A magnetic field is a vector field that describes the magnetic influence on moving ^{2} per ampere), which is equivalent to newton per meter per ampere. and differ in how they account for magnetization. In

^{2} = 1 T. The SI unit tesla is equal to ( newton·

electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...

, which starts at a positive

electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...

of the particle, is the instantaneous

^{2}, the SI unit of magnetization is ampere per meter, identical to that of the -field.
The magnetization field of a region points in the direction of the average magnetic dipole moment in that region. Magnetization field lines, therefore, begin near the magnetic south pole and ends near the magnetic north pole. (Magnetization does not exist outside the magnet.)
In the Amperian loop model, the magnetization is due to combining many tiny Amperian loops to form a resultant current called '' bound current''. This bound current, then, is the source of the magnetic field due to the magnet. Given the definition of the magnetic dipole, the magnetization field follows a similar law to that of Ampere's law:
$$\backslash oint\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash boldsymbol\; =\; I\_\backslash mathrm,$$
where the integral is a line integral over any closed loop and is the bound current enclosed by that closed loop.
In the magnetic pole model, magnetization begins at and ends at magnetic poles. If a given region, therefore, has a net positive "magnetic pole strength" (corresponding to a north pole) then it has more magnetization field lines entering it than leaving it. Mathematically this is equivalent to:
$$\backslash oint\_S\; \backslash mu\_0\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf\; =\; -\; q\_\backslash mathrm,$$
where the integral is a closed surface integral over the closed surface and is the "magnetic charge" (in units of

electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...

(and therefore tends to drive a current in such a coil). This is known as ''Faraday's law'' and forms the basis of many magnetic flux
In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the webe ...

—the product of the area times the magnetic field normal to that area. (This definition of magnetic flux is why is often referred to as ''magnetic flux density''.) The negative sign represents the fact that any current generated by a changing magnetic field in a coil produces a magnetic field that ''opposes'' the ''change'' in the magnetic field that induced it. This phenomenon is known as Lenz's law. This integral formulation of Faraday's law can be converted
A complete expression for Faraday's law of induction in terms of the electric and magnetic fields can be written as:
$$\backslash mathcal\; =\; -\; \backslash frac\; =\; \backslash oint\_\; \backslash left(\; \backslash mathbf(\; \backslash mathbf,\backslash \; t)\; +\backslash mathbf\; \backslash times\; \backslash mathbf(\backslash mathbf,\backslash \; t)\backslash right)\; \backslash cdot\; d\backslash boldsymbol\backslash \; =-\backslash frac\; \backslash iint\_\; d\; \backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; (\backslash mathbf,\backslash \; t)$$
where is the moving closed path bounding the moving surface , and is an element of surface area of . The first integral calculates the work done moving a charge a distance based upon the Lorentz force law. In the case where the bounding surface is stationary, the Kelvin–Stokes theorem can be used to show this equation is equivalent to the Maxwell–Faraday equation.
into a differential form, which applies under slightly different conditions.
$$\backslash nabla\; \backslash times\; \backslash mathbf\; =\; -\backslash frac$$

^{−12} (and limited by experimental errors); for details see precision tests of QED. This makes QED one of the most accurate physical theories constructed thus far.
All equations in this article are in the classical approximation, which is less accurate than the quantum description mentioned here. However, under most everyday circumstances, the difference between the two theories is negligible.

^{14} T. ^{8} to 10^{11} T).

Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside 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.
...

, which explained and united all of classical electricity and magnetism. The first set of these equations was published in a paper entitled ''

Galileo Ferraris (March 1888) ''Rotazioni elettrodinamiche prodotte per mezzo di correnti alternate'' (Electrodynamic rotations by means of alternating currents), memory read at Accademia delle Scienze, Torino, in ''Opere di Galileo Ferraris'', Hoepli, Milano,1902 vol I pages 333 to 348

/ref> The twentieth century showed that classical electrodynamics is already consistent with special relativity, and extended classical electrodynamics to work with quantum mechanics.

Electromagnetism

'". * Nave, R., "

'". HyperPhysics. * "''Magnetism''"

theory.uwinnipeg.ca. * Hoadley, Rick, "

''" 17 July 2005. {{DEFAULTSORT:Magnetic Field Magnetism Physical quantities

electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...

s, electric currents
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving ...

, 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 material
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...

s such as iron
Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in fro ...

, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior ...

, diamagnetism
Diamagnetic materials are repelled by a magnetic field; an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast, paramagnetic and ferromagnetic materials are attract ...

, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in electromagnet
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 ...

s, and by electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...

s varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orien ...

assigning a vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...

to each point of space, called a vector field.
In electromagnetics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...

, the term "magnetic field" is used for two distinct but closely related vector fields denoted by the symbols and . In the International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...

, the unit of , magnetic field strength, is the ampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to ele ...

per meter (A/m). The unit of , the magnetic flux
In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the webe ...

density, is the tesla (in SI base units: kilogram per secondvacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...

, the two fields are related through the vacuum permeability, $\backslash mathbf/\backslash mu\_0\; =\; \backslash mathbf$; but in a magnetized material, the quantities on each side of this equation differ by the magnetization field of the material.
Magnetic fields are produced by moving electric charges and the intrinsic magnetic moments of elementary particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, antiq ...

s associated with a fundamental quantum property, their spin. Magnetic fields and electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...

s are interrelated and are both components of the electromagnetic force
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...

, one of the four fundamental force
In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electr ...

s of nature.
Magnetic fields are used throughout modern technology, particularly in electrical engineering and electromechanics
In engineering, electromechanics combines processes and procedures drawn from electrical engineering and mechanical engineering. Electromechanics focuses on the interaction of electrical and mechanical systems as a whole and how the two syste ...

. Rotating magnetic fields are used in both electric motors and generators. The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as magnetic circuits. Magnetic forces give information about the charge carriers in a material through the Hall effect. The Earth produces its own magnetic field, which shields the Earth's ozone layer from the solar wind
The solar wind is a stream of charged particles released from the upper atmosphere of the Sun, called the corona. This plasma mostly consists of electrons, protons and alpha particles with kinetic energy between . The composition of the ...

and is important in navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...

using a compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...

.
Description

The force on an electric charge depends on its location, speed, and direction; two vector fields are used to describe this force. The first is theelectric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...

