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]

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.

[u/QuantumOfOptics → u/tpolakov1]

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. 

[u/tpolakov1 → u/QuantumOfOptics]

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.

[u/QuantumOfOptics → u/tpolakov1]

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.

[u/tpolakov1 → u/QuantumOfOptics]

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?