TheInfoList

A magnetic field is a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

that describes the magnetic influence on moving
electric charge Electric charge is the physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of th ...
s,
electric currents An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, ...
, 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 A magnet is a material or object that produces a magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vec ...
's magnetic field pulls on
ferromagnetic material Ferromagnetism is the basic mechanism by which certain materials (such as iron Iron () is a chemical element with Symbol (chemistry), symbol Fe (from la, Wikt:ferrum, ferrum) and atomic number 26. It is a metal that belongs to the first tr ...
s such as
iron Iron () is a with Fe (from la, ) and 26. It is a that belongs to the and of the . It is, on , right in front of (32.1% and 30.1%, respectively), forming much of Earth's and . It is the fourth most common . In its metallic state, iron ...

, and attracts or repels other magnets. In addition, a magnetic field that varies with location will exert a force on a range of non-magnetic materials by affecting the motion of their outer atomic electrons. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in
electromagnet File:VFPt Solenoid correct2.svg, Magnetic field produced by a solenoid (coil of wire). This drawing shows a cross section through the center of the coil. The crosses are wires in which current is moving into the page; the dots are wires in whi ...

s, and by
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

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 function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
assigning a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
to each point of space, called a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

. In
electromagnetics Electromagnetism is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics ...
, 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 International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' (Kevin Michael album), 2011 * International (New Order album), '' ...
, , magnetic field strength, is measured in the SI base units of
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 base unit of electric current An electric current is a stream of charged particles, such as ele ...

per meter (A/m). ,
magnetic flux In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

density, is measured in tesla (in SI base units: kilogram per second2 per ampere), which is equivalent to
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
per meter per ampere. and differ in how they account for magnetization. In a
vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Gree ...

, the two fields are related through the
vacuum permeability Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field prod ...
, $\mathbf/\mu_0 = \mathbf$; but in a magnetized material, the terms differ by the material's
magnetization In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass per uni ...
at each point. Magnetic fields are produced by moving electric charges and the intrinsic
magnetic moment The magnetic moment is the magnetic strength and orientation of a or other object that produces a . Examples of objects that have magnetic moments include: loops of (such as s), permanent magnets, s (such as s), various s, and many astronomical ...

s of
elementary particle In , an elementary particle or fundamental particle is a that is not composed of other particles. Particles currently thought to be elementary include the fundamental s (s, s, s, and s), which generally are " particles" and " particles", as well ...
s associated with a fundamental quantum property, their
spin Spin or spinning may refer to: Businesses * or South Pacific Island Network * , an American scooter-sharing system * , a chain of table tennis lounges Computing * , 's tool for formal verification of distributed software systems * , a Mach-like ...
. Magnetic fields and
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

s are interrelated and are both components of the
electromagnetic force Electromagnetism is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ...
, one of the four
fundamental force#REDIRECT Fundamental interaction In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics ...
s of nature. Magnetic fields are used throughout modern technology, particularly in
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics The field of electronics is a branch of physics and electrical enginee ...

and
electromechanics In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encomp ...
. Rotating magnetic fields are used in both s and generators. The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as
magnetic circuit A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and ...
s. Magnetic forces give information about the charge carriers in a material through the
Hall effect The Hall effect is the production of a voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', p ...

. 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 solar corona, corona. This plasma (physics), plasma mostly consists of electrons, protons and alpha particles with kinetic energy between . ...

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 direction The four cardinal directions, or cardinal points, are the directions north, east, south, and west, commonly denoted by their initials N, E, S, and W. East and west are perpendicular ( ...

.

# 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 the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

, 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 field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

s 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 ., 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 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 ...
, 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 A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of th ...
, , is the particle's
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

, and × denotes the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

. 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 Memory is the faculty of the by which or is , stored, and retrieved when needed. It is the ...

