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

Gęstość prądu to całkowita ilość prądu płynącego na jednostkę pola przekroju poprzecznego. Tak więc całkowity prąd przepływający przez pole przekroju poprzecznego można wyrazić jako całkę powierzchniową gęstości prądu.
Teraz zachowanie ładunku wymaga, aby całkowity prąd płynący z danej objętości był równy szybkości spadku ładunku w tej objętości.
Wyrażając całkowity ładunek w kategoriach gęstości ładunku objętościowego, równanie ulega modyfikacji. Ponieważ gęstość ładunku jest funkcją przestrzenną, za pomocą reguły Leibniza jej pochodną czasową można przesunąć wewnątrz całki.
Zastosowanie twierdzenia o dywergencji po lewej stronie równania przekształca całkę z zamkniętej powierzchni w całkę objętościową. Ta zależność dotyczy każdej objętości; W rezultacie całki są równe.
Rozbieżność gęstości prądu jest równa ujemnemu wskaźnikowi zmiany gęstości ładunku objętościowego. To matematyczne stwierdzenie zachowania ładunku lokalnego nazywa się równaniem ciągłości.
W przypadku prądów stałych gęstość ładunku jest niezmienna w czasie. Tak więc rozbieżność gęstości prądu wynosi zero.

<ol>
	<li>Griffiths, David J., Wprowadzenie do elektrodynamiki, CA: Pearson. S. 222</li>
	<li>Ida, Nathan, Inżynieria elektromagnetyczna, wydanie 3, Springer. str. 355-356.</li>
</ol>

równanie ciągłości

Całkowita ilość prądu płynącego na jednostkę powierzchni przekroju poprzecznego nazywana jest gęstością prądu. W związku z tym prąd przepływający przez pole przekroju poprzecznego można zapisać jako całkę powierzchniową gęstości prądu.

<img alt="Równanie1" src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" />

Zgodnie z zasadą zachowania ładunku całkowity prąd wypływający z danej objętości jest równy szybkości zmniejszania się ładunku w tej objętości.

<img alt="Równanie2" src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" />

Teraz całkowity ładunek można zapisać w formie objętościowej gęstości ładunku.

<img alt="Równanie3" src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" />

Gęstość ładunku jest funkcją przestrzenną. Zatem stosując regułę Leibniza, pochodną po czasie można przesunąć wewnątrz całki.

<img alt="Równanie4" src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" />

Gdy twierdzenie o rozbieżności zostanie zastosowane do lewej strony powyższego równania, całka powierzchniowa zamknięta zostanie przekształcona w całkę objętościową.

<img alt="Równanie5" src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" />

Ponieważ ta zależność dotyczy dowolnej objętości, całki są równe, a uzyskaną zależność nazywa się równaniem ciągłości. Oznacza to, że rozbieżność gęstości prądu jest równa ujemnej szybkości zmiany objętościowej gęstości ładunku.

<img alt="Równanie6" src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" />

W przypadku prądów stałych gęstość ładunku jest niezmienna w czasie. Zatem rozbieżność gęstości prądu wynosi zero.

Równanie ciągłości

Całkowita ilość prądu płynącego na jednostkę pola przekroju poprzecznego nazywana jest gęstością prądu. Stąd prąd przepływający przez pole przekroju poprzecznego ...

Education

JoVE Core

Physics

Fizyka

Unit 1

Jednostka 1

Current and Resistance

Natężenie prądu i rezystancja

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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

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...

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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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