Magnetic field

Magnetic field

Overview
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

.Technically, a magnetic field is a pseudo vector; pseudo-vectors, which also include torque and rotational velocity
Rotational speed
Rotational speed tells how many complete rotations there are per time unit. It is therefore a cyclic frequency, measured in hertz in the SI System...

, are similar to vectors except that they remain unchanged when the coordinates are inverted.
The magnetic field is most commonly defined in terms of the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

it exerts on moving electric charges.
Discussion

Encyclopedia
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

.Technically, a magnetic field is a pseudo vector; pseudo-vectors, which also include torque and rotational velocity
Rotational speed
Rotational speed tells how many complete rotations there are per time unit. It is therefore a cyclic frequency, measured in hertz in the SI System...

, are similar to vectors except that they remain unchanged when the coordinates are inverted.
The magnetic field is most commonly defined in terms of the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

it exerts on moving electric charges. There are two separate but closely related fields to which the name 'magnetic field' can refer: a magnetic field and a magnetic field.

Magnetic fields are produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, their spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

. In special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, electric and magnetic fields are two interrelated aspects of a single object, called the electromagnetic field tensor; the aspect of the electromagnetic field that is seen as a magnetic field is dependent on the reference frame
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

of the observer. In quantum physics, the electromagnetic field is quantized and electromagnetic interactions result from the exchange of photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s.

Magnetic fields have had many uses in ancient and modern society. The Earth produces its own magnetic field, which is important in navigation. Rotating magnetic fields are utilized in both electric motor
Electric motor
An electric motor converts electrical energy into mechanical energy.Most electric motors operate through the interaction of magnetic fields and current-carrying conductors to generate force...

s and generators. Magnetic forces give information about the charge carriers in a material through the Hall effect
Hall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...

. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuit
Magnetic circuit
A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other...

s.

History

Although magnets and magnetism were known much earlier, one of the first descriptions of the magnetic field was produced in 1269 by the French scholar Petrus PeregrinusHis Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete, which is often shortened to Epistola de magnete, is dated 1269 C.E. who mapped out the magnetic field on the surface of a spherical magnet using iron needles. Noting that the resulting field lines crossed at two points he named those points 'poles' in analogy to Earth's poles. Almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus' work and was the first to state explicitly that Earth itself was a magnet. Gilbert's great work De Magnete
De Magnete
De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure is a scientific work published in 1600 by the English physician and scientist William Gilbert and his partner Aaron Dowling...

was published in 1600 and helped to establish the study of magnetism as a science.

One of the first successful models of the magnetic field was developed in 1824 by Siméon-Denis Poisson (1781–1840). Poisson assumed that magnetism was due to 'magnetic charges' such that like 'magnetic charges' repulse while opposites attract. The model he created is exactly analogous to modern electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....

with a magnetic H-field being produced by 'magnetic charges' in the same way that an electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

E-field is produced by electric charges. It predicts the correct H-field for permanent magnets. It predicts the forces between magnets. And, it predicts the correct energy stored in the magnetic fields.By the definition of magnetization, in this model, and in analogy to the physics of springs, the work done per unit volume, in stretching and twisting the bonds between magnetic charge to increment the magnetization by μ0δM is W = H · μ0δM. In this model, B = μ0 (H + M ) is an effective magnetization which includes the H-field term to account for the energy of setting up the magnetic field in a vacuum. Therefore the total energy density increment needed to increment the magnetic field is W = H · δB.

Three remarkable discoveries though, would challenge Poisson's model. First, in 1819, Hans Christian Oersted discovered that an electric current generates a magnetic field encircling it. Then, André-Marie Ampère
André-Marie Ampère
André-Marie Ampère was a French physicist and mathematician who is generally regarded as one of the main discoverers of electromagnetism. The SI unit of measurement of electric current, the ampere, is named after him....

showed that parallel wires having currents in the same direction attract one another. Finally Jean-Baptiste Biot
Jean-Baptiste Biot
Jean-Baptiste Biot was a French physicist, astronomer, and mathematician who established the reality of meteorites, made an early balloon flight, and studied the polarization of light.- Biography :...

and Félix Savart
Félix Savart
Félix Savart became a professor at Collège de France in 1836 and was the co-originator of the Biot-Savart Law, along with Jean-Baptiste Biot. Together, they worked on the theory of magnetism and electrical currents. Their law was developed about 1820. The Biot-Savart Law relates magnetic fields to...

discovered the Biot–Savart law which correctly predicts the magnetic field around any current-carrying wire.

Together, these discoveries suggested a model in which the magnetic B field of a material is produced by microscopic current loops. In this model, these current loops (called magnetic dipole
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...

s) would replace the dipoles of charge of the Poisson's model.

It is a remarkable fact that from the 'outside' the field of a dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

of magnetic charge has the exact same form as that of an elementary current loop called a magnetic dipole
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...

. It is therefore only for the physics of magnetism 'inside' of magnetic material that the two models differ.

No magnetic charges are needed which has the additional benefit of explaining why magnetic charge can not be isolated; cutting a magnet in half does not result in two separate poles but in two separate magnets, each of which has both poles.

The next decade saw two developments that help lay the foundation for the full theory of electromagnetism. In 1825, Ampère published his Ampère's law
Ampère's law
In classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère in 1826, relates the integrated magnetic field around a closed loop to the electric current passing through the loop...

which like the Biot–Savart law correctly described the magnetic field generated by a steady current but was more general. And, in 1831, Michael Faraday
Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....

showed that a changing magnetic field generates an encircling electric field and thereby demonstrated that electricity and magnetism are even more tightly knitted.

