Electromagnetism

- Electricity
- Magnetism

A **magnetic field** is a captivating outcome of electric currents and captivating materials. The captivating margin during any given indicate is specified by both a *direction* and a *magnitude* (or strength); as such it is a matrix field.^{[nb 1]} The tenure is used for dual graphic yet closely compared fields denoted by a black **B** and **H**, where **H** is totalled in units of amperes per scale (symbol: A·m^{−1} or A/m) in a SI. **B** is totalled in teslas (symbol: T; note that nonetheless a pitch is collateral T, “tesla” is sum in reduce box in a SI system) and newtons per scale per ampere (symbol: N·m^{−1}·A^{−1} or N/(m·A)) in a SI. **B** is many ordinarily tangible in terms of a Lorentz force it exerts on relocating electric charges.

Magnetic fields can be assembled by relocating electric charges and a unique captivating moments of facile particles compared with a elemental quantum property, their spin.^{[1]}^{[2]} In special relativity, electric and captivating fields are dual associated aspects of a singular object, called a electromagnetic tensor; a apart of this tensor into electric and captivating fields depends on a family quickness of a spectator and charge. In quantum physics, a electromagnetic margin is quantized and electromagnetic interactions outcome from a sell of photons.

In bland life, captivating fields are many mostly encountered as a force sum by permanent magnets, that lift on ferromagnetic materials such as iron, cobalt, or nickel, and attract or repel other magnets. Magnetic fields are widely used around formidable technology, utterly in electrical engineering and electromechanics. The Earth produces a possess captivating field, that is vicious in navigation, and it shields a Earth’s atmosphere from solar wind. Rotating captivating fields are used in both electric motors and generators. Magnetic army give information about a assign carriers in a member by a Hall effect. The communication of captivating fields in electric inclination such as transformers is formidable in a fortify of captivating circuits.

## History[edit]

Although magnets and draw were famous many earlier, a investigate of captivating fields began in 1269 when French academician Petrus Peregrinus de Maricourt mapped out a captivating margin on a aspect of a round magnet regulating iron needles.^{[nb 2]} Noting that a ensuing margin lines crossed during dual points he named those points ‘poles’ in analogy to Earth’s poles. He also clearly articulated a member that magnets always have both a north and south pole, no matter how finely one slices them.

Almost 3 centuries later, William Gilbert of Colchester replicated Petrus Peregrinus’ work and was a initial to state categorically that Earth is a magnet.^{[3]} Published in 1600, Gilbert’s work, *De Magnete*, helped to settle draw as a science.

In 1750, John Michell settled that captivating poles attract and repel in suitability with an conflicting block law.^{[4]}Charles-Augustin de Coulomb experimentally accurate this in 1785 and settled categorically that a north and south poles can't be separated.^{[5]} Building on this force between poles, Siméon Denis Poisson (1781–1840) sum a initial successful indication of a captivating field, that he presented in 1824.^{[6]} In this model, a captivating **H**-field is assembled by ‘magnetic poles’ and draw is due to tiny pairs of north/south captivating poles.

Three discoveries challenged this substructure of magnetism, though. First, in 1819, Hans Christian Ørsted detected that an electric stream generates a captivating margin surrounding it. Then in 1820, André-Marie Ampère showed that together wires carrying currents in a same instruction attract one another. Finally, Jean-Baptiste Biot and Félix Savart detected a Biot–Savart law in 1820, that rightly predicts a captivating margin around any current-carrying wire.

Extending these experiments, Ampère published his possess successful indication of draw in 1825. In it, he showed a equilibrium of electrical currents to magnets^{[7]} and due that draw is due to eternally issuing loops of stream instead of a dipoles of captivating assign in Poisson’s model.^{[nb 3]} This has a additional advantage of explaining given captivating assign can not be isolated. Further, Ampère subsequent both Ampère’s force law describing a force between dual currents and Ampère’s law, which, like a Biot–Savart law, rightly described a captivating margin generated by a plain current. Also in this work, Ampère introduced a tenure electrodynamics to news a attribute between electricity and magnetism.

In 1831, Michael Faraday detected electromagnetic initiation when he found that a changing captivating margin generates an surrounding electric field. He described this materialisation in what is famous as Faraday’s law of induction. Later, Franz Ernst Neumann stream that, for a relocating conductor in a captivating field, initiation is a outcome of Ampère’s force law.^{[8]} In a routine he introduced a captivating matrix potential, that was after shown to be homogeneous to a underlying resource due by Faraday.

In 1850, Lord Kelvin, afterwards famous as William Thomson, renowned between dual captivating fields now denoted **H** and **B**. The former practical to Poisson’s indication and a latter to Ampère’s indication and induction.^{[9]} Further, he subsequent how **H** and **B** news to any other.

The reason **H** and **B** are used for a dual captivating fields has been a source of some discuss among scholarship historians. Most establish that Kelvin avoided **M** to forestall difficulty with a SI elemental territory of length, a Metre, condensed “m”. Others trust a choices were utterly random.^{[10]}^{[11]}

Between 1861 and 1865, James Clerk Maxwell grown and published Maxwell’s equations, that explained and joined all of exemplary electricity and magnetism. The initial set of these equations was published in a paper entitled *On Physical Lines of Force* in 1861. These equations were stream nonetheless incomplete. Maxwell finished his set of equations in his after 1865 paper *A Dynamical Theory of a Electromagnetic Field* and demonstrated a fact that light is an electromagnetic wave. Heinrich Hertz experimentally reliable this fact in 1887.

The twentieth century extended electrodynamics to embody relativity and quantum mechanics. Albert Einstein, in his paper of 1905 that dynamic relativity, showed that both a electric and captivating fields are prejudiced of a same phenomena noticed from conflicting anxiety frames. (See relocating magnet and conductor problem for sum about a suspicion examination that eventually helped Albert Einstein to rise special relativity.) Finally, a emergent margin of quantum mechanics was joined with electrodynamics to form quantum electrodynamics (QED).

## Definitions, units, and measurement[edit]

### The B-field[edit]

The captivating margin can be tangible in several homogeneous ways shaped on a effects it has on a environment.

Often a captivating margin is tangible by a force it exerts on a relocating charged particle. It is famous from experiments in electrostatics that a molecule of assign *q* in an electric margin **E** use a force **F** = *q***E**. However, in other situations, such as when a charged molecule moves in a closeness of a current-carrying wire, a force also depends on a quickness of that particle. Fortunately, a quickness contingent apportionment can be detached out such that a force on a molecule satisfies a *Lorentz force law*,

F=q(E+v×B).{displaystyle mathbf {F} =q(mathbf {E} +mathbf {v} times mathbf {B} ).}

Here **v** is a particle’s quickness and × denotes a cranky product. The matrix **B** is termed a captivating field, and it is *defined* as a matrix margin required to make a Lorentz force law rightly news a suit of a charged particle. This clarification allows a integrity of **B** in a following way^{[14]}

[T]he command, “Measure a instruction and bulk of a matrix

Bduring such and such a place,” calls for a following operations: Take a molecule of famous assignq. Measure a force onqduring rest, to establishE. Then bulk a force on a molecule when a quickness isv; repeat withvin some other direction. Now find aBthat creates a Lorentz force law fit all these results—that is a captivating margin during a place in question.

Alternatively, a captivating margin can be tangible in terms of a torque it produces on a captivating dipole (see captivating torque on permanent magnets below).

### The H-field[edit]

In further to **B**, there is a apportion **H**, that is also infrequently called a *magnetic field*.^{[nb 4]} In a vacuum, **B** and **H** are proportional to any other, with a multiplicative consistent depending on a earthy units. Inside a member they are conflicting (see H and B inside and outmost of captivating materials). The tenure “magnetic field” is historically indifferent for **H** while regulating other terms for **B**. Informally, though, and rigourously for some new textbooks mostly in physics, a tenure ‘magnetic field’ is used to news **B** as good as or in place of **H**.^{[nb 5]} There are many choice names for both (see sidebar).

