Eddy current

Eddy current

Overview
Eddy currents are electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

s induced in conductors when a conductor is exposed to a changing magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

; due to relative motion of the field source and conductor or due to variations of the field with time. This can cause a circulating flow of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s, or current, within the body of the conductor. These circulating eddies
Eddy (fluid dynamics)
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid flows past an obstacle. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object...

of current have inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

and thus induce magnetic fields. These fields can cause repulsive, attractive, propulsion and drag effects. The stronger the applied magnetic field, or the greater the electrical conductivity of the conductor, or the faster the field changes, then the greater the currents that are developed and the greater the fields produced.
Discussion

Encyclopedia
Eddy currents are electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

s induced in conductors when a conductor is exposed to a changing magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

; due to relative motion of the field source and conductor or due to variations of the field with time. This can cause a circulating flow of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s, or current, within the body of the conductor. These circulating eddies
Eddy (fluid dynamics)
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid flows past an obstacle. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object...

of current have inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

and thus induce magnetic fields. These fields can cause repulsive, attractive, propulsion and drag effects. The stronger the applied magnetic field, or the greater the electrical conductivity of the conductor, or the faster the field changes, then the greater the currents that are developed and the greater the fields produced.

The term eddy current comes from analogous currents seen in water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

when dragging an oar
Oar
An oar is an implement used for water-borne propulsion. Oars have a flat blade at one end. Oarsmen grasp the oar at the other end. The difference between oars and paddles are that paddles are held by the paddler, and are not connected with the vessel. Oars generally are connected to the vessel by...

breadthwise: localised areas of turbulence known as eddies give rise to persistent vortices. Somewhat analogously, eddy currents can take time to build up and can persist for very short times in conductors due to their inductance.

Eddy currents, like all electric currents, generate heat as well as electromagnetic forces. The heat can be harnessed for induction heating
Induction heating
Induction heating is the process of heating an electrically conducting object by electromagnetic induction, where eddy currents are generated within the metal and resistance leads to Joule heating of the metal...

. The electromagnetic forces can be used for levitation, creating movement, or to give a strong braking effect. Eddy currents can also have undesirable effects, for instance power loss in transformer
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...

s. In this application, they are minimised with thin plates, by lamination of conductors or other details of conductor shape.

Self-induced eddy currents are responsible for the skin effect
Skin effect
Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

in conductors. The latter can be used for non-destructive testing of materials for geometry features, like micro-cracks. A similar effect is the proximity effect
Proximity effect (electromagnetism)
In a conductor carrying alternating current, if currents are flowing through one or more other nearby conductors, such as within a closely wound coil of wire, the distribution of current within the first conductor will be constrained to smaller regions. The resulting current crowding is termed the...

, which is caused by externally-induced eddy currents.

History

The first person to observe current eddies was François Arago
François Arago
François Jean Dominique Arago , known simply as François Arago , was a French mathematician, physicist, astronomer and politician.-Early life and work:...

(1786–1853), the 25th Prime Minister of France, who was also a mathematician, physicist and astronomer. In 1824 he observed what has been called rotatory magnetism, and the fact that most conductive bodies could be magnetized; these discoveries were completed and explained by Michael Faraday
Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....

(1791–1867).

In 1834, Heinrich Lenz
Heinrich Lenz
Heinrich Friedrich Emil Lenz was a Russian physicist of Baltic German ethnicity. He is most noted for formulating Lenz's law in electrodynamics in 1833....

stated Lenz's law
Lenz's law
Lenz's law is a common way of understanding how electromagnetic circuits must always obey Newton's third law and The Law of Conservation of Energy...

, which says that the direction of induced current flow in an object will be such that its magnetic field will oppose the magnetic field that caused the current flow. Eddy currents develop secondary flux that cancels a part of the external flux.

French physicist Léon Foucault
Léon Foucault
Jean Bernard Léon Foucault was a French physicist best known for the invention of the Foucault pendulum, a device demonstrating the effect of the Earth's rotation...

