Magnetic field

Amagnetic field(sometimes calledB-field^[1]) is aphysical fieldthat describes the magnetic influence on movingelectric charges,electric currents,^[2]^: ch1^[3]and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field.^[2]^: ch13^[4]^: 278Apermanent magnet's magnetic field pulls onferromagnetic materialssuch asiron,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,andantiferromagnetism,although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, electric currents, andelectric fieldsvarying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by afunctionassigning avectorto each point of space, called avector field(more precisely, apseudovectorfield).

Apermanent magnet,a piece of magnetized metal alloy

Asolenoid(electromagnet), a coil of wire with an electric current through it

The shape of the magnetic fields of a permanent magnet and an electromagnet are revealed by the orientation of iron filings sprinkled on pieces of paper

Inelectromagnetics,the termmagnetic fieldis used for two distinct but closely related vector fields denoted by the symbols $B$ and $H$ .In theInternational System of Units,the unit of $B$ ,magnetic fluxdensity, is thetesla(in SI base units: kilogram per second²per ampere),^[5]^: 21which is equivalent tonewtonper meter per ampere. The unit of $H$ ,magnetic field strength, isampereper meter (A/m).^[5]^: 22 $B$ and $H$ differ in how they take the medium and/or magnetization into account. Invacuum,the two fields are related through thevacuum permeability, $\mathbf {B} /\mu _{0}=\mathbf {H}$ ;in a magnetized material, the quantities on each side of this equation differ by themagnetizationfield of the material.

Magnetic fields are produced by moving electric charges and the intrinsicmagnetic momentsofelementary particlesassociated with a fundamental quantum property, theirspin.^[6]^[2]^: ch1Magnetic fields andelectric fieldsare interrelated and are both components of theelectromagnetic force,one of the fourfundamental forcesof nature.

Magnetic fields are used throughout modern technology, particularly inelectrical engineeringandelectromechanics.Rotating magnetic fields are used in bothelectric motorsandgenerators.The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated asmagnetic circuits.Magnetic forces give information about the charge carriers in a material through theHall effect.The Earth producesits own magnetic field,which shields the Earth's ozone layer from thesolar windand is important innavigationusing acompass.

Description

The force on an electric charge depends on its location, speed, and direction; two vector fields are used to describe this force.^[2]^: ch1The first is theelectric 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.^[2]^: ch13The field is defined by theLorentz force lawand 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 $B$ and $H$ .^{[note 1]}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.^[7]Historically, the term "magnetic field" was reserved for $H$ while using other terms for $B$ ,but many recent textbooks use the term "magnetic field" to describe $B$ as well as or in place of $H$ .^{[note 2]} There are many alternative names for both (see sidebars).

The B-field

Finding the magnetic force

A charged particle that is moving with velocityvin a magnetic fieldBwill feel a magnetic forceF.Since the magnetic force always pulls sideways to the direction of motion, the particle moves in a circle.

Since these three vectors are related to each other by across product,the direction of this force can be found using theright hand rule.

Alternative names forB^[8]
Magnetic flux density^[5]^: 138 Magnetic induction^[9] Magnetic field (ambiguous)

The magnetic field vector $B$ at any point can be defined as the vector that, when used in theLorentz force law,correctly predicts the force on a charged particle at that point:^[10]^[11]^: 204

Lorentz force law(vectorform,SI units)

$\mathbf {F} =q\mathbf {E} +q(\mathbf {v} \times \mathbf {B} )$

Here $F$ is the force on the particle, $q$ is the particle'selectric charge, $v$ ,is the particle'svelocity,and × denotes thecross product.The direction of force on the charge can be determined by amnemonicknown as theright-hand rule(see the figure).^{[note 3]}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 fieldcandistinguish between these, seeHall effectbelow.

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

The second term is the magnetic force:^[11] $\mathbf {F} _{\text{magnetic}}=q(\mathbf {v} \times \mathbf {B} ).$

Using the definition of the cross product, the magnetic force can also be written as ascalarequation:^[10]^: 357 $F_{\text{magnetic}}=qvB\sin(\theta )$ where $F magnetic$ , $v$ ,and $B$ are thescalar magnitudeof their respective vectors, and $θ$ is the angle between the velocity of the particle and the magnetic field. The vector $B$ isdefinedas the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle. In other words,^[10]^: 173–4

[T]he command, "Measure the direction and magnitude of the vector $B$ at such and such a place, "calls for the following operations: Take a particle of known charge $q$ .Measure the force on $q$ at rest, to determine $E$ .Then measure the force on the particle when its velocity is $v$ ;repeat with $v$ in some other direction. Now find a $B$ that makes the Lorentz force law fit all these results—that is the magnetic field at the place in question.

The $B$ field can also be defined by the torque on a magnetic dipole, $m$ .^[12]^: 174

Magnetic torque(vectorform,SI units)

${\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B}$

TheSIunit of $B$ istesla(symbol: T).^{[note 4]}TheGaussian-cgs unitof $B$ is thegauss(symbol: G). (The conversion is 1 T ≘ 10000 G.^[13]^[14]) One nanotesla corresponds to 1 gamma (symbol: γ).^[14]

The H-field

Alternative names forH^[8]
Magnetic field intensity^[9] Magnetic field strength^[5]^: 139 Magnetic field Magnetizing field Auxiliary magnetic field

The magnetic $H$ field is defined:^[11]^: 269^[12]^: 192^[2]^: ch36

Definition of the $H$ field(vectorform,SI units)

$\mathbf {H} \equiv {\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M}$

where $\mu _{0}$ is thevacuum permeability,and $M$ is themagnetization vector.In a vacuum, $B$ and $H$ are proportional to each other. Inside a material they are different (seeH and B inside and outside magnetic materials). The SI unit of the $H$ -field is theampereper metre (A/m),^[15]and the CGS unit is theoersted(Oe).^[13]^[10]^: 286

Measurement

An instrument used to measure the local magnetic field is known as amagnetometer.Important classes of magnetometers include usinginduction magnetometers(or search-coil magnetometers) which measure only varying magnetic fields,rotating coil magnetometers,Hall effectmagnetometers,NMR magnetometers,SQUID magnetometers,andfluxgate magnetometers.The magnetic fields of distantastronomical objectsare measured through their effects on local charged particles. For instance, electrons spiraling around a field line producesynchrotron radiationthat is detectable inradio waves.The finest precision for a magnetic field measurement was attained byGravity Probe Bat5 aT(5×10⁻¹⁸T).^[16]

Visualization

Visualizing magnetic fields

Left: the direction of magneticfield linesrepresented byiron filingssprinkled on paper placed above a bar magnet.
Right:compassneedles point in the direction of the local magnetic field, towards a magnet's south pole and away from its north pole.

The field can be visualized by a set ofmagnetic field lines,that follow the direction of the field at each point. The lines can be constructed by measuring the strength and direction of the magnetic field at a large number of points (or at every point in space). Then, mark each location with an arrow (called avector) 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 likestreamlinesinfluid 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 thesurface integralof the magnetic field.^[10]^: 237

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".^{[note 5]}Magnetic field "lines" are also visually displayed inpolar auroras,in whichplasmaparticle 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. Inferromagneticsubstances likeironand in plasmas, magnetic forces can be understood by imagining that the field lines exert atension,(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 magnetsare objects that produce their own persistent magnetic fields. They are made offerromagneticmaterials, such as iron andnickel,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 small^{[note 6]}straight magnet is proportional to the magnet'sstrength(called itsmagnetic dipole moment $m$ ). Theequationsare non-trivial and depend on the distance from the magnet and the orientation of the magnet. For simple magnets, $m$ points in the direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its $m$ by 180 degrees.

The magnetic field of larger magnets can be obtained by modeling them as a collection of a large number of small magnets calleddipoleseach having their own $m$ .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: themagnetic pole modeland theAmperian loop model.These two models produce two different magnetic fields, $H$ and $B$ .Outside a material, though, the two are identical (to a multiplicative constant) so that in many cases the distinction can be ignored. This is particularly true for magnetic fields, such as those due to electric currents, that are not generated by magnetic materials.

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 thespin magnetic momentof electrons (which is not explained by either model) is also a significant contribution to the total moment of magnets.

Magnetic pole model

The magnetic pole model: two opposing poles, North (+) and South (−), separated by a distance d produce a

H

-field (lines).

