Magnetic field

Electromagnetism

Electricity · Magnetism

Magnetic field lines shown by iron filings.The high permeability of individual iron fil-ings causes the magnetic field to be larger atthe ends of the filings. This causes individualfilings to attract each other, forming elong-ated clusters that trace out the appearance oflines. It would not be expected that these"lines" be precisely accurate field lines forthis magnet; rather, the magnetization of theiron itself would be expected to alter the fieldsomewhat.

A magnetic field is a vector field whichsurrounds magnets and electric currents, andis detected by the force it exerts on movingelectric charges and on magnetic materials.When placed in a magnetic field, magneticdipoles tend to align their axes parallel to themagnetic field. Magnetic fields also havetheir own energy with an energy density pro-portional to the square of the field intensity.

For the physics of magnetic materials, seemagnetism and magnet, and more specific-ally ferromagnetism, paramagnetism, anddiamagnetism. For constant magnetic fields,such as are generated by magnetic materialsand steady currents, see magnetostatics. Achanging electric field results in a magneticfield, and a changing magnetic field also

generates a electric field (seeelectromagnetism).

In special relativity, the electric field andmagnetic field are two interrelated aspects ofa single object, called the electromagneticfield. A pure electric field in one referenceframe is observed as a combination of bothan electric field and a magnetic field in amoving reference frame.

B and HSee also: Magnetization

Alternative names for B and HB

name used by

magnetic flux density electrical engineers

magnetic induction electrical engineers

magnetic field physicists

H

name used by

magnetic field intensity electrical engineers

magnetic field strength electrical engineers

auxiliary magnetic field physicists

magnetizing field physicists

The term magnetic field is used for two dif-ferent vector fields, denoted B and H,[1] al-though there are many alternative names forboth (see sidebar). To avoid confusion, thisarticle uses B-field and H-field for thesefields, and uses magnetic field where eitheror both fields apply.

The B-field can be defined in many equi-valent ways based on the effects it has on itsenvironment. For instance, a particle havingan electric charge, q, and moving in a B-fieldwith a velocity, v, experiences a force, F,called the Lorentz force (see below). In SIunits, the Lorentz force equation is

where × is the vector cross product. The B-field is measured in teslas in SI units and ingauss in cgs units.

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Although views have shifted over theyears, B is now understood as being the fun-damental quantity, while H is a derived field.It is defined as a modification of B due to ma-terial media such that (in SI):

(definition of H )

where M is the magnetization of the materialand μ0 is the magnetic constant.[2] The H-field is measured in amperes per meter (A/m)in SI units and in oersteds (Oe) in cgsunits.[3]

In materials for which M is proportional toB the relationship between B and H can becast into the simpler form: H = B ⁄ μ, where μis a material dependent parameter called thepermeability. In free space, there is no mag-netization M so that H = B ⁄ μ0 (free space).For many materials, though, there is nosimple relationship between B and M. Forexample, ferromagnetic materials and super-conductors have a magnetization that is amultiple-valued function of B due to hyster-esis.[4]

See History of B and H below for furtherdiscussion.

The magnetic field andpermanent magnetsPermanent magnets are objects that producetheir own persistent magnetic fields. All per-manent magnets have both a north and asouth pole. Like poles repel and oppositepoles attract. Permanent magnets are madeof ferromagnetic materials such as iron andnickel that have been magnetized. For moredetails about magnets see magnetization be-low and the article ferromagnetism.

Force on a magnet due to a non-uniform BSee also: Magnet#Two models for magnets:magnetic poles and atomic currents andMagnetic momentThe most commonly experienced effect of themagnetic field is the force between two mag-nets. This force is often described as ’likepoles repel while opposites attract’. A moregeneral description, that also applies to mag-netic fields that have no poles (such as thatdue to the current through a straight wire), isthat a magnet experiences a force, when

placed in a non-uniform external magneticfield.