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 The Hall effect is the production of a voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', ...
below. The first term in the Lorentz equation is from the theory of
electrostatics Electrostatics is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related enti ...
, and says that a particle of charge in an electric field experiences an electric force: :$\mathbf_ = q \mathbf.$ The second term is the magnetic force: :$\mathbf_ = q\left(\mathbf \times \mathbf\right).$ Using the definition of the cross product, the magnetic force can also be written as a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
equation: :$F_ = q v B \sin\left(\theta\right)$ 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, . In units, is measured in teslas (symbol: T). (
magnetic flux In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

) is measured in webers (symbol: Wb) so that a flux density of 1 Wb/m2 is 1  tesla. The SI unit of tesla is equivalent to (
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
·
second The second (symbol: s, also abbreviated: sec) is the of in the (SI) (french: Système International d’unités), commonly understood and historically defined as of a – this factor derived from the division of the day first into 24 s, th ...
)/(
coulomb The coulomb (symbol: C) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...
·
metre The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English o ...
). This can be seen from the magnetic part of the Lorentz force law.
In Gaussian-cgs units, is measured in
gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...
(symbol: G). (The conversion is 1 T = 10000 G.) One nanotesla is equivalent to 1 gamma (symbol: γ).

## The H-field

The magnetic field is defined: Where $\mu_0$ is the
vacuum permeability Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field prod ...
, 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 -field is measured in
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 base unit of electric current An electric current is a stream of charged particles, such as ele ...

s per metre (A/m) in SI units, and in
oersted The oersted (symbol Oe) is the coherent derived unit of the auxiliary magnetic field H in the centimetre–gram–second system of units The centimetre–gram–second system of units (abbreviated CGS or cgs) is a variant of the metric syste ...
s (Oe) in cgs units.

## Measurement

An instrument used to measure the local magnetic field is known as a
magnetometer A magnetometer is a device that measures magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in ...
. Important classes of magnetometers include using induction magnetometers (or search-coil magnetometers) which measure only varying magnetic fields, rotating coil magnetometers,
Hall effect The Hall effect is the production of a voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', p ...

magnetometers, NMR magnetometers, , and fluxgate magnetometers. The magnetic fields of distant
astronomical object In , an astronomical object or celestial object is a naturally occurring , association, or structure that exists in the . In , the terms ''object'' and ''body'' are often used interchangeably. However, an astronomical body or celestial body i ...
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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physic ...
that is detectable in
radio waves Radio waves are a type of electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space a ...

. The finest precision for a magnetic field measurement was attained by
Gravity Probe B Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of s ...

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 a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
) 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 physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and h ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral) or heliotrope, a mineral aggregate * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood 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 Ferromagnetism is the basic mechanism by which certain materials (such as iron Iron () is a with Fe (from la, ) and 26. It is a that belongs to the and of the . It is, on , right in front of (32.1% and 30.1%, respectively), formi ...
substances like
iron Iron () is a with Fe (from la, ) and 26. It is a that belongs to the and of the . It is, on , right in front of (32.1% and 30.1%, respectively), forming much of Earth's and . It is the fourth most common . In its metallic state, iron ...

and in plasmas, magnetic forces can be understood by imagining that the field lines exert a
tension Tension may refer to: Science * Psychological stress * Tension (physics), a force related to the stretching of an object (the opposite of compression) * Tension (geology), a stress which stretches rocks in two opposite directions * Voltage or elect ...
, (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 Ferromagnetism is the basic mechanism by which certain materials (such as iron Iron () is a with Fe (from la, ) and 26. It is a that belongs to the and of the . It is, on , right in front of (32.1% and 30.1%, respectively), formi ...
materials, such as iron and
nickel Nickel is a chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same nu ...

, 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 The magnetic moment is the magnetic strength and orientation of a magnet Magnetic field lines of a solenoid electromagnet, which are similar to a bar magnet as illustrated below with the iron filings A magnet is a material or object that ...

