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

K12 Standards

north-carolina-k-12-science-standard-course-of-study

North Carolina K-12 Science Standard Course of Study

9-12

Physical Science

motion-and-stability-forces-and-interactions

Motion and Stability- Forces and Interactions

PS.PSc.6.3

Continuity Equation

Die Stromdichte ist die Gesamtmenge des pro Querschnittseinheit fließenden Stroms. Der Gesamtstrom, der durch eine Querschnittsfläche fließt, kann also als Oberflächenintegral der Stromdichte ausgedrückt werden.
Die Ladungserhaltung erfordert nun, dass der Gesamtstrom, der aus einem gegebenen Volumen fließt, der Rate der Ladungsabnahme innerhalb dieses Volumens entspricht.
Drückt man die Gesamtladung in Bezug auf die Volumenladungsdichte aus, wird die Gleichung modifiziert. Da es sich bei der Ladungsdichte um eine Raumfunktion handelt, kann ihre Zeitableitung nach der Leibniz-Regel innerhalb des Integrals verschoben werden.
Wendet man den Divergenzsatz auf die linke Seite der Gleichung an, wird das Integral der geschlossenen Fläche in ein Volumenintegral umgewandelt. Diese Beziehung gilt für jeden Band; Infolgedessen sind die Integranden gleich.
Die Divergenz der Stromdichte entspricht der negativen Änderungsrate der Volumenladungsdichte. Diese mathematische Aussage der lokalen Ladungserhaltung wird als Kontinuitätsgleichung bezeichnet.
Bei stationären Strömen ist die Ladungsdichte mit der Zeit invariant. Die Divergenz der Stromdichte ist also Null.

<ol>
	<li>Griffiths, David J., Einführung in die Elektrodynamik, CA: Pearson. S. 222</li>
	<li>Ida, Nathan, Engineering Electromagnetics, 3. Auflage, Springer. S. 355-356.</li>
</ol>

kontinuitätsgleichung

Die Gesamtmenge des fließenden Stroms pro Einheitsquerschnittsfläche wird als Stromdichte bezeichnet. Daher kann der durch einen Querschnitt fließende Strom als Oberflächenintegral der Stromdichte geschrieben werden.

<img alt="Equation1" src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" />

Laut dem Erhaltungssatz der Ladung entspricht der Gesamtstrom, der aus einem gegebenen Volumen fließt, der Rate des Ladungsabfalls innerhalb dieses Volumens.

<img alt="Equation2" src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" />

Nun kann die Gesamtladung in Bezug auf die Volumenladungsdichte geschrieben werden.

<img alt="Equation3" src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" />

Die Ladungsdichte ist eine Raumfunktion. Daher kann die zeitliche Ableitung mithilfe der Leibniz-Regel innerhalb des Integrals verschoben werden.

<img alt="Equation4" src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" />

Wenn der Divergenzsatz auf die linke Seite der obigen Gleichung angewendet wird, wird das geschlossene Flächenintegral in das Volumenintegral umgewandelt.

<img alt="Equation5" src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" />

Da diese Beziehung für jedes Volumen gilt, sind die Integranden gleich und die erhaltene Beziehung wird als Kontinuitätsgleichung bezeichnet. Dies besagt, dass die Divergenz der Stromdichte der negativen Änderungsrate der Volumenladungsdichte entspricht.

<img alt="Equation6" src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" />

Für stationäre Ströme ist die Ladungsdichte zeitlich invariant. Daher ist die Divergenz der Stromdichte null.

Kontinuitätsgleichung

Die Gesamtmenge des pro Querschnittseinheit fließenden Stroms wird als Stromdichte bezeichnet. Der Strom, der durch eine Querschnittsfläche fließt, ...

Education

Lehre

JoVE Core

Physics

Physik

Unit 1

Einheit 1

Current and Resistance

Strom und Widerstand

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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

electrical current

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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...

boundary conditions for current density

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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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