(1819–1868) is credited with having discovered Eddy currents. In September, 1855, he discovered that the force required for the rotation of a copper disc becomes greater when it is made to rotate with its rim between the poles of a magnet, the disc at the same time becoming heated by the eddy current induced in the metal.
The first use of eddy current for Non-destructive testing occurred in 1879 when D. E. Hughes used the principles to conduct metallurgical sorting tests.

Explanation

When a conductor moves relative to the field generated by a source, electromotive force
Electromotive force
In physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...

s (EMFs) can be generated around loops within the conductor. These EMFs acting on the resistivity
Resistivity
Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...

of the material generate a current around the loop, in accordance with Faraday's law of induction
Faraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...

. These currents dissipate energy, and create a magnetic field that tends to oppose changes in the current- they have inductance.

Eddy currents are created when a conductor experiences changes in the magnetic field. If either the conductor is moving through a steady magnetic field, or the magnetic field is changing around a stationary conductor, eddy currents will occur in the conductor. Both effects are present when a conductor moves through a varying magnetic field, as is the case at the top and bottom edges of the magnetized region shown in the diagram. Eddy currents will be generated wherever a conducting object experiences a change in the intensity or direction of the magnetic field at any point within it, and not just at the boundaries.

The swirling current set up in the conductor is due to electrons experiencing a Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

that is perpendicular to their motion. Hence, they veer to their right, or left, depending on the direction of the applied field and whether the strength of the field is increasing or declining. The resistivity of the conductor acts to damp the amplitude of the eddy currents, as well as straighten their paths. Lenz's law
Lenz's law
Lenz's law is a common way of understanding how electromagnetic circuits must always obey Newton's third law and The Law of Conservation of Energy...

encapsulates the fact that the current swirls in such a way as to create an induced magnetic field that opposes the phenomenon that created it. In the case of a varying applied field, the induced field will always be in the opposite direction to that applied. The same will be true when a varying external field is increasing in strength. However, when a varying field is falling in strength, the induced field will be in the same direction as that originally applied, in order to oppose the decline.

An object or part of an object experiences steady field intensity and direction where there is still relative motion of the field and the object (for example in the center of the field in the diagram), or unsteady fields where the currents cannot circulate due to the geometry of the conductor. In these situations charges collect on or within the object and these charges then produce static electric potentials that oppose any further current. Currents may be initially associated with the creation of static potentials, but these may be transitory and small.

Eddy currents generate resistive losses that transform some forms of energy, such as kinetic energy, into heat. This Joule heating
Joule heating
Joule heating, also known as ohmic heating and resistive heating, is the process by which the passage of an electric current through a conductor releases heat. It was first studied by James Prescott Joule in 1841. Joule immersed a length of wire in a fixed mass of water and measured the temperature...

reduces efficiency of iron-core transformer
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...

s and electric motors and other devices that use changing magnetic fields. Eddy currents are minimized in these devices by selecting magnetic core
Magnetic core
A magnetic core is a piece of magnetic material with a high permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, inductors and magnetic assemblies. It is made of ferromagnetic metal such...

materials that have low electrical conductivity (e.g., ferrites
Ferrite (magnet)
Ferrites are chemical compounds consisting of ceramic materials with iron oxide as their principal component. Many of them are magnetic materials and they are used to make permanent magnets, ferrite cores for transformers, and in various other applications.Many ferrites are spinels with the...

) or by using thin sheets of magnetic material, known as laminations. Electrons cannot cross the insulating gap between the laminations and so are unable to circulate on wide arcs. Charges gather at the lamination boundaries, in a process analogous to the Hall effect
Hall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...

, producing electric fields that oppose any further accumulation of charge and hence suppressing the eddy currents. The shorter the distance between adjacent laminations (i.e., the greater the number of laminations per unit area, perpendicular to the applied field), the greater the suppression of eddy currents.