Between 1861 and 1865, James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

developed and published a set of Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

which explained and united all of classical electricity and magnetism. The first set of these equations was published in a paper entitled On Physical Lines of Force in 1861. The mechanism that Maxwell proposed to underlie these equations in this paper was fundamentally incorrect, which is not surprising since it predated the modern understanding even of the atom. Yet, the equations were valid although incomplete. He completed the set of Maxwell's equations in his later 1865 paper A Dynamical Theory of the Electromagnetic Field
A Dynamical Theory of the Electromagnetic Field
"A Dynamical Theory of the Electromagnetic Field" is the third of James Clerk Maxwell's papers regarding electromagnetism, published in 1865. It is the paper in which the original set of four Maxwell's equations first appeared...

and demonstrated the fact that light is an electromagnetic wave. Thus, he theoretically unified not only electricity and magnetism but light as well. This fact was then later confirmed experimentally by Heinrich Hertz in 1887.

Even though the classical theory of electrodynamics was essentially complete with Maxwell's equations, the twentieth century saw a number of improvements and extensions to the theory. Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

, in his great 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. (See moving magnet and conductor problem
Moving magnet and conductor problem
The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant velocity, v, with respect to a magnet is calculated in...

for details about the thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

that eventually helped Albert Einstein to develop special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

.) Finally, the emergent field of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

was merged with electrodynamics to form quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics 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 special relativity is achieved...

or QED.

Definitions, units, and measurement

Magnetic field can be defined in many equivalent ways based on the effects it has on its environment. For instance, a particle having an electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

, q, and moving in a magnetic field with a velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

, v, experiences a force, F, called the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

. See Force on a charged particle below. Alternatively, the magnetic field can be defined in terms of the torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

it produces on a magnetic dipole. See Torque on a magnet due to a B-field below.

Devices used to measure the local magnetic field are called magnetometer
Magnetometer
A magnetometer is a measuring instrument used to measure the strength or direction of a magnetic field either produced in the laboratory or existing in nature...

s. Important classes of magnetometers include using a rotating coil, Hall effect
Hall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...

magnetometers, NMR magnetometers
Proton magnetometer
The proton magnetometer, also known as the proton precession magnetometer , uses the principle of Earth's field nuclear magnetic resonance to measure very small variations in the Earth's magnetic field, allowing ferrous objects on land and at sea to be detected.It is used in land-based archaeology...

, SQUID magnetometers
SQUID
A SQUID is a very sensitive magnetometer used to measure extremely weak magnetic fields, based on superconducting loops containing Josephson junctions....

, and fluxgate magnetometers
Fluxgate compass
The basic fluxgate compass is a simple electromagnetic device that employs two or more small coils of wire around a core of highly permeable magnetic material, to directly sense the direction of the horizontal component of the earth's magnetic field...

. The magnetic fields of distant astronomical object
Astronomical object
Astronomical objects or celestial objects are naturally occurring physical entities, associations or structures that current science has demonstrated to exist in the observable universe. The term astronomical object is sometimes used interchangeably with astronomical body...

s are measured through their effects on local charged particles. For instance, electrons spiraling around a field line produce synchrotron radiation
The electromagnetic radiation emitted when charged particles are accelerated radially is called synchrotron radiation. It is produced in synchrotrons using bending magnets, undulators and/or wigglers...

which is detectable in radio waves
Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. Radio waves have frequencies from 300 GHz to as low as 3 kHz, and corresponding wavelengths from 1 millimeter to 100 kilometers. Like all other electromagnetic waves,...

.
Alternative names for B
• Magnetic flux density
• Magnetic induction
• Magnetic field
Alternative names for H
• Magnetic field intensity
• Magnetic field strength
• Magnetic field
• Magnetizing field

• There are two magnetic fields, H and B. In a vacuum they are indistinguishable, differing only by a multiplicative constant that depends on the physical units. Inside a material they are different (see H and B inside and outside of magnetic materials). The term magnetic field is historically reserved for H while using other terms for B. Informally, though, and formally for some recent textbooks mostly in physics, the term 'magnetic field' is used to describe B as well as or in place of H.

Edward Purcell
Edward Mills Purcell
Edward Mills Purcell was an American physicist who shared the 1952 Nobel Prize for Physics for his independent discovery of nuclear magnetic resonance in liquids and in solids. Nuclear magnetic resonance has become widely used to study the molecular structure of pure materials and the...

, in Electricity and Magnetism, McGraw-Hill, 1963, writes, Even some modern writers who treat B as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by H. 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 B the magnetic field. As for H, although other names have been invented for it, we shall call it "the field H" or even "the magnetic field H." In a similar vein, says: "So we may think of both B and H as magnetic fields, but drop the word 'magnetic' from H 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 to right).

The B-field is measured in teslas
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

in SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units and in gauss
Gauss (unit)
The gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter; it equals 1 tesla...

in cgs units. (1 tesla = 10,000 gauss). The SI unit of tesla is equivalent to (newton·second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

)/(coulomb·metre
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

).This can be seen from the magnetic part of the Lorentz force law Fmag = (qvB). The H-field is measured in ampere-turn
Ampere-turn
The ampere-turn was the MKS unit of magnetomotive force , represented by a direct current of one ampere flowing in a single-turn loop in a vacuum...

per metre (A/m) in SI units, and in oersted
Oersted
Oersted is the unit of magnetizing field in the CGS system of units.-Difference between cgs and SI systems:...

s (Oe) in cgs units.

The smallest precision level for a magnetic field measurement is on the order of attoteslas (10−18 tesla); the largest magnetic field produced in a laboratory is 2,800 T (VNIIEF in Sarov
Sarov
Sarov is a closed town in Nizhny Novgorod Oblast, Russia. Until 1995 it was known as Kremlyov ., while from 1946 to 1991 it was called Arzamas-16 . The town is off limits to foreigners as it is the Russian center for nuclear research. Population: -History:The history of the town can be divided...

, Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

, 1998). The magnetic field of some astronomical objects such as magnetar
Magnetar
A magnetar is a type of neutron star with an extremely powerful magnetic field, the decay of which powers the emission of copious high-energy electromagnetic radiation, particularly X-rays and gamma rays...

s are much higher; magnetars range from 0.1 to 100 GT (108 to 1011 T). See orders of magnitude (magnetic field).

Magnetic field lines

Mapping the magnetic field of an object is simple in principle. First, measure the strength and direction of the magnetic field at a large number of locations. Then, mark each location with an arrow (called a vector) pointing in the direction of the local magnetic field with a length proportional to the strength of the magnetic field.

A simpler way to visualize the magnetic field is to 'connect' the arrows to form "magnetic field lines". Magnetic field lines make it much easier to visualize and understand the complex mathematical relationships underlying magnetic field. If done carefully, a field line diagram contains the same information as the vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

it represents. The magnetic field can be estimated at any point on a magnetic field line diagram (whether on a field line or not) using the direction and density of nearby magnetic 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, due to the large permeability of iron relative to air. A higher density of nearby field lines indicates a stronger magnetic field.

Various phenomena have the effect of "displaying" magnetic field lines as though the field lines are physical phenomena. For example, iron filings placed in a magnetic field line up to form lines that correspond to 'field lines'. Magnetic fields "lines" are also visually displayed in polar auroras
Aurora (astronomy)
An aurora is a natural light display in the sky particularly in the high latitude regions, caused by the collision of energetic charged particles with atoms in the high altitude atmosphere...

, in which plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

particle dipole interactions create visible streaks of light that line up with the local direction of Earth's magnetic field. However, field lines are a visual and conceptual aid only and are no more real than (for example) the contour lines (constant altitude) on a topographic map
Topographic map
A topographic map is a type of map characterized by large-scale detail and quantitative representation of relief, usually using contour lines in modern mapping, but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and man-made features...

. They do not exist in the actual field; a different choice of mapping scale could show twice as many "lines" or half as many.

Field lines can be used as a qualitative tool to visualize magnetic forces. In ferromagnetic substances like iron
Iron
Iron is a chemical element with the symbol Fe and atomic number 26. It is a metal in the first transition series. It is the most common element forming the planet Earth as a whole, forming much of Earth's outer and inner core. It is the fourth most common element in the Earth's crust...

and in plasmas, magnetic forces can be understood by imagining that the field lines exert a tension
Maxwell stress tensor
The Maxwell Stress Tensor is a mathematical object in physics, more precisely it is a second rank tensor used in classical electromagnetism to represent the interaction between electric/magnetic forces and mechanical momentum...

, (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.

Most physical phenomena that "display" magnetic field lines do not include which direction along the lines that the magnetic field is in. A compass
Compass
A compass is a navigational instrument that shows directions in a frame of reference that is stationary relative to the surface of the earth. The frame of reference defines the four cardinal directions – north, south, east, and west. Intermediate directions are also defined...

, though, reveals that magnetic field lines outside of a magnet point from the north pole (compass points away from north pole) to the south (compass points toward the south pole). The magnetic field of a straight current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

-carrying wire
Wire
A wire is a single, usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads and to carry electricity and telecommunications signals. Wire is commonly formed by drawing the metal through a hole in a die or draw plate. Standard sizes are determined by various...

encircles the wire with a direction that depends on the direction of the current and that can be measured with a compass as well.

The magnetic field and permanent magnets

Permanent magnets are objects that produce their own persistent magnetic fields. They are made of ferromagnetic
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...

materials, such as iron and nickel
Nickel
Nickel is a chemical element with the chemical symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel belongs to the transition metals and is hard and ductile...

, that have been magnetized, and they have both a north and a south pole.

Magnetic field of permanent magnets

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 that it can be treated as being infinitesimally small. 'Larger' magnets need to include more complicated terms in the expression and depend on the entire geometry of the magnet not just m. straight magnet is proportional to the magnet's strength (called its magnetic dipole moment m). The equations are non-trivial and also depend on the distance from the magnet and the orientation of the magnet. For simple magnets, m 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 m by 180 degrees.

The magnetic field of larger magnets can be obtain by modelling them as a collection of a large number of small magnets called dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

s each having their own m. The magnetic field produced by the magnet then is the net magnetic field of these dipoles. And, any net force on the magnet is a result of adding up the forces on the individual dipoles.

There are two competing models for the nature of these dipoles. These two models produce two different magnetic fields, H and B. 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.

Magnetic pole model and the H-field

It is sometimes useful to 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 between electric charges. In this model, a magnetic H-field is produced by magnetic charges that are 'smeared' around each pole. The H-field, therefore, is analogous to the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

E which starts at a positive electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

and ends at a negative electric charge. Near the north pole, therefore, all H-field lines point away from the north pole (whether inside the magnet or out) while near the south pole (whether inside the magnet or out) all H-field lines point toward the south pole. A north pole, then, feels a force in the direction of the H-field while the force on the south pole is opposite to the H-field.

In the magnetic pole model, the elementary magnetic dipole m is formed by two opposite magnetic charges (poles) of pole strength q separated by a very small distance d, such that m = q d.

Unfortunately, magnetic poles cannot exist apart from each other; all magnets have north/south pairs which cannot be separated without creating two magnets each having a north/south pair. Further, magnetic charge does not account for magnetism that is produced by electric currents nor the force that a magnetic field applies to moving electric charges.

Amperian loop model and the B-field

After Oersted discovered that electric currents produce a magnetic field and Ampere discovered that electric currents attracted and repelled each other similar to magnets, it was natural to hypothesize that all magnetic fields are due to electric current loops. In this model developed by Ampere, 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 m = I A where A is the area of the loop.

