# Divergence-free polarization of dielectric

## Main Question or Discussion Point

Hey all,
I'm studying laser-matter interactions and was wondering: Is there any physical meaning to a non-vanishing polarization field with non-trivial constitutive relation but vanishing divergence? (By non-trivial I mean the constitutive equation does not stipulate that the polarization and electric fields are directly proportional) Is this a model for anything? Could you think of a situation where one starts out (at a given time) with a non-zero, divergence-free polarization field (and lets it evolve according to the MW equations and the constitutive equation)?

From what I have seen so far in textbooks a divergence-free polarization field implies that the density of bound charges in the dielectric is zero. Can that still be considered a dielectric? Does such a thing ever arise, and if yes, in what context?

Thanks a lot for your help...Cliowa

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DrDu
The divergence of the polarization is by definition the density of the charges in the medium (i.e. those charges which are not external). In optics one usually sets magnetization M=0. Hence polarization has to describe all magnetic effects, i.e. the effects of currents with non-vanishing rotation. Such a transversal non-vanishing current possible even if the medium is neutral in every point.

Thanks for that clarification. When you say
The divergence of the polarization is by definition the density of the charges in the medium (i.e. those charges which are not external).
do you really mean that the process of thought is that 1) when you apply an electric field to a medium bound charges (dipole moments) will be "created" (locally) and 2) you then construct a polarization field that has divergence equal to the density of those bound charges?
Concerning the idea of bound charges: The idea is really just the dipoles created in the material, right? Typically the dipole displacement will be very, very small compared to the size of the medium, I guess. For a standard plate capacitor with a dielectric in between the plates: could one thus assume (on the right scales) that the net effect of the dielectric will be additional charge on the plates (now from the bound charges on the "inside")?

More specifically I was studying a model where the polarization field is given as an anharmonic oscillator driven by the electric field, so I am looking at things in the optics regime.
In optics one usually sets magnetization M=0. Hence polarization has to describe all magnetic effects, i.e. the effects of currents with non-vanishing rotation. Such a transversal non-vanishing current possible even if the medium is neutral in every point.
Could you elaborate a bit on that? Do you really mean that there are no dipoles created (or they're negligible) in the medium but the polarization is not trivially related to the electric field?

In general physical applications: Can one say anything about the geometry of the electric, magnetic and polarization fields? (I think for plane waves orthogonality of the former and electric field parallel to polarization can be inferred easily from the equations)

Thanks a lot for helping me out...Cliowa

DrDu
The concept of bound and free charges stems from old days in the afterlast century. With the advent of quantum mechanics it became clear that one cannot distinguish between bound and free charges and it is also not possible to unambiguously define something like dipole density. There are different resolutions of the problem of how to describe a dielectric, depending on whether one is interested in electro-(or magneto-)statics or in the optical range of frequencies. In the optical region one simply sets $$P(x,t)= \int_{-\infty}^t dt' j(x,t')$$ or, in frequency space, $$P(x, \omega)=j(x,\omega)/(i \omega)$$ where j is the current density due to the charges of the medium (all charges but external ones). You can see that together with the choice H=B, all Maxwell equations are fulfilled. Even in the case of simple isotropic media, epsilon is now a tensor, with a longitudinal and a transversal part. In optics one is mainly interested in the transversal part. The divergence of the transversal part of the polarization (and current ...) always vanishes.