Name: free current and magnetization in ampere’s law
Uploaded: 2023-09-18T17:11:20.000Z
Duration: 1 min 22 s
Channel: JoVE

K12 Standards

Virginia  Standards of Learning

HIgh School

Physical Science

PS.9

PS.9.d

Ampere's Law in Matter

The differential form of Ampere&#39;s Law in a vacuum correlates the curl of the magnetic field with the current density.
In a magnetized material, the current density is the sum of the free and bound current densities.
The free current arises due to the motion of free electrons within the material, while the bound current arises due to the alignment of magnetic dipole moments.
So, Ampere&#39;s Law is modified in a magnetized material to incorporate the bound and free current densities.
The bound volume current density is the curl of the magnetization.
Rearranging the terms involving the curl function, the equation for free current is obtained.
The magnetic field over vacuum permeability minus the magnetization is defined as the magnetic field intensity, H.
So, Ampere&#39;s Law in matter states that the curl of H equals the free current density.
In integral form, the line integral of H along an Amperian loop equals the net free current passing through the loop.
The magnetic field intensity in magnetostatics is analogous to the displacement vector in electrostatics.

ampere's law in matter

The total current density in magnetized material is the sum of the free and bound current densities. The free current arises due to the motion of free electrons within the material, while the bound current arises due to the alignment of magnetic dipole moments.
The differential form of Ampere's law in vacuum states that the curl of the magnetic field equals the permeability times the current density. In a magnetized material, the law is modified to incorporate the free and bound current densities.
<img alt="Static equilibrium diagram, ΣF=0 equations, forces on beam system, torque balance study." src="/files/ftp_upload/14201/14201_Equation_1.png" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation1">
The equation is modified when the bound current density is expressed as the curl of the magnetization vector. The terms involving the curl of the parameters are rearranged as
<img alt="Ball valve diagram; labeled components; flow control mechanism; industrial application." src="/files/ftp_upload/14201/14201_Equation_2.png" style="text-align: center; height: 44px;" data-altupdatedat="1761830439000" data-alt="Equation2">
The terms in the bracket can be expressed as magnetic field intensity, <img alt="Magnetic field vector symbol, denoted as \(\vec{H}\)." src="/files/ftp_upload/14201/14201eq06v2.png" style="margin: auto;" data-altupdatedat="1761830439000" data-alt="Equation 6">. It is the magnetic field vector upon permeability minus the magnetization vector.
<img alt="static equilibrium, ΣFy=0 equation, physics diagram, force analysis setup, educational use" src="/files/ftp_upload/14201/14201_Equation_3.png" style="text-align: center; height: 50px;" data-altupdatedat="1761830439000" data-alt="Equation3">
Thus, Ampere's law in matter states that the curl of the <img alt="Magnetic field vector symbol, denoted as \(\vec{H}\)." src="/files/ftp_upload/14201/14201eq06v2.png" style="margin: auto;" data-altupdatedat="1761830439000" data-alt="Equation 6"> equals the free current density.
<img alt="Equilibrium diagram, ΣFx=0, statics principles, balance forces, vector resolution." src="/files/ftp_upload/14201/14201_Equation_4.png" style="text-align: center; height: 28px;" data-altupdatedat="1761830439000" data-alt="Equation4">
The equation can be written in integral form as
<img alt="Static equilibrium equation diagram; ΣFx=0 ΣFy=0 MA=0; physics balance concept." src="/files/ftp_upload/14201/14201_Equation_5.png" style="text-align: center; height: 45px;" data-altupdatedat="1761830439000" data-alt="Equation5">
The line integral of <img alt="Magnetic field vector symbol, denoted as \(\vec{H}\)." src="/files/ftp_upload/14201/14201eq06v2.png" style="margin: auto;" data-altupdatedat="1761830439000" data-alt="Equation 6"> along a closed Amperian loop equals the net free current passing through the loop.
The magnetic field intensity helps express Ampere's law in matter in terms of the free current. Similarly, the electric displacement vector helps in representing Gauss's law in matter in terms of the free charge. Thus, the magnetic field intensity vector in magnetostatics is analogous to the electric displacement vector in electrostatics.
Since an electromagnet is formed when current is passed through a coil, the free current is the known parameter. On the other hand, the B field is a material characteristic. Hence, the H field is often used in experiments rather than the B field.

The total current density in magnetized material is the sum of the free and bound current densities. The free current arises due to the motion of free ...