, which describes the force acting on a stationary charge and gives the component of the force that is independent of motion. The magnetic field, in contrast, describes the component of the force that is proportional to both the speed and direction of charged particles. The field is defined by the Lorentz force law
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyb ...

and is, at each instant, perpendicular to both the motion of the charge and the force it experiences.
There are two different, but closely related vector fields which are both sometimes called the "magnetic field" written and .The letters B and H were originally chosen by Maxwell in his '' Treatise on Electricity and Magnetism'' (Vol. II, pp. 236–237). For many quantities, he simply started choosing letters from the beginning of the alphabet. See While both the best names for these fields and exact interpretation of what these fields represent has been the subject of long running debate, there is wide agreement about how the underlying physics work. Historically, the term "magnetic field" was reserved for while using other terms for , but many recent textbooks use the term "magnetic field" to describe as well as or in place of . Edward Purcell, in Electricity and Magnetism, McGraw-Hill, 1963, writes, ''Even some modern writers who treat as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by . This seems clumsy and pedantic. If you go into the laboratory and ask a physicist what causes the pion trajectories in his bubble chamber to curve, he'll probably answer "magnetic field", not "magnetic induction." You will seldom hear a geophysicist refer to the Earth's magnetic induction, or an astrophysicist talk about the magnetic induction of the galaxy. We propose to keep on calling the magnetic field. As for , although other names have been invented for it, we shall call it "the field " or even "the magnetic field ."'' In a similar vein, says: "So we may think of both and as magnetic fields, but drop the word 'magnetic' from so as to maintain the distinction ... As Purcell points out, 'it is only the names that give trouble, not the symbols'."
There are many alternative names for both (see sidebar).
The B-field

The magnetic field vector at any point can be defined as the vector that, when used in theLorentz force law
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyb ...

, correctly predicts the force on a charged particle at that point:
Here is the force on the particle, is the particle's electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...

, , is the particle's velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...

, and × denotes the cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...

. The direction of force on the charge can be determined by a mnemonic
A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding.
Mnemonics make use of elaborative encoding, retrieval cues, and imag ...

known as the ''right-hand rule'' (see the figure).An alternative mnemonic to the right hand rule is Fleming's left-hand rule. Using the right hand, pointing the thumb in the direction of the current, and the fingers in the direction of the magnetic field, the resulting force on the charge points outwards from the palm. The force on a negatively charged particle is in the opposite direction. If both the speed and the charge are reversed then the direction of the force remains the same. For that reason a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these cases produce the same current.) On the other hand, a magnetic field combined with an electric field ''can'' distinguish between these, see Hall effect below.
The first term in the Lorentz equation is from the theory of electrostatics
Electrostatics is a branch of physics that studies electric charges at rest (static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for am ...

, and says that a particle of charge in an electric field experiences an electric force:
$$\backslash mathbf\_\; =\; q\; \backslash mathbf.$$
The second term is the magnetic force:
$$\backslash mathbf\_\; =\; q(\backslash mathbf\; \backslash times\; \backslash mathbf).$$
Using the definition of the cross product, the magnetic force can also be written as a scalar equation:
$$F\_\; =\; q\; v\; B\; \backslash sin(\backslash theta)$$
where , , and are the scalar magnitude of their respective vectors, and is the angle between the velocity of the particle and the magnetic field. The vector is ''defined'' as the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle. In other words,
The field can also be defined by the torque on a magnetic dipole, .
The SI unit of is tesla (symbol: T).The SI unit of (magnetic flux
In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the webe ...

) is the weber (symbol: Wb), related to the tesla by 1 Wb/msecond
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds e ...

)/( coulomb·metre
The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pr ...

). This can be seen from the magnetic part of the Lorentz force law. The Gaussian-cgs unit of is the gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...

(symbol: G). (The conversion is 1 T ≘ 10000 G.) One nanotesla corresponds to 1 gamma (symbol: γ).
The H-field

The magnetic field is defined: Where $\backslash mu\_0$ is the vacuum permeability, and is the magnetization vector. In a vacuum, and are proportional to each other. Inside a material they are different (see H and B inside and outside magnetic materials). The SI unit of te -field is theampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to ele ...

per metre (A/m), and the CGS unit is the oersted (Oe).
Measurement

An instrument used to measure the local magnetic field is known as amagnetometer
A magnetometer is a device that measures magnetic field or magnetic dipole moment. Different types of magnetometers measure the direction, strength, or relative change of a magnetic field at a particular location. A compass is one such device, on ...

. Important classes of magnetometers include using induction magnetometers (or search-coil magnetometers) which measure only varying magnetic fields, rotating coil magnetometers, Hall effect magnetometers, NMR magnetometers, SQUID magnetometers, and fluxgate magnetometers. The magnetic fields of distant astronomical object
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often us ...

s are measured through their effects on local charged particles. For instance, electrons spiraling around a field line produce synchrotron radiation
Synchrotron radiation (also known as magnetobremsstrahlung radiation) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in ...

that is detectable in radio waves
Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz ( GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (s ...

. The finest precision for a magnetic field measurement was attained by Gravity Probe B at ().
Visualization

The field can be visualized by a set of ''magnetic field lines'', that follow the direction of the field at each point. The lines can be constructed by measuring the strength and direction of the magnetic field at a large number of points (or at every point in space). Then, mark each location with an arrow (called avector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...

) pointing in the direction of the local magnetic field with its magnitude proportional to the strength of the magnetic field. Connecting these arrows then forms a set of magnetic field lines. The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and the local density of field lines can be made proportional to its strength. Magnetic field lines are like streamlines in fluid flow, in that they represent a continuous distribution, and a different resolution would show more or fewer lines.
An advantage of using magnetic field lines as a representation is that many laws of magnetism (and electromagnetism) can be stated completely and concisely using simple concepts such as the "number" of field lines through a surface. These concepts can be quickly "translated" to their mathematical form. For example, the number of field lines through a given surface is the surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one m ...

of the magnetic field.
Various phenomena "display" magnetic field lines as though the field lines were physical phenomena. For example, iron filings placed in a magnetic field form lines that correspond to "field lines".The use of iron filings to display a field presents something of an exception to this picture; the filings alter the magnetic field so that it is much larger along the "lines" of iron, because of the large permeability of iron relative to air. Magnetic field "lines" are also visually displayed in polar auroras, in which plasma particle dipole interactions create visible streaks of light that line up with the local direction of Earth's magnetic field.
Field lines can be used as a qualitative tool to visualize magnetic forces. In ferromagnetic substances like iron
Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in fro ...

and in plasmas, magnetic forces can be understood by imagining that the field lines exert a tension, (like a rubber band) along their length, and a pressure perpendicular to their length on neighboring field lines. "Unlike" poles of magnets attract because they are linked by many field lines; "like" poles repel because their field lines do not meet, but run parallel, pushing on each other.
Magnetic field of permanent magnets

''Permanent magnets'' are objects that produce their own persistent magnetic fields. They are made of ferromagnetic materials, such as iron andnickel
Nickel is a chemical element with symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and ductile transition metal. Pure nickel is chemically reactive but large pieces are slow to ...

, that have been magnetized, and they have both a north and a south pole.
The magnetic field of permanent magnets can be quite complicated, especially near the magnet. The magnetic field of a smallHere, "small" means that the observer is sufficiently far away from the magnet, so that the magnet can be considered as infinitesimally small. "Larger" magnets need to include more complicated terms in the and depend on the entire geometry of the magnet not just . straight magnet is proportional to the magnet's ''strength'' (called its magnetic dipole moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagn ...

). The equations
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in Fr ...

are non-trivial and also depend on the distance from the magnet and the orientation of the magnet. For simple magnets, points in the direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its by 180 degrees.
The magnetic field of larger magnets can be obtained by modeling them as a collection of a large number of small magnets called dipole
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...

s each having their own . The magnetic field produced by the magnet then is the net magnetic field of these dipoles; any net force on the magnet is a result of adding up the forces on the individual dipoles.
There were two simplified models for the nature of these dipoles. These two models produce two different magnetic fields, and . Outside a material, though, the two are identical (to a multiplicative constant) so that in many cases the distinction can be ignored. This is particularly true for magnetic fields, such as those due to electric currents, that are not generated by magnetic materials.
A realistic model of magnetism is more complicated than either of these models; neither model fully explains why materials are magnetic. The monopole model has no experimental support. Ampere's model explains some, but not all of a material's magnetic moment. Like Ampere's model predicts, the motion of electrons within an atom are connected to those electrons' orbital magnetic dipole moment, and these orbital moments do contribute to the magnetism seen at the macroscopic level. However, the motion of electrons is not classical, and the spin magnetic moment of electrons (which is not explained by either model) is also a significant contribution to the total moment of magnets.
Magnetic pole model

Historically, early physics textbooks would model the force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as theCoulomb force
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...

between electric charges. At the microscopic level, this model contradicts the experimental evidence, and the pole model of magnetism is no longer the typical way to introduce the concept. However, it is still sometimes used as a macroscopic model for ferromagnetism due to its mathematical simplicity.
In this model, a magnetic -field is produced by fictitious ''magnetic charges'' that are spread over the surface of each pole. These ''magnetic charges'' are in fact related to the magnetization field . The -field, therefore, is analogous to the electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...

and ends at a negative electric charge. Near the north pole, therefore, all -field lines point away from the north pole (whether inside the magnet or out) while near the south pole all -field lines point toward the south pole (whether inside the magnet or out). Too, a north pole feels a force in the direction of the -field while the force on the south pole is opposite to the -field.
In the magnetic pole model, the elementary magnetic dipole is formed by two opposite magnetic poles of pole strength separated by a small distance vector , such that . The magnetic pole model predicts correctly the field both inside and outside magnetic materials, in particular the fact that is opposite to the magnetization field inside a permanent magnet.
Since it is based on the fictitious idea of a ''magnetic charge density'', the pole model has limitations. Magnetic poles cannot exist apart from each other as electric charges can, but always come in north–south pairs. If a magnetized object is divided in half, a new pole appears on the surface of each piece, so each has a pair of complementary poles. The magnetic pole model does not account for magnetism that is produced by electric currents, nor the inherent connection between angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sys ...

and magnetism.
The pole model usually treats magnetic charge as a mathematical abstraction, rather than a physical property of particles. However, a magnetic monopole is a hypothetical particle (or class of particles) that physically has only one magnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogous to an electric charge. Magnetic field lines would start or end on magnetic monopoles, so if they exist, they would give exceptions to the rule that magnetic field lines neither start nor end. Some theories (such as Grand Unified Theories) have predicted the existence of magnetic monopoles, but so far, none have been observed.
Amperian loop model

In the model developed byAmpere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to ele ...

, the elementary magnetic dipole that makes up all magnets is a sufficiently small Amperian loop of current I. The dipole moment of this loop is where is the area of the loop.
These magnetic dipoles produce a magnetic -field.
The magnetic field of a magnetic dipole is depicted in the figure. From outside, the ideal magnetic dipole is identical to that of an ideal electric dipole of the same strength. Unlike the electric dipole, a magnetic dipole is properly modeled as a current loop having a current and an area . Such a current loop has a magnetic moment of:
$$m\; =\; Ia,$$
where the direction of is perpendicular to the area of the loop and depends on the direction of the current using the right-hand rule. An ideal magnetic dipole is modeled as a real magnetic dipole whose area has been reduced to zero and its current increased to infinity such that the product is finite. This model clarifies the connection between angular momentum and magnetic moment, which is the basis of the Einstein–de Haas effect ''rotation by magnetization'' and its inverse, the Barnett effect or ''magnetization by rotation''.See magnetic moment and Rotating the loop faster (in the same direction) increases the current and therefore the magnetic moment, for example.
Interactions with magnets

Force between magnets

Specifying the force between two small magnets is quite complicated because it depends on the strength and orientation of both magnets and their distance and direction relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic fieldEither or may be used for the magnetic field outside the magnet. of the other. To understand the force between magnets, it is useful to examine the ''magnetic pole model'' given above. In this model, the ''-field'' of one magnet pushes and pulls on ''both'' poles of a second magnet. If this -field is the same at both poles of the second magnet then there is no net force on that magnet since the force is opposite for opposite poles. If, however, the magnetic field of the first magnet is ''nonuniform'' (such as the near one of its poles), each pole of the second magnet sees a different field and is subject to a different force. This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque. This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field, whether caused by permanent magnets or electric currents, exerts a force on a small magnet in this way. The details of the Amperian loop model are different and more complicated but yield the same result: that magnetic dipoles are attracted/repelled into regions of higher magnetic field. Mathematically, the force on a small magnet having a magnetic moment due to a magnetic field is: $$\backslash mathbf\; =\; \backslash boldsymbol\; \backslash left(\backslash mathbf\backslash cdot\backslash mathbf\backslash right),$$ where thegradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...

is the change of the quantity per unit distance and the direction is that of maximum increase of . The dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...

, where and represent the magnitude
Magnitude may refer to:
Mathematics
* Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order o ...

of the and vectors and is the angle between them. If is in the same direction as then the dot product is positive and the gradient points "uphill" pulling the magnet into regions of higher -field (more strictly larger ). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions each having their own then summing up the forces on each of these very small regions.
Magnetic torque on permanent magnets

If two like poles of two separate magnets are brought near each other, and one of the magnets is allowed to turn, it promptly rotates to align itself with the first. In this example, the magnetic field of the stationary magnet creates a ''magnetic torque'' on the magnet that is free to rotate. This magnetic torque tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field. In terms of the pole model, two equal and opposite magnetic charges experiencing the same also experience equal and opposite forces. Since these equal and opposite forces are in different locations, this produces a torque proportional to the distance (perpendicular to the force) between them. With the definition of as the pole strength times the distance between the poles, this leads to , where is a constant called the vacuum permeability, measuring V· s/( A· m) and is the angle between and . Mathematically, the torque on a small magnet is proportional both to the applied magnetic field and to the magnetic moment of the magnet: $$\backslash boldsymbol=\backslash mathbf\backslash times\backslash mathbf\; =\; \backslash mu\_0\backslash mathbf\backslash times\backslash mathbf,\; \backslash ,$$ where × represents the vectorcross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...