Historically, early physics textbooks would model the force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as theCoulomb forcebetween 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.^[11]^: 258However, it is still sometimes used as a macroscopic model for ferromagnetism due to its mathematical simplicity.^[17]

In this model, a magnetic $H$ -field is produced by fictitiousmagnetic chargesthat are spread over the surface of each pole. Thesemagnetic chargesare in fact related to the magnetization field $M$ .The $H$ -field, therefore, is analogous to theelectric field $E$ ,which starts at a positiveelectric chargeand ends at a negative electric charge. Near the north pole, therefore, all $H$ -field lines point away from the north pole (whether inside the magnet or out) while near the south pole all $H$ -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 $H$ -field while the force on the south pole is opposite to the $H$ -field.

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

Since it is based on the fictitious idea of amagnetic 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 betweenangular momentumand magnetism.

The pole model usually treats magnetic charge as a mathematical abstraction, rather than a physical property of particles. However, amagnetic monopoleis 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 asGrand Unified Theories) have predicted the existence of magnetic monopoles, but so far, none have been observed.

Amperian loop model

The Amperian loop model

A current loop (ring) that goes into the page at the x and comes out at the dot produces a

B

-field (lines). As the radius of the current loop shrinks, the fields produced become identical to an abstract "magnetostatic dipole" (represented by an arrow pointing to the right).

In the model developed byAmpere,the elementary magnetic dipole that makes up all magnets is a sufficiently small Amperian loop with current $I$ and loop area $A$ .The dipole moment of this loop is $m = IA$ .

These magnetic dipoles produce a magnetic $B$ -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 $I$ and an area $a$ .Such a current loop has a magnetic moment of $m=Ia,$ where the direction of $m$ is perpendicular to the area of the loop and depends on the direction of the current using the right-hand rule. An ideal magnetic dipole is modeled as a real magnetic dipole whose area $a$ has been reduced to zero and its current $I$ increased to infinity such that the product $m = Ia$ is finite. This model clarifies the connection between angular momentum and magnetic moment, which is the basis of theEinstein–de Haas effectrotation by magnetizationand its inverse, theBarnett effectormagnetization by rotation.^[18]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 theforce between two small magnetsis quite complicated because it depends on the strength andorientationof 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 field^{[note 7]}of the other.

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

This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field, whether caused by permanent magnets or 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 $m$ due to a magnetic field $B$ is:^[19]^{: Eq. 11.42}

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

where thegradient $\nabla$ is the change of the quantity $m \cdot B$ per unit distance and the direction is that of maximum increase of $m \cdot B$ .Thedot product $m \cdot B = mB cos(θ)$ ,where $m$ and $B$ represent themagnitudeof the $m$ and $B$ vectors and $θ$ is the angle between them. If $m$ is in the same direction as $B$ then the dot product is positive and the gradient points "uphill" pulling the magnet into regions of higher $B$ -field (more strictly larger $m \cdot B$ ). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions each having their own $m$ thensumming 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 amagnetic torqueon the magnet that is free to rotate. This magnetic torque $τ$ tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field.

Torque on a dipole

In the pole model of a dipole, an

H

field (to right) causes equal but opposite forces on a N pole (

+ q

) and a S pole (

- q

) creating a torque.

Equivalently, a

B

field induces the same torque on a current loop with the same magnetic dipole moment.

In terms of the pole model, two equal and opposite magnetic charges experiencing the same $H$ 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 $m$ as the pole strength times the distance between the poles, this leads to $τ = μ 0 m H sin θ$ ,where $μ 0$ is a constant called thevacuum permeability,measuring4π×10⁻⁷V·s/(A·m) and $θ$ is the angle between $H$ and $m$ .

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

${\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} =\mu _{0}\mathbf {m} \times \mathbf {H},\,$

where × represents the vectorcross product.This equation includes all of the qualitative information included above. There is no torque on a magnet if $m$ is in the same direction as the magnetic field, 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

Right hand grip rule:a current flowing in the direction of the white arrow produces a magnetic field shown by the red arrows.

All moving charged particles produce magnetic fields. Movingpointcharges, such aselectrons,produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.^[20]

Magnetic field lines form inconcentriccircles around acylindricalcurrent-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.)

ASolenoidwith electric current running through it behaves like a magnet.

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 ironcoremay act as anelectromagnet,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 $I$ (a constant flow of electric charges, in which charge neither accumulates nor is depleted at any point)^{[note 8]}is described by theBiot–Savart law:^[21]^: 224 $\mathbf {B} ={\frac {\mu _{0}I}{4\pi }}\int _{\mathrm {wire} }{\frac {\mathrm {d} {\boldsymbol {\ell }}\times \mathbf {\hat {r}} }{r^{2}}},$ where the integral sums over the wire length where vector $d ℓ$ is the vectorline elementwith direction in the same sense as the current $I$ , $μ 0$ is themagnetic constant, $r$ is the distance between the location of $d ℓ$ and the location where the magnetic field is calculated, and $r̂$ is a unit vector in the direction of $r$ .For example, in the case of a sufficiently long, straight wire, this becomes: $|\mathbf {B} |={\frac {\mu _{0}}{2\pi r}}I$ where $r = | r |$ .The direction is tangent to a circle perpendicular to the wire according to the right hand rule.^[21]^: 225

A slightly more general^[22]^{[note 9]}way of relating the current $I$ to the $B$ -field is throughAmpère's law: $\oint \mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}I_{\mathrm {enc} },$ where theline integralis over any arbitrary loop and $I_{\text{enc}}$ is the current enclosed by that loop. Ampère's law is always valid for steady currents and can be used to calculate the $B$ -field for certain highly symmetric situations such as an infinite wire or an infinite solenoid.

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

Force on moving charges and current

Force on a charged particle

Acharged particlemoving in a $B$ -field experiences asidewaysforce 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 theLorentz force,and is given by $\mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B},$ where $F$ is theforce, $q$ is theelectric chargeof the particle, $v$ is the instantaneousvelocityof the particle, and $B$ is the magnetic field (inteslas).

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 noworkon an isolated charge.^[23]^[24]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-elementarymagnetic dipole,or to charged particles whose motion is constrained by other forces, but this is incorrect^[25]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 theLaplace force. Consider a conductor of length $ℓ$ ,cross section $A$ ,and charge $q$ due to electric current $i$ .If this conductor is placed in a magnetic field of magnitude $B$ that makes an angle $θ$ with the velocity of charges in the conductor, the force exerted on a single charge $q$ is $F=qvB\sin \theta,$ so, for $N$ charges where $N=n\ell A,$ the force exerted on the conductor is $f=FN=qvBn\ell A\sin \theta =Bi\ell \sin \theta,$ where $i = nqvA$ .

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 ownbound current,which can be a challenge to calculate. (This bound current is due to the sum of atomic sized current loops and thespinof the subatomic particles such as electrons that make up the material.) The $H$ -field as defined above helps factor out this bound current; but to see how, it helps to introduce the concept ofmagnetizationfirst.

Magnetization

Themagnetizationvector field $M$ represents how strongly a region of material is magnetized. It is defined as the netmagnetic dipole momentper unit volume of that region. The magnetization of a uniform magnet is therefore a material constant, equal to the magnetic moment $m$ of the magnet divided by its volume. Since the SI unit of magnetic moment is A⋅m²,the SI unit of magnetization $M$ is ampere per meter, identical to that of the $H$ -field.

The magnetization $M$ 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 calledbound current.This bound current, then, is the source of the magnetic $B$ 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:^[26] $\oint \mathbf {M} \cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {b} },$ where the integral is a line integral over any closed loop and $I b$ 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: $\oint _{S}\mu _{0}\mathbf {M} \cdot \mathrm {d} \mathbf {A} =-q_{\mathrm {M} },$ where the integral is a closed surface integral over the closed surface $S$ and $q M$ is the "magnetic charge" (in units ofmagnetic flux) enclosed by $S$ .(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

Comparison of

B

,

H

and

M

inside and outside a cylindrical bar magnet.

In SI units, the H-field is related to the B-field by $\mathbf {H} \ \equiv \ {\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M}.$

In terms of the H-field, Ampere's law is $\oint \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=\oint \left({\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} \right)\cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {tot} }-I_{\mathrm {b} }=I_{\mathrm {f} },$ where $I f$ represents the 'free current' enclosed by the loop so that the line integral of $H$ does not depend at all on the bound currents.^[27]

For the differential equivalent of this equation seeMaxwell's equations.Ampere's law leads to the boundary condition $\left(\mathbf {H_{1}^{\parallel }} -\mathbf {H_{2}^{\parallel }} \right)=\mathbf {K} _{\mathrm {f} }\times {\hat {\mathbf {n} }},$ where $K f$ is the surface free current density and the unit normal ${\hat {\mathbf {n} }}$ points in the direction from medium 2 to medium 1.^[28]

Similarly, asurface integralof $H$ over anyclosed surfaceis independent of the free currents and picks out the "magnetic charges" within that closed surface: $\oint _{S}\mu _{0}\mathbf {H} \cdot \mathrm {d} \mathbf {A} =\oint _{S}(\mathbf {B} -\mu _{0}\mathbf {M} )\cdot \mathrm {d} \mathbf {A} =0-(-q_{\mathrm {M} })=q_{\mathrm {M} },$

which does not depend on the free currents.

