} d\mathcal{V}'$ will get really ugly, and then, mostly, is when the multipole approximation will be useful.
First of all, don't think of multipole moments as separate things that have their own individual meaning. Instead, think of them as parts of one thing. Once we have all the parts written down, we can start naming and organizing each one to determine its contribution to the whole.
Now, for your question
Is there a physical interpretation for multipole moments?
Yes! And they don't necessarily have to do anything with electrostatics, spherical harmonics or geometric series. A multipole expansion of some object in some basis is saying Hmmm, I have this funny shaped thing that isn't an elementary mathematical function, but I want to express it as a sum of elementary functions. (Fourier or someone said you can always do this with enough ink and parchment.)
So you first pick your basis, whether it be sine waves or exponentials or polynomials or the like, and then you start adding more and more terms of that basis, beginning with the lowest order (simplest) component.
For a non-mathematical example, consider this drawing of a cartoon sheep:
Step 1 is very simple since it's basically an oval with ripples. Right off the bat, with the very first thing we drew we have expressed some 75% of what the sheep will look like. This is important in multipole expansions: the lowest order terms dominate. If Step 1 were a square or a triangle then the whole sheep would likely be unrecognizable.
Step 2 does two things: it adds to the drawing and it slightly modifies some of what Step 1 did. You may have heard this called a corrective term or a higher order term. This would be the second term in your multipole expansion.
Step 3 adds less to the picture than Step 2, but look how far we've come. With just the first three multipoles, I'm betting a large percentage of people would recognize our animal as a sheep already. If instead of drawing a sheep with blobs we were constructing an E&M field with Legendre Polynomials, this is about where we stop since we have a good physical view of what's happening (this is the trade off of simplicity vs. accuracy present in all multipole expansions).
Additional steps just add more detail, filling out the gaps in the data at the cost of doing more work and keeping track of more pencil marks.
What is the physical interpretation of higher order multipole moments?
In E&M, we break the arbitrary looking charge distribution into multipoles with the hopes of drawing as much of the sheep as we need with as few details as possible. The multipoles look like:
0. the monopole, the offset that affects the E field in all directions the same way
1. the dipole, the description of how different two halves of the field would be if you drew a symmetry line right down the center
2. the quadrupole, a similar concept to the dipole, but instead of affecting two directions differently, you affect four directions
And you can keep going to as high of a multipole as you want (technically, you need to go to infinity to perfectly redraw an arbitrary sheep, but this isn't useful when we're just trying to predict system certain properties to within a finite precision to begin with).
Summary
A multipole expansion of anything is just breaking it down into a preferred basis. If we picked a good basis, we only need the first few multipoles because after that we're just touching up details we'll never need. Some multipoles are so useful we give them names, like the E&M charge distribution that affects everything isometrically (total charge) and antisymmetrically (dipole).
Best Answer
Yes, that's correct. It is similar to the Fourier series where a periodic function is rebuilt using a basis of oscillating functions. The Fourier decomposition translates a periodic function of one continuous variable into a function of a discrete variable (an index).
If you consider a function of the two angles $\vartheta$ and $\varphi$, that is, a function whose argument is a direction pointing somewhere on a unit sphere, you can use the spherical harmonics as a basis. Here we have a function which is again periodic in its two variables and translate it into a function of two indices. This is used, for example, when analysing the cosmic microwave background which is described by a function $T(\vartheta, \varphi)$ giving the temperature of the background radiation depending on the direction in the night sky, you are pointing your detector at. You can get a set of coefficients $a_{lm}$ from a decomposition in spherical harmonics $Y^m_l(\vartheta,\varphi)$.
If you are interested in a function which is defined on whole $\mathbb{R}^3$, the different $l$-modes scale with some particular power of $r$. But this is specific to the particular function we are expanding, typically the function $\frac{1}{\vec r-\vec r'}$ which is the Green's function of the Laplace operator.