. This equation includes all of the qualitative information included above. There is no torque on a magnet if is in the same direction as the magnetic field, since the cross product is zero for two vectors that are in the same direction. Further, all other orientations feel a torque that twists them toward the direction of magnetic field.
Interactions with electric currents

Currents of electric charges both generate a magnetic field and feel a force due to magnetic B-fields.Magnetic field due to moving charges and electric currents

All moving charged particles produce magnetic fields. Moving point charges, such aselectron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...

s, produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.
Magnetic field lines form in concentric
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center po ...

circles around a cylindrical current-carrying conductor, such as a length of wire. The direction of such a magnetic field can be determined by using the "right-hand grip rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of ...

" (see figure at right). The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength is inversely proportional to the distance.)
Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely spaced loops to form a coil or "solenoid
upright=1.20, An illustration of a solenoid
upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines
A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whose ...

" enhances this effect. A device so formed around an iron core may act as an ''electromagnet'', generating a strong, well-controlled magnetic field. An infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet, with its strength and polarity determined by the current flowing through the coil.
The magnetic field generated by a steady current (a constant flow of electric charges, in which charge neither accumulates nor is depleted at any point) is described by the '' Biot–Savart law'':
$$\backslash mathbf\; =\; \backslash frac\backslash int\_\backslash frac,$$
where the integral sums over the wire length where vector is the vector line element
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...

with direction in the same sense as the current , is the magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constan ...

, is the distance between the location of and the location where the magnetic field is calculated, and is a unit vector in the direction of . For example, in the case of a sufficiently long, straight wire, this becomes:
$$,\; \backslash mathbf,\; =\; \backslash fracI$$
where . The direction is tangent to a circle perpendicular to the wire according to the right hand rule.
A slightly more general
The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of being zero, which is always valid. (There are no magnetic charges.) way of relating the current $I$ to the -field is through Ampère's law:
$$\backslash oint\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash boldsymbol\; =\; \backslash mu\_0\; I\_,$$
where the line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, ...

is over any arbitrary loop and $I\_\backslash text$ is the current enclosed by that loop. Ampère's law is always valid for steady currents and can be used to calculate the -field for certain highly symmetric situations such as an infinite wire or an infinite solenoid.
In a modified form that accounts for time varying electric fields, Ampère's law is one of four Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside 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.
...

that describe electricity and magnetism.
Force on moving charges and current

Force on a charged particle

Acharged particle
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...

moving in a -field experiences a ''sideways'' force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the ''Lorentz force'', and is given by
$$\backslash mathbf\; =\; q\backslash mathbf\; +\; q\; \backslash mathbf\; \backslash times\; \backslash mathbf,$$
where is the force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...

, is the velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...

of the particle, and is the magnetic field (in teslas).
The Lorentz force is always perpendicular to both the velocity of the particle and the magnetic field that created it. When a charged particle moves in a static magnetic field, it traces a helical path in which the helix axis is parallel to the magnetic field, and in which the speed of the particle remains constant. Because the magnetic force is always perpendicular to the motion, the magnetic field can do no work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal ...

on an isolated charge. It can only do work indirectly, via the electric field generated by a changing magnetic field. It is often claimed that the magnetic force can do work to a non-elementary magnetic dipole, or to charged particles whose motion is constrained by other forces, but this is incorrect
because the work in those cases is performed by the electric forces of the charges deflected by the magnetic field.
Force on current-carrying wire

The force on a current carrying wire is similar to that of a moving charge as expected since a current carrying wire is a collection of moving charges. A current-carrying wire feels a force in the presence of a magnetic field. The Lorentz force on a macroscopic current is often referred to as the ''Laplace force''. Consider a conductor of length , cross section , and charge due to electric current . If this conductor is placed in a magnetic field of magnitude that makes an angle with the velocity of charges in the conductor, the force exerted on a single charge is $$F\; =\; qvB\; \backslash sin\backslash theta,$$ so, for charges where $$N\; =\; n\; \backslash ell\; A\; ,$$ the force exerted on the conductor is $$f\; =\; F\; N\; =\; q\; v\; B\; n\backslash ell\; A\; \backslash sin\backslash theta\; =\; Bi\backslash ell\; \backslash sin\backslash theta,$$ where .Relation between H and B

The formulas derived for the magnetic field above are correct when dealing with the entire current. A magnetic material placed inside a magnetic field, though, generates its own bound current, which can be a challenge to calculate. (This bound current is due to the sum of atomic sized current loops and the spin of the subatomic particles such as electrons that make up the material.) The -field as defined above helps factor out this bound current; but to see how, it helps to introduce the concept of ''magnetization'' first.Magnetization

The ''magnetization'' vector field represents how strongly a region of material is magnetized. It is defined as the netmagnetic dipole moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagn ...

per unit volume of that region. The magnetization of a uniform magnet is therefore a material constant, equal to the magnetic moment of the magnet divided by its volume. Since the SI unit of magnetic moment is A⋅mmagnetic flux
In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the webe ...

) enclosed by . (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because the magnetization field moves from south to north.
H-field and magnetic materials

In SI units, the H-field is related to the B-field by $$\backslash mathbf\backslash \; \backslash equiv\; \backslash \; \backslash frac\; -\; \backslash mathbf.$$ In terms of the H-field, Ampere's law is $$\backslash oint\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash boldsymbol\; =\; \backslash oint\; \backslash left(\backslash frac\; -\; \backslash mathbf\backslash right)\; \backslash cdot\; \backslash mathrm\backslash boldsymbol\; =\; I\_\backslash mathrm\; -\; I\_\backslash mathrm\; =\; I\_\backslash mathrm,$$ where represents the 'free current' enclosed by the loop so that the line integral of does not depend at all on the bound currents. For the differential equivalent of this equation seeMaxwell's equations
Maxwell's equations, or Maxwell–Heaviside 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.
...