The $H$ -field, therefore, can be separated into two^{[note 10]}independent parts: $\mathbf {H} =\mathbf {H} _{0}+\mathbf {H} _{\mathrm {d} },$

where $H 0$ is the applied magnetic field due only to the free currents and $H d$ is thedemagnetizing fielddue only to the bound currents.

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

Magnetism

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

Diamagnetic materials^[29]produce a magnetization that opposes the magnetic field.
Paramagnetic materials^[29]produce a magnetization in the same direction as the applied magnetic field.
Ferromagnetic materialsand the closely relatedferrimagnetic materialsandantiferromagnetic materials^[30]^[31]can have a magnetization independent of an applied B-field with a complex relationship between the two fields.
Superconductors(andferromagnetic superconductors)^[32]^[33]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-namedmixed state) under which they exhibit a complex hysteretic dependence of $M$ on $B$ .

In the case of paramagnetism and diamagnetism, the magnetization $M$ is often proportional to the applied magnetic field such that: $\mathbf {B} =\mu \mathbf {H},$ where $μ$ is a material dependent parameter called thepermeability.In some cases the permeability may be a second ranktensorso that $H$ may not point in the same direction as $B$ .These relations between $B$ and $H$ are examples ofconstitutive equations.However, superconductors and ferromagnets have a more complex $B$ -to- $H$ relation; seemagnetic 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 $B = μ H$ where $μ$ is frequency-independent), theenergy densityis:

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

If there are no magnetic materials around then $μ$ can be replaced by $μ 0$ .The above equation cannot be used for nonlinear materials, though; a more general expression given below must be used.

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

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

Once the relationship between $H$ and $B$ is known this equation is used to determine the work needed to reach a given magnetic state. Forhysteretic materialssuch 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 itssources.(For $B$ thesourcesare currents and changing electric fields.) These two properties, along with the two corresponding properties of the electric field, make upMaxwell's Equations.Maxwell's Equations together with the Lorentz force law form a complete description ofclassical electrodynamicsincluding both electricity and magnetism.

The first property is thedivergenceof a vector field $A$ , $\nabla \cdot A$ ,which represents how $A$ "flows" outward from a given point. As discussed above, a $B$ -field line never starts or ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of $B$ is zero. (Such vector fields are calledsolenoidal vector fields.) This property is calledGauss's law for magnetismand is equivalent to the statement that there are no isolated magnetic poles ormagnetic monopoles.

The second mathematical property is called thecurl,such that $\nabla \times A$ represents how $A$ curls or "circulates" around a given point. The result of the curl is called a "circulation source". The equations for the curl of $B$ and of $E$ are called theAmpère–Maxwell equationandFaraday's lawrespectively.

Gauss' law for magnetism

One important property of the $B$ -field produced this way is that magnetic $B$ -field lines neither start nor end (mathematically, $B$ is asolenoidal 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.^[34]Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet $B$ -field lines continue through the magnet from the south pole back to the north.^{[note 11]}If a $B$ -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"^{[note 12]}of field lines that enter the region from the number that exit gives identically zero. Mathematically this is equivalent toGauss's law for magnetism: $\oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$ where the integral is asurface integralover theclosed surface $S$ (a closed surface is one that completely surrounds a region with no holes to let any field lines escape). Since $d A$ points outward, the dot product in the integral is positive for $B$ -field pointing out and negative for $B$ -field pointing in.

Faraday's Law

A changing magnetic field, such as a magnet moving through a conducting coil, generates anelectric field(and therefore tends to drive a current in such a coil). This is known asFaraday's lawand forms the basis of manyelectrical generatorsandelectric motors.Mathematically, Faraday's law is: ${\mathcal {E}}=-{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}$

where ${\mathcal {E}}$ is theelectromotive force(orEMF,thevoltagegenerated around a closed loop) and $Φ$ is themagnetic flux—the product of the area times the magnetic fieldnormalto that area. (This definition of magnetic flux is why $B$ is often referred to asmagnetic flux density.)^[35]^: 210The negative sign represents the fact that any current generated by a changing magnetic field in a coil produces a magnetic field thatopposesthechangein the magnetic field that induced it. This phenomenon is known asLenz's law.This integral formulation of Faraday's law can be converted^{[note 13]}into a differential form, which applies under slightly different conditions.

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

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 asMaxwell's correction to Ampère's lawand 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 termiscritically important in the creation and propagation of electromagnetic waves. Maxwell's correction to Ampère's Law together with Faraday's law of induction describes how mutually changing electric and magnetic fields interact to sustain each other and thus to formelectromagnetic waves,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 $J$ is the complete microscopiccurrent density,and $ε 0$ is thevacuum permittivity.

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

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

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 $B$ and $H$ as well as that between $E$ and $D$ .On the other hand, for simple relationships between these quantities this form of Maxwell's equations can circumvent the need to calculate the bound charges and currents.

Formulation in special relativity and quantum electrodynamics

Relativistic electrodynamics

As different aspects of the same phenomenon

According tothe special theory of relativity,the partition of theelectromagnetic forceinto separate electric and magnetic components is not fundamental, but varies with theobservational 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 fromLorentz transformationof four force fromCoulomb's Lawin particle's rest frame withMaxwell's lawsconsidering definition of fields fromLorentz forceand for non accelerating condition. The form of magnetic field hence obtained byLorentz transformationoffour-forcefrom the form ofCoulomb's lawin source's initial frame is given by:^[36] $\mathbf {B} ={\frac {q}{4\pi \varepsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}{\frac {\mathbf {v} \times \mathbf {r} }{c^{2}}}={\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}$ where $q$ is the charge of the point source, $\varepsilon _{0}$ is thevacuum permittivity, $\mathbf {r}$ is the position vector from the point source to the point in space, $\mathbf {v}$ is the velocity vector of the charged particle, $\beta$ is the ratio of speed of the charged particle divided by the speed of light and $\theta$ is the angle between $\mathbf {r}$ and $\mathbf {v}$ .This form of magnetic field can be shown to satisfy maxwell's laws within the constraint of particle being non accelerating.^[37]The above reduces toBiot-Savart lawfor non relativistic stream of current ( $\beta \ll 1$ ).

Formally, special relativity combines the electric and magnetic fields into a rank-2tensor,called theelectromagnetic tensor.Changing reference framesmixesthese components. This is analogous to the way that special relativitymixesspace and time intospacetime,and mass, momentum, and energy intofour-momentum.^[38]Similarly, theenergy stored in a magnetic fieldis mixed with the energy stored in an electric field in theelectromagnetic stress–energy tensor.

Magnetic vector potential

In advanced topics such asquantum mechanicsandrelativityit is often easier to work with a potential formulation of electrodynamics rather than in terms of the electric and magnetic fields. In this representation, themagnetic vector potential $A$ ,and theelectric scalar potential $φ$ ,are defined usinggauge fixingsuch that: ${\begin{aligned}\mathbf {B} &=\nabla \times \mathbf {A},\\\mathbf {E} &=-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}.\end{aligned}}$

The vector potential, $A$ given by this form may be interpreted as ageneralized potentialmomentumper unit charge^[39]just as $φ$ is interpreted as ageneralizedpotential energyper 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 inLorenz gaugecan be cast into a form that agrees withspecial relativity.^[40]In relativity, $A$ together with $φ$ forms afour-potentialregardless of the gauge condition, analogous to thefour-momentumthat combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler—and it can be easily modified to work with quantum mechanics.

