1. The Displacement field
It is useful to separate the charge density into the bound charge rb and free charge rf:
The displacement field:
Note that that the electric field is irrotational, so E can be written as a gradient of a scalar. However, in general
The vector D obeys different boundary conditions than E.
field term
matter term

2. polarization vector points in the same direction as E
A common strategy: first find D for the specified free charge. Then determine P and E.
Example: A long straight wire, carrying uniform line charge l, is surrounded by a rubber insulation out to a radius a. Find the electric displacement.
We note that this expression holds everywhere: inside and outside the insulation. Outside, P=0; hence,
What about electric field inside the insulation?
How the dielectrics respond to an applied field? The answer to this question can be in principle obtained by quantum-mechanical calculations of electron-molecular structures. In most cases, however, this is not practical. Instead, we will introduce an empirical connection between P and E (or D and E), called the constitutive equation.
polarization vector points in the same direction as E P

3. Isotropic Linear Dielectrics: Electric field proportional to polarization vector (dilute gasses, glass, plastics…). Remember: crystals are different!
permittivity
susceptibility
dielectric constant
Note that for dielectrics
dielectric constant

4. Boundary conditions above below
the normal component of the D-field is discontinuous across any interface:
the tangential component of E is continuous across any interface

5. If both dielectrics are isotropic:1 2