Magnetic Field Intensity

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a 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, and are created by electric currents such as those used in electromagnets, and by electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, called a vector 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 field strength, is the ampere per meter (A/m). The unit of , the magnetic flux density, is the tesla (in SI base units: kilogram per second² per ampere), which is equivalent to newton per meter per ampere. and differ in how they account for magnetization. In vacuum, the two fields are related through the vacuum permeability, $\mathbf/\mu_0 = \mathbf$; but in a magnetized material, the quantities on each side of this equation differ by the magnetization field of the material.

Magnetic fields are produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. Magnetic fields and electric fields 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 electric motors and generators. The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as magnetic circuits. Magnetic forces give information about the charge carriers in a material through the Hall effect. The Earth produces its own magnetic field, which shields the Earth's ozone layer from the solar wind and is important in navigation using a compass.

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

The B-field

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

Here is the force on the particle, is the particle's electric charge, , 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:
$\mathbf_ = q \mathbf.$

The second term is the magnetic force:
$\mathbf_ = q(\mathbf \times \mathbf).$

Using the definition of the cross product, the magnetic force can also be written as a scalar equation:
$F_ = q v B \sin(\theta)$

where , , and are the scalar magnitude of their respective vectors, and is the angle between the velocity of the particle and the magnetic field. The vector is ''defined'' as the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle. In other words,

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

The SI unit of is tesla (symbol: T).The SI unit of ( magnetic flux) 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-cgs unit of is the gauss (symbol: G). (The conversion is 1 T ≘ 10000 G.) One nanotesla corresponds to 1 gamma (symbol: γ).

The H-field

The magnetic field is defined:

Where $\mu_0$ is the vacuum permeability, and is the magnetization vector. In a vacuum, and are proportional to each other. Inside a material they are different (see H and B inside and outside magnetic materials). The SI unit of te -field is the ampere per metre (A/m), and the CGS unit is the oersted (Oe).

Measurement

An instrument used to measure the local magnetic field is known as a magnetometer. Important classes of magnetometers include using induction magnetometers (or search-coil magnetometers) which measure only varying magnetic fields, rotating coil magnetometers, Hall effect magnetometers, NMR magnetometers, SQUID magnetometers, and fluxgate magnetometers. The magnetic fields of distant astronomical 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 ().

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 flow, in that they represent a continuous distribution, and a different resolution would show more or fewer lines.

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

Magnetic field of permanent magnets

''Permanent magnets'' are objects that produce their own persistent magnetic fields. They are made of ferromagnetic materials, such as iron 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 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 also depend on the distance from the magnet and the orientation of the magnet. For simple magnets, points in the direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its by 180 degrees.

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

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

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

Magnetic pole model

Historically, early physics textbooks would model the force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as 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.

Amperian loop model

In the 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 where is the area of the loop.

These magnetic dipoles produce a magnetic -field.

The magnetic field of a magnetic dipole is depicted in the figure. From outside, the ideal magnetic dipole is identical to that of an ideal electric dipole of the same strength. Unlike the electric dipole, a magnetic dipole is properly modeled as a current loop having a current and an area . Such a current loop has a magnetic moment of:

$m = Ia,$
where the direction of is perpendicular to the area of the loop and depends on the direction of the current using the right-hand rule. An ideal magnetic dipole is modeled as a real magnetic dipole whose area has been reduced to zero and its current increased to infinity such that the product is finite. This model clarifies the connection between angular momentum and magnetic moment, which is the basis of the Einstein–de Haas effect ''rotation by magnetization'' and its inverse, the Barnett effect or ''magnetization by rotation''.See magnetic moment and Rotating the loop faster (in the same direction) increases the current and therefore the magnetic moment, for example.

Interactions with magnets

Force between magnets

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

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

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

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

$\mathbf = \boldsymbol \left(\mathbf\cdot\mathbf\right),$

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 summing up the forces on each of these very small regions.

Magnetic torque on permanent magnets

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

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

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

$\boldsymbol=\mathbf\times\mathbf = \mu_0\mathbf\times\mathbf, \,$

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.

Interactions with electric currents

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

Magnetic field due to moving charges and electric currents

All moving charged particles produce magnetic fields. Moving point charges, such as electrons, 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 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'':
$\mathbf = \frac\int_\frac,$
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:
$, \mathbf, = \fracI$
where . The direction is tangent to a circle perpendicular to the wire according to the right hand rule.

