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.

.

(SI units),                          (Gaussian units).

For linear, isotropic, homogeneous (lih) dielectrics we have

,                                            ,

with c_e constant.

,             ,

,              .

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 e₁.  Then placing an image charge  at  z' = -d on the z-axis in a medium with e₁ gives the potential and field in dielectric 1.  And placing an image charge at z’’ = d  on the z-axis  in a medium with e₂ gives the potential and field in dielectric 2.

Energy in Dielectrics

The electrostatic energy stored in a charge distribution is given by 

in SI units.

In a linear isotopic, homogeneous material the total work done in assembling the free charges into a charge distribution is

    or    .

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.