. Ampere's law leads to the boundary condition
$$\backslash left(\backslash mathbf\; -\; \backslash mathbf\backslash right)\; =\; \backslash mathbf\_\backslash mathrm\; \backslash times\; \backslash hat,$$
where is the surface free current density and the unit normal $\backslash hat$ points in the direction from medium 2 to medium 1.
Similarly, a surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one m ...

of over any closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...

is independent of the free currents and picks out the "magnetic charges" within that closed surface:
$$\backslash oint\_S\; \backslash mu\_0\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf\; =\; \backslash oint\_S\; (\backslash mathbf\; -\; \backslash mu\_0\; \backslash mathbf)\; \backslash cdot\; \backslash mathrm\backslash mathbf\; =\; 0\; -\; (-q\_\backslash mathrm)\; =\; q\_\backslash mathrm,$$
which does not depend on the free currents.
The -field, therefore, can be separated into twoA third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below. independent parts:
$$\backslash mathbf\; =\; \backslash mathbf\_0\; +\; \backslash mathbf\_\backslash mathrm,$$
where is the applied magnetic field due only to the free currents and is the demagnetizing field due only to the bound currents.
The magnetic -field, therefore, re-factors the bound current in terms of "magnetic charges". The field lines loop only around "free current" and, unlike the magnetic field, begins and ends near magnetic poles as well.
Magnetism

Most materials respond to an applied -field by producing their own magnetization and therefore their own -fields. Typically, the response is weak and exists only when the magnetic field is applied. The term ''magnetism'' describes how materials respond on the microscopic level to an applied magnetic field and is used to categorize the magnetic phase of a material. Materials are divided into groups based upon their magnetic behavior: * Diamagnetic materials produce a magnetization that opposes the magnetic field. * Paramagnetic materials produce a magnetization in the same direction as the applied magnetic field. *Ferromagnetic materials
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...

and the closely related ferrimagnetic materials and antiferromagnetic materials can have a magnetization independent of an applied B-field with a complex relationship between the two fields.
* Superconductors (and ferromagnetic superconductors) are materials that are characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures and magnetic fields (the so-named mixed state) under which they exhibit a complex hysteretic dependence of on .
In the case of paramagnetism and diamagnetism, the magnetization is often proportional to the applied magnetic field such that:
$$\backslash mathbf\; =\; \backslash mu\; \backslash mathbf,$$
where is a material dependent parameter called the permeability. In some cases the permeability may be a second rank tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

so that may not point in the same direction as . These relations between and are examples of constitutive equations. However, superconductors and ferromagnets have a more complex -to- relation; see magnetic hysteresis.
Stored energy

Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field. For non-dispersive materials, this same energy is released when the magnetic field is destroyed so that the energy can be modeled as being stored in the magnetic field. For linear, non-dispersive, materials (such that where is frequency-independent), theenergy density
In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or .
Often only the ''useful'' or ex ...

is:
$$u\; =\; \backslash frac=\; \backslash frac\; =\; \backslash frac.$$
If there are no magnetic materials around then can be replaced by . The above equation cannot be used for nonlinear materials, though; a more general expression given below must be used.
In general, the incremental amount of work per unit volume needed to cause a small change of magnetic field is:
$$\backslash delta\; W\; =\; \backslash mathbf\backslash cdot\backslash delta\backslash mathbf.$$
Once the relationship between and is known this equation is used to determine the work needed to reach a given magnetic state. For hysteretic materials such as ferromagnets and superconductors, the work needed also depends on how the magnetic field is created. For linear non-dispersive materials, though, the general equation leads directly to the simpler energy density equation given above.
Appearance in Maxwell's equations

Like all vector fields, a magnetic field has two important mathematical properties that relates it to its ''sources''. (For the ''sources'' are currents and changing electric fields.) These two properties, along with the two corresponding properties of the electric field, make up ''Maxwell's Equations''. Maxwell's Equations together with the Lorentz force law form a complete description of classical electrodynamics including both electricity and magnetism. The first property is thedivergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of th ...

of a vector field , , which represents how "flows" outward from a given point. As discussed above, a -field line never starts or ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of is zero. (Such vector fields are called solenoidal vector fields.) This property is called Gauss's law for magnetism
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. I ...

and is equivalent to the statement that there are no isolated magnetic poles or magnetic monopoles.
The second mathematical property is called the curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...

, such that represents how curls or "circulates" around a given point. The result of the curl is called a "circulation source". The equations for the curl of and of are called the Ampère–Maxwell equation and Faraday's law respectively.
Gauss' law for magnetism

One important property of the -field produced this way is that magnetic -field lines neither start nor end (mathematically, is a solenoidal vector field); a field line may only extend to infinity, or wrap around to form a closed curve, or follow a never-ending (possibly chaotic) path. Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet -field lines continue through the magnet from the south pole back to the north.To see that this must be true imagine placing a compass inside a magnet. There, the north pole of the compass points toward the north pole of the magnet since magnets stacked on each other point in the same direction. If a -field line enters a magnet somewhere it has to leave somewhere else; it is not allowed to have an end point. More formally, since all the magnetic field lines that enter any given region must also leave that region, subtracting the "number"As discussed above, magnetic field lines are primarily a conceptual tool used to represent the mathematics behind magnetic fields. The total "number" of field lines is dependent on how the field lines are drawn. In practice, integral equations such as the one that follows in the main text are used instead. of field lines that enter the region from the number that exit gives identically zero. Mathematically this is equivalent toGauss's law for magnetism
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field has divergence equal to zero, in other words, that it is a solenoidal vector field. I ...

:
$$\backslash oint\_S\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf\; =\; 0$$
where the integral is a surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one m ...

over the closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...

(a closed surface is one that completely surrounds a region with no holes to let any field lines escape). Since points outward, the dot product in the integral is positive for -field pointing out and negative for -field pointing in.
Faraday's Law

A changing magnetic field, such as a magnet moving through a conducting coil, generates anelectrical generator
In electricity generation, a generator is a device that converts motive power (mechanical energy) or fuel-based power (chemical energy) into electric power for use in an external circuit. Sources of mechanical energy include steam turbines, ...

s and electric motors. Mathematically, Faraday's law is:
$$\backslash mathcal\; =\; -\; \backslash frac$$
where $\backslash mathcal$ is the electromotive force
In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical '' t ...

(or ''EMF'', the voltage generated around a closed loop) and is the Ampère's Law and Maxwell's correction

Similar to the way that a changing magnetic field generates an electric field, a changing electric field generates a magnetic field. This fact is known as ''Maxwell's correction to Ampère's law'' and is applied as an additive term to Ampere's law as given above. This additional term is proportional to the time rate of change of the electric flux and is similar to Faraday's law above but with a different and positive constant out front. (The electric flux through an area is proportional to the area times the perpendicular part of the electric field.) The full law including the correction term is known as the Maxwell–Ampère equation. It is not commonly given in integral form because the effect is so small that it can typically be ignored in most cases where the integral form is used. The Maxwell term ''is'' critically important in the creation and propagation of electromagnetic waves. Maxwell's correction to Ampère's Law together with Faraday's law of induction describes how mutually changing electric and magnetic fields interact to sustain each other and thus to formelectromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ligh ...

, such as light: a changing electric field generates a changing magnetic field, which generates a changing electric field again. These, though, are usually described using the differential form of this equation given below.
$$\backslash nabla\; \backslash times\; \backslash mathbf\; =\; \backslash mu\_0\backslash mathbf\; +\; \backslash mu\_0\; \backslash varepsilon\_0\; \backslash frac$$
where is the complete microscopic current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ar ...

