Divergence-free polarization of dielectric

In summary, the conversation discusses the physical meaning of a non-vanishing polarization field with non-trivial constitutive relation and vanishing divergence, and whether it can be considered a dielectric. The concept of bound and free charges and the definition of polarization in optics is also mentioned. An example of a material with non-trivial polarization is a free electron gas. The Landau, Lifshetz, Electrodynamics of continuous media is recommended for further understanding.
  • #1
cliowa
191
0
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 let's 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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  • #2
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.
 
  • #3
Thanks for that clarification. When you say
DrDu said:
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.
DrDu said:
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
 
  • #4
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 [tex] P(x,t)= \int_{-\infty}^t dt' j(x,t')[/tex] or, in frequency space, [tex] P(x, \omega)=j(x,\omega)/(i \omega)[/tex] 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.
An example of a material with a highly non-trivial dependence of the polarization on the field is a free electron gas. Look for Lindhard dielectric function in that context.
The whole concept how to define polarization in optics is nicely discussed in Landau, Lifshetz, Electrodynamics of continuous media.
 
  • #5


I can provide some insight into the concept of divergence-free polarization of dielectric materials. This phenomenon is often observed in the study of laser-matter interactions, where the electric field of a laser can induce a polarization in a dielectric material.

The physical meaning behind a non-vanishing polarization field with a vanishing divergence is that the polarization is not a result of the presence of bound charges in the material. In other words, the polarization is not directly proportional to the electric field, but rather has a more complex relationship dictated by the constitutive equation.

This type of polarization can be seen as a model for certain materials where the polarization is induced by the alignment of dipoles rather than the presence of bound charges. One example is ferroelectric materials, where the alignment of dipoles can be controlled by an external electric field.

In terms of starting with a non-zero, divergence-free polarization field and letting it evolve according to the Maxwell's equations and constitutive equation, this can be seen as a way to study the behavior of such materials under different conditions and external influences.

It is important to note that a dielectric material with a divergence-free polarization field may still be considered a dielectric, as the polarization is still a result of the material's response to an electric field. However, the specific properties and behavior of such materials may be different from those with a non-zero bound charge density.

In summary, the concept of divergence-free polarization in dielectric materials has important implications in the study of laser-matter interactions and can serve as a model for certain materials with unique properties. Further research and experimentation in this area can provide valuable insights into the behavior of these materials and their potential applications.
 

1. What is divergence-free polarization of dielectric?

Divergence-free polarization of dielectric refers to the state of a dielectric material where the electric dipole moment per unit volume is constant throughout the material and does not change with position. This means that the material has a uniform polarization distribution, with no regions of higher or lower polarization.

2. What is the significance of divergence-free polarization of dielectric?

Divergence-free polarization of dielectric is important in understanding the behavior of materials in electric fields. It allows for the accurate calculation of electric field strength within the material and is crucial in the design and analysis of electronic devices.

3. How is divergence-free polarization of dielectric different from other forms of polarization?

Unlike other forms of polarization, such as dipole polarization or electronic polarization, divergence-free polarization does not involve the movement of charges within the material. Instead, it is a result of the alignment of permanent dipole moments within the material.

4. Can all dielectric materials exhibit divergence-free polarization?

No, not all dielectric materials can exhibit divergence-free polarization. This type of polarization requires a material to have a uniform and constant electric dipole moment per unit volume, which is not the case for all dielectrics. Materials such as ferroelectrics and some piezoelectric materials can exhibit divergence-free polarization.

5. How is divergence-free polarization of dielectric related to Maxwell's equations?

Divergence-free polarization of dielectric is closely tied to Maxwell's equations, specifically the equation for Gauss's law. This equation states that the total electric flux through a closed surface is equal to the charge enclosed by that surface. Divergence-free polarization ensures that this equation is satisfied and allows for the accurate calculation of electric fields within the dielectric material.

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