### Units[edit]

In SI units, **B** is totalled in teslas (symbol: T) and together Φ_{B} (magnetic flux) is totalled in webers (symbol: Wb) so that a suit firmness of 1 Wb/m^{2} is 1 tesla. The SI territory of tesla is homogeneous to (newton·second)/(coulomb·metre).^{[nb 6]} In Gaussian-cgs units, **B** is totalled in gauss (symbol: G). (The acclimatisation is 1 T = 10,000 G.) One nanotesla is also called a gamma (symbol: γ).^{[15]} The **H**-field is totalled in amperes per metre (A/m) in SI units, and in oersteds (Oe) in cgs units.^{[16]}

### Measurement[edit]

The indicating achieved for a captivating margin dimensions for Gravity Probe B examination is 5 attoteslas (6982500000000000000♠5×10^{−18} T);^{[17]} a largest captivating margin assembled in a laboratory is 2.8 kT (VNIIEF in Sarov, Russia, 1998).^{[18]} The captivating margin of some astronomical objects such as magnetars are many higher; magnetars operation from 0.1 to 100 GT (10^{8} to 10^{11} T).^{[19]} See orders of bulk (magnetic field).

Devices used to bulk a inner captivating margin are called magnetometers. Important classes of magnetometers embody regulating a rotating coil, Hall outcome magnetometers, NMR magnetometers, SQUID magnetometers, and fluxgate magnetometers. The captivating fields of detached astronomical objects are totalled by their effects on inner charged particles. For instance, electrons spiraling around a margin line furnish synchrotron deviation that is detectable in radio waves.

## Magnetic margin lines[edit]

Mapping a captivating margin of an intent is facile in principle. First, bulk a strength and instruction of a captivating margin during a vast series of locations (or during any indicate in space). Then, symbol any plcae with an arrow (called a vector) indicating in a instruction of a inner captivating margin with a bulk proportional to a strength of a captivating field.

An choice routine to map a captivating margin is to ‘connect’ a arrows to form captivating *field lines*. The instruction of a captivating margin during any indicate is together to a instruction of circuitously margin lines, and a inner firmness of margin lines can be finished proportional to a strength.

Magnetic margin lines are like streamlines in glass flow, in that they paint something continuous, and a conflicting fortitude would uncover some-more or fewer lines. An advantage of regulating captivating margin lines as a illustration is that many laws of draw (and electromagnetism) can be settled totally and concisely regulating facile concepts such as a ‘number’ of margin lines by a surface. These concepts can be quick ‘translated’ to their mathematical form. For example, a series of margin lines by a given aspect is a aspect constituent of a captivating field.

Various phenomena have a outcome of “displaying” captivating margin lines as yet a margin lines were earthy phenomena. For example, iron filings placed in a captivating field, form lines that conform to ‘field lines’.^{[nb 7]} Magnetic margin “lines” are also visually displayed in frigid auroras, in that plasma molecule dipole interactions emanate manifest streaks of light that line adult with a inner instruction of Earth’s captivating field.

Field lines can be used as a qualitative apparatus to daydream captivating forces. In ferromagnetic substances like iron and in plasmas, captivating army can be accepted by devising that a margin lines strive a tension, (like a rubber band) along their length, and a vigour perpendicular to their length on adjacent margin lines. ‘Unlike’ poles of magnets attract given they are associated by many margin lines; ‘like’ poles repel given their margin lines do not meet, yet run parallel, pulling on any other. The severe form of this judgment is a electromagnetic stress–energy tensor.

## Magnetic margin and permanent magnets[edit]

*Permanent magnets* are objects that furnish their possess dynamic captivating fields. They are finished of ferromagnetic materials, such as iron and nickel, that have been magnetized, and they have both a north and a south pole.

### Magnetic margin of permanent magnets[edit]

The captivating margin of permanent magnets can be utterly complicated, generally nearby a magnet. The captivating margin of a small^{[nb 8]} loyal magnet is proportional to a magnet’s *strength* (called a captivating dipole impulse **m**). The equations are non-trivial and also count on a widen from a magnet and a course of a magnet. For facile magnets, **m** points in a instruction of a line drawn from a south to a north stick of a magnet. Flipping a bar magnet is homogeneous to rotating a **m** by 180 degrees.

The captivating margin of incomparable magnets can be achieved by modelling them as a collection of a vast series of tiny magnets called dipoles any carrying their possess **m**. The captivating margin assembled by a magnet afterwards is a net captivating margin of these dipoles. And, any net force on a magnet is a outcome of adding adult a army on a sold dipoles.

There are dual competing models for a inlet of these dipoles. These dual models furnish dual conflicting captivating fields, **H** and **B**. Outside a material, though, a dual are matching (to a multiplicative constant) so that in many cases a eminence can be ignored. This is utterly loyal for captivating fields, such as those due to electric currents, that are not generated by captivating materials.

### Magnetic stick indication and a H-field[edit]

It is infrequently useful to indication a force and torques between dual magnets as due to captivating poles repulsion or attracting any other in a same demeanour as a Coulomb force between electric charges. This is called a Gilbert indication of magnetism, after William Gilbert. In this model, a captivating **H**-field is assembled by *magnetic charges* that are ‘smeared’ around any pole. These *magnetic charges* are in fact compared to a magnetization margin **M**.

The **H**-field, therefore, is homogeneous to a electric margin **E**, that starts during a certain electric assign and ends during a disastrous electric charge. Near a north pole, therefore, all **H**-field lines indicate divided from a north stick (whether inside a magnet or out) while nearby a south stick (whether inside a magnet or out) all **H**-field lines indicate toward a south pole. A north pole, then, feels a force in a instruction of a **H**-field while a force on a south stick is conflicting to a **H**-field.

In a captivating stick model, a facile captivating dipole **m** is shaped by dual conflicting captivating poles of stick strength *q*_{m} detached by a tiny widen matrix **d**, such that **m** = *q*_{m} **d**. The captivating stick indication predicts rightly a margin **H** both inside and outmost captivating materials, in sold a fact that **H** is conflicting to a magnetization margin **M** inside a permanent magnet.

Since it is shaped on a fictitious suspicion of a *magnetic assign density*, a Gilbert indication has limitations. Magnetic poles can't exist detached from any other as electric charges can, yet always come in north/south pairs. If a magnetized intent is divided in half, a new stick appears on a aspect of any piece, so any has a span of interrelated poles. The captivating stick indication does not comment for draw that is assembled by electric currents.

### Amperian loop indication and a B-field[edit]

After Ørsted detected that electric currents furnish a captivating margin and Ampere detected that electric currents captivated and detered any other matching to magnets, it was healthy to suppose that all captivating fields are due to electric stream loops. In this indication grown by Ampere, a facile captivating dipole that creates adult all magnets is a amply tiny Amperian loop of stream I. The dipole impulse of this loop is *m* = *IA* where *A* is a area of a loop.

These captivating dipoles furnish a captivating **B**-field. One vicious skill of a **B**-field assembled this approach is that captivating **B**-field lines conjunction start nor finish (mathematically, **B** is a solenoidal matrix field); a margin line possibly extends to forever or wraps around to form a sealed curve.^{[nb 9]} To date no disproportion to this order has been found. (See captivating monopole below.) Magnetic margin lines exit a magnet nearby a north stick and enter nearby a south pole, yet inside a magnet **B**-field lines continue by a magnet from a south stick behind to a north.^{[nb 10]} If a **B**-field line enters a magnet somewhere it has to leave somewhere else; it is not authorised to have an finish point. Magnetic poles, therefore, always come in N and S pairs.

More formally, given all a captivating margin lines that enter any given segment contingency also leave that region, subtracting a ‘number’^{[nb 11]} of margin lines that enter a segment from a series that exit gives equally zero. Mathematically this is homogeneous to:

∮SB⋅dA=0,{displaystyle oint _{S}mathbf {B} cdot mathrm {d} mathbf {A} =0,}

where a constituent is a aspect constituent over a sealed aspect *S* (a sealed aspect is one that totally surrounds a segment with no holes to let any margin lines escape). Since d**A** points outward, a dot product in a constituent is certain for **B**-field indicating out and disastrous for **B**-field indicating in.