Propagation of Electric and Magnetic fields

Special theory of relativityimposes the condition for events related bycause and effectto be time-like separated, that is that causal efficacy propagates no faster than light.^[41]Maxwell's equationsfor 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 theprinciple of localityin 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:

$t_{r}=\mathbf {t} -{\frac {|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}{c}}$

where ${\textstyle {t_{r}}}$ isretarded timeor the time at which the source's contribution of the field originated, ${\textstyle {r}_{s}(t)}$ is the position vector of the particle as function of time, ${\textstyle \mathbf {r} }$ is the point in space, ${\textstyle \mathbf {t} }$ is the time at which fields are measured and ${\textstyle c}$ 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 ${\textstyle {t_{r}}}$ for given $\mathbf {t}$ , $\mathbf {r}$ and $r_{s}(t)$ is valid for charged particles moving slower than speed of light.^[42]

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 ofretarded timeor 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:

$\mathbf {A} (\mathbf {r},\mathbf {t} )={\frac {\mu _{0}c}{4\pi }}\left({\frac {q{\boldsymbol {\beta }}_{s}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}={\frac {{\boldsymbol {\beta }}_{s}(t_{r})}{c}}\varphi (\mathbf {r},\mathbf {t} )$

$\mathbf {B} (\mathbf {r},\mathbf {t} )={\frac {\mu _{0}}{4\pi }}\left({\frac {qc({\boldsymbol {\beta }}_{s}\times \mathbf {n} _{s})}{\gamma ^{2}(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|^{2}}}+{\frac {q\mathbf {n} _{s}\times {\Big (}\mathbf {n} _{s}\times {\big (}(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})\times {\dot {{\boldsymbol {\beta }}_{s}}}{\big )}{\Big )}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}={\frac {\mathbf {n} _{s}(t_{r})}{c}}\times \mathbf {E} (\mathbf {r},\mathbf {t} )$

where ${\textstyle \varphi (\mathbf {r},\mathbf {t} )}$ and ${\textstyle \mathbf {A} (\mathbf {r},\mathbf {t} )}$ are electric scalar potential and magnetic vector potential in Lorentz gauge, $q$ is the charge of the point source, ${\textstyle {n}_{s}(\mathbf {r},t)}$ is a unit vector pointing from charged particle to the point in space, ${\textstyle {\boldsymbol {\beta }}_{s}(t)}$ is the velocity of the particle divided by the speed of light and ${\textstyle \gamma (t)}$ is the correspondingLorentz factor.Hence by theprinciple of superposition,the fields of a system of charges also obeyprinciple of locality.

Quantum electrodynamics

The classical electromagnetic field incorporated into quantum mechanics forms what is known as the semi-classical theory of radiation. However, it is not able to make experimentally observed predictions such asspontaneous emission processorLamb shiftimplying the need for quantization of fields. In modern physics, the electromagnetic field is understood to be not aclassicalfield,but rather aquantum field;it is represented not as a vector of threenumbersat each point, but as a vector of threequantum operatorsat each point. The most accurate modern description of the electromagnetic interaction (and much else) isquantum electrodynamics(QED),^[43]which is incorporated into a more complete theory known as theStandard Model of particle physics.

In QED, the magnitude of the electromagnetic interactions between charged particles (and theirantiparticles) is computed usingperturbation theory.These rather complex formulas produce a remarkable pictorial representation asFeynman diagramsin whichvirtual photonsare exchanged.

Predictions of QED agree with experiments to an extremely high degree of accuracy: currently about 10⁻¹²(and limited by experimental errors); for details seeprecision tests of QED.This makes QED one of the most accurate physical theories constructed thus far.

All equations in this article are in theclassical approximation,which is less accurate than the quantum description mentioned here. However, under most everyday circumstances, the difference between the two theories is negligible.

Uses and examples

Earth's magnetic field

A sketch of Earth's magnetic field representing the source of the field as a magnet. The south pole of the magnetic field is near the geographic north pole of the Earth.

The Earth's magnetic field is produced byconvectionof a liquid iron alloy in theouter core.In adynamo process,the movements drive a feedback process in which electric currents create electric and magnetic fields that in turn act on the currents.^[44]

The field at the surface of the Earth is approximately the same as if a giant bar magnet were positioned at the center of the Earth and tilted at an angle of about 11° off the rotational axis of the Earth (see the figure).^[45]The north pole of a magnetic compass needle points roughly north, toward theNorth Magnetic Pole.However, because a magnetic pole is attracted to its opposite, the North Magnetic Pole is actually the south pole of the geomagnetic field. This confusion in terminology arises because the pole of a magnet is defined by the geographical direction it points.^[46]

Earth's magnetic field is not constant—the strength of the field and the location of its poles vary.^[47]Moreover, the poles periodically reverse their orientation in a process calledgeomagnetic reversal.Themost recent reversaloccurred 780,000 years ago.^[48]

Rotating magnetic fields

Therotating magnetic fieldis a key principle in the operation ofalternating-current motors.A permanent magnet in such a field rotates so as to maintain its alignment with the external field.

Magnetic torque is used to driveelectric motors.In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array ofelectromagnets.By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft.

A rotating magnetic field can be constructed using two orthogonal coils with 90 degrees phase difference in their AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents.

This inequality would cause serious problems in standardization of the conductor size and so, to overcome it,three-phasesystems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 degrees create the rotating magnetic field in this case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world'selectrical powersupply systems.

Synchronous motorsuse DC-voltage-fed rotor windings, which lets the excitation of the machine be controlled—andinduction motorsuse short-circuitedrotors(instead of a magnet) following the rotating magnetic field of a multicoiledstator.The short-circuited turns of the rotor developeddy currentsin the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force.

The Italian physicistGalileo Ferrarisand the Serbian-Americanelectrical engineer Nikola Teslaindependently researched the use of rotating magnetic fields in electric motors. In 1888, Ferraris published his research in a paper to theRoyal Academy of SciencesinTurinand Tesla gainedU.S. patent 381,968for his work.

Hall effect

The charge carriers of a current-carrying conductor placed in a transverse magnetic field experience a sideways Lorentz force; this results in a charge separation in a direction perpendicular to the current and to the magnetic field. The resultant voltage in that direction is proportional to the applied magnetic field. This is known as theHall effect.

TheHall effectis often used to measure the magnitude of a magnetic field. It is used as well to find the sign of the dominant charge carriers in materials such as semiconductors (negative electrons or positive holes).

Magnetic circuits

An important use of $H$ is inmagnetic circuitswhere $B = μ H$ inside a linear material. Here, $μ$ is themagnetic permeabilityof the material. This result is similar in form toOhm's law $J = σ E$ ,where $J$ is the current density, $σ$ is the conductance and $E$ is the electric field. Extending this analogy, the counterpart to the macroscopic Ohm's law ( $I = V ⁄ R$ ) is:

$\Phi ={\frac {F}{R}}_{\mathrm {m} },$

where ${\textstyle \Phi =\int \mathbf {B} \cdot \mathrm {d} \mathbf {A} }$ is the magnetic flux in the circuit, ${\textstyle F=\int \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}}$ is themagnetomotive forceapplied to the circuit, and $R m$ is thereluctanceof the circuit. Here the reluctance $R m$ is a quantity similar in nature toresistancefor the flux. Using this analogy it is straightforward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques ofcircuit theory.

Largest magnetic fields

As of October 2018^[update],the largest magnetic field produced over a macroscopic volume outside a lab setting is 2.8 kT (VNIIEFinSarov,Russia,1998).^[49]^[50]As of October 2018, the largest magnetic field produced in a laboratory over a macroscopic volume was 1.2 kT by researchers at theUniversity of Tokyoin 2018.^[50] The largest magnetic fields produced in a laboratory occur in particle accelerators, such asRHIC,inside the collisions of heavy ions, where microscopic fields reach 10¹⁴T.^[51]^[52]Magnetarshave the strongest known magnetic fields of any naturally occurring object, ranging from 0.1 to 100 GT (10⁸to 10¹¹T).^[53]

