Here is a question sent to me recently:
I will look at part (b). Let and be the relative electric permittivity values. The field continuity equation is (where the electric fields are perpendicular to the interface boundary):
Relate the voltage across the capacitor to the electric fields:
Consider the Gaussian surface shown to the right within the dielectric. The formula for bound charge at a dielectric interface is similar to the formula for Gauss's Law, but with a slight variation. Surfaces and are the top and bottom Gaussian sub-surfaces.
Note that the normal vector for points in the opposite direction as the electric field, and the normal vector for points in the same direction as the electric field. Let these surfaces both have area . The equation simplifies to:
Applying the field continuity relationship makes the relative permittivity terms disappear.
Dividing both sides by gives the bound charge density: