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

Плотность тока — это общая величина тока, протекающего на единицу площади поперечного сечения. Таким образом, полный ток, проходящий через площадь поперечного сечения, может быть выражен как поверхностный интеграл от плотности тока.
Теперь сохранение заряда требует, чтобы общий ток, протекающий из данного объема, был равен скорости уменьшения заряда в пределах этого объема.
Выражая общий заряд в терминах объемной плотности заряда, уравнение модифицируется. Поскольку плотность заряда является пространственной функцией, используя правило Лейбница, ее производную по времени можно переместить внутрь интеграла.
Применение теоремы о дивергенции к левой части уравнения преобразует интеграл замкнутой поверхности в интеграл по объему. Это соотношение справедливо для любого объема; В результате интегралы равны.
Расхождение плотности тока равно отрицательной скорости изменения объемной плотности заряда. Эта математическая постановка локального сохранения заряда называется уравнением непрерывности.
Для постоянных токов плотность заряда инвариантна со временем. Таким образом, расхождение плотности тока равно нулю.

<ol>
	<li>Гриффитс, Дэвид Дж., Введение в электродинамику, Калифорния: Пирсон. стр. 222</li>
	<li>Ида, Натан, Инженерная электромагнетика, 3-е издание, Springer. стр. 355-356.</li>
</ol>

Уравнение непрерывности

Общее количество тока, протекающего через единичное сечение, называется плотностью тока. Следовательно, текущий ток через поперечное сечение можно записать как поверхностный интеграл от плотности тока.

<img alt="Equation1" src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" />

Согласно закону сохранения заряда, общий ток, вытекающий из заданного объёма, равен скорости уменьшения заряда внутри этого объёма.

<img alt="Equation2" src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" />

Теперь общий заряд может быть записан в терминах плотности заряда по объёму.

<img alt="Equation3" src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" />

Плотность заряда - это функция пространства. Следовательно, применяя правило Лейбница, производную по времени можно переместить внутри интеграла.

<img alt="Equation4" src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" />

Когда теорема о дивергенции применяется к левой части уравнения выше, замкнутый поверхностный интеграл преобразуется в объёмный интеграл.

<img alt="Equation5" src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" />

Поскольку эта зависимость справедлива для любого объёма, интегрируемые функции равны, и полученное соотношение называется уравнением непрерывности. Это означает, что дивергенция плотности тока равна отрицательной скорости изменения плотности заряда по объёму.

<img alt="Equation6" src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" />

Для установившихся токов плотность заряда неизменна со временем. Следовательно, дивергенция плотности тока равна нулю.

Общая величина тока, протекающего на единицу площади поперечного сечения, называется плотностью тока. Следовательно, ток, проходящий через область поперечного сечения...

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

Current and Resistance

Ток и сопротивление

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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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

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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

physics

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Charge Conservation and Current Flow

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