In this model, each magnetic pole is asource of a magnetic field that is strongernear the pole. Further, an external magneticfield exerts a force in the direction of themagnetic field for a north pole and in the op-posite direction for the south pole. In anonuniform magnetic field, each pole sees adifferent field and consequently is subject toa different force. The difference in the twoforces moves the magnet in the direction ofincreasing magnetic field. (There may also bea net torque.) In contrast, a magnet in a uni-form magnetic field experiences at most atorque, and no net magnetic force, no matterhow strong the field is.

Unfortunately, the idea of "poles" does notaccurately reflect what happens inside amagnet (see ferromagnetism). For instance, asmall magnet placed inside of a larger mag-net will feel a force in the opposite direction.The more physically correct description ofmagnetism involves atomic sized loops of cur-rent distributed throughout the magnet.

Mathematically, the force on a magnethaving a magnetic moment m is:[5]

.

The force on a magnet due to a non-uniformmagnetic field can be determined by sum-ming up all of the forces on the elementarymagnets that make up the entire magnet.

The ability of a nonuniform magnetic fieldto sort differently oriented dipoles is thebasis of the Stern-Gerlach experiment, whichestablished the quantum mechanical natureof the magnetic dipoles associated withatoms and electrons.[6][7]

Torque on a magnet due to a B-fieldSee also: Faraday’s law of inductionA magnet placed in a magnetic field will feela torque that will try to align the magnet withthe magnetic field. The torque on a magnetdue to an external magnetic field is easy toobserve by placing two magnets near eachother while allowing one to rotate.

The alignment of a magnet with the mag-netic field of the Earth is how compasseswork. It is used to determine the direction ofa local magnetic field as well (see below). Asmall magnet is mounted such that it is freeto turn (in a given plane) and its north pole is

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The direction of the magnetic field nearthe poles of a magnet is revealed by pla-cing compasses nearby. As seen here, themagnetic field points towards a magnet’ssouth pole and away from its north pole.

marked. By definition, the direction of thelocal magnetic field is the direction that thenorth pole of a compass (or of any magnet)tends to point.

The magnetic torque also provides thedriving torque for simple electric motors. Anelectric motor changes electrical energy intomechanical energy (motion). In a motor, amagnet is fixed to a shaft free to rotate(forming a rotor). This magnet is subjected toa magnetic field from an array of electromag-nets —called the stator. The polarity of eachindividual electromagnet in the stator easilycan be flipped by switching the direction ofthe current through its coils. By flipping com-ponent electromagnet polarities in sequence,the field of the stator continuously changesto place like poles next to the rotor, subject-ing the rotor to a torque that is transferred to

the shaft. The inverse process, changingmechanical motion to electrical energy, is ac-complished by the inverse of the above mech-anism in the electric generator.

See Rotating magnetic fields below for anexample using this effect withelectromagnets.

Visualizing the magneticfieldMapping out the strength and direction ofthe magnetic field is simple in principle.First, measure the strength and direction ofthe magnetic field at a large number of loca-tions. Then mark each location with an arrow(called a vector) pointing in the direction ofthe local magnetic field with a length propor-tional to the strength of the magnetic field.An alternative method of visualizing the mag-netic field which greatly simplifies the dia-gram while containing the same informationis to ’connect’ the arrows to form "magneticfield lines".

A compass placed near the north pole of amagnet will point away from that pole—likepoles repel. The opposite occurs for a com-pass placed near a magnet’s south pole. Themagnetic field points away from a magnetnear its north pole and towards a magnetnear its south pole. Not all magnetic fieldsare describable in terms of poles, though. Astraight current-carrying wire, for instance,produces a magnetic field that points neithertowards nor away from the wire, but en-circles it instead.

B-field linesVarious physical phenomena have the effectof displaying magnetic field lines. For ex-ample, iron filings placed in a magnetic fieldwill line up in such a way as to visually showthe orientation of the magnetic field (see fig-ure at top). Another place where magneticfields are visually displayed is in the polar au-roras, in which visible streaks of light line upwith the local direction of Earth’s magneticfield (due to plasma particle dipole interac-tions). In these phenomena, lines or curvesappear that follow along the direction of thelocal magnetic field.

