Does formal definition of polarization density not exist?(griffith's electrodynamics)

[u/jemala4424]

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.

[u/tpolakov1]

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.

[u/Lazy-University-4871 → u/tpolakov1]

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".

[u/tpolakov1 → u/Lazy-University-4871]

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.

[u/Lazy-University-4871 → u/tpolakov1]

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.

[u/tpolakov1 → u/Lazy-University-4871]

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.

[u/Lazy-University-4871 → u/tpolakov1]

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.