Magnetic field

 ( Magnetism ) 

A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving , electric currents,

and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field.

(2025). 9781107014022, Cambridge University Press. ISBN 9781107014022

A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism, diamagnetism, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, electric currents, and varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a Euclidean vector to each point of space, called a vector field (more precisely, a pseudovector field).

In electromagnetics, the term magnetic field is used for two distinct but closely related vector fields denoted by the symbols and . In the International System of Units, the unit of , magnetic flux density, is the tesla (in SI base units: kilogram per second squared per ampere), which is equivalent to newton per meter per ampere. The unit of , magnetic field strength, is ampere per meter (A/m). and differ in how they take the medium and/or magnetization into account. In vacuum, the two fields are related through the vacuum permeability, $\backslash mathbf\{B\}/\backslash mu\_0\; =\; \backslash mathbf\{H\}$; in a magnetized material, the quantities on each side of this equation differ by the magnetization field of the material.

Magnetic fields are produced by moving electric charges and the intrinsic of elementary particles associated with a fundamental quantum property, their spin.

David C. Jiles (1998). 9780412798603, CRC. . ISBN 9780412798603

Magnetic fields and are interrelated and are both components of the electromagnetic force, one of the four fundamental forces of nature.

Magnetic fields are used throughout modern technology, particularly in electrical engineering and electromechanics. Rotating magnetic fields are used in both and generators. The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as . Magnetic forces give information about the charge carriers in a material through the Hall effect. The Earth produces its own magnetic field, which shields the Earth's ozone layer from the solar wind and is important in navigation using a compass.

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Description

The force on an electric charge depends on its location, speed, and direction; two vector fields are used to describe this force. The first is the electric field, which describes the force acting on a stationary charge and gives the component of the force that is independent of motion. The magnetic field, in contrast, describes the component of the force that is proportional to both the speed and direction of charged particles. The field is defined by the Lorentz force law and is, at each instant, perpendicular to both the motion of the charge and the force it experiences.

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

M Gerloch (1983). 9780521249393, Cambridge University Press. . ISBN 9780521249393

says: "So we may think of both and as magnetic fields, but drop the word 'magnetic' from so as to maintain the distinction ... As Purcell points out, 'it is only the names that give trouble, not the symbols'." There are many alternative names for both (see sidebars).

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The B-field

Magnetic flux density Magnetic induction Magnetic field (ambiguous)

The magnetic field vector at any point can be defined as the vector that, when used in the Lorentz force law, correctly predicts the force on a charged particle at that point:

David J. Griffiths (1999). 013805326X, Pearson. 013805326X

Here is the force on the particle, is the particle's electric charge, is the external electric field, , is the particle's velocity, and × denotes the cross product. The direction of force on the charge can be determined by a mnemonic known as the right-hand rule (see the figure).An alternative mnemonic to the right hand rule is Fleming's left-hand rule. Using the right hand, pointing the thumb in the direction of the current, and the fingers in the direction of the magnetic field, the resulting force on the charge points outwards from the palm. The force on a negatively charged particle is in the opposite direction. If both the speed and the charge are reversed then the direction of the force remains the same. For that reason a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these cases produce the same current.) On the other hand, a magnetic field combined with an electric field can distinguish between these, see Hall effect below.

The first term in the Lorentz equation is from the theory of electrostatics, and says that a particle of charge in an electric field experiences an electric force: $$\backslash mathbf\{F\}\_\{\backslash text\{electric\}\}\; =\; q\; \backslash mathbf\{E\}.$$

The second term is the magnetic force: $$\backslash mathbf\{F\}\_\{\backslash text\{magnetic\}\}\; =\; q(\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}).$$

Using the definition of the cross product, the magnetic force can also be written as a scalar equation: $$F\_\{\backslash text\{magnetic\}\}\; =\; q\; v\; B\; \backslash sin(\backslash theta)$$ where , , and are the scalar magnitude of their respective vectors, and is the angle between the velocity of the particle and the magnetic field. The vector is defined as the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle. In other words,

E. Purcell (2025). 9781107013605, Cambridge University Press. . ISBN 9781107013605

The field can also be defined by the torque on a magnetic dipole, .

The SI unit of is tesla (symbol: T).The SI unit of (magnetic flux) is the weber (symbol: Wb), related to the tesla by 1 Wb/m² = 1 T. The SI unit tesla is equal to (newton·second)/(coulomb·metre). This can be seen from the magnetic part of the Lorentz force law. The Gaussian units of is the gauss (symbol: G). (The conversion is 1 T ≘ 10000 G.) One nanotesla corresponds to 1 gamma (symbol: γ).

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The H-field

Magnetic field intensity Julius Adams Stratton (2025). 9780070621503, McGraw-Hill. ISBN 9780070621503 Magnetic field strength Magnetic field Magnetizing field Auxiliary magnetic field

The magnetic field is defined:

where $\backslash mu\_0$ is the vacuum permeability, and is the Magnetization. In a vacuum, and are proportional to each other. Inside a material they are different (see H and B inside and outside magnetic materials). The SI unit of the -field is the ampere per metre (A/m), and the CGS unit is the oersted (Oe).