There is also a homogeneous differential form of this equation lonesome in Maxwell’s equations below.

### Force between magnets[edit]

The force between dual tiny magnets is utterly formidable and depends on a strength and course of both magnets and a widen and instruction of a magnets family to any other. The force is utterly supportive to rotations of a magnets due to captivating torque. The force on any magnet depends on a captivating impulse and a captivating field^{[nb 12]} of a other.

To know a force between magnets, it is useful to inspect a *magnetic stick model* given above. In this model, a * H-field* of one magnet pushes and pulls on

*both*poles of a second magnet. If this

**H**-field is a same during both poles of a second magnet afterwards there is no net force on that magnet given a force is conflicting for conflicting poles. If, however, a captivating margin of a initial magnet is

*nonuniform*(such as a

**H**nearby one of a poles), any stick of a second magnet sees a conflicting margin and is theme to a conflicting force. This disproportion in a dual army moves a magnet in a instruction of augmenting captivating margin and might also means a net torque.

This is a specific instance of a ubiquitous order that magnets are captivated (or repulsed depending on a course of a magnet) into regions of aloft captivating field. Any non-uniform captivating field, possibly caused by permanent magnets or electric currents, exerts a force on a tiny magnet in this way.

The sum of a Amperian loop indication are conflicting and some-more formidable yet furnish a same result: that captivating dipoles are attracted/repelled into regions of aloft captivating field. Mathematically, a force on a tiny magnet carrying a captivating impulse **m** due to a captivating margin **B** is:^{[20]}

F=∇(m⋅B),{displaystyle mathbf {F} =mathbf {nabla } left(mathbf {m} cdot mathbf {B} right),}

where a slope **∇** is a change of a apportion **m** · **B** per territory widen and a instruction is that of extent boost of **m** · **B**. To know this equation, note that a dot product **m** · **B** = *mB*cos(*θ*), where *m* and *B* paint a bulk of a **m** and **B** vectors and *θ* is a angle between them. If **m** is in a same instruction as **B** afterwards a dot product is certain and a slope points ‘uphill’ pulling a magnet into regions of aloft **B**-field (more particularly incomparable **m** · **B**). This equation is particularly customarily stream for magnets of 0 size, yet is mostly a good estimation for not too vast magnets. The captivating force on incomparable magnets is dynamic by dividing them into smaller regions any carrying their possess **m** afterwards summing adult a army on any of these really tiny regions.

### Magnetic torque on permanent magnets[edit]

If dual like poles of dual apart magnets are brought nearby any other, and one of a magnets is authorised to turn, it soon rotates to align itself with a first. In this example, a captivating margin of a still magnet creates a *magnetic torque* on a magnet that is giveaway to rotate. This captivating torque **τ** tends to align a magnet’s poles with a captivating margin lines. A compass, therefore, turns to align itself with Earth’s captivating field.

Magnetic torque is used to expostulate electric motors. In one facile engine design, a magnet is firm to a openly rotating missile and subjected to a captivating margin from an array of electromagnets. By invariably switching a electric stream by any of a electromagnets, thereby flipping a polarity of their captivating fields, like poles are kept subsequent to a rotor; a following torque is eliminated to a shaft. See Rotating captivating fields below.

As is a box for a force between magnets, a captivating stick indication leads some-more straightforwardly to a scold equation. Here, dual equal and conflicting captivating charges experiencing a same **H** also knowledge equal and conflicting forces. Since these equal and conflicting army are in conflicting locations, this produces a torque proportional to a widen (perpendicular to a force) between them. With a clarification of **m** as a stick strength times a widen between a poles, this leads to *τ* = *μ*_{0}*mH*sin*θ*, where *μ*_{0} is a consistent called a opening permeability, measuring 6993400000000000000♠4π×10^{−7} V·s/(A·m) and *θ* is a angle between **H** and **m**.

The Amperian loop indication also predicts a same captivating torque. Here, it is a **B** margin interacting with a Amperian stream loop by a Lorentz force described below. Again, a formula are a same nonetheless a models are totally different.

Mathematically, a torque **τ** on a tiny magnet is proportional both to a practical captivating margin and to a captivating impulse **m** of a magnet:

τ=m×B=μ0m×H,{displaystyle {boldsymbol {tau }}=mathbf {m} times mathbf {B} =mu _{0}mathbf {m} times mathbf {H} ,,}

where × represents a matrix cranky product. Note that this equation includes all of a qualitative information enclosed above. There is no torque on a magnet if **m** is in a same instruction as a captivating field. (The cranky product is 0 for dual vectors that are in a same direction.) Further, all other orientations feel a torque that twists them toward a instruction of captivating field.

## Magnetic margin and electric currents[edit]

Currents of electric charges both beget a captivating margin and feel a force due to captivating B-fields.

### Magnetic margin due to relocating charges and electric currents[edit]

All relocating charged particles furnish captivating fields. Moving indicate charges, such as electrons, furnish formidable yet good famous captivating fields that count on a charge, velocity, and acceleration of a particles.^{[21]}

Magnetic margin lines form in concentric circles around a cylindrical current-carrying conductor, such as a length of wire. The instruction of such a captivating margin can be dynamic by regulating a “right palm hold rule” (see figure during right). The strength of a captivating margin decreases with widen from a wire. (For an gigantic length handle a strength is inversely proportional to a distance.)

Bending a current-carrying handle into a loop concentrates a captivating margin inside a loop while weakening it outside. Bending a handle into churned closely spaced loops to form a twist or “solenoid” enhances this effect. A device so shaped around an iron core might act as an *electromagnet*, generating a strong, well-controlled captivating field. An forever prolonged cylindrical electromagnet has a uniform captivating margin inside, and no captivating margin outside. A calculable length electromagnet produces a captivating margin that looks matching to that assembled by a uniform permanent magnet, with a strength and polarity dynamic by a stream issuing by a coil.

The captivating margin generated by a plain stream I (a consistent upsurge of electric charges, in that assign conjunction accumulates nor is depleted during any point)^{[nb 13]} is described by a *Biot–Savart law*:

B=μ0I4π∫wiredℓ×r^r2,{displaystyle mathbf {B} ={frac {mu _{0}I}{4pi }}int _{mathrm {wire} }{frac {mathrm {d} {boldsymbol {ell }}times mathbf {hat {r}} }{r^{2}}},}

where a constituent sums over a handle length where matrix d**ℓ** is a matrix line member with instruction in a same clarity as a stream *I*, *μ*_{0} is a captivating constant, *r* is a widen between a plcae of d**ℓ** and a plcae where a captivating margin is calculated, and **r̂** is a territory matrix in a instruction of **r**.

A somewhat some-more general^{[22]}^{[nb 14]} approach of relating a stream I{displaystyle {I}} to a **B**-field is by Ampère’s law:

∮B⋅dℓ=μ0Ienc,{displaystyle oint mathbf {B} cdot mathrm {d} {boldsymbol {ell }}=mu _{0}I_{mathrm {enc} },}

where a line constituent is over any capricious loop and I{displaystyle {I}}_{enc} is a stream enclosed by that loop. Ampère’s law is always stream for plain currents and can be used to calculate a **B**-field for certain rarely symmetric situations such as an gigantic handle or an gigantic solenoid.

In a mutated form that accounts for time varying electric fields, Ampère’s law is one of 4 Maxwell’s equations that news electricity and magnetism.

### Force on relocating charges and current[edit]

#### Force on a charged particle[edit]

A charged molecule relocating in a **B**-field use a *sideways* force that is proportional to a strength of a captivating field, a member of a quickness that is perpendicular to a captivating margin and a assign of a particle. This force is famous as a *Lorentz force*, and is given by

F=qv×B,{displaystyle mathbf {F} =qmathbf {v} times mathbf {B} ,}

where **F** is a force, *q* is a electric assign of a particle, **v** is a immediate quickness of a particle, and **B** is a captivating margin (in teslas).

