Surface charge density from volume density in Feynman's treatment of dielectrics In the second volume of the Feynman Lectures, chapter Dielectrics, section $10–3$ Polarization charges, which can be found here, Feynman says about the dielectric slab below 


If $A$ is the area of the plate, the number of electrons that appear at the surface is the product of $A$ and $N$, the number (of atoms) per unit volume, and the displacement $\delta$, which we assume here is perpendicular to the surface. The total charge is obtained by multiplying by the electronic charge $q_e$. To get the surface density of the polarization charge induced on the surface, we divide by $A$. The magnitude of the surface charge density is
  $$\sigma_{pol}=Nq_{e}\delta.$$

What is the argument behind his saying the surface charge density can be deduced from the volume charge density? It doesn't seem obvious at all that by dividing the charge volume density $\rho=ANq_{e}\delta$ we will get the surface charge density $\sigma$.
 A: All atoms within distance $\delta$ of the top dotted line will have the positive ends (charge $q$) of their equivalent dipoles displaced over the dotted line, that is on to the surface. So the charge going on to the surface will indeed be $A\delta N q$.
This is analogous to all the $NAvt$ free electrons within distance $vt$ of a certain cross-section of a wire passing through that cross-section in time $t$. Here, $v$ is the drift velocity of the electrons and $N$ is the number of free electrons per unit volume.
Having cheekily attempted to supply the missing step in Feynmann's argument, I think that we mustn't now model the dielectric as planes of atoms parallel to the dielectric's faces. If we do, then, for a particular value of d, all the negative ends of the atomic dipoles in the surface plane of atoms will go over the dotted line at once. So the surface charge will be a step function of the external field producing the polarisation, rather than a smooth function.
Feynman's treatment seems to work very well, though, if we assume that the atoms are randomly distributed in space. Maybe modelling the dielectric as planes of atoms parallel to the dielectric faces shouldn't even have been considered, as there's no such thing as a microscopically plane surface...
Despite these worries I prefer Feynman's 'direct' treatment to the standard textbook one (see, for example, Purcell), which is to represent the atomic dipoles in a volume dV containing a large number of atoms by a macroscopic vector, PdV, and then to derive the surface charges and so on, using P. Feynman does, of course, introduce and wield the vector P. 