Common formulæ


Current configuration	Magnetic field
Finite beam of current	$\mathbf {B} ={\mu _{0}I \over 4\pi x}(\cos \theta _{1}+\cos \theta _{2}){\hat {\mathbf {x} }}$ where $I$ is the uniform current throughout the beam, with the direction of magnetic field as shown.
Infinite wire	$\mathbf {B} ={\mu _{0}I \over 2\pi R}{\hat {\mathbf {x} }}$ where $I$ is the uniform current flowing through the wire with the direction of magnetic field as shown.
Infinite cylindrical wire	$\mathbf {B} ={\mu _{0}I \over 2\pi x}{\hat {\mathbf {x} }}$ outside the wire carrying a current $I$ uniformly, with the direction of magnetic field as shown.	$\mathbf {B} ={\mu _{0}Ix \over 2\pi R^{2}}{\hat {\mathbf {x} }}$ inside the wire carrying a current $I$ uniformly, with the direction of magnetic field as shown.
Circular loop	$\mathbf {B} ={\mu _{0}IR^{2} \over 2(x^{2}+R^{2})^{3/2}}{\hat {\mathbf {x} }}$ along the axis of the loop, where $I$ is the uniform current flowing through the loop.
Solenoid	$\mathbf {B} ={\mu _{0}nI \over 2}(\cos \theta _{1}+\cos \theta _{2}){\hat {\mathbf {x} }}$ 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	$\mathbf {B} =0$ outside the solenoid carrying current $I$ with $n$ ,uniform number of loops of currents per length of solenoid.	$\mathbf {B} =\mu _{0}nI{\hat {\mathbf {x} }}$ 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	$B={\frac {\mu _{0}NI}{2\pi R}}$ 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	$\mathbf {B} =-{\frac {\mu _{0}\mathbf {m} }{4\pi r^{3}}},$ on the equatorial plane, where $\mathbf {m}$ is themagnetic dipole moment.	$\mathbf {B} ={\frac {\mu _{0}\mathbf {m} }{2\pi x^{3}}},$ on the axial plane (given that $x\gg R$ ),where $x$ can also be negative to indicate position at the opposite direction on the axis, and $\mathbf {m}$ is themagnetic 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 $\theta$ and radius $R$ at the center is $B={\mu _{0}\theta I \over 4\pi R}$ ,or that the magnetic field at the center of a N-sided regular polygon of side $a$ is $B={\mu _{0}NI \over \pi a}\sin {\pi \over N}\tan {\pi \over N}$ ,both outside of the plane with proper directions as inferred by right hand thumb rule.

History

One of the first drawings of a magnetic field, byRené Descartes,1644, showing the Earth attractinglodestones.It illustrated his theory that magnetism was caused by the circulation of tiny helical particles, "threaded parts", through threaded pores in magnets.

Early developments

While magnets and some properties of magnetism were known to ancient societies, the research of magnetic fields began in 1269 when French scholarPetrus Peregrinus de Maricourtmapped 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.^[54]^{[note 14]}

Almost three centuries later,William GilbertofColchesterreplicated Petrus Peregrinus' work and was the first to state explicitly that Earth is a magnet.^[55]^: 34Published in 1600, Gilbert's work,De Magnete,helped to establish magnetism as a science.

Mathematical development

Hans Christian Ørsted,Der Geist in der Natur,1854

In 1750,John Michellstated that magnetic poles attract and repel in accordance with aninverse square law^[55]^: 56Charles-Augustin de Coulombexperimentally verified this in 1785 and stated explicitly that north and south poles cannot be separated.^[55]^: 59Building on this force between poles,Siméon Denis Poisson(1781–1840) created the first successful model of the magnetic field, which he presented in 1824.^[55]^: 64In this model, a magnetic $H$ -field is produced bymagnetic polesand magnetism is due to small pairs of north–south magnetic poles.

Three discoveries in 1820 challenged this foundation of magnetism.Hans Christian Ørsteddemonstrated that a current-carrying wire is surrounded by a circular magnetic field.^{[note 15]}^[56]ThenAndré-Marie Ampèreshowed that parallel wires with currents attract one another if the currents are in the same direction and repel if they are in opposite directions.^[55]^: 87^[57]Finally,Jean-Baptiste BiotandFélix Savartannounced 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.^[58]^[55]^: 86Laplacelater deduced a law of force based on the differential action of a differential section of the wire,^[58]^[59]which became known as theBiot–Savart law,as Laplace did not publish his findings.^[60]

Extending these experiments, Ampère published his own successful model of magnetism in 1825. In it, he showed the equivalence of electrical currents to magnets^[55]^: 88and proposed that magnetism is due to perpetually flowing loops of current instead of the dipoles of magnetic charge in Poisson's model.^{[note 16]}Further, Ampère derived bothAmpère's force lawdescribing the force between two currents andAmpère'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 termelectrodynamicsto describe the relationship between electricity and magnetism.^[55]^: 88–92

In 1831,Michael Faradaydiscoveredelectromagnetic inductionwhen he found that a changing magnetic field generates an encircling electric field, formulating what is now known asFaraday's law of induction.^[55]^{: 189–192}Later,Franz Ernst Neumannproved that, for a moving conductor in a magnetic field, induction is a consequence of Ampère's force law.^[55]^: 222In the process, he introduced the magnetic vector potential, which was later shown to be equivalent to the underlying mechanism proposed by Faraday.^[55]^: 225

In 1850,Lord Kelvin,then known as William Thomson, distinguished between two magnetic fields now denoted $H$ and $B$ .The former applied to Poisson's model and the latter to Ampère's model and induction.^[55]^: 224Further, he derived how $H$ and $B$ relate to each other and coined the termpermeability.^[55]^: 245^[61]

Between 1861 and 1865,James Clerk Maxwelldeveloped and publishedMaxwell's equations,which explained and united all ofclassicalelectricity and magnetism. The first set of these equations was published in a paper entitledOn Physical Lines of Forcein 1861. These equations were valid but incomplete. Maxwell completed his set of equations in his later 1865 paperA Dynamical Theory of the Electromagnetic Fieldand demonstrated the fact that light is anelectromagnetic wave.Heinrich Hertzpublished papers in 1887 and 1888 experimentally confirming this fact.^[62]^[63]

Modern developments

In 1887, Tesla developed aninduction motorthat ran onalternating current.The motor usedpolyphasecurrent, which generated arotating magnetic fieldto turn the motor (a principle that Tesla claimed to have conceived in 1882).^[64]^[65]^[66]Tesla received a patent for his electric motor in May 1888.^[67]^[68]In 1885,Galileo Ferrarisindependently researched rotating magnetic fields and subsequently published his research in a paper to theRoyal Academy of SciencesinTurin,just two months before Tesla was awarded his patent, in March 1888.^[69]

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 ofquantum mechanicswas merged with electrodynamics to formquantum electrodynamics,which first formalized the notion that electromagnetic field energy is quantized in the form of photons.

Notes

^The letters B and H were originally chosen by Maxwell in hisTreatise on Electricity and Magnetism(Vol. II, pp. 236–237). For many quantities, he simply started choosing letters from the beginning of the alphabet. SeeRalph Baierlein (2000). "Answer to Question #73. S is for entropy, Q is for charge".American Journal of Physics.68(8): 691.Bibcode:2000AmJPh..68..691B.doi:10.1119/1.19524.
^Edward Purcell,in Electricity and Magnetism, McGraw-Hill, 1963, writes,Even some modern writers who treat $B$ as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by $H$ .This seems clumsy and pedantic. If you go into the laboratory and ask a physicist what causes the pion trajectories in his bubble chamber to curve, he'll probably answer "magnetic field", not "magnetic induction." You will seldom hear a geophysicist refer to the Earth's magnetic induction, or an astrophysicist talk about the magnetic induction of the galaxy. We propose to keep on calling $B$ the magnetic field. As for $H$ ,although other names have been invented for it, we shall call it "the field $H$ "or even" the magnetic field $H$ ."In a similar vein,M Gerloch (1983).Magnetism and Ligand-field Analysis.Cambridge University Press. p. 110.ISBN 978-0-521-24939-3.says: "So we may think of both $B$ and $H$ as magnetic fields, but drop the word 'magnetic' from $H$ so as to maintain the distinction... As Purcell points out, 'it is only the names that give trouble, not the symbols'. "
^An alternative mnemonic to the right hand rule isFleming's left-hand rule.
^The SI unit of $Φ B$ (magnetic flux) is theweber(symbol: Wb), related to theteslaby 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 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 largepermeabilityof iron relative to air.
^Here, "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 $m$ .
^Either $B$ or $H$ may be used for the magnetic field outside the magnet.
^In practice, the Biot–Savart law and other laws of magnetostatics are often used even when a current change in time, as long as it does not change too quickly. It is often used, for instance, for standard household currents, which oscillate sixty times per second.^[21]^: 223
^ The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of $B$ being zero, which is always valid. (There are no magnetic charges.)
^A third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below.
^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.
^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.
^ A complete expression for Faraday's law of induction in terms of the electric $E$ and magnetic fields can be written as: ${\mathcal {E}}=-{\frac {d\Phi }{dt}}=\oint _{\partial \Sigma (t)}\left(\mathbf {E} (\mathbf {r},\ t)+\mathbf {v} \times \mathbf {B} (\mathbf {r},\ t)\right)\cdot d{\boldsymbol {\ell }}\ =-{\frac {d}{dt}}\iint _{\Sigma (t)}d{\boldsymbol {A}}\cdot \mathbf {B} (\mathbf {r},\ t)$ where $\partialΣ (t)$ is the moving closed path bounding the moving surface $Σ (t)$ ,and $d A$ is an element of surface area of $Σ (t)$ .The first integral calculates the work done moving a charge a distance $d ℓ$ based upon the Lorentz force law. In the case where the bounding surface is stationary, theKelvin–Stokes theoremcan be used to show this equation is equivalent to the Maxwell–Faraday equation.
^HisEpistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete,which is often shortened toEpistola de magnete,is dated 1269 C.E.
^During a lecture demonstration on the effects of a current on a campus needle, Ørsted showed that when a current-carrying wire is placed at a right angle with the compass, nothing happens. When he tried to orient the wire parallel to the compass needle, however, it produced a pronounced deflection of the compass needle. By placing the compass on different sides of the wire, he was able to determine the field forms perfect circles around the wire.^[55]^: 85
^From the outside, the field of a dipole of magnetic charge has exactly the same form as a current loop when both are sufficiently small. Therefore, the two models differ only for magnetism inside magnetic material.

