My guess is that your formula is applicable to non-uniform (non homogeneous and/or anisotropic) dielectric materials.

If the material is homogeneous and isotropic I do not see any way a volumetric polarization charge could develop, so your difficulty in understanding why is - from my point of view - justified.

If the material is not homogeneous, you can see how charge can develop by considering it composed of multiple portions of homogeneous materials. Basically, the surface charge at each interface is what constitutes the volume charge in the chunk of material as a whole.

For anisotropic materials, I pass. :-)

Anyway, here is some comfort coming from the Physics Stack Exchange: Density of polarisation charge is zero always for linear isotropic homogeneous dielectrics?

It seems to me that the volumetric density of polarisation charge in a linear homogeneous isotropic dielectric in an external field is always zero. I find this surprising

Why surprising? A uniform external field can't produce charge in the bulk of any neutral, isolated material, whether conducting or dielectric. All the charge will be at the surface. In fact, this is even more true for a dielectric. All the positive and negative charges are tightly bound. The field can displace them slightly into dipoles, but at the macroscopic level there is still no net charge in the volume.

They also give a reference:

[...] certainly [local charge densities] can't [arise] for an isotropic, uniform material. This is given in Jackson (compare 4.39 to 4.33). I do not think the linearity condition is necessary. It would be interested to know if isotropy is.

My guess is that your formula is applicable to non-uniform (non homogeneous and/or anisotropic) dielectric materials.

If the material is homogeneous and isotropic I do not see any way a volumetric polarization charge could develop, so your difficulty in understanding why is - from my point of view - justified.

If the material is not homogeneous, you can see how charge can develop by considering it composed of multiple portions of homogeneous materials. Basically, the surface charge at each interface is what constitutes the volume charge in the chunk of material as a whole.

For anisotropic materials, I pass :-) but you might want to have a look at Purcell's "Electricity and Magnetism", 3rd edition, section 10.11 "The field of a charge in a dielectric medium and Gauss's Law". Figure 10.28 at p. 498 gives you a hint on how a single free charge placed inside an otherwise isotropic dielectric gives rise to a polarization with positive divergence.

Anyway, here is some comfort coming from the Physics Stack Exchange: Density of polarization charge is zero always for linear isotropic homogeneous dielectrics?

It seems to me that the volumetric density of polarisation charge in a linear homogeneous isotropic dielectric in an external field is always zero. I find this surprising

Why surprising? A uniform external field can't produce charge in the bulk of any neutral, isolated material, whether conducting or dielectric. All the charge will be at the surface. In fact, this is even more true for a dielectric. All the positive and negative charges are tightly bound. The field can displace them slightly into dipoles, but at the macroscopic level there is still no net charge in the volume.

They also give a reference:

[...] certainly [local charge densities] can't [arise] for an isotropic, uniform material. This is given in Jackson (compare 4.39 to 4.33). I do not think the linearity condition is necessary. It would be interested to know if isotropy is.

My guess is that your formula is applicable to non-uniform (non homogeneous and/or anisotropic) dielectric materials.

If the material is homogeneous and isotropic I do not see any way a volumetric polarization charge could develop, so your difficulty in understanding why is - from my point of view - justified.

If the material is not homogeneous, you can see how charge can develop by considering it composed of multiple portions of homogeneous materials. Basically, the surface charge at each interface is what constitutes the volume charge in the chunk of material as a whole.

For anisotropic materials, I pass. :-)

Anyway, here is some comfort coming from the Physics Stack Exchange: Density of polarisation charge is zero always for linear isotropic homogeneous dielectrics?

It seems to me that the volumetric density of polarisation charge in a linear homogeneous isotropic dielectric in an external field is always zero. I find this surprising

Why surprising? A uniform external field can't produce charge in the bulk of any neutral, isolated material, whether conducting or dielectric. All the charge will be at the surface. In fact, this is even more true for a dielectric. All the positive and negative charges are tightly bound. The field can displace them slightly into dipoles, but at the macroscopic level there is still no net charge in the volume.

They also give a reference:

I see, but certainly they can't for an isotropic, uniform material. This is given in Jackson (compare 4.39 to 4.33). I do not think the linearity condition is necessary. It would be interested to know if isotropy is.

My guess is that your formula is applicable to non-uniform (non homogeneous and/or anisotropic) dielectric materials.

If the material is homogeneous and isotropic I do not see any way a volumetric polarization charge could develop, so your difficulty in understanding why is - from my point of view - justified.

If the material is not homogeneous, you can see how charge can develop by considering it composed of multiple portions of homogeneous materials. Basically, the surface charge at each interface is what constitutes the volume charge in the chunk of material as a whole.

For anisotropic materials, I pass. :-)

Anyway, here is some comfort coming from the Physics Stack Exchange: Density of polarisation charge is zero always for linear isotropic homogeneous dielectrics?

It seems to me that the volumetric density of polarisation charge in a linear homogeneous isotropic dielectric in an external field is always zero. I find this surprising

 

Why surprising? A uniform external field can't produce charge in the bulk of any neutral, isolated material, whether conducting or dielectric. All the charge will be at the surface. In fact, this is even more true for a dielectric. All the positive and negative charges are tightly bound. The field can displace them slightly into dipoles, but at the macroscopic level there is still no net charge in the volume.

They also give a reference:

[...] certainly [local charge densities] can't [arise] for an isotropic, uniform material. This is given in Jackson (compare 4.39 to 4.33). I do not think the linearity condition is necessary. It would be interested to know if isotropy is.