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 densità di corrente è la quantità totale di corrente che scorre per unità di area della sezione trasversale. Quindi, la corrente totale che passa attraverso un'area della sezione trasversale può essere espressa come l'integrale di superficie della densità di corrente.
Ora, la conservazione della carica richiede che la corrente totale che esce da un dato volume sia uguale al tasso di diminuzione della carica all'interno di quel volume.
Esprimendo la carica totale in termini di densità di carica volumetrica, l'equazione viene modificata. Poiché la densità di carica è una funzione spaziale, utilizzando la regola di Leibniz, la sua derivata temporale può essere spostata all'interno dell'integrale.
Applicando il teorema di divergenza al lato sinistro dell'equazione si converte l'integrale di superficie chiuso in un integrale di volume. Questa relazione vale per qualsiasi volume; Di conseguenza, gli integrandi sono uguali.
La divergenza della densità di corrente è uguale al tasso di variazione negativo della densità di carica del volume. Questa affermazione matematica della conservazione della carica locale è chiamata equazione di continuità.
Per correnti costanti, la densità di carica è invariante nel tempo. Quindi, la divergenza della densità di corrente è zero.

<ol>
	<li>Griffiths, David J., Introduzione all'elettrodinamica, CA: Pearson. Pp. 222</li>
	<li>Ida, Nathan, Ingegneria elettromagnetica, 3a edizione, Springer. Pagine 355-356.</li>
</ol>

equazione di continuità

La quantità totale di corrente che scorre per unità di area trasversale è chiamata densità di corrente. Pertanto, la corrente che attraversa un'area trasversale può essere scritta come l'integrale superficiale della densità di corrente.

<img alt="Equation1" src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" />

Secondo il principio di conservazione della carica, la corrente totale che esce da un dato volume è uguale al tasso di diminuzione della carica all'interno di quel volume.

<img alt="Equation2" src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" />

Ora, la carica totale può essere scritta in termini della densità di carica volumetrica.

<img alt="Equation3" src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" />

La densità di carica è una funzione dello spazio. Pertanto, applicando la regola di Leibniz, la derivata temporale può essere spostata all'interno dell'integrale.

<img alt="Equation4" src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" />

Quando si applica il teorema della divergenza al lato sinistro dell'equazione sopra, l'integrale superficiale chiuso viene convertito in integrale di volume.

<img alt="Equation5" src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" />

Visto che questa relazione vale per qualsiasi volume, gli integrandi sono uguali e la relazione ottenuta è chiamata equazione di continuità. Questa afferma che la divergenza della densità di corrente è uguale al tasso di cambiamento negativo della densità di carica volumetrica.

<img alt="Equation6" src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" />

Per correnti stazionarie, la densità di carica è invariante nel tempo. Pertanto, la divergenza della densità di corrente è zero.

Equazione di continuità

La quantità totale di corrente che scorre per unità di area della sezione trasversale è chiamata densità di corrente. Quindi, la corrente che passa attraverso un'area della sezione trasversale ...

Education

Didattica

JoVE Core

Physics

Fisica

Unit 1

Unità 1

Current and Resistance

Corrente e Resistenza

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

Current Flow in Simple Circuits

Electrical current is the flow of charge in a circuit. A large current means a large amount of charge moves through a wire in a short time. A small current means only a small amount of charge moves through the circuit over a longer time. Examples include the current needed to run a refrigerator and the smaller current used by a handheld calculator. The SI unit for current is the ampere, or amp. It is named for the French physicist André-Marie Ampère (1775–1836). One ampere means one coulomb of...

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resistivity

Resistivity in Conductors and Insulators

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resistance

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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...

electrical conductivity

Resistance and Ohm's Law in Conductors

Resistance and Ohm's law help explain how electric current moves through a conductor. In a perfect conductor, the electric field inside is zero because free electrons move to cancel it. Any extra charge then stays on the surface. In a practical conductor, an applied electric field can remain in place. That field drives electrons through the material and creates a current. The current density, which is current per unit area, is related to the electric field. More generally, it is related to the...

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Electron Drift in Metallic Conduction

Electron drift in metallic conduction explains how free electrons move through a conductor. A classical model can describe this motion closely, even though quantum mechanics gives the best overall picture. This model is called the theory of metallic conduction. In this theory, Newton's second law is used to find the acceleration of an electron when an electric field is applied. The electron's velocity is then found from that acceleration. The electron travels through a crystal of positive ions...

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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