Name: displacement current in an rc circuit
Uploaded: 2023-09-18T17:11:20.000Z
Duration: 1 min 17 s
Channel: JoVE

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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.6.5

Ampere-Maxwell's Law: Problem-Solving

Ein Parallelplattenkondensator im Vakuum befindet sich im Ladezustand. Der Radius jeder Platte beträgt 5 cm. Der Leitungsstrom, der durch die Drähte fließt, beträgt 0,4 A. Wie hoch wird der Verschiebungsstrom zwischen den Platten zu diesem Zeitpunkt sein?
Berechnen Sie außerdem die Verschiebungsstromdichte und bewerten Sie die Änderungsrate des elektrischen Feldes zwischen den Platten. Berechnen Sie das induzierte Magnetfeld zwischen den Platten und zeichnen Sie es für einen Abstand von 1-5 cm von der Achse.
Der Verschiebungsstrom wird in Form des elektrischen Flusses ausgedrückt, der wiederum in Form von Ladung erhalten werden kann.
Der Verschiebungsstrom entspricht dem durch die Drähte fließenden Leitungsstrom, der 0,4 Ampere beträgt.
Des Weiteren kann die Verschiebungsstromdichte berechnet werden.
Wenn sich der Kondensator lädt, nimmt das elektrische Feld zwischen den Platten zu, das anhand der Verschiebungsstromdichte und der Permittivität des freien Raums ausgewertet wird.
Gemäß dem Ampere-Maxwell-Gesetz wird das Magnetfeld zwischen den Kondensatorplatten induziert, das für unterschiedliche Abstände von der Achse berechnet und wie gezeigt dargestellt wird.

<ol>
	<li>OpenStax. (2019). Universitätsphysik Vol. 2. [Webversion], Seiten 94–95. Abgerufen von <a href="https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves">https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves</a></li>
</ol>

das ampere-maxwellsche gesetz: problemlösung

Ein Parallelplattenkondensator mit der Kapazität C, dessen Platten eine Fläche A und einen Abstand d haben, ist mit einem Widerstand R und einer Batterie mit Spannung V verbunden. Der Strom beginnt bei t = 0 zu fließen. Was ist der Verschiebungsstrom zwischen den Kondensatorplatten zur Zeit t? Wie hoch ist der entsprechende reale Strom aufgrund der Eigenschaften des Kondensators?

Um das Problem zu lösen, können wir die Gleichungen aus der Analyse eines RC-Netzwerks und Maxwells Version des Ampèreschen Gesetzes verwenden.

Für den ersten Teil des Problems wird die Spannung zwischen den Platten zur Zeit t durch die folgende Gleichung gegeben:



<img alt="Equation1" src="/files/ftp_upload/14207/14207_Equation_1.svg" style="text-align: center; height: 35px;" />



Angenommen, dass die z-Achse von der positiven Platte zur negativen Platte zeigt. In diesem Fall ist die z-Komponente des elektrischen Feldes zwischen den Platten als Funktion der Zeit t gegeben durch:



<img alt="Equation2" src="/files/ftp_upload/14207/14207_Equation_2.svg" style="text-align: center; height: 38px;" />



Daher kann die z-Komponente des Verschiebungsstroms Id zwischen den Platten des Kondensators berechnet werden als:



<img alt="Equation3" src="/files/ftp_upload/14207/14207_Equation_3.svg" style="text-align: center; height: 37px;" />



wobei die Kapazität <img alt="Equation4" src="/files/ftp_upload/14207/14207eq04.png" style="margin: auto;" /> verwendet wird.

Weiterhin kann zur Lösung des zweiten Teils des oben genannten Problems der Strom in den Kondensator nach dem Schließen des Stromkreises durch die Ladung auf dem Kondensator ermittelt werden, die wiederum durch den Ausdruck für VC bewertet werden kann. Als Ergebnis wird festgestellt, dass der reale Strom in den Kondensator derselbe ist wie der Verschiebungsstrom.

Das Ampere-Maxwellsche Gesetz: Problemlösung

Ein Plattenkondensator mit der Kapazität C, dessen Platten den Bereich A und den Trennungsabstand d aufweisen, ist mit einem Widerstand R und einer Batterie der Spannung ...

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KEY TERMS AND CONCEPTS

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displacement current in an rc circuit

Displacement current in an RC circuit can be found by tracking the charging capacitor. A parallel-plate capacitor with capacitance C, plate area A, and plate separation d is connected to a resistor R and a battery of voltage V. When the current starts to flow at t = 0, the charge and voltage on the capacitor change with time.