A slightly more general
The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of being zero, which is always valid. (There are no magnetic charges.) way of relating the current $I$ to the -field is through Ampère's law:
$\oint \mathbf \cdot \mathrm\boldsymbol = \mu_0 I_,$
where the line integral is over any arbitrary loop and $I_\text$ is the current enclosed by that loop. Ampère's law is always valid for steady currents and can be used to calculate the -field for certain highly symmetric situations such as an infinite wire or an infinite solenoid.

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

Force on moving charges and current

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
$\mathbf = q\mathbf + q \mathbf \times \mathbf,$
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 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.

Force on current-carrying wire

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

Relation between H and B

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

Magnetization

The ''magnetization'' vector field represents how strongly a region of material is magnetized. It is defined as the 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:
$\oint \mathbf \cdot \mathrm\boldsymbol = I_\mathrm,$
where the integral is a line integral over any closed loop and is the bound current enclosed by that closed loop.

In the magnetic pole model, magnetization begins at and ends at magnetic poles. If a given region, therefore, has a net positive "magnetic pole strength" (corresponding to a north pole) then it has more magnetization field lines entering it than leaving it. Mathematically this is equivalent to:
$\oint_S \mu_0 \mathbf \cdot \mathrm\mathbf = - q_\mathrm,$
where the integral is a closed surface integral over the closed surface and is the "magnetic charge" (in units of magnetic flux) enclosed by . (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because the magnetization field moves from south to north.

H-field and magnetic materials

In SI units, the H-field is related to the B-field by
$\mathbf\ \equiv \ \frac - \mathbf.$

In terms of the H-field, Ampere's law is
$\oint \mathbf \cdot \mathrm\boldsymbol = \oint \left(\frac - \mathbf\right) \cdot \mathrm\boldsymbol = I_\mathrm - I_\mathrm = I_\mathrm,$
where represents the 'free current' enclosed by the loop so that the line integral of does not depend at all on the bound currents.

For the differential equivalent of this equation see Maxwell's equations. Ampere's law leads to the boundary condition
$\left(\mathbf - \mathbf\right) = \mathbf_\mathrm \times \hat,$
where is the surface free current density and the unit normal $\hat$ points in the direction from medium 2 to medium 1.

Similarly, a surface integral of over any closed surface is independent of the free currents and picks out the "magnetic charges" within that closed surface:
$\oint_S \mu_0 \mathbf \cdot \mathrm\mathbf = \oint_S (\mathbf - \mu_0 \mathbf) \cdot \mathrm\mathbf = 0 - (-q_\mathrm) = q_\mathrm,$

which does not depend on the free currents.

The -field, therefore, can be separated into twoA third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below. independent parts:
$\mathbf = \mathbf_0 + \mathbf_\mathrm,$

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

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

Magnetism

Most materials respond to an applied -field by producing their own magnetization and therefore their own -fields. Typically, the response is weak and exists only when the magnetic field is applied. The term ''magnetism'' describes how materials respond on the microscopic level to an applied magnetic field and is used to categorize the magnetic phase of a material. Materials are divided into groups based upon their magnetic behavior:
* Diamagnetic materials produce a magnetization that opposes the magnetic field.
* Paramagnetic materials produce a magnetization in the same direction as the applied magnetic field.
* Ferromagnetic materials and the closely related ferrimagnetic materials and antiferromagnetic materials can have a magnetization independent of an applied B-field with a complex relationship between the two fields.
* Superconductors (and ferromagnetic superconductors) are materials that are characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures and magnetic fields (the so-named mixed state) under which they exhibit a complex hysteretic dependence of on .

In the case of paramagnetism and diamagnetism, the magnetization is often proportional to the applied magnetic field such that:
$\mathbf = \mu \mathbf,$
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.

Stored energy

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

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

$u = \frac= \frac = \frac.$

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

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

$\delta W = \mathbf\cdot\delta\mathbf.$

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

Appearance in Maxwell's equations

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

The first property is 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.

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:
$\oint_S \mathbf \cdot \mathrm\mathbf = 0$
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.

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 electric motors. Mathematically, Faraday's law is:
$\mathcal = - \frac$

where $\mathcal$ is the electromotive force (or ''EMF'', the voltage generated around a closed loop) and is the magnetic flux—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:
$\mathcal = - \frac = \oint_ \left( \mathbf( \mathbf,\ t) +\mathbf \times \mathbf(\mathbf,\ t)\right) \cdot d\boldsymbol\ =-\frac \iint_ d \boldsymbol \cdot \mathbf (\mathbf,\ t)$
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 s