Each material is made up of atoms. Each atom consists of a cloud of electrons surrounding a positively charged central nucleus.When no electric field is applied the electron cloud exhibits spherical symmetry as shown in the diagram.
In the presence of an electric field the charge cloud is distorted. The electrons – being negatively charged - are displaced in in a direction opposite to the direction of the applied electric field – as in the diagram below.
This can be reduced to a simple dipole by finding the distance
between the centres of positive and negative charge.
The effect is that the microscopic electric field is reduced by
where
is the charge on an electron andis the distance between the centres of the charges.. When we multiply this by the number of atoms per unit volume we find the charge in the rectangle drawn around each plate is reduced to
where
is the induced charge per unit area of the capacitor plate.
Applying Gauss's Divergence Theorem,
Put
to get