The conversion of input energy to heat is not always undesirable, however, as there are some practical applications. One is in the brakes of some trains known as eddy current brakes. During braking, the metal wheels are exposed to a magnetic field from an electromagnet, generating eddy currents in the wheels. The eddy currents meet resistance as charges flow through the metal, thus dissipating energy as heat, and this acts to slow the wheels down. The faster the wheels are spinning, the stronger the effect, meaning that as the train slows the braking force is reduced, producing a smooth stopping motion. Induction heating
Induction heating
Induction heating is the process of heating an electrically conducting object by electromagnetic induction, where eddy currents are generated within the metal and resistance leads to Joule heating of the metal...

makes use of eddy currents to provide heating of metal objects.

Strength of eddy currents

Under certain assumptions (uniform material, uniform magnetic field, no skin effect
Skin effect
Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

, etc.) the power lost due to eddy currents can be calculated from the following equations:

For thin sheets:

For thin wires:

where: P - power dissipation (W/kg), Bp - peak flux density
Flux density
-Formal Statement:The flux density is simply defined as the amount of flux passing through a unit-area. -Mathematical Statement:The flux density would essentially be the number of field lines passing through a defined unit-area...

(T), d - thickness of the sheet or diameter of the wire (m), f - frequency (Hz), ρ - resistivity (Ωm), D - specific density (kg/m3).

It should be borne in mind that these equations are valid only under the so-called "quasi-static" conditions, where the frequency of magnetisation does not result in the skin effect, i.e. the electromagnetic wave fully penetrates the material.

Therefore, the following things usually increase the size and effects of eddy currents:
• stronger magnetic fields - increases flux density B
• faster changing fields (due to faster relative speeds or otherwise) - increases the frequency f
• thicker materials - increases the thickness d
• lower resistivity
Resistivity
Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...

materials (aluminium, copper, silver etc.)

Some things reduce the effects:
• weaker magnets - lower B
• slower changing fields (slower relative speeds) - lower f
• thinner materials - lower d
• slotted materials so that currents cannot circulate - reduced d or coefficient in the denominator (6, 12, etc.)
• laminated materials so that currents cannot circulate - reduced d
• higher resistance materials (silicon rich iron etc.)

Skin effect

In very fast changing fields due to skin effect the equations shown above are not valid because the magnetic field does not penetrate the material uniformly. However, in any case increased frequency of the same value of field will always increase eddy currents, even with non-uniform field penetration.

The penetration depth can be calculated from the following equation:

where: δ - penetration depth (m), f - frequency (Hz), μ - magnetic permeability (H/m), σ - electrical conductivity (S/m)

Repulsive effects and levitation

In a fast varying magnetic field the induced currents, in good conductors, particularly copper and aluminium, frequently exhibit diamagnetic-like repulsion effects on the magnetic field, and hence on the magnet and can create repulsive effects and even stable levitation, albeit with reasonably high power dissipation due to the high currents this entails.

They can thus be used to induce a magnetic field in aluminum can
Aluminum can
An aluminum can, or can, is a container for packaging made primarily of aluminum .They are commonly used for foods and beverages but also for products such as oil, chemicals, and other liquids. The common 12-ounce size can weighs 15 grams when empty.-Usage:Use of aluminum in cans began in 1957...

s, which allows them to be separated easily from other recyclables (see also eddy current separator
Eddy current separator
An eddy current separator uses a powerful magnetic field to separate metals from non-metals in garbage. The device makes use of eddy currents to effect the separation....

). With a very strong handheld magnet, such as those made from neodymium
Neodymium
Neodymium is a chemical element with the symbol Nd and atomic number 60. It is a soft silvery metal that tarnishes in air. Neodymium was discovered in 1885 by the Austrian chemist Carl Auer von Welsbach. It is present in significant quantities in the ore minerals monazite and bastnäsite...

, one can easily observe a very similar effect by rapidly sweeping the magnet over a coin with only a small separation. Depending on the strength of the magnet, identity of the coin, and separation between the magnet and coin, one may induce the coin to be pushed slightly ahead of the magnet - even if the coin contains no magnetic elements, such as the US penny. Another example involves dropping a strong magnet down a tube of copper -- the magnet falls at a dramatically slow pace.