Education

JoVE Core

Physics

Unit 1

Sources of Magnetic Fields

KEY TERMS AND CONCEPTS

free current and magnetization in ampere’s law

Ampere’s law in matter links magnetic fields to the free current in a magnetized material. In such materials, the total current density has two parts. These are the free current density and the bound current density.

The free current comes from the motion of free electrons inside the material. The bound current comes from the alignment of magnetic dipole moments. In vacuum, the differential form of Ampere’s law says that the curl of the magnetic field equals permeability times current density. In a magnetized material, this form is changed to include both free and bound current densities.

The equation is simplified by writing the bound current density as the curl of the magnetization vector. After rearranging the terms, the bracketed expression becomes the magnetic field intensity, or H field. This quantity is the magnetic field vector divided by permeability minus the magnetization vector. With that change, Ampere’s law in matter says that the curl of H equals the free current density.

The same idea can be written in integral form. The line integral of H around a closed Amperian loop equals the net free current passing through the loop. The H field is useful because it expresses Ampere’s law in matter in terms of free current. It is also analogous to the electric displacement vector in electrostatics, which helps represent Gauss’s law in matter in terms of free charge.

In magnetostatics, the magnetic field intensity vector plays a practical role in experiments. For an electromagnet, current passes through a coil, so the free current is the known quantity. The B field depends on the material, so H is often used instead of B when measurements are made.

Free Current and Magnetization in Ampere’s Law

free-current-and-magnetization-in-amperes-law

biot-savart law

Magnetic Field from a Current Element

The Biot-Savart law links a current-carrying wire to the magnetic field it creates. It gives both the magnitude and the direction of that field. The law is named for Jean-Baptiste Biot and Félix Savart, who studied how a straight wire carrying current interacts with a permanent magnet. A current-carrying wire produces a magnetic field around it. To describe the field from a tiny section of wire, we use an infinitesimal current element, dl. The direction of dl follows the direction of the...

biot-savart law: problem-solving

Calculating Magnetic Fields with Biot-Savart

The Biot-Savart law is used to calculate the magnetic field made by a steady current. A mobile phone battery bank and the wire between the two can be used as an example of that current. At a field point P, the goal is to find the magnitude of the magnetic field created by the current in the wire. The first step is to look at a small current element of length dl. This element is at a distance r from the field point. The known values are the current, the length of the current element, the...

magnetic field due to moving charges

Magnetic Fields from Charges in Motion

A moving charge creates a magnetic field. For a point charge moving at constant velocity, the field at any point depends on the charge and the distance from the source point to the field point. It is directly proportional to the charge magnitude and inversely proportional to the square of that distance. The magnetic field also has a direction. Unlike the electric field, it is always perpendicular to the plane formed by the line from the source to the field point. Its strength depends on the...

magnetic field due to two straight wires

Net Magnetic Field at the Midpoint of Two Wires

Two parallel straight wires can create a net magnetic field at the point halfway between them. In this example, the wires carry currents of 10 A and 20 A and are separated by 20 cm. The midpoint, labeled P2, is 10 cm from each wire. The magnetic field lines around each wire form counterclockwise concentric circles. At the midpoint, the field from one wire can point in the opposite direction from the field from the other wire. The net magnetic field is found by magnetic field superposition,...

magnetic force between two parallel currents

magnetic-force-between-two-parallel-currents

Parallel Currents and the Pinch Effect

Parallel current-carrying wires can pull on each other with a magnetic force. The force has the same size on both conductors. Its direction depends on whether the currents flow in the same direction or in opposite directions. The magnetic field from one long, straight wire acts on a finite length of the other wire. The force on that wire comes from the product of its current and the cross product of its length vector with the field. Using the right-hand rule, the second conductor is pulled...

magnetic field of a current loop

Field Along the Axis of a Current Loop

A circular current loop creates a magnetic field along its axis. For a loop of radius a carrying current I, the field at a point P can be found with the Biot-Savart law. The point P lies a distance x from the center of the loop. The field from a tiny current element dl points at an angle θ. At every point on the loop, the current element and the unit vector toward P are perpendicular. Using that relationship and writing r in terms of x and a gives the size of the field from each element. The...

divergence and curl of magnetic field

Magnetic Field Loops and Ampere's Law

Magnetic field loops and Ampere's law describe how magnetic fields behave around current. For a volume current distribution, the magnetic field can be written using the Biot–Savart law. The divergence and curl of that field can then be found with vector analysis. To find the divergence, the divergence operator is applied to both sides of the Biot–Savart equation. After using the vector product rule, the integrand simplifies. The term with the curl of the current density becomes zero because...