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Measurement An instrument used to measure the local magnetic field is known as a magnetometer. Important classes of magnetometers include using Search coil (or search-coil magnetometers) which measure only varying magnetic fields, rotating coil magnetometers, Hall effect magnetometers, NMR magnetometers, SQUID, and fluxgate magnetometers. The magnetic fields of distant astronomical objects are measured through their effects on local charged particles. For instance, electrons spiraling around a field line produce synchrotron radiation that is detectable in radio waves. The finest precision for a magnetic field measurement was attained by Gravity Probe B at ().

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Visualization The field can be visualized by a set of magnetic field lines, that follow the direction of the field at each point. The lines can be constructed by measuring the strength and direction of the magnetic field at a large number of points (or at every point in space). Then, mark each location with an arrow (called a vector) pointing in the direction of the local magnetic field with its magnitude proportional to the strength of the magnetic field. Connecting these arrows then forms a set of magnetic field lines. The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and the local density of field lines can be made proportional to its strength. Magnetic field lines are like streamlines in Fluid dynamics, in that they represent a continuous distribution, and a different resolution would show more or fewer lines.

An advantage of using magnetic field lines as a representation is that many laws of magnetism (and electromagnetism) can be stated completely and concisely using simple concepts such as the "number" of field lines through a surface. These concepts can be quickly "translated" to their mathematical form. For example, the number of field lines through a given surface is the surface integral of the magnetic field.

Various phenomena "display" magnetic field lines as though the field lines were physical phenomena. For example, iron filings placed in a magnetic field form lines that correspond to "field lines".The use of iron filings to display a field presents something of an exception to this picture; the filings alter the magnetic field so that it is much larger along the "lines" of iron, because of the large permeability of iron relative to air. Magnetic field "lines" are also visually displayed in polar auroras, in which plasma particle dipole interactions create visible streaks of light that line up with the local direction of Earth's magnetic field.

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

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Magnetic field of permanent magnets Permanent magnets are objects that produce their own persistent magnetic fields. They are made of ferromagnetism materials, such as iron and nickel, that have been magnetized, and they have both a north and a south pole.

The magnetic field of permanent magnets can be quite complicated, especially near the magnet. The magnetic field of a smallHere, "small" means that the observer is sufficiently far away from the magnet, so that the magnet can be considered as infinitesimally small. "Larger" magnets need to include more complicated terms in the mathematical expression of the magnetic field and depend on the entire geometry of the magnet not just . straight magnet is proportional to the magnet's strength (called its magnetic dipole moment ). The equations are non-trivial and depend on the distance from the magnet and the orientation of the magnet. For simple magnets, points in the direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its by 180 degrees.

The magnetic field of larger magnets can be obtained by modeling them as a collection of a large number of small magnets called each having their own . The magnetic field produced by the magnet then is the net magnetic field of these dipoles; any net force on the magnet is a result of adding up the forces on the individual dipoles.

There are two simplified models for the nature of these dipoles: the magnetic pole model and the Amperian loop model. These two models produce two different magnetic fields, and . Outside a material, though, the two are identical (to a multiplicative constant) so that in many cases the distinction can be ignored. This is particularly true for magnetic fields, such as those due to electric currents, that are not generated by magnetic materials.

A realistic model of magnetism is more complicated than either of these models; neither model fully explains why materials are magnetic. The monopole model has no experimental support. The Amperian loop model explains some, but not all of a material's magnetic moment. The model predicts that the motion of electrons within an atom are connected to those electrons' orbital magnetic dipole moment, and these orbital moments do contribute to the magnetism seen at the macroscopic level. However, the motion of electrons is not classical, and the spin magnetic moment of electrons (which is not explained by either model) is also a significant contribution to the total moment of magnets.

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Magnetic pole model Historically, early physics textbooks would model the force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as the Coulomb force between electric charges. At the microscopic level, this model contradicts the experimental evidence, and the pole model of magnetism is no longer the typical way to introduce the concept. However, it is still sometimes used as a macroscopic model for ferromagnetism due to its mathematical simplicity.

In this model, a magnetic -field is produced by fictitious magnetic charges that are spread over the surface of each pole. These magnetic charges are in fact related to the magnetization field . The -field, therefore, is analogous to the electric field , which starts at a positive electric charge and ends at a negative electric charge. Near the north pole, therefore, all -field lines point away from the north pole (whether inside the magnet or out) while near the south pole all -field lines point toward the south pole (whether inside the magnet or out). Too, a north pole feels a force in the direction of the -field while the force on the south pole is opposite to the -field.

In the magnetic pole model, the elementary magnetic dipole is formed by two opposite magnetic poles of pole strength separated by a small distance vector , such that . The magnetic pole model predicts correctly the field both inside and outside magnetic materials, in particular the fact that is opposite to the magnetization field inside a permanent magnet.

Since it is based on the fictitious idea of a magnetic charge density, the pole model has limitations. Magnetic poles cannot exist apart from each other as electric charges can, but always come in north–south pairs. If a magnetized object is divided in half, a new pole appears on the surface of each piece, so each has a pair of complementary poles. The magnetic pole model does not account for magnetism that is produced by electric currents, nor the inherent connection between angular momentum and magnetism.

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

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Amperian loop model

In the model developed by Ampere, the elementary magnetic dipole that makes up all magnets is a sufficiently small Amperian loop with current and loop area . The dipole moment of this loop is .

