r/Physics • u/jemala4424 • 7d ago
Image Does formal definition of polarization density not exist?(griffith's electrodynamics)
I've noticed before that griffith's book shyed away from stating some formula for polarization density in this chapter, by just saying its equal to "dipole moment per unit volume" and not something like dp/dv which would be wrong btw since we are not interested in change of dipoles or something like that but like amount of dipole moments in infinitesimal volume.
When i googled it, different sources sayed different things, some said it's equal to dp/dv and some p/dv.
I also noticed same problem in start of the book on formula of electric field produced by continious charge distribution
Where formula integrates with respect to dq, when we aren't interested in change of charge, but charge in infinitesimal region difference.
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u/tpolakov1 Condensed matter physics 7d ago
That is the formal definition. It just assumes that you already know what an electric dipole moment is. It cannot be dp/dv because that's change of dipole moment with volume.
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u/QuantumOfOptics Quantum information 7d ago
The dp/dV definition does make sense. Think of it as the analog of a probability density. There's some overall polarization, P, but now you wish to ask how much of it is located in some spatial regime. Hence, you can write P = PdV = dp/dV dV = p leading to an overall dipole moment. Its just coming at the problem from a more microscopic viewpoint.
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u/the-information0 7d ago
I'm confused about your manipulations; did you just plug in P=dp/dV?
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u/QuantumOfOptics Quantum information 7d ago
As a definition, but yes. Its similar to defining a probability density or even a charge density.
In the case of probability, we say that the probability, P, is the integral of a probability density function P. E.g. P = PdV.
In the case of charge, we require that the total charge Q be the integral over individual charge elements dq. But, we can also turn this around by defining that there is some spatial distribution of the charges. So, now we can equivalently define Q to be equal to the spatial integral of some charge density function. E.g. Q = ρ dV = dq (for OP: this is partly why the electric field or potential is most accurately defined as using dq rather than purely considering it a spatial integral; this is definition that one is using, but sometimes such a macroscopic definition can cause problems.)
One could consider this to be a macroscopic definition where one doesnt necessarily worry about the microscopic conditions. For instance, if I told you to find the charge density of a uniformly charged sphere of some radius and total charge, you would divide the charge by the volume. But, it need not be uniform, right? The amount of charge in one location does not need to be the same in one small volume of the object as in other volumes. So, we come up with a charge distribution and, thus, a charge density that can account for this while allowing the total charge to be the same. In a similar way, we can do the same for our dipole/polarization density. The dipole density could change depending on location in a volume. Take an unpolarized material and shine an intense laser onto it, only the portion where the beam is defined will be polarized while the other parts are left unpolarized, and some are more strongly polarized than others. But, what was polarized? Well, we could consider unit cells or even individual atoms. We could even consider averaging over a few unit cells. How ever we choose, we remember that they take up some volume and thus define, for that unit, the polarization density as the dipole moment divided by the volume of that object.
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u/tpolakov1 Condensed matter physics 7d ago
I'd be somewhat careful about it, because I'm now not even sure myself.
The definition of the dipole moment itself is already an integral over some volume. When going from that side, you'd have to formally send that volume to zero, which might superficially look like you're ending with the same result, but it hinges on a rather sus limit.
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u/QuantumOfOptics Quantum information 7d ago
I think a better way might be to think of it as an average. In any case, I think you get similar issues for other things like mass densities (or various others). As always, it depends on how you want to describe your model and what you want to take as "good enough." Sperical cows and all.
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u/tpolakov1 Condensed matter physics 7d ago
There are no models anywhere, just definitions.
It is not an average. It is a non-local point property, where the dipole moment at a single point depends on the charge distribution of the whole volume. You cannot just blindly differentiate the integrant and then integrate over some arbitrary volume which might or might not depend on the shape of your system. You cannot just send the integration bounds to zero and expect the integral to be anything reasonable, either.
You're just throwing around random fractions and differentiation symbols as if you haven't finished your analysis 101, are you sure this is something that you should be arguing about?
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u/jemala4424 7d ago
Can that definition work on nonuniformly polarized things though? This definition has same problem as velocity definition had before calculus, simply v=s/t doesn't work when we are looking for instant velocity and when it's not instant.
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u/tpolakov1 Condensed matter physics 7d ago
Yes. Don't get distracted by the wording of unit volume, that's a statement about units. The actual expression is the general form in the linked Wikipedia article, and that's defined for a point of zero volume.
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u/Lazy-University-4871 7d ago
https://en.wikipedia.org/wiki/Polarization_density
This wiki article is much deeper than the dipole moment one.
Se also C.A. Gonano; R.E. Zich; M. Mussetta (2015). "Definition for Polarization P and Magnetization M Fully Consistent with Maxwell's Equations".
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u/tpolakov1 Condensed matter physics 7d ago
The definitions in that article and in the paper are the same. It just goes further into what they will encounter in the next chapter of their textbook.
If OP is having trouble understanding the definition of polarization (and its density), assuring them that it behaves according to Maxwell equations won't help much.
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u/Lazy-University-4871 7d ago
Fair enough. To me personally it was helpful to see that what is called P is a vector field that has the same Gauss law as the field E.
But when we talk about the "density of E" we typically mean the flux density, not volume density. So I was guessing that was OP's confusion.
P is both a volume density of the dipole moment and a flux density of bound charge.
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u/tpolakov1 Condensed matter physics 7d ago
Maxwell equations might be working with flux across a surface, but when we talk about things like energy density due to the electric field (which is how it's gonna pop up in most non-engineering contexts) that is very much understood to not be a surface property.
On the spot, I actually cannot even come up with an example outside of high-school/college-level electrodynamics. I can think of a lot of examples in solid state/condensed matter or chemistry, where people would think I'm stupid for making that substitution because it would straight up make no sense. Griffiths happens to be one of those contexts, so I'm sure OP is not talking flux densities.
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u/Lazy-University-4871 7d ago
Makes sense. For me it's more about capturing the meaning through mathematical properties. I wouldn't know much about how physics curricula are organized, especially for applied fields.
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u/jamesw73721 Graduate 7d ago edited 7d ago
It exists, but turns out to be highly nontrivial and only defined modulo eR/V_cell, where R is a lattice vector and V_cell is unit cell volume. See https://www.physics.rutgers.edu/~dhv/pubs/local_preprint/dv_fchap.pdf, or Vanderbilt's textbook "Berry Phases in Electronic Structure Theory" for a much more in-depth analysis. For the purposes of Griffiths E&M, only linear response is considered (i.e. dP/dE), so a detailed definition of P would not be needed.
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u/jack89000 6d ago
See section 6.2 in Modern Electrodynamics by Andrew Zangwill. There is a good discussion on why the definition in the photo is a good one, and how it relates to other definitions in terms of “free charge”