ampere's law

Magnetic Circulation Around a Current

Magnetic circulation around a current is described by Ampère’s law. A static magnetic field is not conservative, unlike an electrostatic field. Instead, the magnetic field is linked to electric current, which is its source. Ampère’s law uses the line integral of the magnetic field around a closed path. This integral is proportional to the current enclosed by the loop. The law applies only when currents are steady and when no magnetic materials or time-varying electric fields are present. The...

ampere's law: problem-solving

Choosing an Amperian Loop for Magnetic Fields

Ampere's law connects a closed loop path with the current enclosed by that loop. The line integral of the magnetic field around the path equals the vacuum permeability times the enclosed current. A right-hand rule helps set the sign of the current. If the fingers curl with the integration path, the thumb gives positive current. Current opposite the thumb direction is negative. Solving magnetic field problems with Ampere's law starts by checking symmetry in the current distribution. If the...

solenoids

Solenoids as Controllable Electromagnets

A solenoid is a tightly wound coil of insulated conducting wire. It acts as an electromagnet, and its magnetic field comes from the combined field of each turn in the coil. In an ideal solenoid, the magnetic field inside is almost uniform and points along the solenoid axis. The magnetic field outside is nearly zero. Each turn can be treated as a circular current-carrying loop, and each loop creates a magnetic dipole moment. The dipole moment from one turn equals the current multiplied by the...

magnetic field of a solenoid

Solenoid Field Strength Example

A solenoid is a conducting wire coated with an insulating material and wound tightly into a helical coil. Its magnetic field comes from the vector sum of the fields from each turn. For an ideal solenoid, the field inside depends directly on the number of turns per unit length and the current. The magnetic field outside an ideal solenoid is zero. In this example, a solenoid has 100 turns wrapped around a cylinder that is 1 cm in diameter and 10 cm long. A current of 0.5 A is passed through the...

toroids

Toroid Magnetic Fields and Turn Count

A toroid is a donut-shaped coil made from a single conducting wire. It usually has many circular loops that sit perpendicular to its axis. When current flows through it, the magnetic field forms circular lines around the axis. The right-hand rule shows the field direction: curl the fingers of your right hand with the current, and your thumb points in the magnetic field direction. Inside a toroid, the magnetic field changes with distance from the axis. It varies inversely with that distance.

magnetic vector potential

Solenoid Vector Potential Explained

The magnetic vector potential can be used to describe the magnetic field inside an ideal solenoid. In magnetostatics, the magnetic field has zero divergence, so it can be written as the curl of a vector potential. This idea connects the magnetic field to a potential in the same way that electrostatics connects the electric field to an electric potential. For an ideal solenoid, the magnetic field inside is given by the vacuum permeability, the number of turns per unit length, and the current.

potential due to a magnetized object

Magnetization and Bound Currents

Magnetization describes how a magnetic material responds to an external magnetic field. In paramagnets and ferromagnets, magnetic dipoles align with the field. In diamagnets, the dipoles align opposite to the field. Magnetization is the dipole moment per unit volume, and it plays a role similar to polarization in electrostatics. The magnetic vector potential of a magnetized material can be written in terms of its aligned dipoles. This form is simplified with vector analysis. One term...

magnetostatic boundary conditions

Magnetic Fields at Material Interfaces

Magnetic fields change at the boundary between two magnetic materials, and the change depends on the surface conditions at the interface. A magnetic field can be discontinuous at a surface current, just as an electric field can change across a surface charge. The component of the magnetic field that points perpendicular to the boundary stays continuous. The component parallel to the boundary behaves differently. It becomes discontinuous by an amount equal to the vacuum permeability times the...

Ampere’s law in matter links magnetic fields to the free current in a magnetized material. In such materials, the total current density has two parts. These are the free current density and the bound current density. The free current comes from the motion of free electrons inside the material. The bound current comes from the alignment of magnetic dipole moments. In vacuum, the differential form of Ampere’s law says that the curl of the magnetic field equals permeability times current density.

magnetic susceptibility and permeability

How Materials Respond to Magnetic Fields

Magnetic susceptibility and permeability describe how materials respond to a magnetic field. In linear magnetic materials, magnetization is proportional to the magnetic field intensity. The proportionality constant is magnetic susceptibility, which is a dimensionless number. The value of susceptibility depends on the material. Diamagnetic materials develop moments opposite to the applied field. Their susceptibility is slightly negative, usually about 10^-5 to 10^-6. This effect comes from the...