.
As discussed above, materials respond to an applied electric field and an applied magnetic field by producing their own internal "bound" charge and current distributions that contribute to and but are difficult to calculate. To circumvent this problem, and fields are used to re-factor Maxwell's equations in terms of the ''free current density'' :
$$\backslash nabla\; \backslash times\; \backslash mathbf\; =\; \backslash mathbf\_\backslash mathrm\; +\; \backslash frac$$
These equations are not any more general than the original equations (if the "bound" charges and currents in the material are known). They also must be supplemented by the relationship between and as well as that between and . On the other hand, for simple relationships between these quantities this form of Maxwell's equations can circumvent the need to calculate the bound charges and currents.
Formulation in special relativity and quantum electrodynamics

Relativistic Electrodynamics

As different aspects of the same phenomenon

According to the special theory of relativity, the partition of theelectromagnetic force
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...

into separate electric and magnetic components is not fundamental, but varies with the observational frame of reference: An electric force perceived by one observer may be perceived by another (in a different frame of reference) as a magnetic force, or a mixture of electric and magnetic forces.
The magnetic field existing as electric field in other frames can be shown by consistency of equations obtained from Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...

of four force from Coulomb's Law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...

in particle's rest frame with Maxwell's laws considering definition of fields from Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an el ...

and for non accelerating condition. The form of magnetic field hence obtained by Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...

of four-force from the form of Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...

in source's initial frame is given by:$$\backslash mathbf\; =\; \backslash frac\; q\; \backslash frac\; \backslash frac\; =\; \backslash frac$$where $q$ is the charge of the point source, $\backslash mathbf$ is the position vector from the point source to the point in space, $\backslash mathbf$ is the velocity vector of the charged particle, $\backslash beta$ is the ratio of speed of the charged particle divided by the speed of light and $\backslash theta$ is the angle between $\backslash mathbf$ and $\backslash mathbf$. This form of magnetic field can be shown to satisfy maxwell's laws within the constraint of particle being non accelerating. Note that the above reduces to Biot-Savart law for non relativistic stream of current ($\backslash beta\backslash ll\; 1$).
Formally, special relativity combines the electric and magnetic fields into a rank-2 tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

, called the '' electromagnetic tensor''. Changing reference frames ''mixes'' these components. This is analogous to the way that special relativity ''mixes'' space and time into spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

, and mass, momentum, and energy into four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum ...

. Similarly, the energy stored in a magnetic field is mixed with the energy stored in an electric field in the electromagnetic stress–energy tensor.
Magnetic vector potential

In advanced topics such asquantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

and relativity it is often easier to work with a potential formulation of electrodynamics rather than in terms of the electric and magnetic fields. In this representation, the '' magnetic vector potential'' , and the electric scalar potential , are defined using gauge fixing such that:
$$\backslash begin\; \backslash mathbf\; \&=\; \backslash nabla\; \backslash times\; \backslash mathbf,\; \backslash \backslash \; \backslash mathbf\; \&=\; -\backslash nabla\; \backslash varphi\; -\; \backslash frac.\; \backslash end$$.
The vector potential, ' given by this form may be interpreted as a ''generalized potential momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...

per unit charge'' just as is interpreted as a ''generalized potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potent ...

per unit charge''. There are multiple choices one can make for the potential fields that satisfy the above condition. However, the choice of potentials is represented by its respective gauge condition.
Maxwell's equations when expressed in terms of the potentials in Lorentz gauge can be cast into a form that agrees with special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...

. In relativity, together with forms a four-potential regardless of the gauge condition, analogous to the four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum ...

that combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler—and it can be easily modified to work with quantum mechanics.
Propagation of Electric and Magnetic fields

Special theory of relativity imposes the condition for events related bycause and effect
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...

to be time-like separated, that is that causal efficacy propagates no faster than light. Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside 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.
...

for electromagnetism are be found to be in favor of this as electric and magnetic disturbances are found to travel at the speed of light in space. Electric and magnetic fields from classical electrodynamics obey the principle of locality in physics and are expressed in terms of retarded time or the time at which the cause of a measured field originated given that the influence of field travelled at speed of light. The retarded time for a point particle is given as solution of:
$t\_r=\backslash mathbf-\backslash frac$
where $$ is retarded time or the time at which the source's contribution of the field originated, $\_s(t)$ is the position vector of the particle as function of time, $\backslash mathbf$ is the point in space, $\backslash mathbf$ is the time at which fields are measured and $c$ is the speed of light. The equation subtracts the time taken for light to travel from particle to the point in space from the time of measurement to find time of origin of the fields. The uniqueness of solution for $$ for given $\backslash mathbf$, $\backslash mathbf$ and $r\_s(t)$ is valid for charged particles moving slower than speed of light.
Magnetic field of arbitrary moving point charge

The solution of maxwell's equations for electric and magnetic field of a point charge is expressed in terms of retarded time or the time at which the particle in the past causes the field at the point, given that the influence travels across space at the speed of light. Any arbitrary motion of point charge causes electric and magnetic fields found by solving maxwell's equations using green's function for retarded potentials and hence finding the fields to be as follows: $\backslash mathbf(\backslash mathbf,\backslash mathbf)\; =\; \backslash frac\; \backslash left(\backslash frac\; \backslash right)\_\; =\; \backslash frac\; \backslash varphi(\backslash mathbf,\; \backslash mathbf)$ $\backslash mathbf(\backslash mathbf,\; \backslash mathbf)\; =\; \backslash frac\; \backslash left(\backslash frac\; +\; \backslash frac\; \backslash right)\_\; =\; \backslash frac\; \backslash times\; \backslash mathbf(\backslash mathbf,\; \backslash mathbf)$ where $\backslash varphi(\backslash mathbf,\; \backslash mathbf)$and $\backslash mathbf(\backslash mathbf,\backslash mathbf)$ are electric scalar potential and magnetic vector potential in Lorentz gauge, $q$ is the charge of the point source, $\_s(\backslash mathbf,t)$ is a unit vector pointing from charged particle to the point in space, $\backslash boldsymbol\_s(t)$ is the velocity of the particle divided by the speed of light and $\backslash gamma\; (t)$ is the correspondingLorentz factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativi ...