These magnetic dipoles produce a magnetic -field.

The magnetic field of a magnetic dipole is depicted in the figure. From outside, the ideal magnetic dipole is identical to that of an ideal electric dipole of the same strength. Unlike the electric dipole, a magnetic dipole is properly modeled as a current loop having a current and an area . Such a current loop has a magnetic moment of $$m\; =\; Ia\; ,$$ where the direction of is perpendicular to the area of the loop and depends on the direction of the current using the right-hand rule. An ideal magnetic dipole is modeled as a real magnetic dipole whose area has been reduced to zero and its current increased to infinity such that the product is finite. This model clarifies the connection between angular momentum and magnetic moment, which is the basis of the Einstein–de Haas effect rotation by magnetization and its inverse, the Barnett effect or magnetization by rotation.See magnetic moment and

(2025). 9780471477419, Wiley-IEEE. . ISBN 9780471477419

Rotating the loop faster (in the same direction) increases the current and therefore the magnetic moment, for example.

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Interactions with magnets

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Force between magnets Specifying the force between two small magnets is quite complicated because it depends on the strength and orientation of both magnets and their distance and direction relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic fieldEither or may be used for the magnetic field outside the magnet. of the other.

To understand the force between magnets, it is useful to examine the magnetic pole model given above. In this model, the -field of one magnet pushes and pulls on both poles of a second magnet. If this -field is the same at both poles of the second magnet then there is no net force on that magnet since the force is opposite for opposite poles. If, however, the magnetic field of the first magnet is nonuniform (such as the near one of its poles), each pole of the second magnet sees a different field and is subject to a different force. This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque.

This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field, whether caused by permanent magnets or electric currents, exerts a force on a small magnet in this way.

The details of the Amperian loop model are different and more complicated but yield the same result: that magnetic dipoles are attracted/repelled into regions of higher magnetic field. Mathematically, the force on a small magnet having a magnetic moment due to a magnetic field is:

$$\backslash mathbf\{F\}\; =\; \backslash boldsymbol\{\backslash nabla\}\; \backslash left(\backslash mathbf\{m\}\backslash cdot\backslash mathbf\{B\}\backslash right),$$

where the gradient is the change of the quantity per unit distance and the direction is that of maximum increase of . The dot product , where and represent the magnitude of the and vectors and is the angle between them. If is in the same direction as then the dot product is positive and the gradient points "uphill" pulling the magnet into regions of higher -field (more strictly larger ). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions each having their own then Integral.

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Magnetic torque on permanent magnets If two like poles of two separate magnets are brought near each other, and one of the magnets is allowed to turn, it promptly rotates to align itself with the first. In this example, the magnetic field of the stationary magnet creates a magnetic torque on the magnet that is free to rotate. This magnetic torque tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field.

In terms of the pole model, two equal and opposite magnetic charges experiencing the same  also experience equal and opposite forces. Since these equal and opposite forces are in different locations, this produces a torque proportional to the distance (perpendicular to the force) between them. With the definition of  as the pole strength times the distance between the poles, this leads to , where  is a constant called the vacuum permeability, measuring  [[V|Volt]]·[[s|Second]]/([[A|Ampere]]·[[m|meter]]) and  is the angle between  and .
     

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

$$\backslash boldsymbol\{\backslash tau\}=\backslash mathbf\{m\}\backslash times\backslash mathbf\{B\}\; =\; \backslash mu\_0\backslash mathbf\{m\}\backslash times\backslash mathbf\{H\},\; \backslash ,$$

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

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

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Magnetic field due to moving charges and electric currents All moving charged particles produce magnetic fields. Moving point particle charges, such as , produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.

Magnetic field lines form in concentric circles around a cylindrical current-carrying conductor, such as a length of wire. The direction of such a magnetic field can be determined by using the "right-hand grip rule" (see figure at right). The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength is inversely proportional to the distance.)

Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely spaced loops to form a coil or "solenoid" enhances this effect. A device so formed around an iron Magnetic core may act as an electromagnet, generating a strong, well-controlled magnetic field. An infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet, with its strength and polarity determined by the current flowing through the coil.

The magnetic field generated by a steady current (a constant flow of electric charges, in which charge neither accumulates nor is depleted at any point) is described by the Biot–Savart law:

$$\backslash mathbf\{B\}\; =\; \backslash frac\{\backslash mu\_0I\}\{4\backslash pi\}\backslash int\_\{\backslash mathrm\{wire\}\}\backslash frac\{\backslash mathrm\{d\}\backslash boldsymbol\{\backslash ell\}\; \backslash times\; \backslash mathbf\{\backslash hat\; r\}\}\{r^2\},$$ where the integral sums over the wire length where vector is the vector line element with direction in the same sense as the current , is the magnetic constant, is the distance between the location of and the location where the magnetic field is calculated, and is a unit vector in the direction of . For example, in the case of a sufficiently long, straight wire, this becomes: $$|\backslash mathbf\{B\}|\; =\; \backslash frac\{\backslash mu\_0\}\{2\backslash pi\; r\}I$$ where . The direction is tangent to a circle perpendicular to the wire according to the right hand rule.

