Interface conditions for electromagnetic fields

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Maxwell's equations describe the behavior of electromagnetic fields; electric field, electric displacement field, magnetic field and magnetic field strength. 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 interface of two different medium with different values for electrical permittivity and magnetic permeability that doesn't 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[edit]

For electric field[edit]

\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[edit]

(\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.

Therefore the normal component of D has a step of surface charge on the interface surface. If there's no surface charge on the interface, the normal component of D is continuous.

Interface conditions for magnetic field vectors[edit]

For magnetic field[edit]

(\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[edit]

\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.

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

See also[edit]

References[edit]

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