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We have a polarised material, suppose. The goal, is to find the bound charges.

DJ Griffiths (4th ed, section 2.1, page 173) "chops" the material into smaller infinitesimal chunks and writes:

$$V(r)=\frac{1}{4\pi\epsilon}\frac{\vec p\cdot \hat s}{s^2}$$ and by writing $ \vec p = \vec P d\tau' $, he arives at the formula he uses:

$$V(r)=\frac{1}{4\pi\epsilon}\iiint_{V}\frac{\vec P(\vec r')\cdot \hat s d\tau'}{s^2}$$ here $\vec s = \vec r-\vec r' $ where $\vec r'$ is the source point, $\vec r$ is the field point. From here he goes on to "derive" the corresponding surface charge and volume charge densities, by comparing some coefficients. The method that follows after the above formula is not of much concern, but there are some issues I have with the justification of this method (see section 2.4, page 179-181)

Griffiths states that the method falls into place as we are, using the above formulas, calculating the macroscopic field , which he says, is defined as the average electric field in a volume, averaged over that volume which is macroscopic ( contains 1000 molecules).

This average $\vec E_{avg}$ at a point comes from both the inside and outside contributions, and hence $$\vec E_{avg}(r)=\vec E_{{in}_{avg}}(r) + \vec E_{{out}_{avg}}(r)$$

He then uses a result from a previous section: the average electric field due to all the charges outside a sphere, averaged over the sphere is equal to the total electric field at the centre of the sphere due to all the charges outside the sphere.

Keeping the centre of the sphere as the field point, and ascribing the primed coordinates to the charges outside the sphere, he writes:

$$V_{outside}(r)=\frac{1}{4\pi\epsilon}\iiint_{V_{outside}}\frac{\vec P(\vec r')\cdot \hat s d\tau'}{s^2}$$

and for the inside volume, where we obviously can't use the dipole potential approximation, another result comes to save the day: $$\vec E_{in} = -\frac{1}{4\pi\epsilon}\frac{\vec p}{R^3}=-\frac{1}{3\epsilon}\vec P$$ where $\vec p$ is the total dipole moment contained in this volume (sphere) and $\vec P$ is the polarisation. He then assumes the polarisation $\vec P$ does not vary inside this volume and asserts (again from a previous result) that in such a case, the field that is left out (not included in the previous integral) is precisely equal to the $\vec E_{in}$ we wrote down above. This completes his justification. There are some issues: The $\vec P$ is the macroscopic, by which we mean we take a macroscopic volume and sum the total dipoles in that volume, and divide by that volume. We do this at each point so that the microscopic variations are settled. Given this, we can say that polarisation is constant throughout this sphere ( if the macroscopic volume we use in averaging the electric field is of the order of this sphere, used for the polarisation).

But then, what exactly should the size of $d\tau'$ be?

If we say it is of the order of the sphere over which we do the field averaging, then how can we perform the integral over the inner sphere? In fact, the outer integral also suffers a similar issue since, suppose I'm taking a chunk outside the sphere, then we cant really say that all the molecules in this sphere have the same distance from the centre and successfully use the integral, since the integral inherently assumes that all molecules in the $d\tau'$ element are approximately at the same distance from the centre.

If we say that $d\tau'$ is actually much smaller than the volume over which we are doing the field average (and as a consequence, much smaller than the volume over which we do the polarisation calculation), then how do we know that the polarisation (which we calculated over a much larger volume, in comparision to $d\tau'$) times $d\tau'$ would give me the net dipole moment contained in this elementary volume $d\tau'$ ?

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It is really nice to see that Griffiths found it important enough to cover this topic within his introductory textbook. JD Jackson covered only the necessity and sufficiency for spatial averaging, that time averaging alone is neither necessary nor sufficient, and left the topic in limbo. No suitable method was covered, essentially throwing the problem to condensed matter physics.

For the other readers, it is important to state here that we still do not have a completely satisfactory answer to this problem. It is absolutely infernal, reader beware. But these days we do have a much more satisfactory solution in the form of the Modern Theory of Polarisation, for the case of crystalline physics, which is not really doing this spatial averaging directly. However, crystalline physics comes with its own fair share of monsters. Not only are we dealing with the implicit meaninglessness of displacing charges of an infinite lattice (Modern Theory of Polarisation handles this beautifully), but we are also having to deal with the fact that the electronic wavefunctions do not actually have a meaningful charge centre, and we have to deal with effective charges, say of Born's scheme, all nebulously defined concepts and difficult to compute, numerically converge, etc.

