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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 (electromagnetism)|permeability]] that doesn't apply.
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| However the interface conditions for the electromagnetic field vectors can be derived from the integral forms of Maxwell's equations.
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| ==Interface conditions for electric field vectors==
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| === For electric field ===
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| :<math>\mathbf{n}_{12} \times (\mathbf{E}_2 - \mathbf{E}_1) = \mathbf{0} </math>
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| where: <br />
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| <math>\mathbf{n}_{12}</math> is [[normal vector]] from medium 1 to medium 2.
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| Therefore the [[tangential component]] of '''E''' is continuous across the interface.
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| === For electric displacement field ===
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| :<math>(\mathbf{D}_2 - \mathbf{D}_1) \cdot \mathbf{n}_{12} = \rho_{s} </math> | |
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| where: <br />
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| <math>\mathbf{n}_{12}</math> is [[normal vector]] from medium 1 to medium 2.<br />
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| <math>\rho_{s}</math> is the [[surface charge]] between the media.
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| 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.
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| ==Interface conditions for magnetic field vectors==
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| === For magnetic field ===
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| :<math>(\mathbf{B}_2 - \mathbf{B}_1) \cdot \mathbf{n}_{12} = 0 </math>
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| where: <br />
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| <math>\mathbf{n}_{12}</math> is [[normal vector]] from medium 1 to medium 2.
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| Therefore the normal component of '''B''' is continuous across the interface.
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| === For magnetic field strength ===
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| :<math>\mathbf{n}_{12} \times (\mathbf{H}_2 - \mathbf{H}_1) = \mathbf{j}_s </math>
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| where: <br />
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| <math>\mathbf{n}_{12}</math> is [[normal vector]] from medium 1 to medium 2.<br />
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| <math>\mathbf{j}_s</math> is the surface [[current density]] between the two media.
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| Therefore the [[tangential component]] of '''H''' is continuous across the surface if there's no surface current present.
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| == See also ==
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| * [[Maxwell's equations]]
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| == References ==
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| *{{cite book | author=John R. Reitz,Frederick J. Milford, Robert W. Christy|title=Foundations of Electromagnetic theory (4th ed.)| publisher=Addison-Wesley |year=1993 |isbn=0-201-52624-7}}
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| [[Category:Electromagnetism]] | |
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| {{Electromagnetism-stub}}
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