Electrostatics in continuous dielectric media

The description of electrostatic phenomena in condensed molecular environments rests on the observation that charges appear in two kinds. First, molecular electrons are confined to the molecular volume so that molecules move as neutral polarizable bodies. Second, free mobile charges (e.g. ions) may exist. In a continuum description the effect of the polarizable background is expressed by the dielectric response of such environments. 

Consider such an infinite environment (in real systems we assume that the effects of the system boundary can be disregarded). Divide it into small nonoverlapping 

To obtain the second equality we have integrated by parts using Eq. (1.30). According to (1.224) the electrostatic potential field is seen to arise from two charge densities the regular p(r) and an additional contribution associated with the dipole density pp(r ) = —Vr P(r). We will refer to p(r) as the external charge density. This reflects a picture of a dielectric solvent with added ionic charges. 

Equation (1.224) together with (1.214) imply that the Poisson equation (1.216) now takes the fonn 

We disregard the fact that there is at least one volume, that surrounding r. for which this assumption cannot be made. 

---