These magnetic dipoles produce a magnetic B field. One important property of the B-field produced this way is that magnetic B field lines neither start nor end (mathematically, B is a solenoidal vector field); a field line either extends to infinity or wraps around to form a closed curve.Magnetic field lines may also wrap around and around without closing but also without ending. These more complicated non-closing non-ending magnetic field lines are moot, though, since the magnetic field of objects that produce them are calculated by adding the magnetic fields of 'elementary parts' having magnetic field lines that do form closed curves or extend to infinity. To date no exception to this rule has been found. (See magnetic monopole below.) Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet B-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 B-field line enters a magnet somewhere it has to leave somewhere else; it is not allowed to have an end point. Magnetic poles, therefore, always come in N and S pairs.

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:
,

where the integral is a surface integral
Surface integral
In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...

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

There is also a corresponding differential form of this equation covered in Maxwell's equations below.

Force Between magnets

The force between two small magnets is quite complicated and depends on the strength and orientation of both magnets and the distance and direction of the magnets 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 B or H may be used for the magnetic field outside of the magnet. of the other.

To understand the force between magnets, it is useful to examine the magnetic charge model given above. In this model, the H-field of one magnet pushes and pulls on both poles of a second magnet. If this H-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 H 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 by electric currents will exert 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 m due to a magnetic field B is:

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B. To understand this equation, note that the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

m · B = mBcos(θ), where m and B represent the magnitude of the m and B vectors and θ is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points 'uphill' pulling the magnet into regions of higher B-field (more strictly larger m · B). 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 having their own m then summing up the forces on each of these 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 will promptly rotate 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 τ tends to align a magnet's poles with the magnetic field lines. A compass
Compass
A compass is a navigational instrument that shows directions in a frame of reference that is stationary relative to the surface of the earth. The frame of reference defines the four cardinal directions – north, south, east, and west. Intermediate directions are also defined...

, therefore, will turn to align itself with earth's magnetic field.

Magnetic torque is used to drive electric motor
Electric motor
An electric motor converts electrical energy into mechanical energy.Most electric motors operate through the interaction of magnetic fields and current-carrying conductors to generate force...

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
Electromagnet
An electromagnet is a type of magnet in which the magnetic field is produced by the flow of electric current. The magnetic field disappears when the current is turned off...

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. See Rotating magnetic fields below.

Mathematically, the torque τ on a small magnet is proportional both to the applied magnetic field and to the magnetic moment m of the magnet:

where × represents the vector cross product. Note that this equation includes all of the qualitative information included above. There is no torque on a magnet if m is in the same direction as the magnetic field. (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.

As is the case for the force between magnets, the magnetic charge model leads more readily to the above equation. In this model, two equal and opposite magnetic charges experiencing the same H 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 m as the magnetic charge times the distance between the charges, this leads directly to the above equation.

The Amperian loop model also predicts the same magnetic torque. Here, it is the B field interacting with the Amperian current loop through a Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

described below. Again, the results are the same although the models are completely different.

The magnetic field and 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
Point particle
A point particle is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension: being zero-dimensional, it does not take up space...

charges, such as electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s, produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.

Magnetic field lines form in concentric
Concentric
Concentric objects share the same center, axis or origin with one inside the other. Circles, tubes, cylindrical shafts, disks, and spheres may be concentric to one another...

circles around a cylindrical
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

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" (see figure at right). The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength decreases 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
Solenoid
A solenoid is a coil wound into a tightly packed helix. In physics, the term solenoid refers to a long, thin loop of wire, often wrapped around a metallic core, which produces a magnetic field when an electric current is passed through it. Solenoids are important because they can create...

" enhances this effect. A device so formed around an iron core
Magnetic core
A magnetic core is a piece of magnetic material with a high permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, inductors and magnetic assemblies. It is made of ferromagnetic metal such...

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 is neither accumulating nor depleting at any point)
In practice, the Biot–Savart law and other laws of magnetostatics are often used even when the currents are changing in time as long as it is not changing too quickly. It is often used, for instance, for standard household currents which oscillate sixty times per second.
is described by the Biot–Savart law:

where the integral sums over the wire length where vector dℓ is the direction of the current, μ0 is the magnetic constant, r is the distance between the location of dℓ and the location at which the magnetic field is being calculated, and r̂ is a unit vector in the direction of r.

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 B being zero, which is always valid. (There are no magnetic charges.)
way of relating the current to the B-field is through Ampère's law:

where the line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

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 B-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
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

that describe electricity and magnetism.

Force on moving charges and current

Force on a charged particle

A charged particle
Charged particle
In physics, a charged particle is a particle with an electric charge. It may be either a subatomic particle or an ion. A collection of charged particles, or even a gas containing a proportion of charged particles, is called a plasma, which is called the fourth state of matter because its...

moving in a B-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

where
F is the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

, q is the electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

of the particle, v is the instantaneous velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

of the particle, and B is the magnetic field (in teslas
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

).

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 will trace out a helical path in which the helix axis is parallel to the magnetic field and in which the speed of the particle will remain constant. Because the magnetic force is always perpendicular to the motion, the magnetic field can do no work
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...

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
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...

, 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 charge carrying wire is a collection of moving charges. A current carrying wire feels a sideways 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 l and area of cross section A and has charge q which is due to electric current i .If a conductor is placed in a magnetic field of induction B which makes an angle θ (theta) with the velocity of charges in the conductor which has i current flowing in it.
then force exerted due to small particle q is
then
for n number of charges it has
then force exerted on the body is
but
that is

Direction of force

The direction of force on a charge or a current can be determined by a mnemonic
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

known as the right-hand rule. See the figure on the left. Using the right hand and pointing the thumb in the direction of the moving positive charge or positive 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.

An alternative mnemonic to the right hand rule is Fleming's left hand rule
Fleming's left hand rule
Fleming's left-hand rule , and Fleming's right-hand rule are a pair of visual mnemonics that is used for working out the direction of motion in an electric motor, or the direction of electric current in an electric generator...

.

