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 = ρ_f + ρ_p,   σ = σ_f + σ_p,   ρ_p = -∇·P,   σ_p = P·n.
Definition: D = ε₀E + P  ---> ∇·D = ρ_f (Gauss' law for D).
 

For linear, isotropic, homogeneous (lih) dielectrics we have
P =  ε₀χ_eE, with χ_e constant.
D = ε₀(1 + χ_e)E = ε₀κ_eE = εE.
∇²Φ = -ρ_f/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 replacing q with 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} E·E 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} E·D dV,  
which becomes
W = (ε/2)∫_{all space} E·E 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.