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

Current density is the total amount of current flowing per unit cross-sectional area. So, the total current passing through a cross-sectional area can be expressed as the surface integral of the current density.
Now, charge conservation requires that the total current flowing out of a given volume equals the rate of decrease of charge within that volume.
Expressing the&#160; total charge in terms of the volume charge density, the equation gets modified. Since the charge density is a space function, using the Leibniz rule, its time derivative can be moved inside the integral.
Applying the divergence theorem to the left-hand side of the equation converts the closed surface integral to a volume integral. This relation holds for any volume; as a result, the integrands are equal.
The divergence of the current density equals the negative rate of change of volume charge density. This mathematical statement of local charge conservation is called the continuity equation.
For steady currents, the charge density is invariant with time. So, the divergence of the current density is zero.

continuity equation

The total amount of current flowing per unit cross-sectional area is called the current density. Hence, the current passing through a cross-sectional area can be written as the surface integral of the current density.
<img alt="Static equilibrium concept, ΣFx=0, diagram with forces and angles, physics education." src="/files/ftp_upload/14186/14186_Equation_1.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation1">
According to the conservation of charge, the total current flowing out of a given volume equals the rate of decrease of charge within that volume.
<img alt="Static equilibrium, ΣFx=0, ΣFy=0, physics diagram, illustrating balanced forces on an inclined plane." src="/files/ftp_upload/14186/14186_Equation_2.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation2">
Now, the total charge can be written in terms of the volume charge density.
<img alt="Static equilibrium; ΣFx=0; ΣFy=0; beam balance diagram; force vectors; mechanical stability." src="/files/ftp_upload/14186/14186_Equation_3.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation3">
The charge density is a space function. Hence, applying the Leibniz rule, the time derivative can be moved inside the integral.
<img alt="Static equilibrium, ΣFx=0, lever diagram, torque balance analysis, educational physics concept." src="/files/ftp_upload/14186/14186_Equation_4.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation4">
When the divergence theorem is applied to the left-hand side of the above equation, the closed surface integral is converted into the volume integral.
<img alt="Static equilibrium diagram; ΣFx=0, ΣFy=0, showing force vectors and angles, educational concept." src="/files/ftp_upload/14186/14186_Equation_5.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation5">
Since this relation holds for any volume, the integrands are equal and the obtained relation is called the continuity equation. This states that the divergence of the current density equals the negative rate of change of volume charge density.
<img alt="Static equilibrium; ΣFx=0; diagram for force analysis." src="/files/ftp_upload/14186/14186_Equation_6.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation6">
For steady currents, the charge density is invariant with time. Hence, the divergence of the current density is zero.

The total amount of current flowing per unit cross-sectional area is called the current density. Hence, the current passing through a cross-sectional area ...

Education

JoVE Core

Physics

Unit 1

Current and Resistance

KEY TERMS AND CONCEPTS

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

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Current Flow in Simple Circuits

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Current Density in Wires and Solutions

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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...

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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

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Charge Conservation and Current Flow

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