The first step is to use the RC-circuit result for the capacitor voltage at time t. With the z-axis pointing from the positive plate to the negative plate, the electric field between the plates has a z-component that follows that same time dependence. The displacement current between the plates is then evaluated from this electric field and the capacitor relation.

The second step is to find the real current into the capacitor after the circuit is closed. This current is obtained from the charge on the capacitor, using the expression for V_C. The result is that the real current into the capacitor is the same as the displacement current.

Displacement Current in an RC Circuit

induction

Magnetic Flux and Induced Current

Magnetic flux and induced current are linked when a magnetic field changes near a coil. Moving a bar magnet into or out of a coil induces an emf, which is an electromotive force. If the motion goes in the opposite direction, the emf has the opposite sign. Reversing the magnet poles also reverses the direction of the induced emf. The same effect appears when the coil moves instead of the magnet. Relative motion is what matters. A faster motion produces a larger emf. If the magnet and coil stay...

ampere-maxwell's law: problem-solving

Displacement current in an RC circuit can be found by tracking the charging capacitor. A parallel-plate capacitor with capacitance C, plate area A, and plate separation d is connected to a resistor R and a battery of voltage V. When the current starts to flow at t = 0, the charge and voltage on the capacitor change with time. The first step is to use the RC-circuit result for the capacitor voltage at time t. With the z-axis pointing from the positive plate to the negative plate, the electric...

differential form of maxwell's equations

Maxwell’s Equations at a Point

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dc generator

How a DC Generator Makes Current

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Eddy Currents and Magnetic Braking

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Magnetic Flux and Induced Emf

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lenz's law

Magnet Motion and Induced Current

Magnet motion and induced current are linked by Lenz’s law. The law says that the induced emf, or electromotive force, drives current around a wire loop in a direction that opposes the change in magnetic flux that caused it. A moving bar magnet can create this effect in a coil. If the magnet moves toward the coil, the magnetic flux increases and an induced current appears. That current produces a magnetic field that pushes against the stronger field from the magnet. The same idea works in...

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Induced Current from Changing Magnetic Flux

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faraday disk dynamo

Faraday Disk Dynamo and DC Output

A Faraday disk dynamo is a DC generator that produces an emf constant in time. It uses a conducting disk that spins at constant angular velocity in a magnetic field. The magnetic field is perpendicular to the plane of the disk. As the disk rotates, the magnetic flux changes and an emf is induced. This causes opposite charges to build up at the rim and at the center of the disk. The direction of the induced emf depends on the direction of the magnetic field and the direction of rotation. If...

induced electric fields

Faraday’s Law and Induced EMF

Faraday’s law explains how a changing magnetic flux can create an induced emf in a circuit. When current is induced, work is being done on the conduction electrons in the wire. That work must come from a source, and the source is not a battery or a magnetic field. A battery does not have to be present in a circuit for current to be induced. Magnetic fields also do not do work on moving charges. Instead, the change in magnetic flux produces an electric field in the wire. This is called an...

induced electric fields: applications

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displacement current

Maxwell’s Fix for Changing Currents

Maxwell’s correction to Ampère’s law explains how changing electric fields can create magnetic fields. The key idea is the displacement current, Id, which Maxwell added to the real conduction current, I. This extra term helps the law work when current is not steady. The displacement current is defined using the permittivity of free space, ε0, and the electric flux, ΦE. It is not a real flow of charge. Instead, it is an added term in Maxwell’s equations that acts like current in Ampère’s law. <br...

significance of displacement current

Maxwell’s Fix for Charging Capacitors

Maxwell’s fix for a charging capacitor is the displacement current in Ampère’s law. A displacement current is not a flow of charge like conduction current. It comes from a changing electric field. That is why it is defined using a time-varying electric field and has its own displacement current density. Maxwell added this term to the original Ampère’s law. The result is called generalized Ampère’s law. The added displacement current term lets the law work in regions where no charges are moving.

electromagnetic fields

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Electric and magnetic fields can come from different sources, but they can act in surprisingly similar ways. Static charges create electrostatic fields. These fields are conservative, which means a test charge does no net work after moving around a complete loop. Fields made by changing magnetic fields behave differently. They are not conservative, and they do net work on a charge. That work is proportional to how fast the magnetic flux changes. These fields also do not follow Gauss’s law for...

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electric generator: alternator

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back emf

Motor Current and Speed Changes

Motor current and speed changes are linked by back emf in an electric motor. A motor sends current through a wire loop in a magnetic field. The magnetic field pushes on the loop and creates torque, which turns a shaft and does mechanical work. As the coil turns, the magnetic flux through it changes. That changing flux induces an emf, or voltage, by Faraday’s law. The motor then acts like a generator whenever its coil rotates, whether the shaft is turned by an external drive or by the motor...

key terms and concepts

See how displacement current is found in a charging parallel-plate capacitor using RC circuit relations and Maxwell’s version of Ampère’s law.