A slightly more general The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of being zero, which is always valid. (There are no magnetic charges.) way of relating the current $I$ to the -field is through Ampère's law: $$\backslash oint\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash boldsymbol\{\backslash ell\}\; =\; \backslash mu\_0\; I\_\{\backslash mathrm\{enc\}\},$$ where the line integral is over any arbitrary loop and $I\_\backslash text\{enc\}$ is the current enclosed by that loop. Ampère's law is always valid for steady currents and can be used to calculate the -field for certain highly symmetric situations such as an infinite wire or an infinite solenoid.

In a modified form that accounts for time varying electric fields, Ampère's law is one of four Maxwell's equations that describe electricity and magnetism.

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Force on moving charges and current

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Force on a charged particle

A charged particle moving in a -field experiences a sideways force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the Lorentz force, and is given by $$\backslash mathbf\{F\}\; =\; q\backslash mathbf\{E\}\; +\; q\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\},$$ where is the force, is the electric charge of the particle, is the instantaneous velocity of the particle, and is the magnetic field (in teslas).

The Lorentz force is always perpendicular to both the velocity of the particle and the magnetic field that created it. When a charged particle moves in a static magnetic field, it traces a helical path in which the helix axis is parallel to the magnetic field, and in which the speed of the particle remains constant. Because the magnetic force is always perpendicular to the motion, the magnetic field can do no mechanical 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, 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.

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Force on current-carrying wire

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

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Relation between H and B

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

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Magnetization

The magnetization vector 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 is therefore a material constant, equal to the magnetic moment of the magnet divided by its volume. Since the SI unit of magnetic moment is A⋅m², the SI unit of magnetization is ampere per meter, identical to that of the -field.

The magnetization field of a region points in the direction of the average magnetic dipole moment in that region. Magnetization field lines, therefore, begin near the magnetic south pole and ends near the magnetic north pole. (Magnetization does not exist outside the magnet.)

In the Amperian loop model, the magnetization is due to combining many tiny Amperian loops to form a resultant current called bound current. This bound current, then, is the source of the magnetic field due to the magnet. Given the definition of the magnetic dipole, the magnetization field follows a similar law to that of Ampere's law: $$\backslash oint\; \backslash mathbf\{M\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash boldsymbol\{\backslash ell\}\; =\; I\_\backslash mathrm\{b\},$$ where the integral is a line integral over any closed loop and is the bound current enclosed by that closed loop.

In the magnetic pole model, magnetization begins at and ends at magnetic poles. If a given region, therefore, has a net positive "magnetic pole strength" (corresponding to a north pole) then it has more magnetization field lines entering it than leaving it. Mathematically this is equivalent to: $$\backslash oint\_S\; \backslash mu\_0\; \backslash mathbf\{M\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; =\; -\; q\_\backslash mathrm\{M\},$$ where the integral is a closed surface integral over the closed surface and is the "magnetic charge" (in units of magnetic flux) enclosed by . (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because the magnetization field moves from south to north.

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H-field and magnetic materials

In SI units, the H-field is related to the B-field by $$\backslash mathbf\{H\}\backslash \; \backslash equiv\; \backslash \; \backslash frac\{\backslash mathbf\{B\}\}\{\backslash mu\_0\}\; -\; \backslash mathbf\{M\}.$$

In terms of the H-field, Ampere's law is $$\backslash oint\; \backslash mathbf\{H\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash boldsymbol\{\backslash ell\}\; =\; \backslash oint\; \backslash left(\backslash frac\{\backslash mathbf\{B\}\}\{\backslash mu\_0\}\; -\; \backslash mathbf\{M\}\backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash boldsymbol\{\backslash ell\}\; =\; I\_\backslash mathrm\{tot\}\; -\; I\_\backslash mathrm\{b\}\; =\; I\_\backslash mathrm\{f\},$$ where represents the 'free current' enclosed by the loop so that the line integral of does not depend at all on the bound currents.

(1969). 9780486622637, Courier Dover Publications. . ISBN 9780486622637

For the differential equivalent of this equation see Maxwell's equations. Ampere's law leads to the boundary condition $$\backslash left(\backslash mathbf\{H\_1^\backslash parallel\}\; -\; \backslash mathbf\{H\_2^\backslash parallel\}\backslash right)\; =\; \backslash mathbf\{K\}\_\backslash mathrm\{f\}\; \backslash times\; \backslash hat\{\backslash mathbf\{n\}\},$$ where is the surface free current density and the unit normal $\backslash hat\{\backslash mathbf\{n\}\}$ points in the direction from medium 2 to medium 1.

Similarly, a surface integral of over any closed surface is independent of the free currents and picks out the "magnetic charges" within that closed surface: $$\backslash oint\_S\; \backslash mu\_0\; \backslash mathbf\{H\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; =\; \backslash oint\_S\; (\backslash mathbf\{B\}\; -\; \backslash mu\_0\; \backslash mathbf\{M\})\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; =\; 0\; -\; (-q\_\backslash mathrm\{M\})\; =\; q\_\backslash mathrm\{M\},$$

which does not depend on the free currents.

The -field, therefore, can be separated into twoA third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below. independent parts: $$\backslash mathbf\{H\}\; =\; \backslash mathbf\{H\}\_0\; +\; \backslash mathbf\{H\}\_\backslash mathrm\{d\},$$

where is the applied magnetic field due only to the free currents and is the demagnetizing field due only to the bound currents.

