# Eddy current

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Eddy currents (also called Foucault's currents) are loops of electrical current induced within conductors by a changing magnetic field in the conductor according to Faraday's law of induction. Eddy currents flow in closed loops within conductors, in planes perpendicular to the magnetic field. They can be induced within nearby stationary conductors by a time-varying magnetic field created by an AC electromagnet or transformer, for example, or by relative motion between a magnet and a nearby conductor. The magnitude of the current in a given loop is proportional to the strength of the magnetic field, the area of the loop, and the rate of change of flux, and inversely proportional to the resistivity of the material. When graphed, these circular currents within a piece of metal look vaguely like eddies or whirlpools in a liquid.

## Contents

By Lenz's law, an eddy current creates a magnetic field that opposes the change in the magnetic field that created it, and thus eddy currents react back on the source of the magnetic field. For example, a nearby conductive surface will exert a drag force on a moving magnet that opposes its motion, due to eddy currents induced in the surface by the moving magnetic field. This effect is employed in eddy current brakes which are used to stop rotating power tools quickly when they are turned off. The current flowing through the resistance of the conductor also dissipates energy as heat in the material. Thus eddy currents are a cause of energy loss in alternating current (AC) inductors, transformers, electric motors and generators, and other AC machinery, requiring special construction such as laminated magnetic cores or ferrite cores to minimize them. Eddy currents are also used to heat objects in induction heating furnaces and equipment, and to detect cracks and flaws in metal parts using eddy-current testing instruments.

## Origin of term

The term eddy current comes from analogous currents seen in water in fluid dynamics, causing localised areas of turbulence known as eddies giving 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.

## History

The first person to observe eddy currents was François Arago (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 that most conductive bodies could be magnetized; these discoveries were completed and explained by Michael Faraday (1791–1867).

In 1834, Heinrich Lenz stated Lenz's law, which says that the direction of induced current flow in an object will be such that its magnetic field will oppose the change of magnetic flux that caused the current flow. Eddy currents produce a secondary field that cancels a part of the external field and causes some of the external flux to avoid the conductor.

French physicist Léon Foucault (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 David E. Hughes used the principles to conduct metallurgical sorting tests.

## Explanation

A magnet induces circular electric currents in a metal sheet moving through its magnetic field. See the diagram at right. It shows a metal sheet (C) moving to the right under a stationary magnet. The magnetic field (B, green arrows) of the magnet's north pole N passes down through the sheet. Since the metal is moving, the magnetic flux through a given area of the sheet is changing. In the part of the sheet moving under the leading edge of the magnet (left side) the magnetic field through a given point on the sheet is increasing as it gets nearer the magnet, ${\displaystyle {dB \over dt}\;>\;0}$. From Faraday's law of induction, this creates a circular electric field in the sheet in a counterclockwise direction around the magnetic field lines. This field induces a counterclockwise flow of electric current (I, red), in the sheet. This is the eddy current. In the part of the sheet under the trailing edge of the magnet (right side) the magnetic field through a given point on the sheet is decreasing as it is moving further away from the magnet, ${\displaystyle {dB \over dt}\;<\;0}$, inducing a second eddy current in a clockwise direction in the sheet.

Another equivalent way to understand the current is to see that the free charge carriers (electrons) in the metal sheet are moving with the sheet to the right, so the magnetic field exerts a sideways force on them due to the Lorentz force. Since the velocity v of the charges is to the right and the magnetic field B is directed down, from the right hand rule the Lorentz force on positive charges F = q(v × B) is toward the rear of the diagram (to the left when facing in the direction of motion v). This causes a current I toward the rear under the magnet, which circles around through parts of the sheet outside the magnetic field, clockwise to the right and counterclockwise to the left, to the front of the magnet again. The mobile charge carriers in the metal, the electrons, actually have a negative charge (q < 0) so their motion is opposite in direction to the conventional current shown.

The magnetic field of the magnet, acting on the electrons moving sideways under the magnet, then exerts a Lorentz force directed to the rear, opposite to the velocity of the metal sheet. The electrons, in collisions with the metal lattice atoms, transfer this force to the sheet, exerting a drag force on the sheet proportional to its velocity. The kinetic energy which is consumed overcoming this drag force is dissipated as heat by the currents flowing through the resistance of the metal, so the metal gets warm under the magnet.