magnetic moment of an electron

Electron Spin and Bohr Magneton

Electrons create magnetic effects through both orbital motion and spin. When an electron moves around a nucleus, it acts like a tiny current loop. That current produces a magnetic dipole moment that is proportional to the electron's orbital angular momentum. The orbital angular momentum is quantized in units of the reduced Planck's constant. Because of that, the magnetic dipole moment is also quantized in the Bohr magneton. The Bohr magneton has a value of 9.27 x 10 -24 Am 2. Electrons also...

diamagnetism

Diamagnetism and Magnetic Repulsion

Diamagnetism is the magnetic response of materials that are repelled by an external magnetic field. These materials contain paired electrons, so they have no net magnetic moment at first. When a magnetic field is applied, moments opposite to the field are induced. Anton Brugmans discovered this effect in 1778 after observing that bismuth is repelled by magnetic fields. He used this observation to suggest that diamagnets are pushed away from magnets. In a non-uniform magnetic field, diamagnets...

paramagnetism

Paramagnetism and Magnetic Alignment

Paramagnetism and magnetic alignment explain how some materials respond to an outside magnetic field. Paramagnets contain unpaired electrons, so they have a finite magnetic moment. When no magnetic field is present, these moments point in random directions, and the net moment is zero. When an external field is applied, a torque acts on the magnetic moments and pulls them toward the field direction. At the same time, thermal motion of the electrons creates a competing torque that tries to...

ferromagnetism

Magnetic Domains and Hysteresis

Ferromagnetic materials like iron, nickel, and cobalt contain magnetic domains. Inside each domain, magnetic dipoles line up in the same direction. Quantum mechanical coupling keeps this alignment strong, so room-temperature thermal motion usually cannot break it. As a result, each domain has a net dipole moment. Some materials have weaker coupling and become ferromagnetic only at lower temperatures. When an external magnetic field is applied, the dipoles rotate to match the field. At the same...

key terms and concepts

Learn how Ampere’s law in matter uses free current, bound current, and magnetization to define the H field and the Amperian loop form.

Free Current and Magnetization in Ampere’s Law | 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.

Electric Charges

From lightning during thunderstorms to electronic devices, the phenomenon of electromagnetism is all around us. The electromagnetic force is one of the ...

sources-and-properties-of-electric-charge

Sources and Properties of Electric Charge

All objects we see around us consist of atoms, which combine to form molecules. The lightest element in the universe is hydrogen, and a hydrogen atom ...

Conductors and Insulators

Some materials may easily let electrical charges pass through them, while others obstruct their flow. The former are called conductors and the latter ...

Charging Conductors By Induction

The Earth is a good conductor of electricity, and it is so big that it can be considered an infinite source or sink of charges. It can easily exchange ...

Coulomb's Law

Experiments with electric charges have shown that if two objects each have an electric charge, they exert an electric force on each other. The magnitude ...

coulomb-s-law-and-the-principle-of-superposition

Coulomb's Law and The Principle of Superposition

Coulomb&#39;s Law describes the force experienced by two point charges under each other&#39;s presence. But what if there are more than two charges? For ...

Electric Field

Consider two point charges, each exerting Coulomb force on the other. It is possible to describe the Coulomb interaction via an intermediate step by ...

electric-field-of-two-equal-and-opposite-charges

Electric Field of Two Equal and Opposite Charges

Atoms generally contain the same number of positively and negatively charged particles, protons, and electrons. Hence, they are electrically neutral. ...

Continuous Charge Distributions

Imagine a bucket of water. It contains many molecules, of the order of 1026 molecules. Thus, although it contains discrete elements (molecules) at the ...

electric-field-of-a-continuous-line-charge

Electric Field of a Continuous Line Charge

In physics, symmetry in a system means that something in the considered system remains unchanged due to a specific operation to which it is subjected. For ...

Electric Field of a Charged Disk

The simplest case of a surface charge distribution is the uniformly charged disk. Calculating its electric field also helps us calculate the electric ...

Electric Field Lines

The three-dimensional representation of the electric field of a positive point charge requires tracing the electric field vectors, whose lengths decrease ...

Charge on a Conductor

An interesting property of a conductor in static equilibrium is that extra charges on the conductor end up on its outer surface, regardless of where they ...

Properties of Electric Field Lines

The definition of electric field lines greatly eases the visualization of electric fields, a vector field, especially in the presence of many charges. The ...

electric-field-at-the-surface-of-a-conductor

Electric Field at the Surface of a Conductor

Consider a conductor in electrostatic equilibrium. The net electric field inside a conductor vanishes, and extra charges on the conductor reside on its ...