The 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
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

of the subatomic particles such as electrons that make up the material.) The H-field as defined above helps factor out this bound current; but in order to see how, it helps to introduce the concept of magnetization first.

Magnetization

The magnetization field represents how strongly a region of material is magnetized. It is defined as the net magnetic dipole moment per unit volume of that region. The magnetization of a uniform magnet, therefore, is a constant in the material equal to its magnetic moment, m, divided by its volume. Since the SI unit of magnetic moment is ampere-turn meter2, the SI unit of magnetization is ampere-turn per meter which is identical to that of the -field.

The magnetization field of a region points in the direction of the average magnetic dipole moment in the region and is in the same direction as the local -field it produces. Therefore, field lines move from near the south pole of a magnet to near its north. Unlike , magnetization only exists inside a magnetic material. Therefore, magnetization field lines begin and end near magnetic poles.

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 B field due to the magnet. (See Magnetic dipoles below and magnetic poles vs. atomic currents for more information.) Given the definition of the magnetic dipole, the magnetization field follows a similar law to that of Ampere's law:

where the integral is a line integral over any closed loop and is the 'bound current' enclosed by that closed loop.

It is also possible to model the magnetization in terms of magnetic charge in which magnetization begins at and ends at magnetic poles. If a given region, therefore, has a net positive 'magnetic charge' then it will have more magnetic field lines entering it than leaving it. Mathematically this is equivalent to:
,

where the integral is a closed surface integral over the closed surface and is the 'magnetic charge' (in units of magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...

) enclosed by . (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because, like B inside a magnet, the magnetization field moves from south to north.

H-field and magnetic materials

The -field is defined as:
(definition of in SI units)

With this definition, Ampere's law becomes:

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. Ampere's law leads to the boundary condition
where is the surface free current density.

Similarly, a surface integral
Surface integral
In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...

of over any closed surface is independent of the free currents and picks out the 'magnetic charges' within that closed surface:

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:

where is the applied magnetic field due only to the free currents and is the demagnetizing field
Demagnetizing field
The demagnetizing field, also called the stray field, is the magnetic field generated by the magnetization in a magnet. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields of the magnets and the magnetic field due to any free currents or displacement...

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 at near magnetic poles as well.

Magnetism

Most materials respond to an applied B-field by producing their own magnetization M and therefore their own B-field. Typically, the response is very 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
Phase (matter)
In the physical sciences, a phase is a region of space , throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, and chemical composition...

of a material. Materials are divided into groups based upon their magnetic behavior:
• Diamagnetic materials
Diamagnetism
Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the...

produce a magnetization that opposes the magnetic field.
• Paramagnetic materials
Paramagnetism
Paramagnetism is a form of magnetism whereby the paramagnetic material is only attracted when in the presence of an externally applied magnetic field. In contrast with this, diamagnetic materials are repulsive when placed in a magnetic field...

produce a magnetization in the same direction as the applied magnetic field.
• Ferromagnetic materials
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...

and the closely related ferrimagnetic materials
Ferrimagnetism
In physics, a ferrimagnetic material is one in which the magnetic moments of the atoms on different sublattices are opposed, as in antiferromagnetism; however, in ferrimagnetic materials, the opposing moments are unequal and a spontaneous magnetization remains...

and antiferromagnetic materials
Antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...

can have a magnetization independent of an applied B-field with a complex relationship between the two fields.
• Superconductors (and ferromagnetic superconductor
Ferromagnetic superconductor
Ferromagnetic superconductors are materials that display intrinsic coexistence of ferromagnetism and superconductivity. They include UGe2, URhGe, and UCoGe. Evidence of ferromagnetic superconductivity was also reported for ZrZn2 in 2001, but later reports question these findings...

s) 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) for which they exhibit a complex hysteretic dependence of M on B.

In the case of paramagnetism and diamagnetism, the magnetization M is often proportional to the applied magnetic field such that:
where μ is a material dependent parameter called the permeability
Permeability (electromagnetism)
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

. In some cases the permeability may be a second rank tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

so that H may not point in the same direction as B. These relations between B and H are examples of constitutive equation
Constitutive equation
In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...

s. However, superconductors and ferromagnets have a more complex B to H relation, see magnetic hysteresis.

Energy stored in magnetic fields

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 this energy can be modeled as being stored in the magnetic field.

For linear, non-dispersive, materials (such that B = μH where μ is frequency-independent), the energy density
Energy density
Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

is:

If there are no magnetic materials around then μ can be replaced by μ0. 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 δW needed to cause a small change of magnetic field δB is:

Once the relationship between H and B is known this equation is used to determine the work needed to reach a given magnetic state. For hysteretic materials
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...

such as ferromagnets and superconductors the work needed will also depend 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.

Electromagnetism: the relationship between magnetic and electric fields

Faraday's Law: Electric force due to a changing B-field

A changing magnetic field, such as a magnet moving through a conducting coil, generates an electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

(and therefore tends to drive a current in the coil). This is known as Faraday's law and forms the basis of many electrical generator
Electrical generator
In electricity generation, an electric generator is a device that converts mechanical energy to electrical energy. A generator forces electric charge to flow through an external electrical circuit. It is analogous to a water pump, which causes water to flow...

s and electric motor
Electric motor
An electric motor converts electrical energy into mechanical energy.Most electric motors operate through the interaction of magnetic fields and current-carrying conductors to generate force...

s.

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

(or EMF, the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

generated around a closed loop) and Φm is the magnetic flux—the product of the area times the magnetic field normal
Tangential and normal components
In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector...

to that area. (This definition of magnetic flux is why B is often referred to as magnetic flux density.)

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

.

This integral formulation of Faraday's law can be converted

A complete expression for Faraday's law of induction in terms of the electric E and magnetic fields can be written as:

where ∂Σ(t) is the moving closed path bounding the moving surface Σ(t), and dA is an element of surface area of Σ(t). The first integral calculates the work done moving a charge a distance dℓ based upon the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

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. This form is covered as one of Maxwell's equations below.

