Interface conditions for electromagnetic fields


Interface conditions describe the behaviour of electromagnetic fields; electric field, electric displacement field, and the magnetic field at the interface of two materials. The differential forms of these equations require that there is always an open neighbourhood around the point to which they are applied, otherwise the vector fields and H are not differentiable. In other words, the medium must be continuous. On the interface of two different media with different values for electrical permittivity and magnetic permeability, that condition does not apply.
However, the interface conditions for the electromagnetic field vectors can be derived from the integral forms of Maxwell's equations.

Interface conditions for electric field vectors

Electric field strength

where:

is normal vector from medium 1 to medium 2.
Therefore, the tangential component of E is continuous across the interface.
Two of our sides are infinitesimally small, leaving only
After dividing by l, and rearranging,
This argument works for any tangential direction. The difference in electric field dotted into any tangential vector is zero, meaning only the components of parallel to the normal vector can change between mediums. Thus, the difference in electric field vector is parallel to the normal vector. Two parallel vectors always have a cross product of zero.

Electric displacement field

is the unit normal vector from medium 1 to medium 2.

is the surface charge density between the media.
This can be deduced by using Gauss's law and similar reasoning as above.
Therefore, the normal component of D has a step of surface charge on the interface surface. If there is no surface charge on the interface, the normal component of D is continuous.

Interface conditions for magnetic field vectors

For magnetic flux density

where:

is normal vector from medium 1 to medium 2.
Therefore, the normal component of B is continuous across the interface.

For magnetic field strength

where:

is the unit normal vector from medium 1 to medium 2.

is the surface current density between the two media.
Therefore, the tangential component of H is discontinuous across the interface by an amount equal to the magnitude of the surface current density. The normal components of H in the two media are in the ratio of the permeabilities.

Discussion according to the media beside the interface

If medium 1 & 2 are perfect [dielectrics]

There are no charges nor surface currents at the interface, and so the tangential component of H and the normal component of D are both continuous.

If medium 1 is a perfect [dielectric] and medium 2 is a perfect [metal]

There are charges and surface currents at the interface, and so the tangential component of H and the normal component of D are not continuous.

Boundary conditions

The boundary conditions must not be confused with the interface conditions. For numerical calculations, the space where the calculation of the electromagnetic field is achieved must be restricted to some boundaries. This is done by assuming conditions at the boundaries which are physically correct and numerically solvable in finite time. In some cases, the boundary conditions resume to a simple interface condition. The most usual and simple example is a fully reflecting boundary - the outer medium is considered as a perfect conductor. In some cases, it is more complicated: for example, the reflection-less boundaries are simulated as perfectly matched layer or magnetic wall that do not resume to a single interface.