Microscopic explanation for the linear relationship between the Coulomb potential and the effective Coulomb potential The Coulomb potential a distance $r$ from a point charge $q$ is
$$
\phi = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}\,.
$$
Here $\varepsilon_0$ is the permittivity of vacuum. Now assume that there are dipolar molecules surrounding the point charge. These dipolar molecules are allowed to orient themselves freely at given temperature $T$. The polarization of the point charge leads to a shielding of the potential from the central ion. Then the effective potential of the Coulomb potential is
$$
\phi^\text{eff} = \frac{1}{4\pi\varepsilon_r\varepsilon_0}\frac{q}{r}\,.
$$
Here $\varepsilon_r$ is the relative permittivity. I know that there exists an empirical argument that this is valid for materials that are electrically linear. In other words, the second equation is valid for materials that satisfy $\mathbf{P}=\varepsilon_r \mathbf{E}$. Here $\mathbf{P}$ is the polarization and $\mathbf{E}$ is the electric field (https://web.mit.edu/6.013_book/www/chapter6/6.4.html). I am, however, not looking for an empirical explanation.
I am struggling with the microscopic intuition for why the effective potential differs from the Coulomb potential by a proportionality constant $1/\varepsilon_r$. Why is there a linear relationship between the two cases (microscopically)? The potential from a point-dipole scales according to $r^{-3}$. How can molecules with potentials that scale according to $r^{-3}$ lead to a linear decrease in the effective Coulomb potential?
I am looking for either an explanation or a good reference.
 A: $\mathbf{P}=\varepsilon_r\mathbf{E}$ is an approximation, which works rather well in many situations, but which is neither general, nor always applicable. In some important cases the media response might be non-local and/or non-linear. I fact, spatial and temporal non-locality is usually accounted for already in the basic linear response theories (such as Kubo formula, Kramers-Krönig relations, etc.) constant susceptibility and permittivity are a feature of the introductory electrodynamic courses... although, it is worth repeating, that in many real-life situations this approximation is sufficient.
I suggest this answer for a bit more detailed discussion (with equations).
Remarks:

*

*calling the effective potential equation microscopic is not quite a standard terminoligy, since screend potential is macroscopic - the polarization is averaged over a physically small volume. Indeed, the real electric field and potential vary greatly on microscopic distances, as we move from one molecule to another.

*You might be interested in reading about Debye screening, which converts Coulomb potential in Yukava-like shape, i.e., manifestly non-linear relationship between the two.

