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

Akım yoğunluğu, birim kesit alanı başına akan toplam akım miktarıdır. Dolayısıyla, bir kesit alanından geçen toplam akım, akım yoğunluğunun yüzey integrali olarak ifade edilebilir.
Şimdi, yük korunumu, belirli bir hacimden akan toplam akımın, o hacim içindeki yükün azalma oranına eşit olmasını gerektirir.
Toplam yükü hacim yük yoğunluğu cinsinden ifade eden denklem değiştirilir. Yük yoğunluğu bir uzay fonksiyonu olduğundan, Leibniz kuralı kullanılarak, zaman türevi integral içinde hareket ettirilebilir.
Diverjans teoremini denklemin sol tarafına uygulamak, kapalı yüzey integralini bir hacim integraline dönüştürür. Bu ilişki herhangi bir hacim için geçerlidir; Sonuç olarak, integrandlar eşittir.
Akım yoğunluğunun sapması, hacim yük yoğunluğunun negatif değişim oranına eşittir. Yerel yük korunumunun bu matematiksel ifadesine süreklilik denklemi denir.
Sabit akımlar için yük yoğunluğu zamanla değişmezdir. Dolayısıyla, akım yoğunluğunun sapması sıfırdır.

<ol>
	<li>Griffiths, David J., Elektrodinamiğe Giriş, CA: Pearson. sayfa 222</li>
	<li>Ida, Nathan, Mühendislik Elektromanyetikleri, 3. Baskı, Springer. s. 355-356.</li>
</ol>

sürekli denklem

Birim kesit alanına akan toplam akım miktarına akım yoğunluğu denir. Dolayısıyla, bir kesit alanından geçen akım, akım yoğunluğunun yüzey integrali olarak yazılabilir.

<img alt="Denklem1" src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" />

Yük korunumu prensibine göre, belirli bir hacimden dışarıya akan toplam akım miktarı, o hacimdeki yükün azalma hızına eşittir.

<img alt="Denklem2" src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" />

şimdi, toplam yük hacim yük yoğunluğu terimleriyle yazılabilir.

<img alt="Denklem3" src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" />

Yük yoğunluğu bir uzay fonksiyonudur. Dolayısıyla, Leibniz kuralı uygulanarak zaman türetilmesi integralin içine taşınabilir.

<img alt="Denklem4" src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" />

Yukarıdaki denklemin sol tarafına çeşitleme teoremi uygulandığında, kapalı yüzey integrali hacim integraline dönüşür.

<img alt="Denklem5" src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" />

Bu ilişki herhangi bir hacim için geçerli olduğundan, integral unsurlar eşittir ve elde edilen ilişki süreklilik denklemi olarak adlandırılır. Bu, akım yoğunluğunun çeşitlemesinin, hacim yük yoğunluğunun negatif değişim hızına eşit olduğunu belirtir.

<img alt="Denklem6" src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" />

Sabit akımlar için, yük yoğunluğu zamanla değişmez. Dolayısıyla, akım yoğunluğunun çeşitlemesi sıfırdır.

Sürekli denklem

Birim kesit alanı başına geçen toplam akım miktarına akım yoğunluğu denir. Bu nedenle, bir kesit alanından geçen akım ...

Education

JoVE Core

Physics

Fizik

Unit 1

Ünite 1

Current and Resistance

Akım ve Direnç

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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...

drift velocity

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Current Density in Wires and Solutions

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resistivity

Resistivity in Conductors and Insulators

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

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ohm's law

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Ohm's law describes the relationship between current, voltage, and resistance in many materials. In these cases, the current through a substance is directly proportional to the voltage applied to it. The idea was first shown experimentally by German physicist Georg Simon Ohm, who studied current in a metal wire. A material, component, or device that follows this pattern is called ohmic. An ohmic component has a current that increases in step with voltage. A material or component that does not...

non-ohmic devices

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Cutting Electric Bills with Efficient Lighting

Electrical energy is the energy transferred when electrons move from one point to another. The longer an appliance is used, the more electrical energy it consumes. If power stays constant, electrical energy equals the power used by the device times the time it runs. That is why longer use leads to a higher electric bill. Electric bills measure this energy in kilowatt-hours. One kilowatt-hour is equal to 3.6 × 10^6 joules. You can lower electrical energy use by cutting the time an appliance...

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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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

physics

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PS.Phy.4.3

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PS.PSc.4.1

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Electrical current is defined as the rate at which charge flows. When there is a large current present, such as that used to run a refrigerator, a large ...

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PS.PSc.6.4

Induction

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Faraday's Law

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PS.PSc.6.5

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

Charge Conservation and Current Flow

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