The magnetic -field, therefore, re-factors the bound current in terms of "magnetic charges". The field lines loop only around "free current" and, unlike the magnetic field, begins and ends near magnetic poles as well.

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Magnetism Most materials respond to an applied -field by producing their own magnetization and therefore their own -fields. Typically, the response is weak and exists only when the magnetic field is applied. The term magnetism describes how materials respond on the microscopic level to an applied magnetic field and is used to categorize the magnetic phase of a material. Materials are divided into groups based upon their magnetic behavior:

• Diamagnetism
  RJD Tilley (2025). 9780470852750, Wiley. . ISBN 9780470852750
  produce a magnetization that opposes the magnetic field.
• Paramagnetism produce a magnetization in the same direction as the applied magnetic field.
• Ferromagnetism and the closely related Ferrimagnetism and antiferromagnetic materials
  (1997). 9780198517764, Oxford University Press. . ISBN 9780198517764
  Amikam Aharoni (2025). 9780198508083, Oxford University Press. . ISBN 9780198508083
  can have a magnetization independent of an applied B-field with a complex relationship between the two fields.
• (and ferromagnetic superconductors)
  M Brian Maple (2025). 9783540732525, Springer. ISBN 9783540732525
  Naoum Karchev (2025). 9781590338612, Nova Publishers. ISBN 9781590338612
  are materials that are characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures and magnetic fields (the so-named mixed state) under which they exhibit a complex hysteretic dependence of on .

In the case of paramagnetism and diamagnetism, the magnetization is often proportional to the applied magnetic field such that: $$\backslash mathbf\{B\}\; =\; \backslash mu\; \backslash mathbf\{H\},$$ where is a material dependent parameter called the permeability. In some cases the permeability may be a second rank tensor so that may not point in the same direction as . These relations between and are examples of constitutive equations. However, superconductors and ferromagnets have a more complex -to- relation; see magnetic hysteresis.

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Stored energy Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field. For non-dispersive materials, this same energy is released when the magnetic field is destroyed so that the energy can be modeled as being stored in the magnetic field.

For linear, non-dispersive, materials (such that where is frequency-independent), the energy density is:

$$u\; =\; \backslash frac\{\backslash mathbf\{B\}\; \backslash cdot\; \backslash mathbf\{H\}\}\{2\}=\; \backslash frac\{\backslash mathbf\{B\}\; \backslash cdot\; \backslash mathbf\{B\}\}\{2\backslash mu\}\; =\; \backslash frac\{\backslash mu\backslash mathbf\{H\}\; \backslash cdot\; \backslash mathbf\{H\}\}\{2\}.$$

If there are no magnetic materials around then can be replaced by . The above equation cannot be used for nonlinear materials, though; a more general expression given below must be used.

In general, the incremental amount of work per unit volume needed to cause a small change of magnetic field is:

$$\backslash delta\; W\; =\; \backslash mathbf\{H\}\backslash cdot\backslash delta\backslash mathbf\{B\}.$$

Once the relationship between and is known this equation is used to determine the work needed to reach a given magnetic state. For Hysteresis such as ferromagnets and superconductors, the work needed also depends on how the magnetic field is created. For linear non-dispersive materials, though, the general equation leads directly to the simpler energy density equation given above.

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Appearance in Maxwell's equations Like all vector fields, a magnetic field has two important mathematical properties that relates it to its sources. (For the sources are currents and changing electric fields.) These two properties, along with the two corresponding properties of the electric field, make up Maxwell's Equations. Maxwell's Equations together with the Lorentz force law form a complete description of classical electrodynamics including both electricity and magnetism.

The first property is the divergence of a vector field , , which represents how "flows" outward from a given point. As discussed above, a -field line never starts or ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of is zero. (Such vector fields are called solenoidal vector fields.) This property is called Gauss's law for magnetism and is equivalent to the statement that there are no isolated magnetic poles or magnetic monopoles.

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

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Gauss' law for magnetism One important property of the -field produced this way is that magnetic -field lines neither start nor end (mathematically, is a solenoidal vector field); a field line may only extend to infinity, or wrap around to form a closed curve, or follow a never-ending (possibly chaotic) path. Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet -field lines continue through the magnet from the south pole back to the north.To see that this must be true imagine placing a compass inside a magnet. There, the north pole of the compass points toward the north pole of the magnet since magnets stacked on each other point in the same direction. If a -field line enters a magnet somewhere it has to leave somewhere else; it is not allowed to have an end point.

More formally, since all the magnetic field lines that enter any given region must also leave that region, subtracting the "number"As discussed above, magnetic field lines are primarily a conceptual tool used to represent the mathematics behind magnetic fields. The total "number" of field lines is dependent on how the field lines are drawn. In practice, integral equations such as the one that follows in the main text are used instead. of field lines that enter the region from the number that exit gives identically zero. Mathematically this is equivalent to Gauss's law for magnetism: $$\backslash oint\_S\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; =\; 0$$ where the integral is a surface integral over the closed surface (a closed surface is one that completely surrounds a region with no holes to let any field lines escape). Since points outward, the dot product in the integral is positive for -field pointing out and negative for -field pointing in.