Due to Ampere's circuital law each of the circular currents in the sheet creates a counter magnetic field (blue arrows). Another way to understand the drag force is to see that due to Lenz's law the counterfields oppose the change in magnetic field through the sheet. At the leading edge of the magnet (left side) by the right hand rule the counterclockwise current creates a magnetic field pointed up, opposing the magnet's field, causing a repulsive force between the sheet and the leading edge of the magnet. In contrast, at the trailing edge (right side), the clockwise current causes a magnetic field pointed down, in the same direction as the magnet's field, creating an attractive force between the sheet and the trailing edge of the magnet. Both of these forces oppose the motion of the sheet.

## Properties

Eddy currents in conductors of non-zero resistivity generate heat as well as electromagnetic forces. The heat can be used for induction heating. 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 transformers. In this application, they are minimized with thin plates, by lamination of conductors or other details of conductor shape.

Self-induced eddy currents are responsible for the skin effect in conductors. [1] The latter can be used for non-destructive testing of materials for geometry features, like micro-cracks. [2] A similar effect is the proximity effect, which is caused by externally induced eddy currents. [3]

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 reduces efficiency of iron-core transformers and electric motors and other devices that use changing magnetic fields. Eddy currents are minimized in these devices by selecting magnetic core materials that have low electrical conductivity (e.g., ferrites) 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, 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. This eddy current is formed by the movement of the wheels. So, by Lenz's law, the magnetic field formed by the eddy current will oppose its cause. Thus the wheel will face a force opposing the initial movement of the wheel. 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 makes use of eddy currents to provide heating of metal objects.

### Power dissipation of eddy currents

Under certain assumptions (uniform material, uniform magnetic field, no skin effect, etc.) the power lost due to eddy currents per unit mass for a thin sheet or wire can be calculated from the following equation: [4]

${\displaystyle P={\frac {\pi ^{2}B_{\text{p}}^{\,2}d^{2}f^{2}}{6k\rho D}},}$

where

P is the power lost per unit mass (W/kg),
Bp is the peak magnetic field (T),
d is the thickness of the sheet or diameter of the wire (m),
f is the frequency (Hz),
k is a constant equal to 1 for a thin sheet and 2 for a thin wire,
ρ is the resistivity of the material (Ω m), and
D is the density of the material (kg/m3).

This equation is valid only under the so-called quasi-static conditions, where the frequency of magnetisation does not result in the skin effect; that is, the electromagnetic wave fully penetrates the material.

### Skin effect

In very fast-changing fields, the magnetic field does not penetrate completely into the interior of the material. This skin effect renders the above equation invalid. However, in any case increased frequency of the same value of field will always increase eddy currents, even with non-uniform field penetration.[ citation needed ]

The penetration depth for a good conductor can be calculated from the following equation: [5]

${\displaystyle \delta ={\frac {1}{\sqrt {\pi f\mu \sigma }}},}$

where δ is the penetration depth (m), f is the frequency (Hz), μ is the magnetic permeability of the material (H/m), and σ is the electrical conductivity of the material (S/m).

### Diffusion equation

The derivation of a useful equation for modelling the effect of eddy currents in a material starts with the differential, magnetostatic form of Ampère's Law, [6] providing an expression for the magnetizing field H surrounding a current density J:

${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} .}$

Taking the curl on both sides of this equation and then using a common vector calculus identity for the curl of the curl results in

${\displaystyle {\color {white}-}\nabla \left(\nabla \cdot \mathbf {H} \right)-\nabla ^{2}\mathbf {H} =\nabla \times \mathbf {J} .}$

From Gauss's law for magnetism, ·H = 0, so

${\displaystyle -\nabla ^{2}\mathbf {H} =\nabla \times \mathbf {J} .}$

Using Ohm's law, J= σE, which relates current density J to electric field E in terms of a material's conductivity σ, and assuming isotropic homogeneous conductivity, the equation can be written as

${\displaystyle -\nabla ^{2}\mathbf {H} =\sigma \nabla \times {\boldsymbol {E}}.}$

Using the differential form of Faraday's law, ×E = −∂B/∂t, this gives

${\displaystyle {\color {white}-}\nabla ^{2}\mathbf {H} =\sigma {\frac {\partial \mathbf {B} }{\partial t}}.}$

By definition, B = μ0(H + M), where M is the magnetization of the material and μ0 is the vacuum permeability. The diffusion equation therefore is

${\displaystyle {\color {white}-}\nabla ^{2}\mathbf {H} =\mu _{0}\sigma \left({\frac {\partial \mathbf {M} }{\partial t}}+{\frac {\partial \mathbf {H} }{\partial t}}\right).}$

## Applications

### Electromagnetic braking

Eddy current brakes use the drag force created by eddy currents as a brake to slow or stop moving objects. Since there is no contact with a brake shoe or drum, there is no mechanical wear. However, an eddy current brake cannot provide a "holding" torque and so may be used in combination with mechanical brakes, for example, on overhead cranes. Another application is on some roller coasters, where heavy copper plates extending from the car are moved between pairs of very strong permanent magnets. Electrical resistance 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. Using electromagnets, as opposed to permanent magnets, the strength of the magnetic field can be adjusted and so the magnitude of braking effect changed.

