Bound volume charge distribution in dielectrics Problem
Calculate the potential produced by a distribution of charge  at a point $r$ in a dielectric material. 
The expression of $V$ at a point outside the charge distribution (a molecule), but inside the dielectric material is $$V(r)=V_{\rho}(r) + V_{\sigma}(r)$$ where  $\rho$ is the distribution charge (of a solute). I do not understand why there is no potential produced for the bond volume charge distribution $\rho_b$.
How do we know if there is a $\rho_b$ in a dielectric (due to an external electric field)? 
I have read that sometimes that $\rho_b$ is zero in some dielectrics but do not know how to prove it.
Any help?
 A: The definition of the polarization density, $\mathbf{P}$, is the volume density of dipole moments. The definition of the displacement field, $\mathbf{D}$, is:
$$\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}.$$
Take the divergence of both sides and you get the relationship among free charge, total charge, and bound charge densities:
$$\rho_{\mathrm{free}} = \rho_{\mathrm{net}} + \nabla\cdot \mathbf{P}.$$
As you can see, the bound charge density, defined as $\rho_{\mathrm{net}} - \rho_{\mathrm{free}}$, is given by $\rho_{\mathrm{bound}} = -\nabla\cdot \mathbf{P}$.
Also, note that just like you can have surface charge densities from discontinuous changes in the electric field, you can have surface bound charge densities. If you recall, the change in the component of the electric field perpendicular to a surface charge is given by $\Delta \mathbf{E}_\perp = \frac{\sigma_{\mathrm{net}}}{\epsilon_0}$, thus you also have 
\begin{align}
\Delta \mathbf{D}_\perp &= \sigma_{\mathrm{free}} \\
\Delta \mathbf{P}_\perp &= -\sigma_{\mathrm{bound}}.
\end{align}
Also, keep in mind that $\mathbf{D}$ and $\mathbf{P}$ don't have vanishing curls, the way $\mathbf{E}$ does. So, if we take the curl of the definition equation we get:
$$\nabla \times \mathbf{D} = \nabla \times \mathbf{P}.$$
This means that the component of $\mathbf{E}$ that is parallel to a surface ($\mathbf{E}_{||}$) cannot change discontinuously, but if $\mathbf{P}_{||}$ changes discontinuously across a surface, then so must $\mathbf{D}_{||}$:
$$\Delta \mathbf{D}_{||} = \Delta \mathbf{P}_{||}.$$
An example of a problem where this last property is important is if you imagine placing a large flat dielectric sheet at an angle with respect to an externally applied electric field and you want to find the resulting $\mathbf{E}$, $\mathbf{D}$, and $\mathbf{P}$.