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Faraday's Law A changing magnetic field, such as a magnet moving through a conducting coil, generates an electric field (and therefore tends to drive a current in such a coil). This is known as Faraday's law and forms the basis of many electrical generators and . Mathematically, Faraday's law is: $$\backslash mathcal\{E\}\; =\; -\; \backslash frac\{\backslash mathrm\{d\}\backslash Phi\}\{\backslash mathrm\{d\}t\}$$

where $\backslash mathcal\{E\}$ is the electromotive force (or EMF, the voltage generated around a closed loop) and is the magnetic flux—the product of the area times the magnetic field normal to that area. (This definition of magnetic flux is why is often referred to as magnetic flux density.)

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

$$$$

 \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}
     

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Ampère's Law and Maxwell's correction Similar to the way that a changing magnetic field generates an electric field, a changing electric field generates a magnetic field. This fact is known as Maxwell's correction to Ampère's law and is applied as an additive term to Ampere's law as given above. This additional term is proportional to the time rate of change of the electric flux and is similar to Faraday's law above but with a different and positive constant out front. (The electric flux through an area is proportional to the area times the perpendicular part of the electric field.)

The full law including the correction term is known as the Maxwell–Ampère equation. It is not commonly given in integral form because the effect is so small that it can typically be ignored in most cases where the integral form is used.

The Maxwell term is critically important in the creation and propagation of electromagnetic waves. Maxwell's correction to Ampère's Law together with Faraday's law of induction describes how mutually changing electric and magnetic fields interact to sustain each other and thus to form electromagnetic waves, such as light: a changing electric field generates a changing magnetic field, which generates a changing electric field again. These, though, are usually described using the differential form of this equation given below.

$$$$

 \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}
     

where is the complete microscopic current density, and is the vacuum permittivity.

As discussed above, materials respond to an applied electric field and an applied magnetic field by producing their own internal "bound" charge and current distributions that contribute to and but are difficult to calculate. To circumvent this problem, and fields are used to re-factor Maxwell's equations in terms of the free current density :

$$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}\_\backslash mathrm\{f\}\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{D\}\}\; \{\backslash partial\; t\}$$

These equations are not any more general than the original equations (if the "bound" charges and currents in the material are known). They also must be supplemented by the relationship between and as well as that between and . On the other hand, for simple relationships between these quantities this form of Maxwell's equations can circumvent the need to calculate the bound charges and currents.

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Formulation in special relativity and quantum electrodynamics

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Relativistic electrodynamics

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As different aspects of the same phenomenon According to the special theory of relativity, 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.

The magnetic field existing as electric field in other frames can be shown by consistency of equations obtained from Lorentz transformation of four force from Coulomb's Law in particle's rest frame with Maxwell's laws considering definition of fields from Lorentz force and for non accelerating condition. The form of magnetic field hence obtained by Lorentz transformation of four-force from the form of Coulomb's law in source's initial frame is given by:

$$\backslash mathbf\{B\}\; =\; \backslash frac\; q\; \{4\; \backslash pi\; \backslash varepsilon\_0\; r^3\}\; \backslash frac\; \{1-\; \backslash beta^2\}\; \{(1-\; \backslash beta^2\; \backslash sin^2\; \backslash theta)^\{3/2\}\}\; \backslash frac\{\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{r\}\}\{c^2\}\; =\; \backslash frac\{\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{E\}\}\{c^2\}$$

where $q$ is the charge of the point source, $\backslash varepsilon\_0$ is the vacuum permittivity, $\backslash mathbf\{r\}$ is the position vector from the point source to the point in space, $\backslash mathbf\{v\}$ is the velocity vector of the charged particle, $\backslash beta$ is the ratio of speed of the charged particle divided by the speed of light and $\backslash theta$ is the angle between $\backslash mathbf\{r\}$ and $\backslash mathbf\{v\}$. This form of magnetic field can be shown to satisfy Maxwell's laws within the constraint of particle being non accelerating.

The above reduces to Biot-Savart law for non relativistic stream of current ($\backslash beta\backslash ll\; 1$).

Formally, special relativity combines the electric and magnetic fields into a rank-2 tensor, called the electromagnetic tensor. Changing reference frames mixes these components. This is analogous to the way that special relativity mixes space and time into spacetime, and mass, momentum, and energy into four-momentum.C. Doran and A. Lasenby (2003) Geometric Algebra for Physicists, Cambridge University Press, p. 233. . Similarly, the Magnetic energy is mixed with the energy stored in an electric field in the electromagnetic stress–energy tensor.

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Magnetic vector potential In advanced topics such as quantum mechanics and relativity it is often easier to work with a potential formulation of electrodynamics rather than in terms of the electric and magnetic fields. In this representation, the magnetic vector potential , and the electric scalar potential , are defined using gauge fixing such that:

$$\backslash begin\{align\}$$

 \mathbf{B} &=  \nabla \times \mathbf{A}, \\
 \mathbf{E} &= -\nabla \varphi - \frac{ \partial \mathbf{A} }{ \partial t }.
     

\end{align}

The vector potential, given by this form may be interpreted as a generalized potential momentum per unit charge just as is interpreted as a generalized potential energy per unit charge. There are multiple choices one can make for the potential fields that satisfy the above condition. However, the choice of potentials is represented by its respective gauge condition.