References

^Nave, Rod."Magnetic Field".HyperPhysics.Retrieved20 May2024.
^^a ^b ^c ^d ^e ^fFeynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963).The Feynman Lectures on Physics.Vol. 2. California Institute of Technology.ISBN 9780465040858.
^Young, Hugh D.; Freedman, Roger A.; Ford, A. Lewis (2008).Sears and Zemansky's university physics: with modern physics.Vol. 2. Pearson Addison-Wesley. pp. 918–919.ISBN 9780321501219.
^Purcell, Edward M.;Morin, David J. (2013).Electricity and Magnetism(3rd ed.). Cambridge University Press.ISBN 9781107014022.
^^a ^b ^c ^dThe International System of Units(PDF)(9th ed.), International Bureau of Weights and Measures, December 2022,ISBN 978-92-822-2272-0
^Jiles, David C. (1998).Introduction to Magnetism and Magnetic Materials(2 ed.). CRC. p. 3.ISBN 978-0412798603.
^John J. Roche (2000). "B and H, the intensity vectors of magnetism: A new approach to resolving a century-old controversy".American Journal of Physics.68(5): 438.Bibcode:2000AmJPh..68..438R.doi:10.1119/1.19459.
^^a ^bE. J. Rothwell and M. J. Cloud (2010)Electromagnetics.Taylor & Francis. p. 23.ISBN 1420058266.
^^a ^bStratton, Julius Adams (1941).Electromagnetic Theory(1st ed.). McGraw-Hill. p. 1.ISBN 978-0070621503.
^^a ^b ^c ^d ^ePurcell, E. (2011).Electricity and Magnetism(2nd ed.). Cambridge University Press.ISBN 978-1107013605.
^^a ^b ^c ^dGriffiths, David J.(1999).Introduction to Electrodynamics(3rd ed.). Pearson.ISBN 0-13-805326-X.
^^a ^bJackson, John David(1998).Classical electrodynamics(3rd ed.). New York: Wiley.ISBN 0-471-30932-X.
^^a ^b"Non-SI units accepted for use with the SI, and units based on fundamental constants (contd.)".SI Brochure: The International System of Units (SI) [8th edition, 2006; updated in 2014].Bureau International des Poids et Mesures. Archived fromthe originalon 8 June 2019.Retrieved19 April2018.
^^a ^bLang, Kenneth R. (2006).A Companion to Astronomy and Astrophysics.Springer. p. 176.ISBN 9780387333670.Retrieved19 April2018.
^"International system of units (SI)".NIST reference on constants, units, and uncertainty.National Institute of Standards and Technology. 12 April 2010.Retrieved9 May2012.
^"Gravity Probe B Executive Summary"(PDF).pp. 10, 21.Archived(PDF)from the original on 9 October 2022.
^Brown, William Fuller (1962).Magnetostatic Principles in Ferromagnetism.North Holland publishing company. p. 12.ASIN B0006AY7F8.
^Seemagnetic moment^{[broken anchor]}andB. D. Cullity; C. D. Graham (2008).Introduction to Magnetic Materials(2 ed.). Wiley-IEEE. p. 103.ISBN 978-0-471-47741-9.
^E. Richard Cohen; David R. Lide; George L. Trigg (2003).AIP physics desk reference(3 ed.). Birkhäuser. p. 381.ISBN 978-0-387-98973-0.
^Griffiths 1999,p. 438
^^a ^b ^cGriffiths, David J.(2017).Introduction to Electrodynamics(4th ed.). Cambridge University Press.ISBN 9781108357142.
^Griffiths 1999,pp. 222–225
^"K. McDonald's Physics Examples - Disk"(PDF).puhep1.princeton.edu.Archived(PDF)from the original on 9 October 2022.Retrieved13 February2021.
^"K. McDonald's Physics Examples - Railgun"(PDF).puhep1.princeton.edu.Archived(PDF)from the original on 9 October 2022.Retrieved13 February2021.
^ Deissler, R.J. (2008)."Dipole in a magnetic field, work, and quantum spin"(PDF).Physical Review E.77(3, pt 2): 036609.Bibcode:2008PhRvE..77c6609D.doi:10.1103/PhysRevE.77.036609.PMID 18517545.Archived(PDF)from the original on 9 October 2022.
^Griffiths 1999,pp. 266–268
^John Clarke Slater; Nathaniel Herman Frank (1969).Electromagnetism(first published in 1947 ed.). Courier Dover Publications. p. 69.ISBN 978-0-486-62263-7.
^Griffiths 1999,p. 332
^^a ^bRJD Tilley (2004).Understanding Solids.Wiley. p.368.ISBN 978-0-470-85275-0.
^Sōshin Chikazumi; Chad D. Graham (1997).Physics of ferromagnetism(2 ed.). Oxford University Press. p. 118.ISBN 978-0-19-851776-4.
^Amikam Aharoni (2000).Introduction to the theory of ferromagnetism(2 ed.). Oxford University Press. p. 27.ISBN 978-0-19-850808-3.
^M Brian Maple; et al. (2008)."Unconventional superconductivity in novel materials".In K. H. Bennemann; John B. Ketterson (eds.).Superconductivity.Springer. p. 640.ISBN 978-3-540-73252-5.
^Naoum Karchev (2003)."Itinerant ferromagnetism and superconductivity".In Paul S. Lewis; D. Di (CON) Castro (eds.).Superconductivity research at the leading edge.Nova Publishers. p. 169.ISBN 978-1-59033-861-2.
^Lieberherr, Martin (6 July 2010)."The magnetic field lines of a helical coil are not simple loops".American Journal of Physics.78(11): 1117–1119.Bibcode:2010AmJPh..78.1117L.doi:10.1119/1.3471233.
^Jackson, John David(1975).Classical electrodynamics(2nd ed.). New York: Wiley.ISBN 9780471431329.
^Rosser, W. G. V. (1968).Classical Electromagnetism via Relativity.pp. 29–42.doi:10.1007/978-1-4899-6559-2.ISBN 978-1-4899-6258-4.
^Purcell, Edward (22 September 2011).Electricity and Magnetism.Cambridge University Press.doi:10.1017/cbo9781139005043.ISBN 978-1-107-01360-5.
^C. Doran and A. Lasenby (2003)Geometric Algebra for Physicists,Cambridge University Press, p. 233.ISBN 0521715954.
^ E. J. Konopinski(1978). "What the electromagnetic vector potential describes".Am. J. Phys.46(5): 499–502.Bibcode:1978AmJPh..46..499K.doi:10.1119/1.11298.
^Griffiths 1999,p. 422
^Naber, Gregory L. (2012).The Geometry of Minkowski spacetime: an introduction to the mathematics of the special theory of relativity.Springer. pp. 4–5.ISBN 978-1-4419-7837-0.OCLC 804823303.
^Rosser, W. G. V. (1968).Classical Electromagnetism via Relativity.doi:10.1007/978-1-4899-6559-2.ISBN 978-1-4899-6258-4.
^ For a good qualitative introduction see: Richard Feynman(2006).QED: the strange theory of light and matter.Princeton University Press.ISBN 978-0-691-12575-6.
^Weiss, Nigel (2002)."Dynamos in planets, stars and galaxies".Astronomy and Geophysics.43(3): 3.09–3.15.Bibcode:2002A&G....43c...9W.doi:10.1046/j.1468-4004.2002.43309.x.
^"What is the Earth's magnetic field?".Geomagnetism Frequently Asked Questions.National Centers for Environmental Information, National Oceanic and Atmospheric Administration.Retrieved19 April2018.
^Raymond A. Serway; Chris Vuille; Jerry S. Faughn (2009).College physics(8th ed.). Belmont, CA: Brooks/Cole, Cengage Learning. p.628.ISBN 978-0-495-38693-3.
^Merrill, Ronald T.; McElhinny, Michael W.; McFadden, Phillip L. (1996). "2. The present geomagnetic field: analysis and description from historical observations".The magnetic field of the earth: paleomagnetism, the core, and the deep mantle.Academic Press.ISBN 978-0-12-491246-5.
^Phillips, Tony (29 December 2003)."Earth's Inconstant Magnetic Field".Science@Nasa.Archived fromthe originalon 1 November 2022.Retrieved27 December2009.
^Boyko, B.A.; Bykov, A.I.; Dolotenko, M.I.; Kolokolchikov, N.P.; Markevtsev, I.M.; Tatsenko, O.M.; Shuvalov, K. (1999). "With record magnetic fields to the 21st Century".Digest of Technical Papers. 12th IEEE International Pulsed Power Conference. (Cat. No.99CH36358).Vol. 2. pp. 746–749.doi:10.1109/PPC.1999.823621.ISBN 0-7803-5498-2.S2CID 42588549.
^^a ^bDaley, Jason."Watch the Strongest Indoor Magnetic Field Blast Doors of Tokyo Lab Wide Open".Smithsonian Magazine.Retrieved8 September2020.
^Tuchin, Kirill (2013)."Particle production in strong electromagnetic fields in relativistic heavy-ion collisions".Adv. High Energy Phys.2013:490495.arXiv:1301.0099.Bibcode:2013arXiv1301.0099T.doi:10.1155/2013/490495.S2CID 4877952.
^Bzdak, Adam; Skokov, Vladimir (29 March 2012). "Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions".Physics Letters B.710(1): 171–174.arXiv:1111.1949.Bibcode:2012PhLB..710..171B.doi:10.1016/j.physletb.2012.02.065.S2CID 118462584.
^Kouveliotou, C.; Duncan, R. C.; Thompson, C. (February 2003). "Magnetars Archived11 June 2007 at theWayback Machine".Scientific American;Page 36.
^Chapman, Allan (2007). "Peregrinus, Petrus (Flourished 1269)".Encyclopedia of Geomagnetism and Paleomagnetism.pp. 808–809.doi:10.1007/978-1-4020-4423-6_261.ISBN 978-1-4020-3992-8.
^^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿWhittaker, E. T.(1910).A History of the Theories of Aether and Electricity.Dover Publications.ISBN 978-0-486-26126-3.
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^Blundell, Stephen J. (2012).Magnetism: A Very Short Introduction.OUP Oxford. p. 31.ISBN 9780191633720.
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^Erlichson, Herman (1998)."The experiments of Biot and Savart concerning the force exerted by a current on a magnetic needle".American Journal of Physics.66(5): 389.Bibcode:1998AmJPh..66..385E.doi:10.1119/1.18878.
^Frankel, Eugene (1972).Jean-Baptiste Biot: The career of a physicist in nineteenth-century France.Princeton University: Doctoral dissertation. p. 334.
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^"Galileo Ferraris (March 1888)Rotazioni elettrodinamiche prodotte per mezzo di correnti alternate(Electrodynamic rotations by means of alternating currents), memory read at Accademia delle Scienze, Torino, inOpere di Galileo Ferraris,Hoepli, Milano,1902 vol I pages 333 to 348 "(PDF).Archived fromthe original(PDF)on 9 July 2021.Retrieved2 July2021.

