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 = P·n.
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.
Assume the z = 0 plane is a plane interface between
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.
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,
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.