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

전류 밀도는 단위 단면적당 흐르는 전류의 총량입니다. 따라서 단면적을 통과하는 총 전류는 전류 밀도의 표면 적분으로 표현될 수 있습니다.
이제 전하 보존은 주어진 부피에서 흘러나오는 총 전류가 해당 부피 내의 전하 감소율과 같아야 합니다.
총 전하를 체적 전하 밀도(volume charge density)로 표현하면 방정식이 수정됩니다. 전하 밀도는 공간 함수이기 때문에 라이프니츠 규칙을 사용하여 시간 도함수를 적분 내부로 이동할 수 있습니다.
발산 정리를 방정식의 왼쪽에 적용하면 닫힌 곡면 적분이 체적 적분으로 변환됩니다. 이 관계는 모든 볼륨에 적용됩니다. 결과적으로 적분은 동일합니다.
전류 밀도의 발산은 체적 전하 밀도의 음의 변화율과 같습니다. 국소 전하 보존에 대한 이러한 수학적 설명을 연속성 방정식(continuity equation)이라고 합니다.
정상 전류의 경우 전하 밀도는 시간에 따라 변하지 않습니다. 따라서 전류 밀도의 발산은 0입니다.

<ol>
	<li>Griffiths, David J., 전기역학 입문, CA: Pearson. 222쪽</li>
	<li>Ida, Nathan, 엔지니어링 전자기학, 3판, Springer. 355-356쪽.</li>
</ol>

연속 방정식

단위 단면적당 흐르는 전류의 총량을 전류밀도라고 합니다. 따라서 단면적을 통과하는 전류는 전류 밀도의 표면 적분으로 쓸 수 있습니다.

<img alt="Equation1" src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" />

전하 보존에 따르면, 주어진 볼륨에서 흘러나오는 총 전류는 해당 볼륨 내에서 전하가 감소하는 비율과 같습니다.

<img alt="Equation2" src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" />

이제 총 전하는 부피 전하 밀도로 나타낼 수 있습니다.

<img alt="Equation3" src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" />

전하 밀도는 공간 함수입니다. 따라서 라이프니츠 규칙을 적용하면 시간 미분은 적분 내부로 이동할 수 있습니다.

<img alt="Equation4" src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" />

위 방정식의 좌변에 발산 정리를 적용하면 닫힌 표면 적분이 부피 적분으로 변환됩니다.

<img alt="Equation5" src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" />

이 관계는 모든 체적에 대해 적용되므로 피적분 함수는 동일하며 얻은 관계를 연속 방정식이라고 합니다. 이는 전류 밀도의 발산이 부피 전하 밀도의 음의 변화율과 동일함을 나타냅니다.

<img alt="Equation6" src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" />

정상 전류의 경우 전하 밀도는 시간에 따라 변하지 않습니다. 따라서 전류밀도의 발산은 0입니다.

단위 단면적당 흐르는 전류의 총량을 전류 밀도라고 합니다. 따라서, 단면적을 통과하는 전류는 ...

Education

교육

JoVE Core

Physics

물리학

Unit 1

제1단원

Current and Resistance

전류와 저항

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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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current density

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resistivity

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Resistivity describes how much a material opposes electric current. When a voltage is applied to a conductor, an electric field forms. The charges in the conductor feel that force, and a current density develops. In some materials, including metals at a given temperature, the current density is approximately proportional to the electric field. In those cases, current density can be modeled with conductivity, written as sigma. Electrical conductivity is measured in (ohm·m)^−1. It tells how well...

resistance

Factors That Affect Electrical Resistance

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non-ohmic devices

Diode Current and Reverse Bias

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Cutting Electric Bills with Efficient Lighting

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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...

theory of metallic conduction

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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