Ambiguity of electrostatic polarization?

1. Jun 3, 2012

Jano L.

Hello everybody,

https://en.wikipedia.org/wiki/Polarization_density

especially the section in the end where the writer claims the polarization is ambiguous.

In the example about Alice, the writer states that the pairing of +/- particles is ambiguous and hence the polarization is ambiguous. I think he incorrectly interprets the meaning of the polarization.

The writer even states that Alice can back up her strange pairing procedure by ascribing the crystal surface a non-zero density of (free!) charge. This is ridiculous. The crystal is a dielectric and there is no free charge. All polarization comes from displacements of the bound charges. There will be only bound surface charge.

I think the proper way to define the polarization of the crystal at $\mathbf x$ is to average the dipole moments of the smallest neutral cells k hitting the averaging volume V, which is centred at $\mathbf x$. The polarization is then

$$\mathbf P(\mathbf x) = \frac{1}{V} \sum_k \mathbf \mu_k$$

What do you think - is not this unambiguous definition?