Name: current density at conductivity boundaries
Uploaded: 2023-09-18T17:11:18.000Z
Duration: 1 min 25 s
Channel: JoVE

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Physical Science

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Motion and Stability- Forces and Interactions

PS.PSc.6.3

Boundary Conditions for Current Density

Die Stromdichte wird über eine Grenzfläche mit unterschiedlichen elektrischen Leitfähigkeiten diskontinuierlich.
Bei stationären Strömen ist die Divergenz der Stromdichte Null. Die normale Komponente der Stromdichte ist also über die Grenze hinweg kontinuierlich.
Erinnern Sie sich daran, dass die tangentiale Komponente des elektrischen Feldes über eine Grenzfläche kontinuierlich ist. Drückt man das elektrische Feld in Bezug auf die Stromdichte und die elektrische Leitfähigkeit aus, erhält man die Randbedingung für die tangentiale Komponente der Stromdichte.
Nun ist die normale Komponente der elektrischen Verschiebung über die Grenzfläche diskontinuierlich. Die normale Komponente des elektrischen Feldes ist also auch über eine Grenzfläche diskontinuierlich.
Auch hier wird unter Verwendung des Ausdrucks des elektrischen Feldes und der Stromdichte die Randbedingung für die Normalkomponente der Stromdichte in Bezug auf Permittivität und Leitfähigkeit erhalten.
Es zeigt, dass eine Oberflächenladungsdichte über einer Grenzfläche mit unterschiedlichen Leitfähigkeiten und/oder Permittivitäten erzeugt wird.
Die Oberflächenladungsdichte ist Null für eine Grenzfläche mit gleichen Leitfähigkeits- und Permittivitätswerten oder mit einem gleichen Verhältnis von Permittivität zu Leitfähigkeit.

<ol>
	<li>Ida, Nathan, Engineering Electromagnetics, 3. Auflage, Springer. S. 355-356</li>
	<li>Cheng, David K., Feld- und Wellenelektromagnetik, Addison-Wesley Publishing Company, S. 183.</li>
</ol>

randbedingungen für die stromdichte

Die Stromdichte wird über eine Schnittstelle von Materialien mit unterschiedlicher elektrischer Leitfähigkeit unstetig. Die normale Komponente der Stromdichte ist über die Grenze hinweg kontinuierlich.

<img alt="Equation1" src="/files/ftp_upload/14187/14187_Equation_1.png" style="text-align: center; height: 24px;" />

Jedoch sind die tangentialen Komponenten der Stromdichte über die Schnittstelle hinweg unstetig.

<img alt="Equation2" src="/files/ftp_upload/14187/14187_Equation_2.png" style="text-align: center; height: 55px;" />

Betrachten Sie eine Schnittstelle, die durch zwei leitende Medien mit Leitfähigkeiten σ1 und σ2 getrennt ist. Die gleichmäßige Stromdichte an der Schnittstelle beträgt <img height="22" src="file:///C:/Users/tirth/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="11" />, in Medium 1 an einem Punkt P1. Sie bildet einen Winkel α1 mit der Normalen. Die Stromdichte <img height="22" src="file:///C:/Users/tirth/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="11" /> am Punkt P2 in Medium 2 bildet einen Winkel α2 mit der Normalen.

<img alt="Figure1" src="/files/ftp_upload/14187/14187_figure_1.png" style="text-align: center" />

Die normale und tangentialen Komponenten der Stromdichte ergeben die folgenden Gleichungen:

<img alt="Equation3" src="/files/ftp_upload/14187/14187_Equation_3.png" style="text-align: center; height: 16px;" />

<img alt="Equation4" src="/files/ftp_upload/14187/14187_Equation_4.png" style="text-align: center; height: 16px;" />

Wenn man das Verhältnis der beiden Gleichungen nimmt, erhält man den Ausdruck für die elektrischen Leitfähigkeiten in beiden Medien.

<img alt="Equation5" src="/files/ftp_upload/14187/14187_Equation_5.png" style="text-align: center; height: 40px;" />

Ist die elektrische Leitfähigkeit des Mediums 1 größer als die des Mediums 2, geht der Winkel α2 gegen null. Dies impliziert, dass die Stromdichte <img height="22" src="file:///C:/Users/tirth/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="11" /> normal zur Oberfläche des Leiters 1 ist.

Randbedingungen für die Stromdichte

Die Stromdichte wird an einer Grenzfläche von Materialien mit unterschiedlicher elektrischer Leitfähigkeit diskontinuierlich. Die normale Komponente des Stroms ...

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current density at conductivity boundaries

Current density at a boundary changes when two conducting materials have different electrical conductivities. The normal component of current density stays continuous across the interface. The tangential component, which runs along the surface, becomes discontinuous across the boundary.

Consider two conducting media with conductivities σ1 and σ2 separated by an interface. At point P1 in medium 1, the steady current density makes an angle α1 with the normal. At point P2 in medium 2, the current density makes an angle α2 with the normal.

The normal and tangential parts of the current density lead to two equations. Taking the ratio of those equations gives the relation for the electrical conductivities in both media. If medium 1 has a greater conductivity than medium 2, angle α2 approaches zero. In that case, the current density becomes normal to the surface of conductor 1.

Current Density at Conductivity Boundaries

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Current density at a boundary changes when two conducting materials have different electrical conductivities. The normal component of current density stays continuous across the interface. The tangential component, which runs along the surface, becomes discontinuous across the boundary. Consider two conducting media with conductivities σ1 and σ2 separated by an interface. At point P1 in medium 1, the steady current density makes an angle α1 with the normal. At point P2 in medium 2, the current...

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Learn how current density behaves at the boundary between two conductors. See why the normal component stays continuous and the tangential component changes.

Current Density at Conductivity Boundaries | JoVE K12

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Current Density at Conductivity Boundaries

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