The Lorentz force is always perpendicular to both a quickness of a molecule and a captivating margin that sum it. When a charged molecule moves in a immobile captivating field, it traces a scrolled trail in that a wind pivot is together to a captivating field, and in that a speed of a molecule stays constant. Because a captivating force is always perpendicular to a motion, a captivating margin can do no work on an removed charge. It can customarily do work indirectly, around a electric margin generated by a changing captivating field. It is mostly claimed that a captivating force can do work to a non-elementary captivating dipole, or to charged particles whose suit is compelled by other forces, yet this is incorrect^{[23]} given a work in those cases is achieved by a electric army of a charges deflected by a captivating field.

#### Force on current-carrying wire[edit]

The force on a stream carrying handle is matching to that of a relocating assign as approaching given a assign carrying handle is a collection of relocating charges. A current-carrying handle feels a force in a participation of a captivating field. The Lorentz force on a perceivable stream is mostly referred to as a *Laplace force*. Consider a conductor of length *ℓ*, cranky territory *A*, and assign *q* due to electric stream *i*. If this conductor is placed in a captivating margin of bulk *B* that creates an angle θ with a quickness of charges in a conductor, a force exerted on a singular assign *q* is

F=qvBsinθ,{displaystyle F=qvBsin theta ,}

so, for *N* charges where

N=nℓA{displaystyle N=nell A},

the force exerted on a conductor is

f=FN=qvBnℓAsinθ=Biℓsinθ{displaystyle f=FN=qvBnell Asin theta =Biell impiety theta },

where *i* = *nqvA*.

#### Direction of force[edit]

The instruction of force on a assign or a stream can be dynamic by a mnemonic famous as a *right-hand rule* (see a figure). Using a right palm and indicating a ride in a instruction of a relocating certain assign or certain stream and a fingers in a instruction of a captivating margin a ensuing force on a assign points outwards from a palm. The force on a negatively charged molecule is in a conflicting direction. If both a speed and a assign are topsy-turvy afterwards a instruction of a force stays a same. For that reason a captivating margin dimensions (by itself) can't heed possibly there is a certain assign relocating to a right or a disastrous assign relocating to a left. (Both of these cases furnish a same current.) On a other hand, a captivating margin sum with an electric margin *can* heed between these, see Hall outcome below.

An choice mnemonic to a right palm order Flemings’s left palm rule.

## Relation between H and B[edit]

The formulas subsequent for a captivating margin above are scold when traffic with a whole current. A captivating member placed inside a captivating field, though, generates a possess firm current, that can be a plea to calculate. (This firm stream is due to a sum of atomic sized stream loops and a spin of a subatomic particles such as electrons that make adult a material.) The **H**-field as tangible above helps means out this firm current; yet to see how, it helps to deliver a judgment of *magnetization* first.

### Magnetization[edit]

The *magnetization* matrix margin **M** represents how strongly a segment of member is magnetized. It is tangible as a net captivating dipole impulse per territory volume of that region. The magnetization of a uniform magnet is therefore a member constant, equal to a captivating impulse **m** of a magnet divided by a volume. Since a SI territory of captivating impulse is A·m^{2}, a SI territory of magnetization **M** is ampere per meter, matching to that of a **H**-field.

The magnetization **M** margin of a segment points in a instruction of a normal captivating dipole impulse in that region. Magnetization margin lines, therefore, start nearby a captivating south stick and ends nearby a captivating north pole. (Magnetization does not exist outmost of a magnet.)

In a Amperian loop model, a magnetization is due to mixing many tiny Amperian loops to form a following stream called *bound current*. This firm current, then, is a source of a captivating **B** margin due to a magnet. (See Magnetic dipoles subsequent and captivating poles vs. atomic currents for some-more information.) Given a clarification of a captivating dipole, a magnetization margin follows a matching law to that of Ampere’s law:^{[24]}

∮M⋅dℓ=Ib,{displaystyle oint mathbf {M} cdot mathrm {d} {boldsymbol {ell }}=I_{mathrm {b} },}

where a constituent is a line constituent over any sealed loop and *I*_{b} is a ‘bound current’ enclosed by that sealed loop.

In a captivating stick model, magnetization starts during and ends during captivating poles. If a given region, therefore, has a net certain ‘magnetic stick strength’ (corresponding to a north pole) afterwards it has some-more magnetization margin lines entering it than withdrawal it. Mathematically this is homogeneous to:

∮Sμ0M⋅dA=−qM{displaystyle oint _{S}mu _{0}mathbf {M} cdot mathrm {d} mathbf {A} =-q_{mathrm {M} }},

where a constituent is a sealed aspect constituent over a sealed aspect *S* and *q*_{M} is a ‘magnetic charge’ (in units of captivating flux) enclosed by *S*. (A sealed aspect totally surrounds a segment with no holes to let any margin lines escape.) The disastrous pointer occurs given a magnetization margin moves from south to north.

### H-field and captivating materials[edit]

In SI units, a H-field is compared to a B-field by

H ≡ Bμ0−M.{displaystyle mathbf {H} equiv {frac {mathbf {B} }{mu _{0}}}-mathbf {M} .}

In terms of a H-field, Ampere’s law is

∮H⋅dℓ=∮(Bμ0−M)⋅dℓ=Itot−Ib=If,{displaystyle 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 a ‘free current’ enclosed by a loop so that a line constituent of **H** does not count during all on a firm currents.^{[25]}

For a differential homogeneous of this equation see Maxwell’s equations. Ampere’s law leads to a operation condition

(H1∥−H2∥)=Kf×n^,{displaystyle left(mathbf {H_{1}^{parallel }} -mathbf {H_{2}^{parallel }} right)=mathbf {K} _{mathrm {f} }times {hat {mathbf {n} }},}

where **K**_{f} is a aspect giveaway stream firmness and a territory normal n^{displaystyle {hat {mathbf {n} }}} points in a instruction from middle 2 to middle 1.^{}[26]

Similarly, a aspect constituent of **H** over any sealed aspect is eccentric of a giveaway currents and picks out a “magnetic charges” within that sealed surface:

∮Sμ0H⋅dA=∮S(B−μ0M)⋅dA=0−(−qM)=qM,{displaystyle 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 count on a giveaway currents.

The **H**-field, therefore, can be detached into two^{[nb 15]} eccentric parts:

H=H0+Hd,{displaystyle mathbf {H} =mathbf {H} _{0}+mathbf {H} _{mathrm {d} },,}

where **H**_{0} is a practical captivating margin due customarily to a giveaway currents and **H**_{d} is a demagnetizing margin due customarily to a firm currents.

The captivating **H**-field, therefore, re-factors a firm stream in terms of “magnetic charges”. The **H** margin lines loop customarily around ‘free current’ and, distinct a captivating **B** field, starts and ends nearby captivating poles as well.

### Magnetism[edit]

Most materials respond to an practical **B**-field by producing their possess magnetization **M** and therefore their possess **B**-field. Typically, a response is diseased and exists customarily when a captivating margin is applied. The tenure *magnetism* describes how materials respond on a tiny spin to an practical captivating margin and is used to specify a captivating proviso of a material. Materials are divided into groups shaped on their captivating behavior:

- Diamagnetic materials
^{[27]}furnish a magnetization that opposes a captivating field. - Paramagnetic materials
^{[27]}furnish a magnetization in a same instruction as a practical captivating field. - Ferromagnetic materials and a closely compared ferrimagnetic materials and antiferromagnetic materials
^{[28]}^{[29]}can have a magnetization eccentric of an practical B-field with a formidable attribute between a dual fields. - Superconductors (and ferromagnetic superconductors)
^{[30]}^{[31]}are materials that are characterized by ideal conductivity subsequent a vicious heat and captivating field. They also are rarely captivating and can be ideal diamagnets subsequent a reduce vicious captivating field. Superconductors mostly have a extended operation of temperatures and captivating fields (the so-named churned state) underneath that they vaunt a formidable hysteretic coherence of**M**on**B**.

