Gauss's Law in Dielectrics

Consider a charged parallel plate capacitor separated by a vacuum. Now, applying Gauss's law to a Gaussian surface, the electric flux is proportional to the charge enclosed by the surface.

When the vacuum is replaced with a dielectric, free charges develop on the surface of the capacitor plates, and bound charges are induced on the dielectric surface, decreasing the net electric field.

So, the magnitude of the electric flux through the Gaussian surface inside this capacitor with the dielectric is proportional to the net charge inside the surface.

The dielectric polarization causes a decrease in the electric field by a factor of the dielectric constant; so, the net charge equals the ratio of free charge to the dielectric constant.

This net charge, when substituted into Gauss's law yields Gauss's law in dielectrics, where the electric field is replaced with the product of the dielectric constant and electric field.

The product of the vacuum permittivity, dielectric constant, and electric field is called the electric displacement. So, Gauss's law can be rewritten in terms of electric displacement.