These field lines provide a simple way todepict or draw the magnetic field (or any oth-er vector field). [8] The magnetic field can beestimated at any point (whether on a field

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line or not) by looking at the direction anddensity of the field lines nearby.

Field lines are also a good tool for visualiz-ing magnetic forces. When dealing with mag-netic fields in ferromagnetic substances likeiron, and in plasmas, the magnetic forces canbe understood by imagining that the fieldlines exert a tension, (like a rubber band)along their length, and a pressure perpendic-ular to their length on neighboring field lines.The ’unlike’ poles of magnets attract becausethey are linked by many field lines, while’like’ poles repel because the field linesbetween them don’t meet, but run parallel,pushing on each other.

B-field lines always form closed loopsField lines are a useful way to represent anyvector field and often reveal sophisticatedproperties of fields quite simply. One import-ant property of the B-field that can be veri-fied with field lines is that magnetic fieldlines always make complete loops. Magneticfield lines neither start nor end (althoughthey can extend to or from infinity). To dateno exception to this rule has been found. (Seemagnetic monopole below.)

Since magnetic field lines always come inloops, magnetic poles always come in N andS pairs. Magnetic field leaves a magnet nearits north pole and enters the magnet near itssouth pole but inside the magnet the magnet-ic field continues from the south pole back tothe north. [9] If a magnetic field line enters amagnet somewhere it has to leave the mag-net somewhere else; it is not allowed to havean end point. For this reason as well, cuttinga magnet in half will result in two separatemagnets each with both a north and a southpole.

Magnetic monopole(hypothetical)A magnetic monopole is a hypotheticalparticle (or class of particles) that has, as itsname suggests, only one magnetic pole(either a north pole or a south pole). In otherwords, it would possess a "magnetic charge"analogous to electric charge.

Modern interest in this concept stemsfrom particle theories, notably Grand UnifiedTheories and superstring theories, that pre-dict either the existence or the possibility ofmagnetic monopoles. These theories and oth-ers have inspired extensive efforts to search

for monopoles. Despite these efforts, no mag-netic monopole has been observed todate.[10]

The magnetic field andelectrical currentsCurrents of electrical charges both generatea magnetic field and feel a force due tomagnetic B-fields.

Electrical currents (movingcharges) as a source of magnet-ic fieldAll moving charges produce a magnetic field.[11] The magnetic field of a moving charge isvery complicated but is well known. (See Jefi-menko’s equations.) It forms closed loopsaround a line that is pointing in the directionthe charge is moving. The magnetic field of acurrent on the other hand is much easier tocalculate.

Magnetic field of a steady current

Current (I) through a wire produces a mag-netic field (B) around the wire. The field isoriented according to the right hand griprule.

The magnetic field generated by a steadycurrent (a continual flow of charges, for ex-ample through a wire, which is constant intime and in which charge is neither buildingup nor depleting at any point), is describedby the Biot-Savart law.[12] This is a

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consequence of Ampere’s law, one of the fourMaxwell’s equations that describe electricityand magnetism. The magnetic field lines gen-erated by a current carrying wire form con-centric circles around the wire. The directionof the magnetic field of the loops is determ-ined by the right hand grip rule. One can alsoimagine driving a bus along the current. Toyour left, the magnetic field points up. Toyour right, it points down. (See figure to theright.) The strength of the magnetic field de-creases with distance from the wire.

A current carrying wire can be bent in aloop such that the field is concentrated (andin the same direction) inside of the loop. Thefield will be weaker outside of the loop.Stacking many such loops to form a solenoid(or long coil) can greatly increase the mag-netic field in the center and decrease themagnetic field outside of the solenoid. Suchdevices are called electromagnets and are ex-tremely important in generating strong andwell controlled magnetic fields. An infinitelylong solenoid will have a uniform magneticfield inside of the loops and no magnetic fieldoutside. A finite length electromagnet willproduce essentially the same magnetic fieldas a uniform permanent magnet of the sameshape and size. An electromagnet has the ad-vantage, though, that you can easily vary thestrength (even creating a field in the oppositedirection) simply by controlling the input cur-rent. One important use is to continuallyswitch the polarity of a stationary electro-magnet to force a rotating permanent mag-net to continually rotate using the fact thatopposite poles attract and like poles repel.This can be used to create an important typeof electrical motor.