### Repulsive effects and levitation

In a varying magnetic field, the induced currents exhibit diamagnetic-like repulsion effects. A conductive object will experience a repulsion force. This can lift objects against gravity, though with continual power input to replace the energy dissipated by the eddy currents. An example application is separation of aluminum cans from other metals in an eddy current separator. Ferrous metals cling to the magnet, and aluminum (and other non-ferrous conductors) are forced away from the magnet; this can separate a waste stream into ferrous and non-ferrous scrap metal.

With a very strong handheld magnet, such as those made from neodymium, 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 [7] – the magnet falls at a dramatically slow pace.

In a perfect conductor with no resistance (a superconductor), surface eddy currents exactly cancel the field inside the conductor, so no magnetic field penetrates the conductor. Since no energy is lost in resistance, eddy currents created when a magnet is brought near the conductor persist even after the magnet is stationary, and can exactly balance the force of gravity, allowing magnetic levitation. Superconductors also exhibit a separate inherently quantum mechanical phenomenon called the Meissner effect in which any magnetic field lines present in the material when it becomes superconducting are expelled, thus the magnetic field in a superconductor is always zero.

Using electromagnets with electronic switching comparable to electronic speed control it is possible to generate electromagnetic fields moving in an arbitrary direction. As described in the section above about eddy current brakes, a non-ferromagnetic conductor surface tends to rest within this moving field. When however this field is moving, a vehicle can be levitated and propelled. This is comparable to a maglev but is not bound to a rail. [8]

### Identification of metals

In some coin-operated vending machines, eddy currents are used to detect counterfeit coins, or slugs. 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 and 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 using vacuum tube circuitry. In the late 1950s, solid-state versions were developed by Donald E. Bently at Bently Nevada 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. 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 (API) Standard 670 and ISO 7919.

A Ferraris acceleration sensor, also called a Ferraris sensor, is a contactless sensor that uses eddy currents to measure relative acceleration. [9] [10] [11]

### Structural testing

Eddy current techniques are commonly used for the nondestructive examination (NDE) and condition monitoring of a large variety of metallic structures, including heat exchanger tubes, aircraft fuselage, and aircraft structural components.

### Skin effects

Eddy currents are the root cause of the skin effect 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 in inductors and transformers having magnetic cores.

## Related Research Articles

An electromagnetic field is a classical field produced by accelerating electric charges. It is the field described by classical electrodynamics and is the classical counterpart to the quantized electromagnetic field tensor in quantum electrodynamics. The electromagnetic field propagates at the speed of light and interacts with charges and currents. Its quantum counterpart is one of the four fundamental forces of nature

The Hall effect is the production of a voltage difference across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discovered by Edwin Hall in 1879.

In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet′s magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a magnetic field that varies with location will exert a force on a range of non-magnetic materials by affecting the motion of their outer atomic electrons. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in electromagnets, and by electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, they are described as a map assigning a vector to each point of space or, more precisely—because of the way the magnetic field transforms under mirror reflection—as a field of pseudovectors.

Electrical resistivity is a fundamental property of a material that quantifies how strongly it resists electric current. Its inverse, called electrical conductivity, quantifies how well a material conducts electricity. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the Greek letter ρ (rho). The SI unit of electrical resistivity is the ohm-meter (Ω⋅m). For example, if a 1 m solid cube of material has sheet contacts on two opposite faces, and the resistance between these contacts is 1 Ω, then the resistivity of the material is 1 Ω⋅m.

Electromagnetic or magnetic induction is the production of an electromotive force across an electrical conductor in a changing magnetic field.

An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. Electromagnets usually consist of wire wound into a coil. A current through the wire creates a magnetic field which is concentrated in the hole, denoting the center of the coil. The magnetic field disappears when the current is turned off. The wire turns are often wound around a magnetic core made from a ferromagnetic or ferrimagnetic material such as iron; the magnetic core concentrates the magnetic flux and makes a more powerful magnet.

Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. The electric current flows mainly at the "skin" of the conductor, between the outer surface and a level called the skin depth. Skin depth depends on the frequency of the alternating current; as frequency increases, current flow moves to the surface, resulting in less skin depth. Skin effect reduces the effective cross-section of the conductor and thus increases its effective resistance. Skin effect is caused by opposing eddy currents induced by the changing magnetic field resulting from the alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies the skin depth becomes much smaller.

The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa via a thermocouple. A thermoelectric device creates a voltage when there is a different temperature on each side. Conversely, when a voltage is applied to it, heat is transferred from one side to the other, creating a temperature difference. At the atomic scale, an applied temperature gradient causes charge carriers in the material to diffuse from the hot side to the cold side.

"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon known as electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

In electrical engineering, electromagnetic shielding is the practice of reducing the electromagnetic field in a space by blocking the field with barriers made of conductive or magnetic materials. Shielding is typically applied to enclosures to isolate electrical devices from their surroundings, and to cables to isolate wires from the environment through which the cable runs. Electromagnetic shielding that blocks radio frequency (RF) electromagnetic radiation is also known as RF shielding.

An eddy current brake, also known as an induction brake, electric brake or electric retarder, is a device used to slow or stop a moving object by dissipating its kinetic energy as heat. Unlike friction brakes, where the drag force that stops the moving object is provided by friction between two surfaces pressed together, the drag force in an eddy current brake is an electromagnetic force between a magnet and a nearby conductive object in relative motion, due to eddy currents induced in the conductor through electromagnetic induction.

Eddy-current testing is one of many electromagnetic testing methods used in nondestructive testing (NDT) making use of electromagnetic induction to detect and characterize surface and sub-surface flaws in conductive materials.

The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant velocity, v, with respect to a magnet is calculated in the frame of reference of the magnet and in the frame of reference of the conductor. The observable quantity in the experiment, the current, is the same in either case, in accordance with the basic principle of relativity, which states: "Only relative motion is observable; there is no absolute standard of rest". However, according to Maxwell's equations, the charges in the conductor experience a magnetic force in the frame of the magnet and an electric force in the frame of the conductor. The same phenomenon would seem to have two different descriptions depending on the frame of reference of the observer.

An inductive sensor is a device that uses the principle of electromagnetic induction to detect or measure objects. An inductor develops a magnetic field when a current flows through it; alternatively, a current will flow through a circuit containing an inductor when the magnetic field through it changes. This effect can be used to detect metallic objects that interact with a magnetic field. Non-metallic substances such as liquids or some kinds of dirt do not interact with the magnetic field, so an inductive sensor can operate in wet or dirty conditions.

Arago's rotations is an observable magnetic phenomenon that involves the interactions between a magnetized needle and a moving metal disk. The effect was discovered by François Arago in 1824. At the time of their discovery, Arago's rotations were surprising effects that were difficult to explain. In 1831, Michael Faraday introduced the theory of electromagnetic induction, which explained how the effects happen in detail.

Lorentz force velocimetry (LFV) is a noncontact electromagnetic flow measurement technique. LFV is particularly suited for the measurement of velocities in liquid metals like steel or aluminium and is currently under development for metallurgical applications. The measurement of flow velocities in hot and aggressive liquids such as liquid aluminium and molten glass constitutes one of the grand challenges of industrial fluid mechanics. Apart from liquids, LFV can also be used to measure the velocity of solid materials as well as for detection of micro-defects in their structures.

Electromagnetism is one of the fundamental forces of nature. Early on, electricity and magnetism were studied separately and regarded as separate phenomena. Hans Christian Ørsted discovered that the two were related – electric currents give rise to magnetism. Michael Faraday discovered the converse, that magnetism could induce electric currents, and James Clerk Maxwell put the whole thing together in a unified theory of electromagnetism. Maxwell's equations further indicated that electromagnetic waves existed, and the experiments of Heinrich Hertz confirmed this, making radio possible. Maxwell also postulated, correctly, that light was a form of electromagnetic wave, thus making all of optics a branch of electromagnetism. Radio waves differ from light only in that the wavelength of the former is much longer than the latter. Albert Einstein showed that the magnetic field arises through the relativistic motion of the electric field and thus magnetism is merely a side effect of electricity. The modern theoretical treatment of electromagnetism is as a quantum field in quantum electrodynamics.

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