Electrostatic Boundary Conditions in Dielectrics

The electrostatic boundary condition gives the discontinuity of an electric field at a surface boundary. Consider the case where both the media across the boundary are dielectric materials.

Rewriting the electric field in terms of electric displacement shows that its tangential component is discontinuous across the interface.

For the normal component, consider a Gaussian pillbox on the interface. Rewriting Gauss's law in terms of electric displacement, the change in normal components at the interface gives the surface free charge density.

If no free charges exist at the boundary, the normal components are continuous across the interface. However, the electric field for the same is discontinuous.

Consider a perfect conductor in place of one of the dielectrics. The electric field inside a perfect conductor is zero. Substituting this in the dielectric-dielectric boundary conditions gives the boundary conditions at a conductor-dielectric interface.

If the other dielectric is removed, the material's permittivity equals the permittivity of free space. This gives the boundary conditions at a conductor-free space interface.