# Surface bound charge density in dielectric displacement

In Griffiths, it derived that the electric potential caused by Polarization $\bf {P}$ is written

$$V({\bf r}) = \frac{1}{4\pi \epsilon_0} \int_{\mathcal{V}} \frac{\hat{\gamma} \cdot {\bf P}{({\bf r'})}}{\gamma ^2}d\tau'$$ here $\bf{\gamma = r - r'}$, when $\bf r$ is field point and $\bf r'$ is source point.

1. Is it jus approximation? I think to get the electric potential more precisely we have to consider not only dipole but also quadrapole etc, but there is no mention about considering quadrapole.

The textbook also introduce the divergence of dielectric displacement is free charge density. However it doesn't think about surface bound charge.

$$\epsilon_0 \nabla \cdot {\bf E} = \rho = \rho_{\textrm{free}} + \rho_{\textrm{bound}}$$

I think we should add surface bound charge $\sigma_b$ in the above equation. The Griffiths wrote why he did not add it but I don't understand the logic of Griffiths.

1. Why should we omit surface bound charge $\sigma_b$?
• For 2 : The bound charge must be included in the divergence of E. However, they are included in the displacement electric field D for which the divergence is only affected by free charges. Feb 2 '16 at 18:47
• For 1 : I think that the higher order electric models such as quadrupole have to be included but I think the polarization may also take them into account. Nevertheless, the dipole approximation is quite a reasonable one usually. Feb 2 '16 at 18:50

As for the point about the surface charge, Griffiths only derives the expression for a gaussian surface in the meat of the dielectric volume, where $$\rho = \rho_{f} +\rho_{b}$$ is obviously true (the surface straddles the volume and does not lie within any gaussian surface inside). Then the steps follow logically.
The other case, where you "would have to" consider the surface charge density, is where the gaussian surface encloses it -- but this is of course outside the dielectric, and $$\mathbf{P}$$ drops quickly to zero, such that $$- \nabla \cdot \mathbf{P}$$ evaluates to zero. So we are evaluating this surface charge after all -- it is just hidden in the $$\mathbf{P}$$. It appears when it needs to, outside the volume, but does not "exist" otherwise (as you would expect from a heaviside function).