). The
equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

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 In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin (physics), spin of elementary particles. For example, the electron is an elementary spin-1/2 fermion. Quantum electrodyna ...
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 the
Coulomb force Coulomb's law, or Coulomb's inverse-square law, is an experimental law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system ...
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 field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

, which starts at a positive
electric charge Electric charge is the physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of th ...
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 , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

and magnetism. The pole model usually treats magnetic charge as a mathematical abstraction, rather than a physical property of particles. However, a
magnetic monopole In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is th ...
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 A Grand Unified Theory (GUT) is a model in particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of natur ...
) have predicted the existence of magnetic monopoles, but so far, none have been observed.

## Amperian loop model

In the model developed by
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 base unit of electric current An electric current is a stream of charged particle In physics ...
, 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 effectThe Einstein–de Haas effect is a physical phenomenon in which a change in the magnetic moment of a free body causes this body to rotate. The effect is a consequence of the conservation of angular momentum. It is strong enough to be observable in fe ...
''rotation by magnetization'' and its inverse, the
Barnett effect The Barnett effect is the magnetization of an uncharged body when spun on its axis. It was discovered by American physicist Samuel Jackson Barnett, Samuel Barnett in 1915. An uncharged object rotating with angular velocity ω tends to spontaneousl ...
or ''magnetization by rotation''.See
magnetic moment The magnetic moment is the magnetic strength and orientation of a or other object that produces a . Examples of objects that have magnetic moments include: loops of (such as s), permanent magnets, s (such as s), various s, and many astronomical ...
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: See Eq. 11.42 in :$\mathbf = \boldsymbol \left\left(\mathbf\cdot\mathbf\right\right),$ where the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
, 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 of ...
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 .

## 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 Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field prod ...
, measuring · s/(·) 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: :$\boldsymbol=\mathbf\times\mathbf = \mu_0\mathbf\times\mathbf, \,$ where × represents the vector
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

. 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
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charges, such as
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

s, produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles. Magnetic field lines form in
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circles around a
cylindrical A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

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 (vector space), orientation of axes in three-dimensional space. Most of the various left-hand and right-hand rules arise from the fact that the th ...

" (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 A solenoid (,) is a type of electromagnet Magnetic field produced by a solenoid (coil of wire). This drawing shows a cross section through the center of the coil. The crosses are wires in which current is moving into the page; the dots a ...

" enhances this effect. A device so formed around an iron
core Core or cores may refer to: Science and technology * Core (anatomy) In common parlance, the core of the body is broadly considered to be the torso. Functional movements are highly dependent on this part of the body, and lack of core muscular dev ...

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 In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
'': $\mathbf = \frac\int_\frac,$ where the integral sums over the wire length where vector is the vector
line element In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

with direction in the same sense as the current , is the
magnetic constant Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field pro ...
, 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: $, \mathbf, = \fracI$ 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  to the -field is through Ampère's law: $\oint \mathbf \cdot \mathrm\boldsymbol = \mu_0 I_,$ where the
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is over any arbitrary loop and enc 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 are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
that describe electricity and magnetism.

## Force on moving charges and current

### Force on a charged particle

A
charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
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 $\mathbf = q\mathbf + q \mathbf \times \mathbf,$ where is the
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

, is the
electric charge Electric charge is the physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of th ...
of the particle, is the instantaneous
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

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 * Work (physics), the product of ...
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 A magnetic dipole is the limit of either a closed loop of electric current 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 ...
, 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 $F = qvB \sin\theta,$ so, for charges where $N = n \ell A ,$ the force exerted on the conductor is $f=FN=qvB n\ell A \sin\theta = Bi\ell \sin\theta,$ 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 In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and ...
, which can be a challenge to calculate. (This bound current is due to the sum of atomic sized current loops and the
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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 net
magnetic dipole moment The magnetic moment is the magnetic strength and orientation of a magnet Magnetic field lines of a solenoid electromagnet, which are similar to a bar magnet as illustrated below with the iron filings A magnet is a material or object that ...