In a box of paramagnetism and diamagnetism, a magnetization **M** is mostly proportional to a practical captivating margin such that:

B=μH,{displaystyle mathbf {B} =mu mathbf {H} ,}

where *μ* is a member contingent parameter called a permeability. In some cases a permeability might be a second arrange tensor so that **H** might not indicate in a same instruction as **B**. These family between **B** and **H** are examples of constitutive equations. However, superconductors and ferromagnets have a some-more formidable **B** to **H** relation; see captivating hysteresis.

## Energy stored in captivating fields[edit]

Energy is indispensable to beget a captivating margin both to work conflicting a electric margin that a changing captivating margin creates and to change a magnetization of any member within a captivating field. For non-dispersive materials this same appetite is expelled when a captivating margin is broken so that this appetite can be modeled as being stored in a captivating field.

For linear, non-dispersive, materials (such that **B** = *μ***H** where *μ* is frequency-independent), a appetite firmness is:

u=B⋅H2=B⋅B2μ=μH⋅H2.{displaystyle u={frac {mathbf {B} cdot mathbf {H} }{2}}={frac {mathbf {B} cdot mathbf {B} }{2mu }}={frac {mu mathbf {H} cdot mathbf {H} }{2}}.}

If there are no captivating materials around afterwards *μ* can be transposed by *μ*_{0}. The above equation can't be used for nonlinear materials, though; a some-more ubiquitous countenance given subsequent contingency be used.

In general, a incremental volume of work per territory volume *δW* indispensable to means a tiny change of captivating margin *δ***B** is:

δW=H⋅δB.{displaystyle delta W=mathbf {H} cdot delta mathbf {B} .}

Once a attribute between **H** and **B** is famous this equation is used to establish a work indispensable to strech a given captivating state. For hysteretic materials such as ferromagnets and superconductors, a work indispensable also depends on how a captivating margin is created. For linear non-dispersive materials, though, a ubiquitous equation leads directly to a easier appetite firmness equation given above.

## Electromagnetism: a attribute between captivating and electric fields[edit]

### Faraday’s Law: Electric force due to a changing B-field[edit]

A changing captivating field, such as a magnet relocating by a conducting coil, generates an electric margin (and therefore tends to expostulate a stream in such a coil). This is famous as *Faraday’s law* and forms a basement of many electrical generators and electric motors.

Mathematically, Faraday’s law is:

E=−dΦmdt,{displaystyle {mathcal {E}}=-{frac {mathrm {d} Phi _{mathrm {m} }}{mathrm {d} t}},}

where E{displaystyle scriptstyle {mathcal {E}}} is a electromotive force (or *EMF*, a voltage generated around a sealed loop) and Φ_{m} is a *magnetic flux*—the product of a area times a captivating margin normal to that area. (This clarification of captivating suit is given **B** is mostly referred to as *magnetic suit density*.)^{[32]}^{:210}

The disastrous pointer represents a fact that any stream generated by a changing captivating margin in a twist produces a captivating margin that *opposes* a *change* in a captivating margin that prompted it. This materialisation is famous as Lenz’s law.

This constituent plan of Faraday’s law can be converted^{[nb 16]} into a differential form, that relates underneath somewhat conflicting conditions. This form is lonesome as one of Maxwell’s equations below.

### Maxwell’s improvement to Ampère’s Law: The captivating margin due to a changing electric field[edit]

Similar to a approach that a changing captivating margin generates an electric field, a changing electric margin generates a captivating field. This fact is famous as *Maxwell’s improvement to Ampère’s law* and is practical as an further tenure to Ampere’s law as given above. This additional tenure is proportional to a time rate of change of a electric suit and is matching to Faraday’s law above yet with a conflicting and certain consistent out front. (The electric suit by an area is proportional to a area times a perpendicular prejudiced of a electric field.)

The full law including a improvement tenure is famous as a Maxwell–Ampère equation. It is not ordinarily given in constituent form given a outcome is so tiny that it can typically be abandoned in many cases where a constituent form is used.

The Maxwell tenure *is* critically vicious in a origination and propagation of electromagnetic waves. Maxwell’s improvement to Ampère’s Law together with Faraday’s law of initiation describes how jointly changing electric and captivating fields correlate to means any other and so to form electromagnetic waves, such as light: a changing electric margin generates a changing captivating field, that generates a changing electric margin again. These, though, are customarily described regulating a differential form of this equation given below.

### Maxwell’s equations[edit]

Like all matrix fields, a captivating margin has dual vicious mathematical properties that relates it to a *sources*. (For **B** a *sources* are currents and changing electric fields.) These dual properties, along with a dual homogeneous properties of a electric field, make adult *Maxwell’s Equations*. Maxwell’s Equations together with a Lorentz force law form a finish outline of exemplary electrodynamics including both electricity and magnetism.

The initial skill is a dissimilarity of a matrix margin **A**, **∇** · **A**, that represents how **A** ‘flows’ outmost from a given point. As discussed above, a **B**-field line never starts or ends during a indicate yet instead forms a finish loop. This is mathematically homogeneous to observant that a dissimilarity of **B** is zero. (Such matrix fields are called solenoidal matrix fields.) This skill is called Gauss’s law for draw and is homogeneous to a matter that there are no removed captivating poles or captivating monopoles. The electric margin on a other palm starts and ends during electric charges so that a dissimilarity is non-zero and proportional to a assign firmness (See Gauss’s law).

The second mathematical skill is called a curl, such that **∇** × **A** represents how **A** curls or ‘circulates’ around a given point. The outcome of a twist is called a ‘circulation source’. The equations for a twist of **B** and of **E** are called a Ampère–Maxwell equation and Faraday’s law respectively. They paint a differential forms of a constituent equations given above.

The finish set of Maxwell’s equations afterwards are:

∇⋅B=0,∇⋅E=ρε0,∇×B=μ0J+μ0ε0∂E∂t,∇×E=−∂B∂t,{displaystyle {begin{aligned}nabla cdot mathbf {B} =0,\nabla cdot mathbf {E} ={frac {rho }{varepsilon _{0}}},\nabla times mathbf {B} =mu _{0}mathbf {J} +mu _{0}varepsilon _{0}{frac {partial mathbf {E} }{partial t}},\nabla times mathbf {E} =-{frac {partial mathbf {B} }{partial t}},end{aligned}}}

where **J** = finish tiny stream firmness and *ρ* is a assign density.

Technically, **B** is a pseudovector (also called an *axial vector*) due to being tangible by a matrix cranky product. (See diagram.)

As discussed above, materials respond to an practical electric **E** margin and an practical captivating **B** margin by producing their possess inner ‘bound’ assign and stream distributions that minister to **E** and **B** yet are formidable to calculate. To by-pass this problem, **H** and **D** fields are used to re-factor Maxwell’s equations in terms of a *free stream density* **J**_{f} and *free assign density* *ρ*_{f}:

∇⋅B=0,∇⋅D=ρf,∇×H=Jf+∂D∂t,∇×E=−∂B∂t.{displaystyle {begin{aligned}nabla cdot mathbf {B} =0,\nabla cdot mathbf {D} =rho _{mathrm {f} },\nabla times mathbf {H} =mathbf {J} _{mathrm {f} }+{frac {partial mathbf {D} }{partial t}},\nabla times mathbf {E} =-{frac {partial mathbf {B} }{partial t}}.end{aligned}}}

These equations are not any some-more ubiquitous than a bizarre equations (if a ‘bound’ charges and currents in a member are known). They also contingency be supplemented by a attribute between **B** and **H** as good as that between **E** and **D**. On a other hand, for facile relations between these quantities this form of Maxwell’s equations can by-pass a need to calculate a firm charges and currents.

### Electric and captivating fields: conflicting aspects of a same phenomenon[edit]

According to a special speculation of relativity, a assign of a electromagnetic force into apart electric and captivating components is not fundamental, yet varies with a observational support of reference: An electric force viewed by one spectator might be viewed by another (in a conflicting support of reference) as a captivating force, or a reduction of electric and captivating forces.

