Dielectrics

Polarization

The polarization P = dp/dV is defined as the dipole moment per unit volume.
The total charge density is due to free and to bound (polarization) charges.
ρ = ρf + ρp,   σ = σf + σp,   ρp = P σp = Pn.
Definition: D = ε0E + P.

For linear, isotropic, homogeneous (lih) dielectrics we have
P =  ε0χeE, with χe constant.
D = ε0(1 + χe)E = ε0κeE = εE.
D = ρf,  ∇2Φ = -ρf/ε.


The method of images

Assume the z = 0 plane is a plane interface between two dielectrics
Consider a charge q at z = d on the z-axis in ε1.  Then placing an image charge q' = -q(ε2 - ε1)/(ε2 + ε1) at z' = -d on the z-axis in a medium with ε1 gives the potential and field in dielectric 1.  And placing an image q' = q(2ε2)/(ε2 + ε1) charge at z''= d  on the z-axis in a medium with ε2 gives the potential and field in dielectric 2.


Energy in Dielectrics

The electrostatic energy stored in a charge distribution is given by 
U = (ε0/2) ∫all space EE dV.
In the presence of a dielectric, the total work done in assembling the free charges into a charge distribution is
W = ½ ∫all space ED dV,  
which becomes
W = (ε/2) ∫all space EE dV.
in a linear, isotopic, homogeneous material .

W > U.  As you do work on the free charges against electrostatic forces, the electric field does work on the bound charges against non-electrostatic forces, thus lowering the total electrostatic potential energy stored in the system.  Some of the external work is stored as non-electrostatic potential energy.