Electrostatic Boundary Conditions in Dielectrics

When an electric field crosses the boundary between two different dielectric materials, the tangential component of the electric field becomes discontinuous due to differences in the materials' permittivity. However, the tangential component of electric displacement remains continuous across the interface. This discontinuity arises because the electric field and electric displacement are related through each material's permittivity, which varies between dielectrics.

At a dielectric-dielectric interface, the normal component of electric displacement is continuous if no free charges exist at the boundary. This is determined by applying Gauss's law in terms of electric displacement across a Gaussian pillbox at the interface. In contrast, the normal component of the electric field itself is discontinuous at the same boundary due to differences in material permittivity.

At a conductor-dielectric interface, the electric field inside the perfect conductor is zero. Applying this condition to the dielectric-dielectric boundary conditions yields the conductor-dielectric boundary conditions. The tangential component of the electric field at the conductor surface is zero, while the normal component of electric displacement equals the surface free charge density on the conductor.

When a perfect conductor meets free space instead of a dielectric, the material's permittivity equals the permittivity of free space, which has a dielectric constant of 1. Substituting this into the conductor-dielectric boundary conditions gives the conductor-free space interface conditions. The electric field inside the conductor remains zero, and the normal component of electric displacement at the surface equals the surface charge density.

The tangential component of electric displacement is continuous across a dielectric-dielectric interface because it is not affected by the boundary conditions derived from Gauss's law. Unlike the electric field, which depends on permittivity and becomes discontinuous, the electric displacement relates directly to free charges. Since no free charges typically exist at the interface itself, the tangential component of displacement remains continuous.

A Gaussian pillbox positioned at the interface allows application of Gauss's law to determine how the normal components of electric field and displacement change across the boundary. By rewriting Gauss's law in terms of electric displacement, the pillbox method reveals that the change in normal displacement components relates to the surface free charge density. This mathematical approach is essential for deriving all electrostatic boundary conditions at material interfaces.

Permittivity determines the relationship between electric field and electric displacement in each material. Since the two materials have different permittivity values, the electric field must be discontinuous to maintain continuity of the tangential displacement component. The greater the difference in permittivity between the two dielectrics, the more pronounced the discontinuity in the electric field at their shared boundary.