Name: current density at conductivity boundaries
Uploaded: 2023-09-18T17:11:18.000Z
Duration: 1 min 25 s
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Boundary Conditions for Current Density

La densité de courant devient discontinue aux bornes d’une interface ayant des conductivités électriques différentes.
Pour les courants stationnaires, la divergence de la densité de courant est nulle. Ainsi, la composante normale de la densité de courant est continue à travers la frontière.
Rappelons que la composante tangentielle du champ électrique est continue aux bornes d’une interface. L’expression du champ électrique en termes de densité de courant et de conductivité électrique donne la condition aux limites de la composante tangentielle de la densité de courant.
Maintenant, la composante normale de la cylindrée électrique est discontinue à travers l’interface. Ainsi, la composante normale du champ électrique est également discontinue à travers une interface.
Encore une fois, en utilisant l’expression du champ électrique et de la densité de courant, la condition aux limites de la composante normale de la densité de courant est obtenue en termes de permittivité et de conductivité.
Il montre qu’une densité de charge de surface est créée à travers une interface ayant des conductivités et/ou des permittivités différentes.
La densité de charge superficielle est nulle pour une interface ayant les mêmes valeurs de conductivité et de permittivité ou avec un rapport permistivite/conductivité égal.

<ol>
	<li>Ida, Nathan, Engineering Electromagnetics, 3e édition, Springer. pp. 355-356</li>
	<li>Cheng, David K., Électromagnétisme des champs et des ondes, Addison-Wesley Publishing Company, p. 183.</li>
</ol>

conditions aux limites pour la densité de courant

La densité de courant devient discontinue à travers une interface de matériaux avec différentes conductivités électriques. La composante normale de la densité de courant est continue à travers la frontière.

<img alt="Equation1" src="/files/ftp_upload/14187/14187_Equation_1.png" style="text-align: center; height: 24px;" />

Cependant, les composantes tangentielles de la densité de courant sont discontinues à travers l'interface.

<img alt="Equation2" src="/files/ftp_upload/14187/14187_Equation_2.png" style="text-align: center; height: 55px;" />

Considérons une interface séparée par deux milieux conducteurs avec des conductivités σ1 et σ2 La densité de courant stationnaire à l'interface est <img height="22" src="file:///C:/Users/tirth/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="11" /> dans le milieu 1 en un point P1. Elle fait un angle α1 avec la normale. La densité de courant <img height="22" src="file:///C:/Users/tirth/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="11" /> au point P2 dans le milieu 2 fait un angle α2 avec la normale.

<img alt="Figure1" src="/files/ftp_upload/14187/14187_figure_1.png" style="text-align: center" />

Les composantes normale et tangentielle de la densité de courant donnent les équations suivantes:

<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;" />

En prenant le rapport des deux équations, l'expression des conductivités électriques dans les deux milieux est obtenue.

<img alt="Equation5" src="/files/ftp_upload/14187/14187_Equation_5.png" style="text-align: center; height: 40px;" />

Si la conductivité électrique du milieu 1 est plus élevée que celle du milieu 2, l'angle α2 tend vers zéro. Cela implique que la densité de courant <img height="22" src="file:///C:/Users/tirth/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="11" /> est normale à la surface du conducteur 1.

Conditions aux limites pour la densité de courant

La densité de courant devient discontinue à travers une interface de matériaux ayant des conductivités électriques différentes. La composante normale du courant ...

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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 Conductivity Boundaries | JoVE K12

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Current Density at Conductivity Boundaries

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