# Dielectric material problem

I read about dielectric just 2 days ago and come across something about what polarization means: how a neutral object can be created to be a dielectric under external electric field, etc.

Then I read about the electric field in a dielectric material and there I found two terms: one is surface polarization charge density ($\sigma$) and the other is volume polarization charge density ($\rho$), where, $\sigma=\mathbf{p}\cdot\mathbf{n}$ and $\rho=-\nabla\cdot\mathbf{p}$, with $\mathbf{p}$ being the polarization vector.

However, I don't understand why they are called surface and volume charge densities, respectively? How does $\nabla\cdot\mathbf{p}$ give a volume charge density and why does a volume charge exist when $\nabla\cdot\mathbf{p}\neq 0$? What is the relationship between the volume charge density and the fact that the polarization vector is a diverging vector?

I think what you are calling ${\bf p}$ is the polarisation field, which is the electric dipole moment per unit volume, with units of charge times length. If you take the divergence of this, you calculate the flux of this quantity, which is a dipole moment multiplied by a closed area divided by a volume, per unit volume. This yields a charge per unit volume.
Whether you have a volume polarisation charge of course depends on the form of ${\bf p}$. If it is uniform, then clearly there is no net volume charge density. The dipole charges are separated, but there is no net charge density anywhere in the volume. A volume charge density will only arise through discontinuities in ${\bf p}$ (perhaps through discontinuities in $\epsilon$). Lines of ${\bf p}$ must begin or end on polarisation charges, but if the divergence is zero then there is no net polarisation charge and there are just as many field lines beginning as there are ending in any considered small volume.
Surface polarisation charge density arises because at some point you reach the surface of the medium and you "expose" one end of all the polarisation dipoles at the surface. If you design a Gaussian surface that cuts through a set of dipoles just inside the surface of the medium, then the total charge enclosed is $\oint {\bf p}\cdot {\bf n}\ dS = \oint \sigma\ dS$, where ${\bf n}$ is a normal unit vector to the surface.