Maxwell's equations when expressed in terms of the potentials in Lorenz gauge can be cast into a form that agrees with special relativity. In relativity, together with forms a four-potential regardless of the gauge condition, analogous to the four-momentum that combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler—and it can be easily modified to work with quantum mechanics.

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Propagation of Electric and Magnetic fields Special theory of relativity imposes the condition for events related by Causality to be time-like separated, that is that causal efficacy propagates no faster than light.

Naber, Gregory L. (2025). 9781441978370, Springer. ISBN 9781441978370

Maxwell's equations for electromagnetism are found to be in favor of this as electric and magnetic disturbances are found to travel at the speed of light in space. Electric and magnetic fields from classical electrodynamics obey the principle of locality in physics and are expressed in terms of retarded time or the time at which the cause of a measured field originated given that the influence of field travelled at speed of light. The retarded time for a point particle is given as solution of:

$$t\_r\; =\; \backslash mathbf\{t\}\; -\; \backslash frac\{\backslash left|\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\}\_s(t\_r)\backslash right|\}\{c\}$$

where $t\_r$ is retarded time or the time at which the source's contribution of the field originated, $r\_s(t)$ is the position vector of the particle as function of time, $\backslash mathbf\{r\}$ is the point in space, $\backslash mathbf\{t\}$ is the time at which fields are measured and $c$ is the speed of light. The equation subtracts the time taken for light to travel from particle to the point in space from the time of measurement to find time of origin of the fields. The uniqueness of solution for $t\_r$ for given $\backslash mathbf\{t\}$, $\backslash mathbf\{r\}$ and $r\_s(t)$ is valid for charged particles moving slower than speed of light.

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Magnetic field of arbitrary moving point charge The solution of maxwell's equations for electric and magnetic field of a point charge is expressed in terms of retarded time or the time at which the particle in the past causes the field at the point, given that the influence travels across space at the speed of light.

Any arbitrary motion of point charge causes electric and magnetic fields found by solving maxwell's equations using green's function for retarded potentials and hence finding the fields to be as follows:

   \frac{q c(\boldsymbol{\beta}_s \times \mathbf{n}_s)}{\gamma^2 {\left(1 - \mathbf{n}_s \cdot \boldsymbol{\beta}_s\right)}^3 {\left|\mathbf{r} - \mathbf{r}_s\right|}^2}
   + \frac{q \mathbf{n}_s \times \left(\mathbf{n}_s \times \left(\left(\mathbf{n}_s - \boldsymbol{\beta}_s\right) \times \dot{\boldsymbol{\beta}_s}\right) \right)}\hat\mathbf x
     

along the axis of the loop, where $I$ is the uniform current flowing through the loop. |- |Solenoid | |colspan="2" | $B=\backslash frac\{\backslash mu\_0nI\}2(cos\backslash theta\_1+\backslash cos\backslash theta\_2)$

along the axis of the solenoid carrying current $I$ with $n$, uniform number of loops of currents per length of solenoid; and the direction of magnetic field as shown. |- |Infinite solenoid | | $\backslash mathbf\; B=0$

outside the solenoid carrying current $I$ with $n$, uniform number of loops of currents per length of solenoid. | $B\; =\; \backslash mu\_0\; n\; I$

inside the solenoid carrying current $I$ with $n$, uniform number of loops of currents per length of solenoid, with the direction of magnetic field as shown. |- |Circular Toroid | | colspan="2" | $B\; =\; \backslash frac\{\backslash mu\_0NI\}\{2\backslash pi\; R\}$

along the bulk of the circular toroid carrying uniform current $I$ through $N$ number of uniformly distributed poloidal loops, with the direction of magnetic field as indicated. |- |Magnetic Dipole | | $\backslash mathbf\; B\; =\; -\backslash frac\{\backslash mu\_0\backslash mathbf\; m\}\{4\backslash pi\; r^3\},$

on the equatorial plane, where $\backslash mathbf\; m$ is the magnetic dipole moment. | $\backslash mathbf\; B\; =\; \backslash frac\{\backslash mu\_0\backslash mathbf\; m\}\{2\backslash pi$

^3},

on the axial plane (given that $x\; \backslash gg\; R$), where $x$ can also be negative to indicate position at the opposite direction on the axis, and $\backslash mathbf\; m$ is the magnetic dipole moment. |}

Additional magnetic field values can be found through the magnetic field of a finite beam, for example, that the magnetic field of an arc of angle $\backslash theta$ and radius $R$ at the center is $B=\{\backslash mu\_0\backslash theta\; I\backslash over\; 4\backslash pi\; R\}$, or that the magnetic field at the center of a N-sided regular polygon of side $a$ is $B=\; \{\backslash mu\_0NI\backslash over\backslash pi\; a\}\; \backslash sin\{\backslash pi\backslash over\; N\}\backslash tan\{\backslash pi\backslash over\; N\}$, both outside of the plane with proper directions as inferred by right hand thumb rule.

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History

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Early developments While magnets and some properties of magnetism were known to ancient societies, the research of magnetic fields began in 1269 when French scholar Petrus Peregrinus de Maricourt mapped out the magnetic field on the surface of a spherical magnet using iron needles. Noting the resulting field lines crossed at two points he named those points "poles" in analogy to Earth's poles. He also articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them.

