Name: magnetic fields at material interfaces
Uploaded: 2023-09-18T17:11:20.000Z
Duration: 1 min 28 s
Channel: JoVE

K12 Standards

Louisiana Student Standards for Science

High School

Earth Science

Earth’s Systems

HS-ESS2-3

Magnetostatic Boundary Conditions

Inside a homogenous magnetic medium, the magnetic field is continuous. However,&#160; on crossing a boundary, it becomes discontinuous.
Consider a Gaussian pillbox across the boundary. The zero divergence of the magnetic field ensures that the total magnetic flux enclosed within the pillbox is zero. So, the normal components of the field are equal across the boundary.
Now, consider an Amperian loop perpendicular to the surface current and apply Ampere&#39;s Law to each loop section.
The line integral of the magnetic field exists only for the loop&#39;s top and bottom sections since the loop width tends to zero. So, the field&#39;s tangential component is discontinuous.
The discontinuity of the tangential components can be expressed in vectorial form.
Now, consider an interface between two media. The divergence of magnetic field intensity equals the negative divergence of the magnetization. So, the normal component of magnetic field intensity is discontinuous.
Consider an Amperian loop perpendicular to the surface current. Applying Ampere&#39;s law, the tangential component of magnetic field intensity is discontinuous. This discontinuity can be expressed in vector form.

magnetostatic boundary conditions

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous across any boundary; hence the derivative of the vector potential is discontinuous across the boundary.
Consider two linear magnetic media with permeability μ1 and μ2, separated by an interface. The magnetic field intensity in medium 1 at a point P1 has a magnitude H1, and makes an angle α1 with the normal. Similarly, the magnetic field intensity in medium 2 at a point P2 has a magnitude H2, and makes an angle α2 with the normal.
<img alt="Static equilibrium diagram; vectors, angles, magnetic fields; physics concept illustration." src="/files/ftp_upload/14202/14202_figure_1.png" style="text-align: center" data-altupdatedat="1761830439000" data-alt="Figure1">
The zero divergence of the magnetic field ensures that the normal component of the magnetic field is continuous across the boundary. The magnetic field is proportional to the magnetic field intensity for linear magnetic media. Thus, the boundary conditions for the normal component of the magnetic field intensity are obtained.
Rewriting the normal components of the magnetic field in terms of the angles α1 and α2,
<img alt="Snell's Law; refraction formula; symbols diagram; optics study; angle of incidence analysis." src="/files/ftp_upload/14202/14202_Equation_1.svg" style="text-align: center; height: 18px;" data-altupdatedat="1761830439000" data-alt="Equation1">
Since both media possess finite conductivities, free surface currents do not exist at the interface. Thus, the tangential component of the magnetic field intensity is continuous across the boundary. Again, the tangential components can be rewritten in terms of the angles α1 and α2:
<img alt="Wave optics formula diagram, demonstrating refraction principles: H2sinα2=H1sinα1." src="/files/ftp_upload/14202/14202_Equation_2.svg" style="text-align: center; height: 18px;" data-altupdatedat="1761830439000" data-alt="Equation2">
Taking the ratio of normal and tangential components, a new expression is derived:
<img alt="Snell's law formula diagram, refraction angles and refractive indices in optics study." src="/files/ftp_upload/14202/14202_Equation_3.svg" style="text-align: center; height: 40px;" data-altupdatedat="1761830439000" data-alt="Equation3">
This expression is similar to the expression obtained for boundary conditions for two dielectric interfaces, except that the permittivity replaces the permeability.
If the first medium is a non-magnetic medium, like air, and the second one is a magnetic medium, the permeability of the second medium is greater than the permeability of the first medium. Thus, the angle α2 approaches ninety degrees. This implies that, for any arbitrary angle α1 that is not close to zero, the magnetic field in a ferromagnetic medium runs parallel to the interface. If the first medium is ferromagnetic and the second is non-magnetic, α2 approaches zero. This implies that if a magnetic field originates in a ferromagnetic medium, the magnetic flux lines emerge into the air medium in a direction almost normal to the interface.

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular ...

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JoVE Core

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Unit 1

Sources of Magnetic Fields

KEY TERMS AND CONCEPTS

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 surface current, with the current direction setting the perpendicular relation. The vector potential also stays continuous across a boundary, like the scalar potential in electrostatics, so its derivative must change across the interface.

For two linear magnetic media with permeabilities μ1 and μ2, the magnetic field intensity H can be described at points on either side of the interface. Let H1 make an angle α1 with the normal in medium 1, and let H2 make an angle α2 with the normal in medium 2. Because the magnetic field has zero divergence, its normal component is continuous across the boundary. Since the field is proportional to the field intensity in linear media, the same boundary condition can be written for the normal part of H.

The tangential component of H is also continuous across the boundary when both media have finite conductivity, because free surface currents do not exist at the interface. Writing the normal and tangential parts in terms of α1 and α2 gives a ratio that links the angles to the permeabilities. This result matches the form used for dielectric boundaries, except that permeability replaces permittivity.

The angle change becomes clear in common cases. If the first medium is non-magnetic, such as air, and the second is magnetic, then the larger permeability of the second medium makes α2 approach 90 degrees. In that case, magnetic field lines in a ferromagnetic medium tend to run parallel to the interface unless α1 is very close to zero. If the field starts in a ferromagnetic medium and enters a non-magnetic medium, then α2 approaches 0 degrees, so the magnetic flux lines leave into air almost normal to the boundary.

Magnetic Fields at Material Interfaces

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...

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

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.

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 magnetic field components behave at the boundary between two magnetic media, including continuity, surface currents, and angle changes in air and ferromagnets.

Magnetic Fields at Material Interfaces | JoVE K12

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HS-ESS2-4

Magnetic Fields at Material Interfaces

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