Name: charge conservation and current flow
Uploaded: 2023-09-18T17:11:18.000Z
Duration: 1 min 20 s
Channel: JoVE

K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Physical Science

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Motion and Stability- Forces and Interactions

PS.PSc.6.3

Continuity Equation

La densité de courant est la quantité totale de courant circulant par unité de section transversale. Ainsi, le courant total passant par une section transversale peut être exprimé comme l’intégrale de surface de la densité de courant.
Or, la conservation de la charge exige que le courant total sortant d’un volume donné soit égal au taux de diminution de la charge à l’intérieur de ce volume.
En exprimant la charge totale en termes de densité de charge volumique, l’équation est modifiée. Comme la densité de charge est une fonction spatiale, en utilisant la règle de Leibniz, sa dérivée temporelle peut être déplacée à l’intérieur de l’intégrale.
L’application du théorème de divergence au membre gauche de l’équation convertit l’intégrale de surface fermée en intégrale de volume. Cette relation est valable pour tout volume ; En conséquence, les intgrands sont égaux.
La divergence de la densité de courant est égale au taux de variation négatif de la densité de charge volumique. Cette affirmation mathématique de la conservation de la charge locale s’appelle l’équation de continuité.
Pour les courants stationnaires, la densité de charge est invariante avec le temps. Ainsi, la divergence de la densité de courant est nulle.

<ol>
	<li>Griffiths, David J., Introduction à l’électrodynamique, CA : Pearson. p. 222</li>
	<li>Ida, Nathan, Ingénierie électromagnétique, 3e édition, Springer. pp. 355-356.</li>
</ol>

Équation de continuité

La quantité totale de courant traversant par unité de surface transversale est appelée densité de courant. Ainsi, le courant traversant une surface transversale peut être écrit comme l'intégrale de surface de la densité de courant.

<img alt="Equation1" src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" />

Selon la conservation de la charge, le courant total sortant d'un volume donné est égal au taux de diminution de la charge à l'intérieur de ce volume.

<img alt="Equation2" src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" />

Maintenant, la charge totale peut être écrite en fonction de la densité de charge volumique.

<img alt="Equation3" src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" />

La densité de charge est une fonction spatiale. Par conséquent, en appliquant la règle de Leibniz, la dérivée temporelle peut être déplacée à l'intérieur de l'intégrale.

<img alt="Equation4" src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" />

Lorsque le théorème de la divergence est appliqué au côté gauche de l'équation ci-dessus, l'intégrale de surface fermée est convertie en intégrale de volume.

<img alt="Equation5" src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" />

Puisque cette relation est valable pour n'importe quel volume, les intégrandes sont égales et la relation obtenue est appelée équation de continuité. Cela indique que la divergence de la densité de courant est égale au taux de changement négatif de la densité de charge volumique.

<img alt="Equation6" src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" />

Pour les courants continus, la densité de charge est invariante dans le temps. Par conséquent, la divergence de la densité de courant est nulle.

La quantité totale de courant circulant par unité de section transversale est appelée densité de courant. Par conséquent, le courant traversant une section transversale ...

Education

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JoVE Core

Physics

Physique

Unit 1

Unité 1

Current and Resistance

Courant et résistance

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

charge conservation and current flow

Charge conservation and current flow are linked through the continuity equation. The current density is the amount of current flowing per unit cross-sectional area. The current through a surface can be written as a surface integral of the current density.

The conservation of charge says that the total current leaving a volume equals the rate at which charge inside that volume decreases. The total charge can be written using volume charge density, which is a space function. Because of this, the time derivative can be moved inside the integral by applying the Leibniz rule.

Next, the divergence theorem is used on the left-hand side. This changes the closed surface integral into a volume integral. Since the relation must hold for any volume, the integrands must be equal. This gives the continuity equation, which states that the divergence of the current density equals the negative rate of change of volume charge density. For steady currents, the charge density does not change with time, so the divergence of the current density is zero.

Charge Conservation and Current Flow

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continuity equation

Charge conservation and current flow are linked through the continuity equation. The current density is the amount of current flowing per unit cross-sectional area. The current through a surface can be written as a surface integral of the current density. The conservation of charge says that the total current leaving a volume equals the rate at which charge inside that volume decreases. The total charge can be written using volume charge density, which is a space function. Because of this, the...

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key terms and concepts

Learn how charge conservation leads to the continuity equation, how current density relates to charge flow, and why steady currents have zero divergence.

Charge Conservation and Current Flow | JoVE K12

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Charge Conservation and Current Flow

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