Problem with understanding polarization

[neworder1]

Suppose we have a ball made of linear dielectric with permittivity $$\epsilon$$, with some initial homogenous polarization $$\vec{P}$$ aligned with z axis. Then we know that inside the ball the polarization generates an electric field $$\vec{E}=\frac{-1}{3\epsilon_{0}}\vec{P}$$ (standard calculation). But we also know that in a linear dielectric we have the relation $$\vec{P}=(\epsilon - \epsilon_{0})\vec{E}$$, and these two equations lead to contradiction since we have $$\vec{E}=\frac{-1}{3\epsilon_{0}}\vec{P} = \frac{-1}{3\epsilon_{0}}(\epsilon - \epsilon_{0})\vec{E}$$. Does it mean that a linear dielectric can't be polarized this way without an external field?

 

[Euclid]

Suppose we have a ball made of linear dielectric with permittivity $$\epsilon$$, with some initial homogenous polarization $$\vec{P}$$ aligned with z axis. Then we know that inside the ball the polarization generates an electric field $$\vec{E}=\frac{-1}{3\epsilon_{0}}\vec{P}$$ (standard calculation). But we also know that in a linear dielectric we have the relation $$\vec{P}=(\epsilon - \epsilon_{0})\vec{E}$$, and these two equations lead to contradiction since we have $$\vec{E}=\frac{-1}{3\epsilon_{0}}\vec{P} = \frac{-1}{3\epsilon_{0}}(\epsilon - \epsilon_{0})\vec{E}$$. Does it mean that a linear dielectric can't be polarized this way without an external field?

Yeah, of course! If there is no field, and the object is polarized... it's not a linear material.

 

[astrokreng]

Thank you for bringing this issue to my attention. The contradiction you have pointed out is indeed a valid concern, and it highlights a common misconception about polarization in dielectric materials.

Firstly, it is important to clarify that the equation \vec{P}=(\epsilon - \epsilon_{0})\vec{E} is only valid in the absence of any external electric field. In other words, this equation describes the intrinsic polarization of a material due to its own electric field, not the polarization induced by an external field.

In the scenario you have described, the ball is initially polarized in the z direction, which means there is already an electric field present within the material. This electric field will influence the polarization of the material, and it is this interaction that leads to the contradiction you have pointed out.

To better understand this, let's consider the case where there is no initial polarization in the ball. In this scenario, the equation \vec{P}=(\epsilon - \epsilon_{0})\vec{E} is valid, and we can see that the induced polarization is directly proportional to the external electric field. However, when there is an initial polarization present, this equation becomes more complicated as the electric field inside the material is no longer solely determined by the external field, but also by the intrinsic polarization of the material.

In short, the contradiction you have pointed out does not mean that a linear dielectric cannot be polarized without an external field. It simply means that the relationship between polarization and electric field in a dielectric material is more complex than what is commonly understood. Further research and experimentation are needed to fully understand this phenomenon.