Perfect conductors allow lossless conduction that allows eddy currents to form on the surface of the conductor that exactly cancel any changes in the magnetic field applied to the object after the material's resistance went to zero, thus allowing magnetic levitation
Magnetic levitation
Magnetic levitation, maglev, or magnetic suspension is a method by which an object is suspended with no support other than magnetic fields...

. Superconductors are a subclass of perfect conductors in that they also exhibit the Meissner Effect
Meissner effect
The Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. The German physicists Walther Meissner and Robert Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin...

, an inherently quantum mechanical phenomenon that is responsible for expelling any magnetic field lines present during the superconducting transition, thus making the magnetic field zero in the bulk of the superconductor.

Attractive effects

In some geometries the overall force of eddy currents can be attractive, for example, where the flux lines are past 90 degrees to a surface, the induced currents in a nearby conductor cause a force that pushes a conductor towards an electromagnet.

Identification of metals

In coin operated vending machine
Vending machine
A vending machine is a machine which dispenses items such as snacks, beverages, alcohol, cigarettes, lottery tickets, consumer products and even gold and gems to customers automatically, after the customer inserts currency or credit into the machine....

s, eddy currents are used to detect counterfeit coins, or slug
Slug (coin)
A slug is a counterfeit coin that is used to make illegal purchases from a coin-operated device, such as a vending machine, pay phone, parking meter, transit farebox, copy machine, coin laundry, gaming machine, or arcade game...

s. The coin rolls past a stationary magnet, and eddy currents slow its speed. The strength of the eddy currents, and thus the retardation, depends on the conductivity of the coin's metal. Slugs are slowed to a different degree than genuine coins, and this is used to send them into the rejection slot.

Vibration | Position Sensing

Eddy currents are used in certain types of proximity sensors to observe the vibration and position of rotating shafts within their bearings. This technology was originally pioneered in the 1930s by researchers at General Electric
General Electric
General Electric Company , or GE, is an American multinational conglomerate corporation incorporated in Schenectady, New York and headquartered in Fairfield, Connecticut, United States...

using vacuum tube circuitry. In the late 1950s, solid-state versions were developed by Donald E. Bently
Donald E. Bently
Donald E. Bently is an American entrepreneur and engineer, best known as the founder and former owner of Bently Nevada Corporation where he performed pioneering work in the field of instrumentation for measuring the mechanical condition of rotating machinery. He founded Bently Nevada in October,...

Bently Nevada is a name long associated with condition monitoring instrumentation and services, most notably sensors, systems, and diagnostic services for monitoring machinery vibration...

Corporation. These sensors are extremely sensitive to very small displacements making them well suited to observe the minute vibrations (on the order of several thousandths of an inch) in modern turbomachinery
Turbomachinery
Turbomachinery, in mechanical engineering, describes machines that transfer energy between a rotor and a fluid, including both turbines and compressors. While a turbine transfers energy from a fluid to a rotor, a compressor transfers energy from a rotor to a fluid...

. A typical proximity sensor used for vibration monitoring has a scale factor of 200 mV/mil. Widespread use of such sensors in turbomachinery has led to development of industry standards that prescribe their use and application. Examples of such standards are American Petroleum Institute
American Petroleum Institute
The American Petroleum Institute, commonly referred to as API, is the largest U.S trade association for the oil and natural gas industry...

(API) Standard 670 and ISO 7919.

Electromagnetic braking

Eddy currents are used for braking at the end of some roller coasters. This mechanism has no mechanical wear and produces a very precise braking force. Typically, heavy copper
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...

plates extending from the car are moved between pairs of very strong permanent magnets. Electrical resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

within the plates causes a dragging effect analogous to friction, which dissipates the kinetic energy of the car. The same technique is used in electromagnetic brakes in railroad cars and to quickly stop the blades in power tools such as circular saws.

Structural testing

Eddy current techniques are commonly used for the nondestructive examination
Nondestructive testing
Nondestructive testing or Non-destructive testing is a wide group of analysis techniques used in science and industry to evaluate the properties of a material, component or system without causing damage....