(2025). 9781402039928, Springer. ISBN 9781402039928

Almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus' work and was the first to state explicitly that Earth is a magnet.

Published in 1600, Gilbert's work, De Magnete, helped to establish magnetism as a science.

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Mathematical development In 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law Charles-Augustin de Coulomb experimentally verified this in 1785 and stated explicitly that north and south poles cannot be separated. Building on this force between poles, Siméon Denis Poisson (1781–1840) created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic -field is produced by magnetic poles and magnetism is due to small pairs of north–south magnetic poles.

Three discoveries in 1820 challenged this foundation of magnetism. Hans Christian Ørsted demonstrated that a current-carrying wire is surrounded by a circular magnetic field. Then André-Marie Ampère showed that parallel wires with currents attract one another if the currents are in the same direction and repel if they are in opposite directions.

(2025). 9780191633720, OUP Oxford. ISBN 9780191633720

Finally, Jean-Baptiste Biot and Félix Savart announced empirical results about the forces that a current-carrying long, straight wire exerted on a small magnet, determining the forces were inversely proportional to the perpendicular distance from the wire to the magnet. Laplace later deduced a law of force based on the differential action of a differential section of the wire, which became known as the Biot–Savart law, as Laplace did not publish his findings.

Extending these experiments, Ampère published his own successful model of magnetism in 1825. In it, he showed the equivalence of electrical currents to magnets and proposed that magnetism is due to perpetually flowing loops of current instead of the dipoles of magnetic charge in Poisson's model. Further, Ampère derived both Ampère's force law describing the force between two currents and Ampère's law, which, like the Biot–Savart law, correctly described the magnetic field generated by a steady current. Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism.

In 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field, formulating what is now known as Faraday's law of induction. Later, Franz Ernst Neumann proved that, for a moving conductor in a magnetic field, induction is a consequence of Ampère's force law. In the process, he introduced the magnetic vector potential, which was later shown to be equivalent to the underlying mechanism proposed by Faraday.

In 1850, Lord Kelvin, then known as William Thomson, distinguished between two magnetic fields now denoted and . The former applied to Poisson's model and the latter to Ampère's model and induction. Further, he derived how and relate to each other and coined the term permeability. Lord Kelvin of Largs. physik.uni-augsburg.de. 26 June 1824

Between 1861 and 1865, James Clerk Maxwell developed and published Maxwell's equations, which explained and united all of classical theory electricity and magnetism. The first set of these equations was published in a paper entitled On Physical Lines of Force in 1861. These equations were valid but incomplete. Maxwell completed his set of equations in his later 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrated the fact that light is an electromagnetic wave. Heinrich Hertz published papers in 1887 and 1888 experimentally confirming this fact.Huurdeman, Anton A. (2003) The Worldwide History of Telecommunications. Wiley. . p. 202

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Modern developments

In 1887, Tesla developed an induction motor that ran on alternating current. The motor used Polyphase system current, which generated a rotating magnetic field to turn the motor (a principle that Tesla claimed to have conceived in 1882).

(1993). 9780801846144, JHU Press. . ISBN 9780801846144

Thomas Parke Hughes, Networks of Power: Electrification in Western Society, 1880–1930, pp. 115–118

(1998). 9780815315612, Taylor & Francis. . ISBN 9780815315612

Tesla received a patent for his electric motor in May 1888. In 1885, Galileo Ferraris independently researched rotating magnetic fields and subsequently published his research in a paper to the Royal Academy of Sciences in Turin, just two months before Tesla was awarded his patent, in March 1888.

The twentieth century showed that classical electrodynamics is already consistent with special relativity, and extended classical electrodynamics to work with quantum mechanics. Albert Einstein, in his paper of 1905 that established relativity, showed that both the electric and magnetic fields are part of the same phenomena viewed from different reference frames. Finally, the emergent field of quantum mechanics was merged with electrodynamics to form quantum electrodynamics, which first formalized the notion that electromagnetic field energy is quantized in the form of photons.

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See also

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General

• Magnetohydrodynamics – the study of the dynamics of electrically conducting fluids
• Magnetic hysteresis – application to ferromagnetism
• Magnetic nanoparticles – extremely small magnetic particles that are tens of atoms wide
• Magnetic reconnection – an effect that causes and auroras
• Magnetic scalar potential
• SI electromagnetism units – common units used in electromagnetism
• Orders of magnitude (magnetic field) – list of magnetic field sources and measurement devices from smallest magnetic fields to largest detected
• Upward continuation
• Moses Effect

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Mathematics

• Magnetic helicity – extent to which a magnetic field wraps around itself

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Applications

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

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Notes

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Further reading

• David Jiles (1994). 9780412495809, Springer. ISBN 9780412495809
• Paul Tipler (2025). 9780716708100, W. H. Freeman. ISBN 9780716708100

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External links

• Crowell, B., " Electromagnetism ".
• Nave, R., " Magnetic Field". HyperPhysics.
• " Magnetism", The Magnetic Field (archived 9 July 2006). theory.uwinnipeg.ca.
• Hoadley, Rick, " What do magnetic fields look like ?" 17 July 2005.

Categories: Magnetism, Electromagnetic Quantities

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