Formally, special relativity combines a electric and captivating fields into a rank-2 tensor, called a *electromagnetic tensor*. Changing anxiety frames *mixes* these components. This is homogeneous to a approach that special relativity *mixes* space and time into spacetime, and mass, movement and appetite into four-momentum.^{[33]}

### Magnetic matrix potential[edit]

In modernized topics such as quantum mechanics and relativity it is mostly easier to work with a intensity plan of electrodynamics rather than in terms of a electric and captivating fields. In this representation, a *vector potential* **A**, and a scalar intensity *φ*, are tangible such that:

B=∇×A,E=−∇φ−∂A∂t.{displaystyle {begin{aligned}mathbf {B} =nabla times mathbf {A} ,\mathbf {E} =-nabla varphi -{frac {partial mathbf {A} }{partial t}}.end{aligned}}}

The matrix intensity **A** might be interpreted as a *generalized intensity movement per territory charge*^{[34]} usually as *φ* is interpreted as a *generalized intensity appetite per territory charge*.

Maxwell’s equations when voiced in terms of a potentials can be expel into a form that agrees with special relativity with tiny effort.^{[35]} In relativity **A** together with *φ* forms a four-potential, homogeneous to a four-momentum that combines a movement and appetite of a particle. Using a 4 intensity instead of a electromagnetic tensor has a advantage of being many simpler—and it can be simply mutated to work with quantum mechanics.

### Quantum electrodynamics[edit]

In formidable physics, a electromagnetic margin is accepted to be not a *classical* field, yet rather a quantum field; it is represented not as a matrix of 3 numbers during any point, yet as a matrix of 3 quantum operators during any point. The many accurate formidable outline of a electromagnetic communication (and many else) is *quantum electrodynamics* (QED),^{[36]} that is incorporated into a some-more finish speculation famous as a *Standard Model of molecule physics*.

In QED, a bulk of a electromagnetic interactions between charged particles (and their antiparticles) is computed regulating distress theory. These rather formidable formulas furnish a conspicuous impressive illustration as Feynman diagrams in that practical photons are exchanged.

Predictions of QED establish with experiments to an intensely high grade of accuracy: now about 10^{−12} (and singular by initial errors); for sum see indicating tests of QED. This creates QED one of a many accurate earthy theories assembled so far.

All equations in this essay are in a exemplary approximation, that is reduction accurate than a quantum outline mentioned here. However, underneath many bland circumstances, a disproportion between a dual theories is negligible.

## Important uses and examples of captivating field[edit]

### Earth’s captivating field[edit]

The Earth’s captivating margin is suspicion to be assembled by convection currents in a outdoor glass of Earth’s core. The Dynamo speculation proposes that these movements furnish electric currents that, in turn, furnish a captivating field.^{[37]}

The participation of this margin causes a compass, placed anywhere within it, to stagger so that a “north pole” of a magnet in a compass points roughly north, toward Earth’s North Magnetic Pole. This is a normal clarification of a “north pole” of a magnet, nonetheless other homogeneous definitions are also possible.

One difficulty that arises from this clarification is that, if Earth itself is deliberate as a magnet, a *south* stick of that magnet would be a one nearer a north captivating pole, and clamp versa. The north captivating stick is so-named not given of a polarity of a margin there yet given of a geographical location. The north and south poles of a permanent magnet are supposed given they are “north-seeking” and “south-seeking”, respectively.^{[38]}^{[39]}

The figure is a blueprint of Earth’s captivating margin represented by margin lines. For many locations, a captivating margin has a poignant up/down member in further to a north/south component. (There is also an east/west component, as Earth’s captivating and geographical poles do not coincide.) The captivating margin can be visualised as a bar magnet buried low in Earth’s interior.

Earth’s captivating margin is not constant—the strength of a margin and a plcae of a poles vary. Moreover, a poles intermittently retreat their course in a routine called geomagnetic reversal. The many new annulment occurred 780,000 years ago.

### Rotating captivating fields[edit]

The *rotating captivating field* is a pivotal member in a operation of alternating-current motors. A permanent magnet in such a margin rotates so as to say a fixing with a outmost field. This outcome was conceptualized by Nikola Tesla, and after employed in his, and others’, early AC (alternating current) electric motors.

A rotating captivating margin can be assembled regulating dual quadratic coils with 90 degrees proviso disproportion in their AC currents. However, in use such a complement would be granted by a three-wire arrangement with unsymmetrical currents.

This inequality would means critical problems in standardization of a conductor widen and so, to overcome it, three-phase systems are used where a 3 currents are equal in bulk and have 120 degrees proviso difference. Three matching coils carrying mutual geometrical angles of 120 degrees emanate a rotating captivating margin in this case. The ability of a three-phase complement to emanate a rotating field, employed in electric motors, is one of a categorical reasons given three-phase systems browbeat a world’s electrical energy supply systems.

Synchronous motors use DC-voltage-fed rotor windings, that lets a excitation of a appurtenance be controlled—and initiation motors use short-circuited rotors (instead of a magnet) following a rotating captivating margin of a multicoiled stator. The short-circuited turns of a rotor rise eddy currents in a rotating margin of a stator, and these currents in spin pierce a rotor by a Lorentz force.

In 1882, Nikola Tesla identified a judgment of a rotating captivating field. In 1885, Galileo Ferraris exclusively researched a concept. In 1888, Tesla gained U.S. Patent 381,968 for his work. Also in 1888, Ferraris published his investigate in a paper to a *Royal Academy of Sciences* in Turin.

### Hall effect[edit]

The assign carriers of a current-carrying conductor placed in a cross captivating margin knowledge a laterally Lorentz force; this formula in a assign subdivision in a instruction perpendicular to a stream and to a captivating field. The following voltage in that instruction is proportional to a practical captivating field. This is famous as a *Hall effect*.

The *Hall effect* is mostly used to bulk a bulk of a captivating field. It is used as good to find a pointer of a widespread assign carriers in materials such as semiconductors (negative electrons or certain holes).

### Magnetic circuits[edit]

An vicious use of **H** is in *magnetic circuits* where **B** = *μ***H** inside a linear material. Here, *μ* is a captivating permeability of a material. This outcome is matching in form to Ohm’s law **J** = *σ***E**, where **J** is a stream density, *σ* is a conductance and **E** is a electric field. Extending this analogy, a reflection to a perceivable Ohm’s law (*I* = *V*⁄*R*) is:

Φ=FRm,{displaystyle Phi ={frac {F}{R}}_{mathrm {m} },}

where Φ=∫B⋅dA{displaystyle Phi =int mathbf {B} cdot mathrm {d} mathbf {A} } is a captivating suit in a circuit, F=∫H⋅dℓ{displaystyle F=int mathbf {H} cdot mathrm {d} {boldsymbol {ell }}} is a magnetomotive force practical to a circuit, and *R*_{m} is a hostility of a circuit. Here a hostility *R*_{m} is a apportion matching in inlet to insurgency for a flux.

Using this analogy it is candid to calculate a captivating suit of formidable captivating margin geometries, by regulating all a accessible techniques of circuit theory.