Maxwell's correction to Ampère's Law: The magnetic field due to a changing electric field

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'. Maxwell's correction to Ampère's Law bootstrap
Bootstrapping
Bootstrapping or booting refers to a group of metaphors that share a common meaning: a self-sustaining process that proceeds without external help....

together with Faraday's law of induction to form electromagnetic waves, such as light. Thus, a changing electric field generates a changing magnetic field which generates a changing electric field again.

Maxwell's correction to Ampère law is applied as an additive term to Ampere's law given above. This additive 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.)

This full Ampère 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. These, though, are usually described using the differential form of this equation given below.

Maxwell's equations

Like all vector fields the B-field has two important mathematical properties that relates it to its sources. (For magnetic fields 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
Classical electromagnetism
Classical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...

including both electricity and magnetism.

The first property is the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of a vector field A, ∇ · A which represents how A 'flows' outward from a given point. As discussed above, a B-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 B 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 magnetic charges or magnetic monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...

s. The electric field on the other hand begins and ends at electric charges so that its divergence is non-zero and proportional to the charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

(See Gauss's law
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...

).

The second mathematical property is called the curl, such that ∇ × A represents how A curls or 'circulates' around a given point. The result of the curl is called a 'circulation source'. The equations for the curl of B and of E are called the Ampère–Maxwell equation and Faraday's law
Faraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...

respectively. They represent the differential forms of the integral equations given above.
The complete set of Maxwell's equations then are:

where J = complete microscopic current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...

and ρ is the charge density.

Technically, B is a pseudovector
Pseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...

(also called an axial vector) due to being defined by a vector cross product. (See diagram to right.)

As discussed above, materials respond to an applied electric E field and an applied magnetic B field by producing their own internal 'bound' charge and current distributions that contribute to E and B but are difficult to calculate. To circumvent this problem the auxiliary H and D fields are defined so that Maxwell's equations can be re-factored in terms of the free current density Jf and free charge density ρf:

These equations are not any more general than the original equations (if the 'bound' charges and currents in the material are known'). They also need to be supplemented by the relationship between B and H as well as that between E and D. 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.

Electric and magnetic fields: different aspects of the same phenomenon

According to the special theory of relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, the partition of the electromagnetic force 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.

Formally, special relativity combines the electric and magnetic fields into a rank-2 tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

, called the electromagnetic tensor
Electromagnetic tensor
The electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field of a physical system in Maxwell's theory of electromagnetism...

. Changing reference frames mixes these components. This is analogous to the way that special relativity mixes space and time into spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

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

.

Magnetic vector potential

In advanced topics such as quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

and relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

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 vector potential, A, and the scalar potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

, φ, are defined such that:
The vector potential A may be interpreted as a generalized potential momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

per unit charge just as φ is interpreted as a generalized potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

per unit charge.

Maxwell's equations when expressed in terms of the potentials can be cast into a form that agrees with special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

with little effort. In relativity A together with φ forms the four-potential analogous to the four-momentum which combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler; further 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
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

, but rather a quantum field; it is represented not as a vector of three numbers
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

at each point, but as a vector of three quantum operators
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

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
Antiparticle
Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...

s) is computed using perturbation theory
Perturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...

; these rather complex formulas have a remarkable pictorial representation as Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...

s 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
Precision tests of QED
Quantum electrodynamics , a relativistic quantum field theory of electrodynamics, is among the most stringently tested theories in physics....

. This makes QED one of the most accurate physical theories constructed thus far.

Classical electromagnetism
Classical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...

, which is less accurate than the quantum description mentioned here. However, under most everyday circumstances, the difference between the two theories is negligible.

Earth's magnetic field

The Earth's magnetic field is thought to be produced by convection currents in the outer liquid of Earth's core. The Dynamo theory
Dynamo theory
In geophysics, dynamo theory proposes a mechanism by which a celestial body such as the Earth or a star generates a magnetic field. The theory describes the process through which a rotating, convecting, and electrically conducting fluid can maintain a magnetic field over astronomical time...

proposes that these movements produce electric currents which, in turn, produce the magnetic field.

The presence of this field causes a compass
Compass
A compass is a navigational instrument that shows directions in a frame of reference that is stationary relative to the surface of the earth. The frame of reference defines the four cardinal directions – north, south, east, and west. Intermediate directions are also defined...

, placed anywhere within it, to rotate so that the "north pole" of the magnet in the compass points roughly north, toward Earth's north magnetic pole
North Magnetic Pole
The Earth's North Magnetic Pole is the point on the surface of the Northern Hemisphere at which the Earth's magnetic field points vertically downwards . Though geographically in the north, it is, by the direction of the magnetic field lines, physically the south pole of the Earth's magnetic field...

. This is the traditional definition of the "north pole" of a magnet, although other equivalent definitions are also possible.

One confusion that arises from this definition is that, if Earth itself is considered as a magnet, the south pole of that magnet would be the one nearer the north magnetic pole, and vice-versa (opposite poles attract, so the north pole of the compass magnet is attracted to the south pole of Earth's interior magnet).

The north magnetic pole is so-named not because of the polarity of the field there but because of its geographical location. The north and south poles of a permanent magnet are so-called because they are "north-seeking" and "south-seeking", respectively.

The figure to the right is a sketch of Earth's magnetic field represented by field lines. For most locations, the magnetic field has a significant up/down component in addition to the North/South component. (There is also an East/West component; Earth's magnetic poles do not coincide exactly with Earth's geological pole.) The magnetic field can be visualised as a bar magnet buried deep in Earth's interior.