Displacement Current in an RC Circuit | JoVE K12

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PS.9.e

Position and Displacement

The position of an object defines its location relative to a convenient frame of reference at any particular time. A frame of reference is an arbitrary ...

Average Velocity

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Instantaneous Velocity - I

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Average Acceleration

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Acceleration is in the direction of the change in velocity, but it is not always in the direction of motion. When an object slows down, its acceleration ...

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Average and Instantaneous Velocity Vectors

To calculate other physical quantities in kinematics, the time variable must be introduced. The time variable not only allows us to state where an object ...

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The term momentum is used in various ways in everyday language, most of which are consistent with the precise scientific definition. Generally, momentum ...

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Analyze and interpret data to explain the motion of an object moving with a constant velocity or that is accelerating.

PS.PSc.4.1

Electrical Current

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 ...

Drift Velocity

The high speed of electrical signals results from the fact that the force between charges acts rapidly at a distance. Thus, when a free charge is forced ...

Current Density

The total amount of current flowing through one unit value of a cross-sectional area is referred to as current density. If the current flow is uniform, ...

Resistivity

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Resistance

When a current moves through any conductor, the conductor causes some level of difficulty for the current to flow. The measure of that difficulty is known ...

Ohm's Law

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Non-ohmic Devices

In most substances, the current flow is proportional to the voltage applied to it. A simple relationship between the values of current, voltage, and ...

Construct an explanation to compare simple series and parallel circuits in terms of Ohm's Law.

PS.PSc.6.2

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 ...

Boundary Conditions for Current Density

Current density becomes discontinuous across an interface of materials with different electrical conductivities. The normal component of the current ...

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Theory of Metallic Conduction

The conduction of free electrons inside a conductor is best described by quantum mechanics. However, a classical model makes predictions close to the ...

Obtain, evaluate, and communicate information to explain how current is affected by changes in composition, length, temperature, and diameter of wire.

PS.PSc.6.3

Magnetism

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Magnetic Field Lines

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Magnetic Flux

The magnetic flux measures the number of magnetic field lines passing through a given surface area. The SI unit for magnetic flux is the weber (Wb). ...

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Motion Of A Charged Particle In A Magnetic Field

A charged particle experiences a force when moving through a magnetic field. Consider the field to be uniform and the charged particle to move ...

Magnetic Force

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Magnetic Force On A Current-Carrying Conductor

Moving charges experience a force in a magnetic field. Since the magnetic fields produced by moving charges are proportional to the current, a conductor ...

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Magnetic Force On Current-Carrying Wires: Example

In a magnetic field, moving charges encounter a force. If a wire contains these moving charges, i.e., if the wire is carrying a current, then a force acts ...

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Force On A Current Loop In A Magnetic Field

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Torque On A Current Loop In A Magnetic Field

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The Hall Effect

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Consider a point charge moving with a ...

Biot-Savart Law

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Biot-Savart Law: Problem-Solving

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Consider a mobile phone battery ...

Magnetic Field Due to Two Straight Wires

Consider two parallel straight wires carrying a current of 10 A and 20 A in the same direction and separated by a distance of 20 cm. Calculate the ...

magnetic-force-between-two-parallel-currents

Magnetic Force Between Two Parallel Currents

Two long, straight, and parallel current-carrying conductors exert a force of equal magnitude on one another. The direction of the force depends on the ...

Magnetic Field Of A Current Loop

Consider a circular loop with a radius a, that carries a current I. The magnetic field due to the current at an arbitrary point P along the axis of the ...

Divergence and Curl of Magnetic Field

The magnetic field due to a volume current distribution given by the Biot&#8211;Savart Law can be expressed as follows:

To evaluate the divergence of the ...

Ampere's Law

A fundamental property of a static magnetic field is that it is not conservative, unlike an electrostatic field. Instead, there is a relationship between ...

Ampere's Law: Problem-Solving

Ampere&#x27;s law states that for any closed looped path, the line integral of the magnetic field along the path equals the vacuum permeability times the ...

Solenoids

A solenoid is a conducting wire coated with an insulating material, wound tightly in the form of a helical coil. The magnetic field for a solenoid is the ...

Use models to explain magnetism in terms of domains, interactions of poles, and magnetic fields.

PS.PSc.6.4

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

Displacement Current in an RC Circuit

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