### Magnetic margin figure descriptions[edit]

- An
*azimuthal*captivating margin is one that runs east–west. - A
*meridional*captivating margin is one that runs north–south. In a solar hustler indication of a Sun, differential revolution of a solar plasma causes a meridional captivating margin to widen into an azimuthal captivating field, a routine called a*omega-effect*. The retreat routine is called a*alpha-effect*.^{[40]} - A
*dipole*captivating margin is one seen around a bar magnet or around a charged facile molecule with nonzero spin. - A
*quadrupole*captivating margin is one seen, for example, between a poles of 4 bar magnets. The margin strength grows linearly with a radial widen from a longitudinal axis. - A
*solenoidal*captivating margin is matching to a dipole captivating field, solely that a plain bar magnet is transposed by a vale electromagnetic twist magnet. - A
*toroidal*captivating margin occurs in a doughnut-shaped coil, a electric stream spiraling around a tube-like surface, and is found, for example, in a tokamak. - A
*poloidal*captivating margin is generated by a stream issuing in a ring, and is found, for example, in a tokamak. - A
*radial*captivating margin is one in that margin lines are destined from a core outwards, matching to a spokes in a bicycle wheel. An instance can be found in a loudspeaker transducers (driver).^{[41]} - A
*helical*captivating margin is corkscrew-shaped, and infrequently seen in space plasmas such as a Orion Molecular Cloud.^{[42]}

### Magnetic dipoles[edit]

The captivating margin of a captivating dipole is decorated in a figure. From outside, a ideal captivating dipole is matching to that of an ideal electric dipole of a same strength. Unlike a electric dipole, a captivating dipole is scrupulously modeled as a stream loop carrying a stream *I* and an area *a*. Such a stream loop has a captivating impulse of:

m=Ia,{displaystyle m=Ia,,}

where a instruction of **m** is perpendicular to a area of a loop and depends on a instruction of a stream regulating a right-hand rule. An ideal captivating dipole is modeled as a genuine captivating dipole whose area *a* has been reduced to 0 and a stream *I* increasing to forever such that a product *m* = *Ia* is finite. This indication clarifies a tie between bony movement and captivating moment, that is a basement of a Einstein–de Haas outcome *rotation by magnetization* and a inverse, a Barnett outcome or *magnetization by rotation*.^{[43]} Rotating a loop faster (in a same direction) increases a stream and therefore a captivating moment, for example.

It is infrequently useful to indication a captivating dipole matching to a electric dipole with dual equal yet conflicting captivating charges (one south a other north) detached by widen *d*. This indication produces an **H**-field not a **B**-field. Such a indication is deficient, though, both in that there are no captivating charges and in that it obscures a couple between electricity and magnetism. Further, as discussed above it fails to explain a fundamental tie between bony movement and magnetism.

### Magnetic monopole (hypothetical)[edit]

A *magnetic monopole* is a suppositious molecule (or category of particles) that has, as a name suggests, customarily one captivating stick (either a north stick or a south pole). In other words, it would possess a “magnetic charge” homogeneous to an electric charge. Magnetic margin lines would start or finish on captivating monopoles, so if they exist, they would give exceptions to a order that captivating margin lines conjunction start nor end.

Modern seductiveness in this judgment stems from molecule theories, particularly Grand Unified Theories and superstring theories, that envision possibly a existence, or a possibility, of captivating monopoles. These theories and others have desirous endless efforts to hunt for monopoles. Despite these efforts, no captivating monopole has been celebrated to date.^{[nb 17]}

In new research, materials famous as spin ices can copy monopoles, yet do not enclose tangible monopoles.^{[44]}^{[45]}

## See also[edit]

### General[edit]

- Magnetohydrodynamics – a investigate of a dynamics of electrically conducting fluids
- Magnetic nanoparticles – intensely tiny captivating particles that are tens of atoms wide
- Magnetic reconnection – an outcome that causes solar flares and auroras
- Magnetic potential – a matrix and scalar intensity illustration of magnetism
- SI electromagnetism units – common units used in electromagnetism
- Orders of bulk (magnetic field) – list of captivating margin sources and dimensions inclination from smallest captivating fields to largest detected
- Upward continuation

### Mathematics[edit]

- Magnetic helicity – border to that a captivating margin wraps around itself

### Applications[edit]

- Dynamo theory – a due resource for a origination of a Earth’s captivating field
- Helmholtz coil – a device for producing a segment of scarcely uniform captivating field
- Magnetic margin observation film – Film used to perspective a captivating margin of an area
- Maxwell coil – a device for producing a vast volume of an roughly consistent captivating field
- Stellar captivating field – a contention of a captivating margin of stars
- Teltron tube – device used to arrangement an nucleus lamp and demonstrates outcome of electric and captivating fields on relocating charges

## Notes[edit]

**^**Strictly speaking, a captivating margin is a pseudo vector; pseudo-vectors, that also embody torque and rotational velocity, are matching to vectors solely that they sojourn unvaried when a coordinates are inverted.**^**His*Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete*, that is mostly condensed to*Epistola de magnete*, is antiquated 1269 C.E.**^**From a outside, a margin of a dipole of captivating assign has accurately a same form as a stream loop when both are amply small. Therefore, a dual models differ customarily for draw inside captivating material.**^**The letters B and H were creatively selected by Maxwell in his*Treatise on Electricity and Magnetism*(Vol. II, pp. 236–237). For many quantities, he simply started selecting letters from a commencement of a alphabet. See Ralph 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 formidable writers who provide*In a matching vein, M Gerloch (1983).**B**as a primary margin feel thankful to call it a captivating initiation given a name captivating margin was historically preempted by**H**. This seems awkward and pedantic. If we go into a laboratory and ask a physicist what causes a pion trajectories in his burble cover to curve, he’ll substantially answer “magnetic field”, not “magnetic induction.” You will occasionally hear a geophysicist impute to a Earth’s captivating induction, or an astrophysicist speak about a captivating initiation of a galaxy. We introduce to keep on job**B**a captivating field. As for**H**, nonetheless other names have been invented for it, we shall call it “the margin**H**” or even “the captivating margin**H**.”*Magnetism and Ligand-field Analysis*. Cambridge University Press. p. 110. ISBN 0-521-24939-2. says: “So we might consider of both**B**and**H**as captivating fields, yet dump a word ‘magnetic’ from**H**so as to say a eminence … As Purcell points out, ‘it is customarily a names that give trouble, not a symbols’.”**^**This can be seen from a captivating prejudiced of a Lorentz force law*F*=*qvB*sinθ.**^**The use of iron filings to arrangement a margin presents something of an disproportion to this picture; a filings change a captivating margin so that it is many incomparable along a “lines” of iron, due to a vast permeability of iron family to air.**^**Here ‘small’ means that a spectator is amply distant divided that it can be treated as being infinitesimally small. ‘Larger’ magnets need to embody some-more formidable terms in a countenance and count on a whole geometry of a magnet not usually**m**.**^**Magnetic margin lines might also hang around and around yet shutting yet also yet ending. These some-more formidable non-closing non-ending captivating margin lines are moot, though, given a captivating margin of objects that furnish them are distributed by adding a captivating fields of ‘elementary parts’ carrying captivating margin lines that do form sealed curves or extend to infinity.**^**To see that this contingency be loyal suppose fixation a compass inside a magnet. There, a north stick of a compass points toward a north stick of a magnet given magnets built on any other indicate in a same direction.**^**As discussed above, captivating margin lines are essentially a unpractical apparatus used to paint a arithmetic behind captivating fields. The sum ‘number’ of margin lines is contingent on how a margin lines are drawn. In practice, constituent equations such as a one that follows in a categorical content are used instead.**^**Either**B**or**H**might be used for a captivating margin outmost of a magnet.**^**In practice, a Biot–Savart law and other laws of magnetostatics are mostly used even when a stream change in time, as prolonged as it does not change too quickly. It is mostly used, for instance, for customary domicile currents, that teeter sixty times per second.**^**The Biot–Savart law contains a additional limitation (boundary condition) that a B-field contingency go to 0 quick adequate during infinity. It also depends on a dissimilarity of**B**being zero, that is always valid. (There are no captivating charges.)**^**A third tenure is indispensable for changing electric fields and polarization currents; this banishment stream tenure is lonesome in Maxwell’s equations below.**^**A finish countenance for Faraday’s law of initiation in terms of a electric**E**and captivating fields can be sum as: E=−dΦmdt{displaystyle textstyle {mathcal {E}}=-{frac {dPhi _{m}}{dt}}} =∮∂Σ(t)(E(r, t)+v×B(r, t))⋅dℓ {displaystyle textstyle =oint _{partial Sigma (t)}left(mathbf {E} (mathbf {r} , t)+mathbf {vtimes B} (mathbf {r} , t)right)cdot d{boldsymbol {ell }} } =−ddt∬Σ(t)dA⋅B(r, t),{displaystyle textstyle =-{frac {d}{dt}}iint _{Sigma (t)}d{boldsymbol {A}}cdot mathbf {B} (mathbf {r} , t),} where**∂Σ**(*t*) is a relocating sealed trail bounding a relocating aspect**Σ**(*t*), and d**A**is an member of aspect area of**Σ**(*t*). The initial constituent calculates a work finished relocating a assign a widen d**ℓ**shaped on a Lorentz force law. In a box where a bounding aspect is stationary, a Kelvin–Stokes postulate can be used to uncover this equation is homogeneous to a Maxwell–Faraday equation.**^**Two experiments assembled claimant events that were primarily interpreted as monopoles, yet these are now deliberate inconclusive. For sum and references, see captivating monopole.