. Hence by the principle of superposition, the fields of a system of charges also obey principle of locality.
Quantum electrodynamics

In modern physics, the electromagnetic field is understood to be not a '' classical'' field, but rather a quantum field; it is represented not as a vector of threenumbers
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...

at each point, but as a vector of three quantum operators at each point. The most accurate modern description of the electromagnetic interaction (and much else) is ''quantum electrodynamics'' (QED), which is incorporated into a more complete theory known as the ''Standard Model of particle physics''.
In QED, the magnitude of the electromagnetic interactions between charged particles (and their antiparticle
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...

s) is computed using perturbation theory. These rather complex formulas produce a remarkable pictorial representation as Feynman diagrams in which virtual photons are exchanged.
Predictions of QED agree with experiments to an extremely high degree of accuracy: currently about 10Uses and examples

Earth's magnetic field

The Earth's magnetic field is produced byconvection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the conv ...

of a liquid iron alloy in the outer core
Earth's outer core is a fluid layer about thick, composed of mostly iron and nickel that lies above Earth's solid inner core and below its mantle. The outer core begins approximately beneath Earth's surface at the core-mantle boundary an ...

. In a dynamo process, the movements drive a feedback process in which electric currents create electric and magnetic fields that in turn act on the currents.
The field at the surface of the Earth is approximately the same as if a giant bar magnet were positioned at the center of the Earth and tilted at an angle of about 11° off the rotational axis of the Earth (see the figure). The north pole of a magnetic compass needle points roughly north, toward the North Magnetic Pole. However, because a magnetic pole is attracted to its opposite, the North Magnetic Pole is actually the south pole of the geomagnetic field. This confusion in terminology arises because the pole of a magnet is defined by the geographical direction it points.
Earth's magnetic field is not constant—the strength of the field and the location of its poles vary. Moreover, the poles periodically reverse their orientation in a process called geomagnetic reversal. The most recent reversal occurred 780,000 years ago.
Rotating magnetic fields

The ''rotating magnetic field'' is a key principle in the operation of alternating-current motors. A permanent magnet in such a field rotates so as to maintain its alignment with the external field. This effect was conceptualized byNikola Tesla
Nikola Tesla ( ; ,"Tesla"

'' alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...

) electric motors.
Magnetic torque is used to drive electric motors. In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array of '' alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...

electromagnet
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 ...

s. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft.
A rotating magnetic field can be constructed using two orthogonal coils with 90 degrees phase difference in their AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents.
This inequality would cause serious problems in standardization of the conductor size and so, to overcome it, three-phase systems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 degrees create the rotating magnetic field in this case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's electrical power supply systems.
Synchronous motor
A synchronous electric motor is an AC electric motor in which, at steady state,
the rotation of the shaft is synchronized with the frequency of the supply current; the rotation period is exactly equal to an integral number of AC cycles. Sync ...

s use DC-voltage-fed rotor windings, which lets the excitation of the machine be controlled—and induction motor
An induction motor or asynchronous motor is an AC electric motor in which the electric current in the rotor needed to produce torque is obtained by electromagnetic induction from the magnetic field of the stator winding. An induction moto ...

s use short-circuited rotors (instead of a magnet) following the rotating magnetic field of a multicoiled stator
The stator is the stationary part of a rotary system, found in electric generators, electric motors, sirens, mud motors or biological rotors. Energy flows through a stator to or from the rotating component of the system. In an electric ...

. The short-circuited turns of the rotor develop eddy current
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 or by the relative motion of a conductor in a mag ...

s in the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force.
In 1882, Nikola Tesla identified the concept of the rotating magnetic field. In 1885, Galileo Ferraris
Galileo Ferraris (31 October 1847 – 7 February 1897) was an Italian university professor, physicist and electrical engineer, one of the pioneers of AC power system and inventor of the induction motor although he never patented his work. Many ...

independently researched the concept. In 1888, Tesla gained for his work. Also in 1888, Ferraris published his research in a paper to the ''Royal Academy of Sciences'' in Turin.
Hall effect

The charge carriers of a current-carrying conductor placed in a transverse magnetic field experience a sideways Lorentz force; this results in a charge separation in a direction perpendicular to the current and to the magnetic field. The resultant voltage in that direction is proportional to the applied magnetic field. This is known as the ''Hall effect''. The ''Hall effect'' is often used to measure the magnitude of a magnetic field. It is used as well to find the sign of the dominant charge carriers in materials such as semiconductors (negative electrons or positive holes).Magnetic circuits

An important use of is in ''magnetic circuits'' where inside a linear material. Here, is the magnetic permeability of the material. This result is similar in form toOhm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equa ...

, where is the current density, is the conductance and is the electric field. Extending this analogy, the counterpart to the macroscopic Ohm's law () is:
$$\backslash Phi\; =\; \backslash frac\; F\; R\_\backslash mathrm,$$
where $\backslash Phi\; =\; \backslash int\; \backslash mathbf\backslash cdot\; \backslash mathrm\backslash mathbf$ is the magnetic flux in the circuit, $F\; =\; \backslash int\; \backslash mathbf\backslash cdot\; \backslash mathrm\backslash boldsymbol$ is the magnetomotive force applied to the circuit, and is the reluctance of the circuit. Here the reluctance is a quantity similar in nature to resistance for the flux. Using this analogy it is straightforward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of circuit theory.
Largest magnetic fields

, the largest magnetic field produced over a macroscopic volume outside a lab setting is 2.8 kT ( VNIIEF in Sarov, Russia, 1998). As of October 2018, the largest magnetic field produced in a laboratory over a macroscopic volume was 1.2 kT by researchers at theUniversity of Tokyo
, abbreviated as or UTokyo, is a public research university located in Bunkyō, Tokyo, Japan. Established in 1877, the university was the first Imperial University and is currently a Top Type university of the Top Global University Project b ...

in 2018.
The largest magnetic fields produced in a laboratory occur in particle accelerators, such as RHIC, inside the collisions of heavy ions, where microscopic fields reach 10Magnetar
A magnetar is a type of neutron star with an extremely powerful magnetic field (∼109 to 1011 T, ∼1013 to 1015 G). The magnetic-field decay powers the emission of high- energy electromagnetic radiation, particularly X-rays and gamma rays. ...

s have the strongest known magnetic fields of any naturally occurring object, ranging from 0.1 to 100 GT (10History

Early developments

While magnets and some properties of magnetism were known to ancient societies, the research of magnetic fields began in 1269 when French scholar Petrus Peregrinus de Maricourt mapped out the magnetic field on the surface of a spherical magnet using iron needles. Noting the resulting field lines crossed at two points he named those points "poles" in analogy to Earth's poles. He also articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them. Almost three centuries later, William Gilbert ofColchester
Colchester ( ) is a city in Essex, in the East of England. It had a population of 122,000 in 2011. The demonym is Colcestrian.
Colchester occupies the site of Camulodunum, the first major city in Roman Britain and its first capital. Col ...

replicated Petrus Peregrinus's work and was the first to state explicitly that Earth is a magnet. Published in 1600, Gilbert's work, ''De Magnete
''De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure'' (''On the Magnet and Magnetic Bodies, and on That Great Magnet the Earth'') is a scientific work published in 1600 by the English physician and scientist William Gilbert. A h ...

