Name: maxwell’s equations at a point
Uploaded: 2025-05-22T09:06:01.000Z
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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

Differential Form of Maxwell's Equations

As quatro equações de Maxwell explicam os fundamentos do eletromagnetismo.
Aplicando o teorema da divergência à lei de Gauss e reescrevendo a carga fechada em termos da densidade total de carga dá a forma diferencial da lei de Gauss.
Da mesma forma, a aplicação do teorema da divergência à lei de Gauss para o campo magnético fornece a forma diferencial da lei de Gauss no magnetismo.
Além disso, reorganizar a lei de Faraday e aplicar o teorema de Stoke a ela dá a lei de Faraday na forma diferencial.
Considere a equação de Ampère-Maxwell, na qual a corrente fechada pode ser expressa em termos da integral da densidade da corrente.
Agora, reorganizar os termos e aplicar o teorema de Stoke a ele dá a forma diferencial da equação de Ampère-Maxwell.
Agrupar os termos de campos elétricos e magnéticos de um lado e as fontes que produzem esses campos do outro lado sugere que todos os campos eletromagnéticos são produzidos por cargas e correntes.
A forma integral da equação de Maxwell se aplica a campos em uma região contendo carga ou corrente, enquanto a forma diferencial se aplica a um determinado ponto com densidades de carga e corrente.

<ol>
	<li>David J. Griffiths (2013) Introdução à Eletrodinâmica, Quarta Edição: Pearson. Pág. 338-339.</li>
	<li>OpenStax. (2019). Física Universitária Vol. 2. [Versão Web]. seção 16.1; páginas 700–703. Obtido de <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>

forma diferencial das equações de maxwell

James Clerk Maxwell (1831-1879) foi um dos principais contribuidores para a física no século XIX. Ele é, provavelmente, mais conhecido por ter combinado o conhecimento existente das leis da eletricidade e das leis do magnetismo, utilizando suas novas ideias, para formar uma teoria eletromagnética abrangente completa, representada pelas equações de Maxwell. As quatro leis básicas da eletricidade e do magnetismo foram descobertas experimentalmente pelo trabalho de físicos como Oersted, Coulomb, Gauss e Faraday. Maxwell descobriu inconsistências lógicas nesses resultados anteriores e identificou a incompletude da lei de Ampère como sua causa. As equações de Maxwell e a lei da força de Lorentz abrangem todas as leis da eletricidade e do magnetismo.

As formas integrais das equações de Maxwell contêm todas as informações sobre a interdependência dos campos e fontes em uma determinada região do espaço. No entanto, essas equações não permitem estudar a interação entre os vetores de campo e suas relações com as densidades de fontes em pontos individuais. As equações de Maxwell na forma diferencial podem ser derivadas aplicando as equações de Maxwell na forma integral a caminhos, superfícies e volumes fechados infinitesimais, de forma que o limite encolha para pontos. As equações diferenciais relacionam as variações espaciais dos vetores de campo elétrico e magnético em um determinado ponto com suas variações temporais.

Além disso, a forma diferencial das equações de Maxwell também correlaciona as variações espaciais de ambos os campos com as densidades de carga e de corrente em um determinado ponto. Agrupando os termos dos campos elétrico e magnético em um lado e as fontes que produzem esses campos do outro sugere que as cargas e correntes produzem todos os campos eletromagnéticos. As equações de Maxwell mostram que as cargas produzem campos eletromagnéticos, e as leis da força afirmam que os campos afetam as cargas.

Forma Diferencial das Equações de Maxwell

James Clerk Maxwell (1831–1879) foi um dos contribuintes significativos para a física no século XIX. Ele é provavelmente mais conhecido por ter ...

Education

JoVE Core

Physics

Física

Unit 1

Unidade 1

Electromagnetic Induction

Indução Eletromagnética

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

maxwell’s equations at a point

Maxwell’s equations at a point describe how electric and magnetic fields change in space and time. They connect field behavior to charge and current density at a single location. This point-by-point form is useful when studying electromagnetic fields in detail.

James Clerk Maxwell was a major physicist of the nineteenth century. He combined earlier laws of electricity and magnetism into one electromagnetic theory. The basic experimental laws came from scientists such as Oersted, Coulomb, Gauss, and Faraday. Maxwell also found that earlier results contained logical problems, especially because Ampère’s law was incomplete.

Maxwell’s equations and the Lorentz force law together cover the laws of electricity and magnetism. The integral forms of Maxwell’s equations describe the relationship between fields and sources over a region in space. They do not, however, show how field vectors relate to source densities at individual points.

The differential form is found by applying the integral equations to very small closed paths, surfaces, and volumes. As these shapes shrink to points, the equations become local. This form links spatial changes in the electric and magnetic fields to their time changes. It also links those field changes to charge and current densities at the same point.

When the equations are grouped this way, the electric and magnetic fields appear on one side and the sources appear on the other. This arrangement shows that charges and currents produce electromagnetic fields. The force laws then show that fields act back on charges.

