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

A densidade de corrente é a quantidade total de corrente que flui por unidade de área da seção transversal. Assim, a corrente total que passa por uma área de seção transversal pode ser expressa como a integral da superfície da densidade de corrente.
Agora, a conservação de carga requer que a corrente total que flui para fora de um determinado volume seja igual à taxa de diminuição da carga dentro desse volume.
Expressando a carga total em termos da densidade de carga de volume, a equação é modificada. Como a densidade de carga é uma função espacial, usando a regra de Leibniz, sua derivada de tempo pode ser movida para dentro da integral.
A aplicação do teorema da divergência ao lado esquerdo da equação converte a integral de superfície fechada em uma integral de volume. Essa relação vale para qualquer volume; Como resultado, os integrandos são iguais.
A divergência da densidade de corrente é igual à taxa negativa de variação da densidade de carga de volume. Essa afirmação matemática da conservação de carga local é chamada de equação de continuidade.
Para correntes constantes, a densidade de carga é invariante com o tempo. Portanto, a divergência da densidade de corrente é zero.

<ol>
	<li>Griffiths, David J., Introdução à Eletrodinâmica, CA: Pearson. pág. 222</li>
	<li>Ida, Nathan, Engenharia Eletromagnética, 3ª Edição, Springer. pág. 355-356.</li>
</ol>

equação de continuidade

A quantidade total de corrente que flui por unidade de área transversal é chamada de densidade de corrente. Portanto, a corrente que passa por uma área transversal pode ser escrita como a integral de superfície da densidade de corrente.

<img alt="Equation1" src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" />

De acordo com a conservação da carga, a corrente total que flui para fora de um determinado volume é igual à taxa de diminuição da carga dentro desse volume.

<img alt="Equation2" src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" />

Assim, a carga total pode ser escrita em termos da densidade de carga volumétrica.

<img alt="Equation3" src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" />

A densidade de carga é uma função do espaço. Portanto, aplicando a regra de Leibniz, a derivada temporal pode ser movida para dentro da integral.

<img alt="Equation4" src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" />

Quando o teorema da divergência é aplicado ao lado esquerdo da equação acima, a integral de superfície fechada é convertida em uma integral de volume.

<img alt="Equation5" src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" />

Uma vez que essa relação vale para qualquer volume, os integrandos são iguais e a relação obtida é chamada de equação de continuidade. Isso afirma que a divergência da densidade de corrente é igual à taxa negativa de mudança da densidade de carga volumétrica.

<img alt="Equation6" src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" />

Para correntes contínuas, a densidade de carga é invariante com o tempo. Portanto, a divergência da densidade de corrente é zero.

Equação de Continuidade

A quantidade total de corrente que flui por unidade de área da seção transversal é chamada de densidade de corrente. Assim, a corrente que passa por uma área de seção transversal ...

Education

JoVE Core

Physics

Física

Unit 1

Unidade 1

Current and Resistance

Corrente e Resistência

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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

physics

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PS.Phy.4.3

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PS.PSc.6.2

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PS.PSc.6.4

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Analyze motion in terms of speed, velocity, acceleration, and momentum.

Charge Conservation and Current Flow

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