Electric Flux

The concept of flux describes how much of something goes through a given area. More formally, it is the dot product of a vector field within an area. For ...

electric-field-of-a-non-uniformly-charged-sphere

Electric Field of a Non Uniformly Charged Sphere

Gauss&#39;s law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for ...

Calculation of Electric Flux

Consider the electric field of an oppositely charged, parallel-plate system and an imaginary box between those plates. Let the bottom face of the box be ...

electric-field-of-parallel-conducting-plates

Electric Field of Parallel Conducting Plates

Gauss&#39; law relates the electric flux through a closed surface to the net charge enclosed by that surface. Gauss&#39;s law can be applied to find the ...

Gauss's Law

If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one ...

Electric Potential Energy

When an electric field accelerates a free positive charge q, it is given kinetic energy. The process is analogous to an object accelerated by a ...

Gauss's Law: Problem-Solving

Gauss&#39;s law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain ...

electric-potential-energy-in-a-uniform-electric-field

Electric Potential Energy in a Uniform Electric Field

When an electric field accelerates a free positive charge, it acquires kinetic energy. This process is analogous to an object being accelerated by a ...

Gauss's Law: Spherical Symmetry

A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In ...

electric-potential-energy-of-two-point-charges

Electric Potential Energy of Two Point Charges

The electric potential energy of a test charge in a uniform eclectic field can be generalized to any electric field produced by static charge ...

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along ...

electric-potential-and-potential-difference

Electric Potential and Potential Difference

Suppose a positive test charge moves away from a positive static charge, then the Coulomb force does positive work, and its electric potential energy ...

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane ...

finding-electric-potential-from-electric-field

Finding Electric Potential From Electric Field

For a system of charges, it is easy to calculate the system&#39;s potential because potential is a scalar quantity. However, in some instances where ...

Electric Field Inside a Conductor

When a conductor is placed in an external electric field, the free charges in the conductor redistribute and very quickly reach electrostatic equilibrium. ...

Calculations of Electric Potential I

Consider a ring of radius R with a uniform charge density &#955;. What will the electric potential be at point M, which is located on the axis of the ring ...

Calculations of Electric Potential II

An electric dipole is a system of two equal but opposite charges, separated by a fixed distance. This system is used to model many real-world systems, ...

Induction

An emf is induced when the magnetic field in a coil is changed by pushing a bar magnet into or out of the coil. emfs of opposite signs are produced by ...

Equipotential Surfaces and Field Lines

Electric potential can be pictorially represented as a three-dimensional surface. On such a surface, the electric potential is constant everywhere. The ...

Faraday's Law

Faraday&#39;s law state that the induced emf is the negative change in the magnetic flux per unit of time. Any change in the magnetic field or change in ...

Equipotential Surfaces and Conductors

For a conductor in which all charges are at rest, the conductor&#39;s surface is equipotential. The electric field is always perpendicular to ...

Capacitors and Capacitance

A device consisting of two electrical conductors that are separated by a distance and used to store electrical charges is called a capacitor. The space ...

Spherical and Cylindrical Capacitor

A spherical capacitor consists of two concentric conducting spherical shells of radii R1&#160;(inner shell) and R2&#160;(outer shell). The shells ...

Capacitors in Series and Parallel

Multiple capacitors connected serve as electrical components in various applications. These multiple capacitors behave as a single equivalent capacitor, ...

Equivalent Capacitance

Multiple capacitors can be connected in a circuit in series or parallel configuration. When the capacitor combination is connected to a battery, the ...

Energy Stored in a Capacitor

When an archer pulls the string in a bow, he saves the work done in the form of elastic potential energy. When he releases the string, the potential ...

energy-stored-in-a-capacitor-problem-solving

Energy Stored in a Capacitor: Problem Solving

In 1749, Benjamin Franklin coined the word battery for a series of capacitors connected to store energy. Capacitors store electric potential energy that ...

Capacitor With A Dielectric

Parallel plate capacitors consist of two conducting plates separated by a certain distance. However, it is mechanically difficult to hold the large plates ...

The student will investigate and understand that there are basic principles of electricity and magnetism. Key ideas include an imbalance of charge generates static electricity;

PS.9.a

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 ...

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 ...

The student will investigate and understand that there are basic principles of electricity and magnetism. Key ideas include materials have different conductive properties;

PS.9.b

Free Current and Magnetization in Ampere’s Law

Description

Transcript

Tags

More videos by Standards