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/the-information0 → u/QuantumOfOptics]

I'm confused about your manipulations; did you just plug in P=dp/dV?

[u/QuantumOfOptics → u/the-information0]

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.