Name: maxwell’s equations at a point
Uploaded: 2025-05-22T09:06:01.000Z
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K12 Standards

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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.6.5

Differential Form of Maxwell's Equations

Четыре уравнения Максвелла объясняют основы электромагнетизма.
Применение теоремы о дивергенции к закону Гаусса и переписывание вложенного заряда в терминах полной плотности заряда дает дифференциальную форму закона Гаусса.
Аналогично, применение теоремы о дивергенции к закону Гаусса для магнитного поля дает дифференциальную форму закона Гаусса в магнетизме.
Более того, перестановка закона Фарадея и применение к нему теоремы Стока дает закон Фарадея в дифференциальной форме.
Рассмотрим уравнение Ампера-Максвелла, в котором замкнутый ток может быть выражен в терминах интеграла плотности тока.
Теперь, переставляя члены и применяя к ним теорему Стока, мы получаем дифференциальную форму уравнения Ампера-Максвелла.
Группировка членов электрических и магнитных полей с одной стороны и источников, создающих эти поля, с другой стороны, предполагает, что все электромагнитные поля порождаются зарядами и токами.
Интегральная форма уравнения Максвелла применима к полям в области, содержащей заряд или ток, тогда как дифференциальная форма применяется в данной точке с плотностями заряда и тока.

<ol>
	<li>Дэвид Гриффитс (2013) Введение в электродинамику, четвертое издание: Пирсон. Стр. 338-339.</li>
	<li>OpenStax. (2019). Университетская физика Том 2. [Веб-версия]. раздел 16.1; стр. 700–703. Получено из <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>

Дифференциальная форма уравнений Максвелла

Джеймс Клерк Максвелл (1831-1879) был одним из значимых учёных XIX века. Он, вероятно, наиболее известен тем, что объединил существующие знания о законах электричества и магнетизма с своими идеями, чтобы сформировать полную, всеобъемлющую электромагнитную теорию, представленную уравнениями Максвелла. Четыре основных закона электричества и магнетизма были открыты экспериментально благодаря работе физиков, таких как Эрстед, Кулон, Гаусс и Фарадей. Максвелл обнаружил логические несоответствия в этих ранних результатах и идентифицировал неполные законы Ампера как их причину. Уравнения Максвелла и закон Лоренца описывают все законы электричества и магнетизма.

Интегральные формы уравнений Максвелла содержат всю информацию обо взаимосвязи между полевыми и исходными величинами в заданной области пространства. Однако эти уравнения не позволяют изучать взаимодействие между векторами полей и их взаимосвязь с плотностями источников в отдельных точках. Уравнения Максвелла в дифференциальной форме могут быть получены путём применения интегральных уравнений Максвелла к бесконечно малым замкнутым путям, поверхностям и объёмам, так что предел сжимается до точек. Дифференциальные уравнения связывают пространственные вариации векторов электрического и магнитного поля в заданной точке с их временными вариациями.

Более того, дифференциальная форма уравнений Максвелла также связывает пространственные вариации обоих полей с плотностями зарядов и токов в заданной точке. Группировка термов электрических и магнитных полей на одной стороне и источников, создающих эти поля, на другой, предполагает, что заряды и токи создают все электромагнитные поля. Уравнения Максвелла показывают, что заряды создают электромагнитные поля, а законы сил утверждают, что поля воздействуют на заряды.

Джеймс Клерк Максвелл (1831–1879) был одним из значительных вкладчиков в физику девятнадцатого века. Он, вероятно, наиболее известен тем, что у него ...

