Interface conditions for electromagnetic fields

Maxwell's equations describe the behavior of electric field, magnetic field, electric flux density and magnetic flux density. The differential forms of these equations require that there's always an open neighbourhood around the point they're applied to, otherwise the vector fields E, D, B and H are not differentiable. In other words the medium must be continuous. On the boundary of two different medium with different values for electrical permittivity and magnetic permeability that doesn't apply.

However the boundary conditions for the elecromagnetic field vectors can be derived from the integral forms of Maxwell's equations.

Boundary conditions for electric field vectors

For electric flux density

$(\mathbf {D} _{2}-\mathbf {D} _{1})\cdot \mathbf {n} _{12}=\rho _{s}$

where:
$\mathbf {n} _{12}$ is normal vector from medium 1 to medium 2.
$\rho _{s}$ is the surface charge between the media.

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.

Boundary conditions for magnetic field vectors

For magnetic flux density

$(\mathbf {B} _{2}-\mathbf {B} _{1})\cdot \mathbf {n} _{12}=0$

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

For magnetic field

$\mathbf {n} _{12}\times (\mathbf {H} _{2}-\mathbf {H} _{1})=\mathbf {j} _{s}$

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.

See also

• Maxwell's equations

References

• John R. Reitz,Frederick J. Milford, Robert W. Christy (1993). Foundations of Electromagnetic theory (4rd ed.). Addison-Wesley. ISBN 0-201-52624-7.{{cite book}}: CS1 maint: multiple names: authors list (link)