(NDE) and condition monitoring of a large variety of metallic structures, including heat exchanger
Heat exchanger
A heat exchanger is a piece of equipment built for efficient heat transfer from one medium to another. The media may be separated by a solid wall, so that they never mix, or they may be in direct contact...

tubes, aircraft fuselage, and aircraft structural components..

Side effects

Eddy currents are the root cause of the skin effect
Skin effect
Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...

in conductors carrying AC current.

Similarly, in magnetic materials of finite conductivity eddy currents cause the confinement of the majority of the magnetic fields to only a couple skin depths of the surface of the material. This effect limits the flux linkage
Flux linkage is a property of a coil of conducting wire and the magnetic field through which it passes. It is determined by the number of turns in the coil and the flux of the magnetic field....

in inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

s and transformer
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...

s having magnetic core
Magnetic core
A magnetic core is a piece of magnetic material with a high permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, inductors and magnetic assemblies. It is made of ferromagnetic metal such...

s.

Other applications

• Metal detector
Metal detector
A metal detector is a device which responds to metal that may not be readily apparent.The simplest form of a metal detector consists of an oscillator producing an alternating current that passes through a coil producing an alternating magnetic field...

s
• Eddy-current testing
Eddy-current testing
Eddy-current testing uses electromagnetic induction to detect flaws in conductive materials. There are several limitations, among them: only conductive materials can be tested, the surface of the material must be accessible, the finish of the material may cause bad readings, the depth of...

• Electric meters
Electricity meter
An electricity meter or energy meter is a device that measures the amount of electric energy consumed by a residence, business, or an electrically powered device....

(Electromechanical Induction Meters)
• Eddy current brakes
• Induction heating
Induction heating
Induction heating is the process of heating an electrically conducting object by electromagnetic induction, where eddy currents are generated within the metal and resistance leads to Joule heating of the metal...

• Proximity sensor
Proximity sensor
A proximity sensor is a sensor able to detect the presence of nearby objects without any physical contact.A proximity sensor often emits an electromagnetic or a beam of electromagnetic radiation , and looks for changes in the field or return signal. The object being sensed is often referred to as...

(Displacement sensors)
• Traffic Detection Systems
• Vending machine
Vending machine
A vending machine is a machine which dispenses items such as snacks, beverages, alcohol, cigarettes, lottery tickets, consumer products and even gold and gems to customers automatically, after the customer inserts currency or credit into the machine....

s (detection of coins)
• Coating Thickness Measurements
• Sheet Resistance Measurement
• Eddy current separator
Eddy current separator
An eddy current separator uses a powerful magnetic field to separate metals from non-metals in garbage. The device makes use of eddy currents to effect the separation....

for metal separation
• Mechanical speedometer
Speedometer
A speedometer is a gauge that measures and displays the instantaneous speed of a land vehicle. Now universally fitted to motor vehicles, they started to be available as options in the 1900s, and as standard equipment from about 1910 onwards. Speedometers for other vehicles have specific names...

s
• Safety Hazard and defect detection applications

Diffusion Equation

The derivation of a useful equation for modeling the effect of eddy currents in a material starts with the differential, magnetostatic form of Ampère's Law, providing an expression for the magnetic field surrounding a current density ,.
The curl is taken on both sides of the equation,,
and using a common vector calculus identity for the curl of the curl results in.
From Gauss's law for magnetism, , which drops a term from the expression and gives.
Using Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

, , which relates current density to electric field in terms of a material's conductivity , and assuming isotropic homogeneous conductivity, the equation can be written as.
The differential form of Faraday's law
Electromagnetic induction
Electromagnetic induction is the production of an electric current across a conductor moving through a magnetic field. It underlies the operation of generators, transformers, induction motors, electric motors, synchronous motors, and solenoids....

, , provides an equivalence for the change in magnetic flux in place of the curl of the electric field, so that the equation can be simplified to.
By definition, , where is the magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...

of a material, is the vacuum permeability
Vacuum permeability
The physical constant μ0, commonly called the vacuum permeability, permeability of free space, or magnetic constant is an ideal, physical constant, which is the value of magnetic permeability in a classical vacuum...

, and the diffusion equation finally appears as