External links

Media related toMagnetic fieldsat Wikimedia Commons
Crowell, B., "Electromagnetism Archived30 April 2010 at theWayback Machine".
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.

[7] The letters B and H were originally chosen by Maxwell in hisTreatise on Electricity and Magnetism(Vol. II, pp. 236–237). For many quantities, he simply started choosing letters from the beginning of the alphabet. SeeRalph Baierlein (2000). "Answer to Question #73. S is for entropy, Q is for charge".American Journal of Physics.68(8): 691.Bibcode:2000AmJPh..68..691B.doi:10.1119/1.19524.

[ex03-9] Edward Purcell,in Electricity and Magnetism, McGraw-Hill, 1963, writes,Even some modern writers who treat $B$ as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by $H$ .This seems clumsy and pedantic. If you go into the laboratory and ask a physicist what causes the pion trajectories in his bubble chamber to curve, he'll probably answer "magnetic field", not "magnetic induction." You will seldom hear a geophysicist refer to the Earth's magnetic induction, or an astrophysicist talk about the magnetic induction of the galaxy. We propose to keep on calling $B$ the magnetic field. As for $H$ ,although other names have been invented for it, we shall call it "the field $H$ "or even" the magnetic field $H$ ."In a similar vein,M Gerloch (1983).Magnetism and Ligand-field Analysis.Cambridge University Press. p. 110.ISBN 978-0-521-24939-3.says: "So we may think of both $B$ and $H$ as magnetic fields, but drop the word 'magnetic' from $H$ so as to maintain the distinction... As Purcell points out, 'it is only the names that give trouble, not the symbols'. "

[14] An alternative mnemonic to the right hand rule isFleming's left-hand rule.

[16] The SI unit of $Φ B$ (magnetic flux) is theweber(symbol: Wb), related to theteslaby 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.

[ex07-21] 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 largepermeabilityof iron relative to air.

[ex05-22] Here, "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 $m$ .

[ex04-25] Either $B$ or $H$ may be used for the magnetic field outside the magnet.

[ex12-29] In practice, the Biot–Savart law and other laws of magnetostatics are often used even when a current change in time, as long as it does not change too quickly. It is often used, for instance, for standard household currents, which oscillate sixty times per second.^[21]^: 223

[ex13-31] The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of $B$ being zero, which is always valid. (There are no magnetic charges.)

[38] A third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below.

[ex08-45] 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.

[ex09-46] 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.

[ex14-48] A complete expression for Faraday's law of induction in terms of the electric $E$ and magnetic fields can be written as: ${\mathcal {E}}=-{\frac {d\Phi }{dt}}=\oint _{\partial \Sigma (t)}\left(\mathbf {E} (\mathbf {r},\ t)+\mathbf {v} \times \mathbf {B} (\mathbf {r},\ t)\right)\cdot d{\boldsymbol {\ell }}\ =-{\frac {d}{dt}}\iint _{\Sigma (t)}d{\boldsymbol {A}}\cdot \mathbf {B} (\mathbf {r},\ t)$ where $\partialΣ (t)$ is the moving closed path bounding the moving surface $Σ (t)$ ,and $d A$ is an element of surface area of $Σ (t)$ .The first integral calculates the work done moving a charge a distance $d ℓ$ based upon the Lorentz force law. In the case where the bounding surface is stationary, theKelvin–Stokes theoremcan be used to show this equation is equivalent to the Maxwell–Faraday equation.

[68] HisEpistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete,which is often shortened toEpistola de magnete,is dated 1269 C.E.

[70] During a lecture demonstration on the effects of a current on a campus needle, Ørsted showed that when a current-carrying wire is placed at a right angle with the compass, nothing happens. When he tried to orient the wire parallel to the compass needle, however, it produced a pronounced deflection of the compass needle. By placing the compass on different sides of the wire, he was able to determine the field forms perfect circles around the wire.^[55]^: 85

[76] From the outside, the field of a dipole of magnetic charge has exactly the same form as a current loop when both are sufficiently small. Therefore, the two models differ only for magnetism inside magnetic material.

[1] Nave, Rod."Magnetic Field".HyperPhysics.Retrieved20 May2024.

[feynman-2] Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963).The Feynman Lectures on Physics.Vol. 2. California Institute of Technology.ISBN 9780465040858.

[3] Young, Hugh D.; Freedman, Roger A.; Ford, A. Lewis (2008).Sears and Zemansky's university physics: with modern physics.Vol. 2. Pearson Addison-Wesley. pp. 918–919.ISBN 9780321501219.

[Purcell3ed-4] Purcell, Edward M.;Morin, David J. (2013).Electricity and Magnetism(3rd ed.). Cambridge University Press.ISBN 9781107014022.

[:1-5] The International System of Units(PDF)(9th ed.), International Bureau of Weights and Measures, December 2022,ISBN 978-92-822-2272-0

[Jiles-6] Jiles, David C. (1998).Introduction to Magnetism and Magnetic Materials(2 ed.). CRC. p. 3.ISBN 978-0412798603.