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⋅m2, 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 In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and ...
''. 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: $\oint \mathbf \cdot \mathrm\boldsymbol = I_\mathrm,$ 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: $\oint_S \mu_0 \mathbf \cdot \mathrm\mathbf = -q_\mathrm$, where the integral is a closed surface integral over the closed surface and is the "magnetic charge" (in units of
magnetic flux In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

) 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 $\mathbf\ \equiv \ \frac - \mathbf.$ In terms of the H-field, Ampere's law is $\oint \mathbf \cdot \mathrm\boldsymbol = \oint \left(\frac - \mathbf\right) \cdot \mathrm\boldsymbol = I_\mathrm - I_\mathrm = I_\mathrm,$ 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 see
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
. Ampere's law leads to the boundary condition $\left(\mathbf - \mathbf\right) = \mathbf_\mathrm \times \hat,$ where is the surface free current density and the unit normal $\hat$ points in the direction from medium 2 to medium 1. Similarly, a
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of over any closed surface is independent of the free currents and picks out the "magnetic charges" within that closed surface: $\oint_S \mu_0 \mathbf \cdot \mathrm\mathbf = \oint_S (\mathbf - \mu_0 \mathbf) \cdot \mathrm\mathbf = 0 - (-q_\mathrm) = q_\mathrm,$ 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: $\mathbf = \mathbf_0 + \mathbf_\mathrm,$ 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 (matter), phase of a material. Materials are divided into groups based upon their magnetic behavior: * Diamagnetism, Diamagnetic materials produce a magnetization that opposes the magnetic field. * Paramagnetism, Paramagnetic materials produce a magnetization in the same direction as the applied magnetic field. * Ferromagnetism, Ferromagnetic materials and the closely related Ferrimagnetism, ferrimagnetic materials and Antiferromagnetism, 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 Type II superconductor#Mixed state, 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: $\mathbf = \mu \mathbf,$ where is a material dependent parameter called the permeability (electromagnetism), permeability. In some cases the permeability may be a second rank tensor 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 hysteresis#Magnetic hysteresis, 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), the energy density is: $u = \frac= \frac = \frac.$ 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: $\delta W = \mathbf\cdot\delta\mathbf.$ Once the relationship between and is known this equation is used to determine the work needed to reach a given magnetic state. For Hysteresis, 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 electromagnetism, classical electrodynamics including both electricity and magnetism. The first property is the divergence 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 and is equivalent to the statement that there are no isolated magnetic poles or
magnetic monopole In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is th ...
s. The second mathematical property is called the curl (mathematics), curl, 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 of induction, 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 to Gauss's law for magnetism: : where the integral is a
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

over the closed surface (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.

A changing magnetic field, such as a magnet moving through a conducting coil, generates an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

(and therefore tends to drive a current in such a coil). This is known as ''Faraday's law'' and forms the basis of many electrical generators and s. Mathematically, Faraday's law is: $\mathcal = - \frac$ where $\mathcal$ is the electromotive force (or ''EMF'', the voltage generated around a closed loop) and is the
magnetic flux In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

—the product of the area times the magnetic field Tangential and normal components, 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: $\textstyle\mathcal = - d\Phi/dt$$\textstyle= \oint_\left\left( \mathbf\left( \mathbf,\ t\right) +\mathbf\left(\mathbf,\ t\right)\right\right) \cdot d\boldsymbol\$$\textstyle=-\frac \iint_ d \boldsymbol \cdot \mathbf \left(\mathbf,\ t\right)$ 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. $\nabla \times \mathbf = -\frac$

## 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 form electromagnetic waves, 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. $\nabla \times \mathbf = \mu_0\mathbf + \mu_0 \varepsilon_0 \frac$ where is the complete microscopic current density. 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'' : $\nabla \times \mathbf = \mathbf_\mathrm + \frac$ 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