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
Geomagnetic reversal
A geomagnetic reversal is a change in the Earth's magnetic field such that the positions of magnetic north and magnetic south are interchanged. The Earth's field has alternated between periods of normal polarity, in which the direction of the field was the same as the present direction, and reverse...

. 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 by Nikola Tesla
Nikola Tesla
Nikola Tesla was a Serbian-American inventor, mechanical engineer, and electrical engineer...

, and later utilized in his, and others', early AC (alternating-current
Alternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....

) electric motors.

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, in order to overcome it, three-phase
Three-phase
In electrical engineering, three-phase electric power systems have at least three conductors carrying voltage waveforms that are radians offset in time...

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.

Because magnets degrade with time, synchronous motor
Synchronous motor
A synchronous electric motor is an AC motor distinguished by a rotor spinning with coils passing magnets at the same rate as the power supply frequency and resulting rotating magnetic field which drives it....

s use DC voltage fed rotor windings which allows the excitation of the machine to be controlled and induction motor
Induction motor
An induction or asynchronous motor is a type of AC motor where power is supplied to the rotor by means of electromagnetic induction. These motors are widely used in industrial drives, particularly polyphase induction motors, because they are robust and have no brushes...

s use short-circuited rotors
Rotor (electric)
The rotor is the non-stationary part of a rotary electric motor, electric generator or alternator, which rotates because the wires and magnetic field of the motor are arranged so that a torque is developed about the rotor's axis. In some designs, the rotor can act to serve as the motor's armature,...

(instead of a magnet) following the rotating magnetic field of a multicoiled stator
Stator
The stator is the stationary part of a rotor system, found in an electric generator, electric motor and biological rotors.Depending on the configuration of a spinning electromotive device the stator may act as the field magnet, interacting with the armature to create motion, or it may act as the...

. The short-circuited turns of the rotor develop eddy current
Eddy current
Eddy currents are electric currents induced in conductors when a conductor is exposed to a changing magnetic field; due to relative motion of the field source and conductor or due to variations of the field with time. This can cause a circulating flow of electrons, or current, within the body of...

s in the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...

.

In 1882, Nikola Tesla identified the concept of the rotating magnetic field. In 1885, Galileo Ferraris
Galileo Ferraris
Galileo Ferraris was an Italian physicist and electrical engineer, noted mostly for the studies and independent discovery of the rotating magnetic field, a basic working principle of the induction motor...

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
Turin
Turin is a city and major business and cultural centre in northern Italy, capital of the Piedmont region, located mainly on the left bank of the Po River and surrounded by the Alpine arch. The population of the city proper is 909,193 while the population of the urban area is estimated by Eurostat...

.

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 H is in magnetic circuits where inside a linear material B = μ H. Here, μ is the magnetic permeability of the material. This result is similar in form to Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

J = σ E, where J is the current density, σ is the conductance and E is the electric field. Extending this analogy, the counterpart to the macroscopic Ohm's law ( I = V ⁄ R ) is:

where is the magnetic flux in the circuit, is the magnetomotive force
Magnetomotive force
Magnetomotive force is any physical driving force that produces magnetic flux. In this context, the expression "driving force" is used in a general sense of "work potential", and is analogous, but distinct from force measured in newtons...

applied to the circuit, and is the reluctance of the circuit. Here the reluctance is a quantity similar in nature to resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

for the flux.

Using this analogy it is straight-forward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of circuit theory.

Magnetic field shape descriptions

• An azimuthal magnetic field is one that runs east-west.
• A meridional magnetic field is one that runs north-south. In the solar dynamo
Solar dynamo
The solar dynamo is the physical process that generates the Sun's magnetic field. The Sun is permeated by an overall dipole magnetic field, as are many other celestial bodies such as the Earth. The dipole field is produced by a circular electric current flowing deep within the star, following...

model of the Sun, differential rotation
Differential rotation
Differential rotation is seen when different parts of a rotating object move with different angular velocities at different latitudes and/or depths of the body and/or in time. This indicates that the object is not solid. In fluid objects, such as accretion disks, this leads to shearing...

of the solar plasma causes the meridional magnetic field to stretch into an azimuthal magnetic field, a process called the omega-effect. The reverse process is called the alpha-effect.
• A dipole magnetic field
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...

is one seen around a bar magnet or around a charged
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

with nonzero spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

.
Quadrupole magnets consist of groups of four magnets laid out so that in the multipole expansion of the field the dipole terms cancel and where the lowest significant terms in the field equations are quadrupole. Quadrupole magnets are useful as they create a magnetic field whose magnitude grows...

is one seen, for example, between the poles of four bar magnets. The field strength grows linearly with the radial distance from its longitudinal axis.
• A solenoidal magnetic field is similar to a dipole magnetic field, except that a solid bar magnet is replaced by a hollow electromagnetic coil magnet.
• A toroidal magnetic field occurs in a doughnut-shaped coil, the electric current spiraling around the tube-like surface, and is found, for example, in a tokamak
Tokamak
A tokamak is a device using a magnetic field to confine a plasma in the shape of a torus . Achieving a stable plasma equilibrium requires magnetic field lines that move around the torus in a helical shape...

.
• A poloidal magnetic field is generated by a current flowing in a ring, and is found, for example, in a tokamak
Tokamak
A tokamak is a device using a magnetic field to confine a plasma in the shape of a torus . Achieving a stable plasma equilibrium requires magnetic field lines that move around the torus in a helical shape...

.
• A radial magnetic field is one in which the field lines are directed from the center outwards, similar to the spokes in a bicycle wheel. An example can be found in a loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...

transducers (driver).
• A helical magnetic field is corkscrew-shaped, and sometimes seen in space plasmas such as the Orion Molecular Cloud
Orion Molecular Cloud Complex
The Orion Molecular Cloud Complex refers to a large group of bright nebula, dark clouds, and young stars located in the constellation of Orion. The cloud itself is between 1,500 and 1,600 light-years away and is hundreds of light-years across...