[8] John J. Roche (2000). "B and H, the intensity vectors of magnetism: A new approach to resolving a century-old controversy".American Journal of Physics.68(5): 438.Bibcode:2000AmJPh..68..438R.doi:10.1119/1.19459.

[Electromagnetics-10] E. J. Rothwell and M. J. Cloud (2010)Electromagnetics.Taylor & Francis. p. 23.ISBN 1420058266.

[Stratton-11] Stratton, Julius Adams (1941).Electromagnetic Theory(1st ed.). McGraw-Hill. p. 1.ISBN 978-0070621503.

[purcell2ed-12] Purcell, E. (2011).Electricity and Magnetism(2nd ed.). Cambridge University Press.ISBN 978-1107013605.

[Griffiths3ed-13] Griffiths, David J.(1999).Introduction to Electrodynamics(3rd ed.). Pearson.ISBN 0-13-805326-X.

[jackson3ed-15] Jackson, John David(1998).Classical electrodynamics(3rd ed.). New York: Wiley.ISBN 0-471-30932-X.

[BIPMTab9-17] "Non-SI units accepted for use with the SI, and units based on fundamental constants (contd.)".SI Brochure: The International System of Units (SI) [8th edition, 2006; updated in 2014].Bureau International des Poids et Mesures. Archived fromthe originalon 8 June 2019.Retrieved19 April2018.

[KLang-18] Lang, Kenneth R. (2006).A Companion to Astronomy and Astrophysics.Springer. p. 176.ISBN 9780387333670.Retrieved19 April2018.

[19] "International system of units (SI)".NIST reference on constants, units, and uncertainty.National Institute of Standards and Technology. 12 April 2010.Retrieved9 May2012.

[20] "Gravity Probe B Executive Summary"(PDF).pp. 10, 21.Archived(PDF)from the original on 9 October 2022.

[23] Brown, William Fuller (1962).Magnetostatic Principles in Ferromagnetism.North Holland publishing company. p. 12.ASIN B0006AY7F8.

[Graham-24] Seemagnetic moment^{[broken anchor]}andB. D. Cullity; C. D. Graham (2008).Introduction to Magnetic Materials(2 ed.). Wiley-IEEE. p. 103.ISBN 978-0-471-47741-9.

[Trigg-26] E. Richard Cohen; David R. Lide; George L. Trigg (2003).AIP physics desk reference(3 ed.). Birkhäuser. p. 381.ISBN 978-0-387-98973-0.

[27] Griffiths 1999,p. 438

[Griffiths4ed-28] Griffiths, David J.(2017).Introduction to Electrodynamics(4th ed.). Cambridge University Press.ISBN 9781108357142.

[30] Griffiths 1999,pp. 222–225

[32] "K. McDonald's Physics Examples - Disk"(PDF).puhep1.princeton.edu.Archived(PDF)from the original on 9 October 2022.Retrieved13 February2021.

[33] "K. McDonald's Physics Examples - Railgun"(PDF).puhep1.princeton.edu.Archived(PDF)from the original on 9 October 2022.Retrieved13 February2021.

[Deissler-34] Deissler, R.J. (2008)."Dipole in a magnetic field, work, and quantum spin"(PDF).Physical Review E.77(3, pt 2): 036609.Bibcode:2008PhRvE..77c6609D.doi:10.1103/PhysRevE.77.036609.PMID 18517545.Archived(PDF)from the original on 9 October 2022.

[35] Griffiths 1999,pp. 266–268

[Slater-36] John Clarke Slater; Nathaniel Herman Frank (1969).Electromagnetism(first published in 1947 ed.). Courier Dover Publications. p. 69.ISBN 978-0-486-62263-7.

[37] Griffiths 1999,p. 332

[Tilley-39] RJD Tilley (2004).Understanding Solids.Wiley. p.368.ISBN 978-0-470-85275-0.

[Chikazumi-40] Sōshin Chikazumi; Chad D. Graham (1997).Physics of ferromagnetism(2 ed.). Oxford University Press. p. 118.ISBN 978-0-19-851776-4.

[Aharoni-41] Amikam Aharoni (2000).Introduction to the theory of ferromagnetism(2 ed.). Oxford University Press. p. 27.ISBN 978-0-19-850808-3.

[Bennemann-42] M Brian Maple; et al. (2008)."Unconventional superconductivity in novel materials".In K. H. Bennemann; John B. Ketterson (eds.).Superconductivity.Springer. p. 640.ISBN 978-3-540-73252-5.

[Lewis-43] Naoum Karchev (2003)."Itinerant ferromagnetism and superconductivity".In Paul S. Lewis; D. Di (CON) Castro (eds.).Superconductivity research at the leading edge.Nova Publishers. p. 169.ISBN 978-1-59033-861-2.

[lieb-44] Lieberherr, Martin (6 July 2010)."The magnetic field lines of a helical coil are not simple loops".American Journal of Physics.78(11): 1117–1119.Bibcode:2010AmJPh..78.1117L.doi:10.1119/1.3471233.

[47] Jackson, John David(1975).Classical electrodynamics(2nd ed.). New York: Wiley.ISBN 9780471431329.

[49] Rosser, W. G. V. (1968).Classical Electromagnetism via Relativity.pp. 29–42.doi:10.1007/978-1-4899-6559-2.ISBN 978-1-4899-6258-4.

[50] Purcell, Edward (22 September 2011).Electricity and Magnetism.Cambridge University Press.doi:10.1017/cbo9781139005043.ISBN 978-1-107-01360-5.

[51] C. Doran and A. Lasenby (2003)Geometric Algebra for Physicists,Cambridge University Press, p. 233.ISBN 0521715954.

[52] E. J. Konopinski(1978). "What the electromagnetic vector potential describes".Am. J. Phys.46(5): 499–502.Bibcode:1978AmJPh..46..499K.doi:10.1119/1.11298.

[53] Griffiths 1999,p. 422

[54] Naber, Gregory L. (2012).The Geometry of Minkowski spacetime: an introduction to the mathematics of the special theory of relativity.Springer. pp. 4–5.ISBN 978-1-4419-7837-0.OCLC 804823303.

[55] Rosser, W. G. V. (1968).Classical Electromagnetism via Relativity.doi:10.1007/978-1-4899-6559-2.ISBN 978-1-4899-6258-4.

[56] For a good qualitative introduction see: Richard Feynman(2006).QED: the strange theory of light and matter.Princeton University Press.ISBN 978-0-691-12575-6.

[Weiss-57] Weiss, Nigel (2002)."Dynamos in planets, stars and galaxies".Astronomy and Geophysics.43(3): 3.09–3.15.Bibcode:2002A&G....43c...9W.doi:10.1046/j.1468-4004.2002.43309.x.

[58] "What is the Earth's magnetic field?".Geomagnetism Frequently Asked Questions.National Centers for Environmental Information, National Oceanic and Atmospheric Administration.Retrieved19 April2018.

[59] Raymond A. Serway; Chris Vuille; Jerry S. Faughn (2009).College physics(8th ed.). Belmont, CA: Brooks/Cole, Cengage Learning. p.628.ISBN 978-0-495-38693-3.

[60] Merrill, Ronald T.; McElhinny, Michael W.; McFadden, Phillip L. (1996). "2. The present geomagnetic field: analysis and description from historical observations".The magnetic field of the earth: paleomagnetism, the core, and the deep mantle.Academic Press.ISBN 978-0-12-491246-5.

[61] Phillips, Tony (29 December 2003)."Earth's Inconstant Magnetic Field".Science@Nasa.Archived fromthe originalon 1 November 2022.Retrieved27 December2009.

[62] Boyko, B.A.; Bykov, A.I.; Dolotenko, M.I.; Kolokolchikov, N.P.; Markevtsev, I.M.; Tatsenko, O.M.; Shuvalov, K. (1999). "With record magnetic fields to the 21st Century".Digest of Technical Papers. 12th IEEE International Pulsed Power Conference. (Cat. No.99CH36358).Vol. 2. pp. 746–749.doi:10.1109/PPC.1999.823621.ISBN 0-7803-5498-2.S2CID 42588549.

[smithsonianMagnetRecord-63] Daley, Jason."Watch the Strongest Indoor Magnetic Field Blast Doors of Tokyo Lab Wide Open".Smithsonian Magazine.Retrieved8 September2020.

[64] Tuchin, Kirill (2013)."Particle production in strong electromagnetic fields in relativistic heavy-ion collisions".Adv. High Energy Phys.2013:490495.arXiv:1301.0099.Bibcode:2013arXiv1301.0099T.doi:10.1155/2013/490495.S2CID 4877952.

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