# Problem with understanding polarization

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?

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?