.

Magnetic dipoles

The magnetic field of a magnetic dipole is depicted on the right. 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 I and an area a. Such a current loop has a magnetic moment of:

where the direction of m 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 a has been reduced to zero and its current I increased to infinity such that the product m = Ia is finite. In this model it is easy to see the connection between angular momentum and magnetic moment which is the basis of the Einstein-de Haas effect
Einstein-de Haas effect
The Einstein–de Haas effect, or the Richardson effect , is a physical phenomenon delineated by Albert Einstein and Wander Johannes de Haas in the mid 1910's, that exposes a relationship between magnetism, angular momentum, and the spin of elementary particles...

"rotation by magnetization" and its inverse, the Barnett effect
Barnett Effect
The Barnett effect is the magnetization of an uncharged body when spun on its axis. It was discovered by American physicist Samuel Barnett in 1915....

or "magnetization by rotation". Rotating the loop faster (in the same direction) increases the current and therefore the magnetic moment, for example.

It is sometimes useful to model the magnetic dipole similar to the electric dipole with two equal but opposite magnetic charges (one south the other north) separated by distance d. This model produces an H-field not a B-field. Such a model is deficient, though, both in that there are no magnetic charges and in that it obscures the link between electricity and magnetism. Further, as discussed above it fails to explain the inherent connection between angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

and magnetism.

Magnetic monopole (hypothetical)

A magnetic monopole is a hypothetical particle (or class of particles) that has, as its name suggests, 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.

Modern interest in this concept stems from particle theories, notably Grand Unified Theories and superstring theories
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...

, that predict either the existence, or the possibility, of magnetic monopoles. These theories and others have inspired extensive efforts to search for monopoles. Despite these efforts, no magnetic monopole has been observed to date.Two experiments produced candidate events that were initially interpreted as monopoles, but these are now regarded to be inconclusive. For details and references, see magnetic monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...

.

In recent research, materials known as spin ice
Spin ice
A spin ice is a substance that is similar to water ice in that it can never be completely frozen. This is because it does not have a single minimal-energy state. A spin ice has "spin" degrees of freedom , with frustrated interactions which prevent it freezing...

s can simulate monopoles, but do not contain actual monopoles.

General
• Magnetohydrodynamics
Magnetohydrodynamics
Magnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...

– the study of the dynamics of electrically conducting fluids.
• Magnetic nanoparticles
Magnetic nanoparticles
Magnetic nanoparticles are a class of nanoparticle which can be manipulated using magnetic field. Such particles commonly consist of magnetic elements such as iron, nickel and cobalt and their chemical compounds. While nanoparticles are smaller than 1 micrometer in diameter , the larger microbeads...

– extremely small magnetic particles that are tens of atoms wide
• Magnetic reconnection
Magnetic reconnection
Magnetic reconnection is a physical process in highly conducting plasmas in which the magnetic topology is rearranged and magnetic energy is converted to kinetic energy, thermal energy, and particle acceleration...

– an effect which causes solar flare
Solar flare
A solar flare is a sudden brightening observed over the Sun surface or the solar limb, which is interpreted as a large energy release of up to 6 × 1025 joules of energy . The flare ejects clouds of electrons, ions, and atoms through the corona into space. These clouds typically reach Earth a day...

s and auroras.
• Magnetic potential
Magnetic potential
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...

– the vector and scalar potential
Scalar potential
A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...

representation of magnetism.
• 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
Upward continuation
Upward continuation is a method used in oil exploration and geophysics to estimate the values of a gravitational or magnetic field by using measurements at a lower elevation and extrapolating upward, assuming continuity. This technique is commonly used to merge different measurements to a common...

Mathematics
• Magnetic helicity
Magnetic helicity
In plasma physics, magnetic helicity is the extent to which a magnetic field "wraps around itself". It is a generalization of the topological concept of linking number to the differential quantities required to describe the magnetic field...

– extent to which a magnetic field "wraps around itself".

Applications
• Dynamo theory
Dynamo theory
In geophysics, dynamo theory proposes a mechanism by which a celestial body such as the Earth or a star generates a magnetic field. The theory describes the process through which a rotating, convecting, and electrically conducting fluid can maintain a magnetic field over astronomical time...

– a proposed mechanism for the creation of the Earth's magnetic field.
• Helmholtz coil
Helmholtz coil
A Helmholtz coil is a device for producing a region of nearly uniform magnetic field. It is named in honor of the German physicist Hermann von Helmholtz.- Description :A Helmholtz pair consists of two identical circular magnetic...

– a device for producing a region of nearly uniform magnetic field.
• Magnetic field viewing film
Magnetic field viewing film
Magnetic field viewing film is used to show stationary or slowly-changing magnetic fields; it shows their location and direction. It is a translucent thin flexible sheet, coated with micro-capsules containing nickel flakes suspended in oil...

– Film used to view the magnetic field of an area.
• Maxwell coil
Maxwell coil
A Maxwell coil is a device for producing a large volume of almost constant magnetic field.-Description:A constant-field Maxwell coil set consists of three coils oriented on the surface of a virtual sphere...

– a device for producing a large volume of an almost constant magnetic field.
• Stellar magnetic field
Stellar magnetic field
A stellar magnetic field is a magnetic field generated by the motion of conductive plasma inside a star. This motion is created through convection, which is a form of energy transport involving the physical movement of material. A localized magnetic field exerts a force on the plasma, effectively...

– a discussion of the magnetic field of stars.
• Teltron Tube
Teltron Tube
A teltron tube is a type of cathode ray tube used to demonstrate the properties of electrons. It usually contains two electron guns, which can project two thin electron beams at right angles. The beams can be bent by applying voltages to various electrodes in the tube...

– device used to display an electron beam and demonstrates effect of electric and magnetic fields on moving charges.