'', helped to establish magnetism as a science.
Mathematical development

In 1750, John Michell stated that magnetic poles attract and repel in accordance with aninverse square law
In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be unders ...

Charles-Augustin de Coulomb
Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attra ...

experimentally verified this in 1785 and stated explicitly that north and south poles cannot be separated. Building on this force between poles, Siméon Denis Poisson (1781–1840) created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic -field is produced by ''magnetic poles'' and magnetism is due to small pairs of north–south magnetic poles.
Three discoveries in 1820 challenged this foundation of magnetism. Hans Christian Ørsted
Hans Christian Ørsted ( , ; often rendered Oersted in English; 14 August 17779 March 1851) was a Danish physicist and chemist who discovered that electric currents create magnetic fields, which was the first connection found between electrici ...

demonstrated that a current-carrying wire is surrounded by a circular magnetic field. Then André-Marie Ampère showed that parallel wires with currents attract one another if the currents are in the same direction and repel if they are in opposite directions. Finally, Jean-Baptiste Biot and Félix Savart announced empirical results about the forces that a current-carrying long, straight wire exerted on a small magnet, determining the forces were inversely proportional to the perpendicular distance from the wire to the magnet. Laplace later deduced a law of force based on the differential action of a differential section of the wire, which became known as the Biot–Savart law, as Laplace did not publish his findings.
Extending these experiments, Ampère published his own successful model of magnetism in 1825. In it, he showed the equivalence of electrical currents to magnets and proposed that magnetism is due to perpetually flowing loops of current instead of the dipoles of magnetic charge in Poisson's model. Further, Ampère derived both Ampère's force law describing the force between two currents and Ampère's law, which, like the Biot–Savart law, correctly described the magnetic field generated by a steady current. Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism.
In 1831, Michael Faraday discovered electromagnetic induction
Electromagnetic or magnetic induction is the production of an electromotive force (emf) across an electrical conductor in a changing magnetic field.
Michael Faraday is generally credited with the discovery of induction in 1831, and James Cle ...

when he found that a changing magnetic field generates an encircling electric field, formulating what is now known as Faraday's law of induction. Later, Franz Ernst Neumann proved that, for a moving conductor in a magnetic field, induction is a consequence of Ampère's force law. In the process, he introduced the magnetic vector potential, which was later shown to be equivalent to the underlying mechanism proposed by Faraday.
In 1850, Lord Kelvin, then known as William Thomson, distinguished between two magnetic fields now denoted and . The former applied to Poisson's model and the latter to Ampère's model and induction. Further, he derived how and relate to each other and coined the term ''permeability''.
Between 1861 and 1865, James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...

developed and published On Physical Lines of Force
"On Physical Lines of Force" is a four-part paper written by James Clerk Maxwell published in 1861. In it, Maxwell derived the equations of electromagnetism in conjunction with a "sea" of "molecular vortices" which he used to model Faraday's lin ...

'' in 1861. These equations were valid but incomplete. Maxwell completed his set of equations in his later 1865 paper '' A Dynamical Theory of the Electromagnetic Field'' and demonstrated the fact that light is an electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ligh ...

. Heinrich Hertz
Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. The unit o ...

published papers in 1887 and 1888 experimentally confirming this fact.Huurdeman, Anton A. (2003) ''The Worldwide History of Telecommunications''. Wiley. . p. 202
Modern developments

In 1887, Tesla developed aninduction motor
An induction motor or asynchronous motor is an AC electric motor in which the electric current in the rotor needed to produce torque is obtained by electromagnetic induction from the magnetic field of the stator winding. An induction moto ...

that ran on alternating current
Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...

. The motor used polyphase current, which generated a rotating magnetic field A rotating magnetic field is the resultant magnetic field produced by a system of coils symmetrically placed and supplied with polyphase currents. A rotating magnetic field can be produced by a poly-phase (two or more phases) current or by a sing ...

to turn the motor (a principle that Tesla claimed to have conceived in 1882). Tesla received a patent for his electric motor in May 1888. In 1885, Galileo Ferraris
Galileo Ferraris (31 October 1847 – 7 February 1897) was an Italian university professor, physicist and electrical engineer, one of the pioneers of AC power system and inventor of the induction motor although he never patented his work. Many ...

independently researched rotating magnetic fields and subsequently published his research in a paper to the ''Royal Academy of Sciences'' in Turin, just two months before Tesla was awarded his patent, in March 1888./ref> The twentieth century showed that classical electrodynamics is already consistent with special relativity, and extended classical electrodynamics to work with quantum mechanics.

Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

, in his paper of 1905 that established relativity, showed that both the electric and magnetic fields are part of the same phenomena viewed from different reference frames. Finally, the emergent field of quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

was merged with electrodynamics to form quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...

, which first formalized the notion that electromagnetic field energy is quantized in the form of photons.
See also

General

* Magnetohydrodynamics – the study of the dynamics of electrically conducting fluids * Magnetic hysteresis – application to ferromagnetism * Magnetic nanoparticles – extremely small magnetic particles that are tens of atoms wide * Magnetic reconnection – an effect that causessolar flare
A solar flare is an intense localized eruption of electromagnetic radiation in the Sun's atmosphere. Flares occur in active regions and are often, but not always, accompanied by coronal mass ejections, solar particle events, and other solar ...

s and auroras
* Magnetic scalar potential
Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electric potential. It is used to specify the magnetic H-field in cases when there are no free currents, in a manner analogous to using the electric p ...

* SI electromagnetism units
See also
* SI
* Speed of light
* List of electromagnetism equations
This article summarizes equations in the theory of electromagnetism.
Definitions
Here subscripts ''e'' and ''m'' are used to differ between electric and magnetic charges. ...

– common units used in electromagnetism
* Orders of magnitude (magnetic field) – list of magnetic field sources and measurement devices from smallest magnetic fields to largest detected
* Upward continuation
* Moses Effect
Mathematics

* Magnetic helicity – extent to which a magnetic field wraps around itselfApplications

* Dynamo theory – a proposed mechanism for the creation of the Earth's magnetic field * Helmholtz coil – a device for producing a region of nearly uniform magnetic field * Magnetic field viewing film – Film used to view the magnetic field of an area * Magnetic pistol – a device on torpedoes or naval mines that detect the magnetic field of their target * Maxwell coil – a device for producing a large volume of an almost constant magnetic field * Stellar magnetic field – a discussion of the magnetic field of stars * Teltron tube – device used to display an electron beam and demonstrates effect of electric and magnetic fields on moving chargesNotes

References

Further reading

* *External links

* * Crowell, B., "Electromagnetism

'". * Nave, R., "

'". HyperPhysics. * "''Magnetism''"

theory.uwinnipeg.ca. * Hoadley, Rick, "

''" 17 July 2005. {{DEFAULTSORT:Magnetic Field Magnetism Physical quantities