## Electric and magnetic fields: different aspects of the same phenomenon

According to special relativity, the special theory of relativity, the partition of the
electromagnetic force Electromagnetism is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ...
into separate electric and magnetic components is not fundamental, but varies with the Frame of reference#Observational frames of reference, 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. Formally, special relativity combines the electric and magnetic fields into a rank-2 tensor, 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, and mass, momentum, and energy into four-momentum. Similarly, the Magnetic energy, 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 as quantum mechanics and Theory of relativity, 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 potential, electric scalar potential , are defined such that: :$\begin \mathbf &= \nabla \times \mathbf, \\ \mathbf &= -\nabla \varphi - \frac. \end$ The vector potential may be interpreted as a ''generalized potential momentum per unit charge'' just as is interpreted as a ''generalized potential energy per unit charge''. Maxwell's equations when expressed in terms of the potentials can be cast into a form that agrees with special relativity with little effort. In relativity together with forms the four-potential, analogous to the Four-vector#Four-momentum, 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.

## Quantum electrodynamics

In modern physics, the electromagnetic field is understood to be not a ''classical physics, classical'' field (physics), field, but rather a quantum field; it is represented not as a vector of three real number, numbers at each point, but as a vector of three operator (physics), 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 antiparticles) is computed using perturbation theory (quantum mechanics), 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 10−12 (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 electromagnetism, 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.

# Uses and examples

## Earth's magnetic field

The Earth's magnetic field is produced by convection of a liquid iron alloy in the outer core. In a Dynamo theory, 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 Brunhes–Matuyama reversal, most recent reversal occurred 780,000 years ago.

## Rotating magnetic fields

The ''rotating magnetic field'' is a key principle in the operation of electric motor#AC motors, 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 by Nikola Tesla, and later utilized in his and others' early AC (alternating current) electric motors. Magnetic torque is used to drive s. In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array of
electromagnet File:VFPt Solenoid correct2.svg, Magnetic field produced by a solenoid (coil of wire). This drawing shows a cross section through the center of the coil. The crosses are wires in which current is moving into the page; the dots are wires in whi ...

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 electric power, 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 motors use DC-voltage-fed rotor windings, which lets the excitation of the machine be controlled—and induction motors use short-circuited Rotor (electric), rotors (instead of a magnet) following the rotating magnetic field of a multicoiled stator. The short-circuited turns of the rotor develop eddy currents 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 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 to Ohm's law , 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: $\Phi = \frac F R_\mathrm,$ where $\Phi = \int \mathbf\cdot \mathrm\mathbf$ is the magnetic flux in the circuit, $F = \int \mathbf\cdot \mathrm\boldsymbol$ 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 Electrical resistance, 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

As of October 2018, 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 the University of Tokyo in 2018. The largest magnetic fields produced in a laboratory occur in particle accelerators, such as Relativistic Heavy Ion Collider, RHIC, inside the collisions of heavy ions, where microscopic fields reach 1014 T. Magnetars have the strongest known magnetic fields of any naturally occurring object, ranging from 0.1 to 100 GT (108 to 1011 T).

# History

## 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 (astronomer), William Gilbert of Colchester 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'', helped to establish magnetism as a science.

## Mathematical development

In 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law Charles-Augustin de Coulomb 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 A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge ...

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 In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
, 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 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 developed and published
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
, which explained and united all of classical theory, classical electricity and magnetism. The first set of these equations was published in a paper entitled '':CCommons:File:On Physical Lines of Force.pdf, On Physical Lines of Force'' 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. Heinrich Hertz 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 an induction motor that ran on alternating current. The motor used Polyphase system, polyphase current, which generated a rotating magnetic field 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 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. 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. Albert Einstein, 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 was merged with electrodynamics to form quantum electrodynamics, which first formalized the notion that electromagnetic field energy is quantized in the form of photons.

## 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 causes solar flares and auroras * Magnetic scalar potential * SI electromagnetism units – 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 itself

## Applications

* 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 charges

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