Maxwell’s Equations at a Point

induction

Magnetic Flux and Induced Current

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ampere-maxwell's law: problem-solving

Displacement Current in an RC Circuit

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differential form of maxwell's equations

Maxwell’s equations at a point describe how electric and magnetic fields change in space and time. They connect field behavior to charge and current density at a single location. This point-by-point form is useful when studying electromagnetic fields in detail. James Clerk Maxwell was a major physicist of the nineteenth century. He combined earlier laws of electricity and magnetism into one electromagnetic theory. The basic experimental laws came from scientists such as Oersted, Coulomb,...

dc generator

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magnetic damping

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superconductor

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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...

motional emf

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

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induced electric fields: applications

Electric Fields from Changing Magnetism

An electric field can be induced by a changing magnetic field. This induced electric field is nonconservative, which means it does not do net work when a charge moves around a closed path. That behavior is different from an electrostatic field, which comes from a fixed charge distribution. An electrostatic field is conservative, so it does no net work over a closed path. Because of this difference, electric potential can be linked to the electrostatic field, but not to the induced field. When...

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Eddy Currents in Recycling, Brakes, and Heat

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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...

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electromagnetic fields

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maxwell's equation of electromagnetism

Maxwell's Equations and Light Waves

Maxwell's equations bring electricity, magnetism, and light together in one electromagnetic theory. James Clerk Maxwell was a major nineteenth-century physicist, and his work helped shape modern physics. He also made important contributions to the kinetic theory of gases, color vision, and the study of Saturn's rings. Maxwell built on earlier experiments in electricity and magnetism from Oersted, Coulomb, Gauss, and Faraday. These physicists had discovered the basic laws separately. Maxwell...

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Alternators produce alternating current by rotating a conducting loop in a magnetic field. As the loop turns inside the field, the magnetic flux through the coil changes with time. That changing flux induces an emf, or electric voltage, in the circuit. A simple alternator uses a loop placed in a uniform magnetic field. The loop connects to split rings, and brushes carry the current to the external circuit. As the loop rotates, the magnetic flux changes sinusoidally, so the induced emf also...

back emf

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key terms and concepts

Learn how Maxwell’s equations in differential form describe electric and magnetic fields at a single point, and how charges and currents create them.

Maxwell’s Equations at a Point | JoVE K12

physics

JoVE Core - Fisica apresenta a teoria e a aplicação de tópicos fundamentais da física, incluindo cinemática, dinâmica, mecânica dos fluidos, estática, ondas e acústica, com vídeos animados concisos e de fácil compreensão. Além disso, os vídeos de cientistas em ação demonstram a aplicação de conceitos relacionados em experimentos de pesquisa clássica e original realizados nos laboratórios de todo o mundo hoje em dia.

PS.Phy.5.2

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HIgh School

PS.9

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

To calculate the other physical quantities in kinematics, we must introduce the time variable. The time variable allows us not only to state the position ...

Instantaneous Velocity - I

The average velocity during a time interval cannot tell us how fast or in what direction a particle is moving at any given time during the interval. To ...

Instantaneous Velocity - II

Instantaneous velocity is the quantity that measures how fast an object is moving along its path. In other words, the instantaneous velocity of an object ...

Average Acceleration

The importance of understanding acceleration spans our day-to-day experiences, as well as the vast reaches of outer space and the tiny world of subatomic ...

Instantaneous Acceleration

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 ...

Position and Displacement Vectors

To describe the motion of an object, one should first be able to describe its position (where it is at any particular time). More precisely, the position ...

average-and-instantaneous-velocity-vectors

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 ...

Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, &#916;v, therefore the ...

Direction of Acceleration Vectors

Acceleration occurs when velocity changes in magnitude (an increase or decrease in speed), direction, or both. Although acceleration is in the direction ...

Linear Momentum

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

Force and 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

When a voltage is applied to a conductor, an electrical field is generated, and charges in the conductor feel the force due to the electrical field. The ...

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 ...

Electrical Conductivity

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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

Magnets are commonly found in everyday objects, such as toys, hangers, elevators, doorbells, and computer devices. Experimentation on these magnets shows ...

Magnetic Fields

A moving charge or a current creates a magnetic field in the surrounding space, in addition to its electric field. The magnetic field exerts a force on ...

Magnetic Field Lines

The representation of magnetic fields by magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. Each of the ...

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). ...

motion-of-a-charged-particle-in-a-magnetic-field

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

In addition to the electric forces between electric charges, moving electric charges exert magnetic forces on each other. A magnetic field is created by a ...

magnetic-force-on-a-current-carrying-conductor

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 ...

magnetic-force-on-current-carrying-wires-example

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 ...

force-on-a-current-loop-in-a-magnetic-field

Force On A Current Loop In A Magnetic Field

Magnetic forces on wires carrying current are most frequently applied in motors. A DC motor is a device that converts electrical energy into mechanical ...

torque-on-a-current-loop-in-a-magnetic-field

Torque On A Current Loop In A Magnetic Field

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The Hall Effect

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Magnetic Field due to Moving Charges

A stationary charge creates and interacts with the electric field, while a moving charge creates a magnetic field.
Consider a point charge moving with a ...

Biot-Savart Law

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Biot-Savart Law: Problem-Solving

The magnitude and direction of a magnetic field created by a steady current can be calculated using the Biot-Savart law.
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.

Maxwell’s Equations at a Point

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