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

Electromagnetic Induction

Электромагнитная индукция

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

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

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 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

How a DC Generator Makes Current

A DC generator makes electrical energy from mechanical motion. It produces DC pulses with the same polarity, unlike an alternator, which gives sinusoidal AC current. The two machines have similar construction, but a DC generator uses a single split ring instead of a pair of slip rings. That split ring is called a commutator. It works like a rotary switch that changes the contact with the brushes after each half turn of the conducting loop. This action keeps the induced emf at the same...

magnetic damping

Eddy Currents and Magnetic Braking

Eddy currents can create strong drag on moving objects. This drag is called magnetic damping. A metallic pendulum bob swinging between the poles of a strong magnet feels this force as it enters and leaves the field, so its motion dies out quickly. The effect is much smaller for a slotted metal plate. As the plate enters the magnetic field, a changing flux induces an emf, or electromotive force. The slots limit the size of the current loops, and nearby loops can carry currents in opposite...

superconductor

Superconductor Critical Temperature

Superconductivity is the state in which a material has no electrical resistance and magnetic fields cannot penetrate it. A material that reaches this state is called a superconductor. The key idea in the transcript is the critical temperature, which is the temperature at which the change happens. Heike Kamerlingh Onnes of Leiden University first observed this behavior in mercury in 1911. He cooled a mercury sample in liquid helium to study how resistance changed with temperature. As the...

types of superconductors

Superconductors and Magnetic Levitation

Superconductors are materials that can carry electric current with zero resistance when they are cooled below a critical temperature. At this point, they also show the Meissner effect, which means they push magnetic fields out of their interior. A light permanent magnet can then levitate in a stable position above the superconducting sample. This magnetic levitation has real uses. High-speed trains have been built with strong superconducting magnets so they can float above the track and avoid...

faraday's law

Magnetic Flux and Induced Emf

Faraday’s law connects magnetic flux changes to induced emf, or voltage. A change in the magnetic field or in the coil’s orientation with respect to the field can produce this voltage. Magnetic flux describes how many magnetic field lines pass through a given surface area.

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

Induced current from changing magnetic flux depends on how a magnetic field interacts with an area. Magnetic flux changes when the field strength changes, when the area it passes through changes, or when the field turns relative to the surface. When any of these factors vary, the magnetic flux also varies. A change in area can happen in two common ways. The flux through a rectangular loop increases as the loop moves into a magnetic field. The magnetic flux through a rotating coil also changes...

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

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...

eddy currents

Eddy Currents in Recycling, Brakes, and Heat

Eddy currents affect how metals move, stop, and heat up. They form only in conductors, so they are useful for separating metals from nonmetals and for other practical devices. In recycling centers, trash can be sent down a ramp under a strong magnet. Metals in the trash slow down because eddy currents resist their motion. Nonmetals keep moving, so the materials separate. This method works for metals in general, not only ferromagnetic ones. Eddy currents also help in metal detectors. A...

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

Electric and Magnetic Fields in Motion

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...

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...

symmetry in maxwell's equations

Electric and Magnetic Field Symmetry

Electric and magnetic field symmetry is a key part of Maxwell's equations. After the fields are calculated with Maxwell's four equations, the Lorentz force equation gives the force those fields exert on a charged particle moving at a certain velocity. The Lorentz force law combines the effect of the electric field and the magnetic field on the moving charge. Maxwell's equations and the Lorentz force law together describe electricity and magnetism. The symmetry in Maxwell's framework is easier...

electric generator: alternator

How Alternators Produce AC Power

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

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

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 Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

PS.Phy.5.2

Virginia  Standards of Learning

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

Force and momentum are intimately related. Force acting over time can change momentum, and Newton&#39;s second law of motion can be stated in its most ...

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

Many materials exhibit a simple relationship between the values of current, voltage, and resistance, known as Ohm&#8217;s law. The current that flows ...

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

In perfect conductors, the electric field inside is always zero due to the abundance of free electrons, which nullify any field by flowing. As a result, ...

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

The most common application of magnetic force on current-carrying wires is in electric motors. These consist of loops of wire, which are placed between ...

The Hall Effect

Edwin H. Hall, in the year 1879, devised an experiment that could be used to identify the polarity of the predominant charge carriers in a conducting ...

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

The Biot-Savart law gives the magnitude and direction of the magnetic field produced by a current. This empirical law was named in honor of two ...

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 ...

Magnetic Field of a Solenoid

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

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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