Polarization charge density of homogeneous dielectric

In summary, the conversation discusses the expression for polarization in a dielectric material under a conservative and non-uniform electric field. The textbook introduces the electric induction vector and states that there are no free charges in the volume being integrated, resulting in zero divergence. The characteristic of electrostatic conservative non-uniform fields to make divergence zero is also mentioned, along with examples of geometries where this occurs. However, for most cases, the resulting polarization is not uniform and the electric field and polarization become quite complex. Outside the material, the electric field from polarization is zero for a dielectric slab, but has a complex form for other geometries. This is typically covered in advanced E&M courses.
  • #1
Roadtripper
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Hi everyone,
there's something that I can't comprehend: when a homogeneous is in a conservative and non-uniform in module electric field polarization expression is given by P0χE. Supposing the most general situation there's: divPp where ρp is the polarization charge density in the dielectric. When my textbook introduces the electric induction vector D0E + P it says that divD=0 if in the volume (the dielectric material) you are going to integrate the divergence there are no free charges and it makes absolutely sense. Troubles have come in my mind when it states that this means divPp=0 too because "every electrostatic field makes the variation end up with 0". I mean what's the characteristic of electrostatic conservative non-uniform (in module) fields to make divP=0? I think I missed some logics here. I mean, why does it jump to the conclusion that polarization charges are only located on the surfaces of the dielectric material?
 
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  • #2
In general ## \nabla \cdot P=\rho_p ## is not equal to zero in a dielectric material when an electric field is applied. There are a couple of geometries where a uniform polarization ## P ## in the material results from an applied uniform external electric field , because the polarization charge forms in such a manner on the surface of the solid, that its electric field, ## E_p ##, (inside the material), added to the applied electric field ## E_o ##, results in a uniform ## E_i ## in the material and thereby a uniform ## P ##. A uniform ## P ## will have zero divergence,(and thereby ## \rho_p=0 ##), and surface polarization charge density is ## \sigma_p =P \cdot \hat{n} ##. ## \\ ##This self-consistent uniform ## P ## case happens in the case of a dielectric slab, where the electric field from the surface polarization charges ## E_p=-\frac{P}{\epsilon_o} ## , and also for a dielectric sphere, where ## E_p=-\frac{P}{3 \epsilon_o} ##. For most geometric shapes, in an applied electric field that is uniform, the resulting ## P ## is not uniform, and the resulting electric field, along with the resulting polarization will be quite complex.## \\ ## One other case where a simple self-consistent solution occurs is a cylinder turned sideways. For that case ## E_p=-\frac{P}{2 \epsilon_o} ##. These simple cases can be readily solved by writing ## E_i=E_o+E_p ##, and ## P=\epsilon_o \chi E_i ##. Since ## E_p=-\frac{D P}{\epsilon_o } ##, where ## D ## is the geometric factor for a particular geometry, it is a simple matter of solving two equations for the two unknowns: ## E_i ## and ## P ##. ## \\ ## For most cases, there is no constant ## D ##, like there is for the 3 cases mentioned above. (## D=1 ## for a slab, ## D=\frac{1}{3} ## for a sphere, and ## D=\frac{1}{2} ## for a cylinder that is turned sideways). ## \\ ## Additional note: Outside the material, ## E_p =0 ## for the dielectric slab. For the sphere and cylinder geometries, ## E_p ## outside the material has a somewhat complex form, and the solution for those cases is presented in advanced E&M courses.
 
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  • #3
Thank you so much!
 
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One typo/correction: The first line of post 2 should read ## -\nabla \cdot P=\rho_p ## with a minus sign in the equation.
 
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1. What is polarization charge density?

Polarization charge density is the measure of the amount of charge per unit volume that is induced in a dielectric material due to the presence of an external electric field. It is a result of the displacement of bound charges within the dielectric material.

2. How is polarization charge density calculated?

Polarization charge density can be calculated by dividing the total dipole moment per unit volume of the dielectric material by the volume of the material. This can be represented by the equation: ρ_p = P/V, where ρ_p is the polarization charge density, P is the total dipole moment, and V is the volume of the material.

3. What factors affect the polarization charge density of a homogeneous dielectric?

The polarization charge density of a homogeneous dielectric is affected by the magnitude of the external electric field, the properties of the dielectric material (such as its dielectric constant and polarizability), and the temperature of the material. It is also influenced by the molecular structure and orientation of the material's molecules.

4. How does the polarization charge density affect the behavior of a dielectric material?

The polarization charge density plays a crucial role in determining the behavior of a dielectric material in an electric field. It affects the dielectric constant, which is a measure of the material's ability to store electrical energy. A higher polarization charge density results in a higher dielectric constant, indicating a stronger ability to store charge and resist the electric field.

5. Can the polarization charge density be negative?

Yes, the polarization charge density can be negative. This occurs when the direction of the induced dipole moment is opposite to the direction of the external electric field. In this case, the polarization charge density contributes to the overall electric field in the opposite direction, resulting in a decrease in the net electric field within the material.

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