Now that we have gotten that out of the way, we can start answering your question. First of all, $\mathrm d\tau^\prime$ has to be infinitesimal for the integral and the choice of symbols to make sense. You are not averaging over these infinitesimal volume elements. You are averaging over some volume, and coming from crystalline physics, it is exceedingly clear that you are averaging a multiple of the (primitive) unit cell. (Needless to say, we are all lazy, and non-primitive unit cell will be much less of a conceptual headache, and much more likely to be implemented correctly in computer codes.)

The rest of this answer will follow Kittel Intro to SSP instead of Griffiths, and in fact, coming from memory, about a decade later. Needless to say, not with source material at hand. Pardon any mistakes.

When we want to study polarisation of materials, we start with a macroscopic sample, to put between a parallel plate capacitor. The capacitor sets up an electric field, that is, hopefully, approximately uniform in the sample. The sample gets polarised, and we have the first contribution to consider. For the purposes of the theory, we can assume that the sample is uniform and regularly shaped and aligned with the geometry, so that analysis will be most simple. All of these assumptions are physically realisable.

This means that we have a polarisation of the material, that we assume to be uniform enough inside the material, that we can study. Experimenters would extract the susceptibility or dielectric constant, and so forth, at least as a rough estimate, from which we continue the theoretical investigation.

The smart thing to do in the next step of the theoretical study, is to assume that this polarised material is infinitely large. After all, we are trying to study the intrinsic properties of the sample, and finite sample size is just an unnecessary detail at this point. The geometrical annoyances of the finite macroscopic sample had already been sorted out by the above considerations, so it is safe for us to work with an infinite sample, with E and P fields pre-imposed.

Then, we want to stop making the smooth, averaged field approximation, so that we can redo it in greater detail. To that end, we imagine that inside our treatment of the infinite medium as being a uniformly polarised jelly, we dig up a cavity of free space, and then repopulate it with the underlying ions and electrons, so that we can average out this cavity.

Because of the evacuation of the cavity, this means that the uniformly polarised medium would now have a surface charge density on the cavity wall. This will make the electric field as felt by stuff inside the cavity bigger, just the opposite of the original medium getting polarised would cause the electric field as seen by whatever stuff is in the medium, see a smaller effective electric field.

And then we consider the polarisation of all the ions and electrons around whatever is at the centre of the cavity, find their contribution to the electric field felt at the centre, and derive a relationship between the polarisation of the centre of the cavity, with the electric field felt at that very same centre. Then, separately in quantum theory, we can compute the atomic polarisability, say, of what is at the centre, and use that to finally derive the theoretically correct value of the dielectric constant.

We have to be careful here: the shape of the cavity, and also of the initial macroscopic sample, is actually important. It would change the details of the calculation, even though the final result ought to not be affected. A spherical cavity would involve different factors as that of a cylindrical cavity, say. But otherwise, the scheme is great at this point. We might need the polarisability along different crystal directions, so it might be nicer to use a spherical shape.

The size of the cavity cannot be too small, because the outside of the cavity is approximated as a uniform, continuous medium. If the cavity size is so small as to make the continuum approximation be inapplicable, obviously the theory cannot be trusted. So, this cavity has to be in one of the standard physics trickery: microscopically large, but macroscopically small. This is, of course, easy to practically achieve theoretically, because of how tremendously large Avogadro's number is. This is the volume you are averaging over, the solution to your problem. It has to be consistent; you have to pick the size of the averaging sphere to be equal to the size of the cavity.

The obvious next step is to theoretically let the size of this cavity go to infinity if we so wish to be exact. But in practice, the averaging will converge within a few unit cell sizes, and the averaging gets computationally expensive to do as the size gets larger. It is thus not needed to actually be very big.

The above is a prescription of a scheme that is successful. However, it is definitely not the last word on the subject. There are known theoretical issues with this. For example, the scheme can be done with a variety of dielectric constants, and even from one to the other. It does not have to be just comparing with the vacuum. But when you actually perform the prescription, you will realise that the result by going from one dielectric constant to the other, disagrees with the reverse choice, and so it is somewhat theoretically internally inconsistent. This, and other problems, and other prescriptions, are covered in a nice little book by TC Choy, titled Effective Medium Theory, and readers not yet disgusted by the complications, might find it nice to read.

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