## References[edit]

**^**Jiles, David C. (1998).*Introduction to Magnetism and Magnetic Materials*(2 ed.). CRC. p. 3. ISBN 0412798603.**^**Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew (1964).*The Feynman Lectures on Physics*.**2**. California Institute of Technology. pp. 1.7–1.8. ISBN 0465079989.**^**Whittaker 1951, p. 34**^**Whittaker 1951, p. 56**^**Whittaker 1951, p. 59**^**Whittaker 1951, p. 64**^**Whittaker 1951, p. 88**^**Whittaker 1951, p. 222**^**Whittaker 1951, p. 244**^**Kelvin (1900). “Kabinett physikalischer Raritäten.” Page 200**^**Lord Kelvin of Largs. physik.uni-augsburg.de. 26 Jun 1824- ^
^{a}^{b}E. J. Rothwell and M. J. Cloud (2010)*Electromagnetics*. Taylor Francis. p. 23. ISBN 1420058266. **^**R.P. Feynman; R.B. Leighton; M. Sands (1963).*The Feynman Lectures on Physics, volume 2*.**^**Purcell, E. (2011).*Electricity and Magnetism*(2nd ed.). Cambridge University Press. pp. 173–4. ISBN 1107013607.**^**“Geomagnetism Frequently Asked Questions”. National Geophysical Data Center. Retrieved 21 October 2013.**^**“International complement of units (SI)”.*NIST anxiety on constants, units, and uncertainty*. National Institute of Standards and Technology. Retrieved 9 May 2012.**^**“Gravity Probe B Executive Summary” (PDF). pp. 10, 21.**^**“With record captivating fields to a 21st Century”.*IEEE Xplore*.**^**Kouveliotou, C.; Duncan, R. C.; Thompson, C. (February 2003). “Magnetars”.*Scientific American*; Page 36.**^**See Eq. 11.42 in E. Richard Cohen; David R. Lide; George L. Trigg (2003).*AIP production table reference*(3 ed.). Birkhäuser. p. 381. ISBN 0-387-98973-0.**^**Griffiths 1999, p. 438**^**Griffiths 1999, pp. 222–225**^**Deissler, R.J. (2008). “Dipole in a captivating 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.**^**Griffiths 1999, pp. 266–268**^**John Clarke Slater; Nathaniel Herman Frank (1969).*Electromagnetism*(first published in 1947 ed.). Courier Dover Publications. p. 69. ISBN 0-486-62263-0.**^**Griffiths 1999, p. 332- ^
^{a}^{b}RJD Tilley (2004).*Understanding Solids*. Wiley. p. 368. ISBN 0-470-85275-5. **^**Sōshin Chikazumi; Chad D. Graham (1997).*Physics of ferromagnetism*(2 ed.). Oxford University Press. p. 118. ISBN 0-19-851776-9.**^**Amikam Aharoni (2000).*Introduction to a speculation of ferromagnetism*(2 ed.). Oxford University Press. p. 27. ISBN 0-19-850808-5.**^**M Brian Maple; et al. (2008). “Unconventional superconductivity in novel materials”. In K. H. Bennemann; John B. Ketterson.*Superconductivity*. Springer. p. 640. ISBN 3-540-73252-7.**^**Naoum Karchev (2003). “Itinerant ferromagnetism and superconductivity”. In Paul S. Lewis; D. Di (CON) Castro.*Superconductivity investigate during a heading edge*. Nova Publishers. p. 169. ISBN 1-59033-861-8.**^**Jackson, John David (1975).*Classical electrodynamics*(2nd ed.). New York: Wiley. ISBN 9780471431329.**^**C. Doran and A. Lasenby (2003)*Geometric Algebra for Physicists*, Cambridge University Press, p. 233. ISBN 0521715954.**^**E. J. Konopinski (1978). “What a electromagnetic matrix intensity describes”.*Am. J. Phys*.**46**(5): 499–502. Bibcode:1978AmJPh..46..499K. doi:10.1119/1.11298.**^**Griffiths 1999, p. 422**^**For a good qualitative introduction see: Richard Feynman (2006).*QED: a bizarre speculation of light and matter*. Princeton University Press. ISBN 0-691-12575-9.**^**Yahreas Herbert (June 1954). “What creates a earth Wobble”.*Popular Science*. New York: Godfrey Hammond: 266.**^**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.**^**Ron Kurtus (2004). “Magnets”.*School for champions: Physics topics*. Retrieved 17 July 2010.**^**The Solar Dynamo. Retrieved 15 Sep 2007.**^**I. S. Falconer and M. I. Large (edited by I. M. Sefton), “Magnetism: Fields and Forces” Lecture E6, The University of Sydney. Retrieved 3 Oct 2008**^**Robert Sanders (12 Jan 2006) “Astronomers find captivating Slinky in Orion“, UC Berkeley.**^**See captivating impulse and B. D. Cullity; C. D. Graham (2008).*Introduction to Magnetic Materials*(2 ed.). Wiley-IEEE. p. 103. ISBN 0-471-47741-9.**^**“‘Magnetricity’ Observed And Measured For First Time”.*Science Daily*. 15 Oct 2009. Retrieved 10 June 2010.**^**M.J.P. Gingras (2009). “Observing Monopoles in a Magnetic Analog of Ice”.*Science*.**326**(5951): 375–376. doi:10.1126/science.1181510. PMID 19833948.

## Further reading[edit]

- Durney, Carl H. Johnson, Curtis C. (1969).
*Introduction to formidable electromagnetics*. McGraw-Hill. ISBN 0-07-018388-0. - Furlani, Edward P. (2001).
*Permanent Magnet and Electromechanical Devices: Materials, Analysis and Applications*. Academic Press Series in Electromagnetism. ISBN 0-12-269951-3. OCLC 162129430. - Griffiths, David J. (1999).
*Introduction to Electrodynamics*(3rd ed.). Prentice Hall. p. 438. ISBN 0-13-805326-X. OCLC 40251748. - Jiles, David (1994).
*Introduction to Electronic Properties of Materials*(1st ed.). Springer. ISBN 0-412-49580-5. - Kraftmakher, Yaakov (2001). “Two experiments with rotating captivating field”.
*Eur. J. Phys*.**22**(5): 477–482. Bibcode:2001EJPh…22..477K. doi:10.1088/0143-0807/22/5/302. - Melle, Sonia; Rubio, Miguel A.; Fuller, Gerald G. (2000). “Structure and dynamics of magnetorheological fluids in rotating captivating fields”.
*Phys. Rev. E*.**61**(4): 4111–4117. Bibcode:2000PhRvE..61.4111M. doi:10.1103/PhysRevE.61.4111. - Rao, Nannapaneni N. (1994).
*Elements of engineering electromagnetics (4th ed.)*. Prentice Hall. ISBN 0-13-948746-8. OCLC 221993786. - Mielnik, Bogdan; FernáNdez c., David J. Fernández C. (1989). “An nucleus trapped in a rotating captivating field”.
*Journal of Mathematical Physics*.**30**(2): 537–549. Bibcode:1989JMP….30..537M. doi:10.1063/1.528419. - Thalmann, Julia K. (2010).
*Evolution of Coronal Magnetic Fields*. uni-edition. ISBN 978-3-942171-41-0. - Tipler, Paul (2004).
*Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.)*. W. H. Freeman. ISBN 0-7167-0810-8. OCLC 51095685. - Whittaker, E. T. (1951).
*A History of a Theories of Aether and Electricity*. Dover Publications. p. 34. ISBN 0-486-26126-3.

## External links[edit]

Article source: https://en.wikipedia.org/wiki/Magnetic_field

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