Interface conditions for electromagnetic fields

Maxwell's equations describe the behaviour of electromagnetic fields; electric field, electric displacement field, magnetic field and magnetic field strength. 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 E, D, B and H are not differentiable. In other words, the medium must be continuous. On the interface of two different medium with different values for electrical permittivity and magnetic permeability that 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

For electric field

$\mathbf{n}_{12} \times (\mathbf{E}_2 - \mathbf{E}_1) = \mathbf{0}$

where:
$\mathbf{n}_{12}$ is normal vector from medium 1 to medium 2.

Therefore the tangential component of E is continuous across the interface.

For electric displacement field

$(\mathbf{D}_2 - \mathbf{D}_1) \cdot \mathbf{n}_{12} = \rho_{s}$

where:
$\mathbf{n}_{12}$ is normal vector from medium 1 to medium 2.
$\rho_{s}$ is the surface charge between the media (unbounded charges only, not coming from polarization of the materials).

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 field

$(\mathbf{B}_2 - \mathbf{B}_1) \cdot \mathbf{n}_{12} = 0$

where:
$\mathbf{n}_{12}$ is normal vector from medium 1 to medium 2.

Therefore the normal component of B is continuous across the interface.

For magnetic field strength

$\mathbf{n}_{12} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{j}_s$

where:
$\mathbf{n}_{12}$ is normal vector from medium 1 to medium 2.
$\mathbf{j}_s$ is the surface current density between the two media (unbounded current only, not coming from polarisation of the materials).

Therefore the tangential component of H is continuous across the surface if there's no surface current present.

See also

• Maxwell's equations

References

• John R. Reitz,Frederick J. Milford, Robert W. Christy (1993). Foundations of Electromagnetic theory (4th ed.). Addison-Wesley. ISBN 0-201-52624-7.