Force due to a B-field on a mov-ing chargeForce on a charged particle

A charged particle moving in a B-field willfeel a sideways force that is proportional tothe strength of the magnetic field, the com-ponent of the velocity that is perpendicular tothe magnetic field and the charge of theparticle. This force is known as the Lorentzforce, and is given by

whereF is the force (in newtons)

Charged particle drifts in a homogeneousmagnetic field. (A) No disturbing force (B)With an electric field, E (C) With an inde-pendent force, F (e.g. gravity) (D) In an in-homogeneous magnetic field, grad H

Beam of electrons moving in a circle. Light-ing is caused by excitation of atoms of gas ina bulb.

q is the electric charge of the particle (incoulombs)

v is the instantaneous velocity of theparticle (in meters per second)

B is the magnetic field (in teslas).

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The force is always perpendicular to both thevelocity of the particle and the magnetic fieldthat created it. Neither a stationary particlenor one moving in the direction of the mag-netic field lines will experience a force. Forthat reason, charged particles move in acircle (or more generally, in a helix) aroundmagnetic field lines; this is called cyclotronmotion. Because the magnetic force is alwaysperpendicular to the motion, the magneticfields can do no work on an isolated charge.It can and does, however, change theparticle’s direction, even to the extent that aforce applied in one direction can cause theparticle to drift in a perpendicular direction.(See figure.) The magnetic force can do workto a magnetic dipole, or to a charged particlewhose motion is constrained by other forces.

Force on current-carrying wireThe force on a current carrying wire is simil-ar to that of a moving charge as expectedsince a charge carrying wire is a collection ofmoving charges. A current carrying wire willfeel a sideways force in the presence of amagnetic field. The Lorentz force on a macro-scopic current is often referred to as the La-place force.

Direction of force

The right-hand rule: With the thumb of theright hand pointing in the direction of theconventional current or moving positivecharge and the fingers pointing in the direc-tion of the magnetic field the force on thecurrent will be in a direction out of the palm.The direction of the force is reversed for anegative charge.

The direction of force on a positive chargeor a current is determined by the right-handrule. See the figure on the right. Using theright hand and pointing the thumb in the dir-ection of the moving positive charge or posit-ive current and the fingers in the direction ofthe magnetic field the resulting force on the

charge will point outwards from the palm.The force on a negative charged particle is inthe opposite direction. If both the speed andthe charge are reversed then the direction ofthe force remains the same. For that reason amagnetic field measurement (by itself) can-not distinguish whether there is a positivecharge moving to the right or a negativecharge moving to the left. (Both of these willproduce the same current.) On the otherhand, a magnetic field combined with anelectric field can distinguish between these,see Hall effect below.

An alternative, similar trick to the righthand rule is Fleming’s left hand rule.

Electromagnetism: therelationship betweenmagnetic and electricfieldsThe magnetic field due to achanging electric fieldSee also: Ampere’s Law and Maxwell’sequationsA changing electric field generates a magnet-ic field proportional to the time rate of thechange of the electric field. This fact isknown as Maxwell’s correction to Ampere’sLaw. Therefore the full Ampere’s Law is:

where J is the current density, and partial de-rivatives indicate spatial location is fixedwhen the time derivative is taken. The lastterm is Maxwell’s correction. This equation isvalid even when magnetic materials are in-volved, but in practice it is often easier to usean alternate equation .

Electric force due to a changingB-fieldAbove is a discussion of how a changing E-field can cause a B-field. The inverse processalso occurs: a magnet moving through a sta-tionary coil will generate an electric field(and therefore tend to drive a current) in thecoil. (These two effects bootstrap together toform electromagnetic waves, such as light.)Both these phenomena play a part in

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Faraday’s Law, which forms the basis ofmany electric generators and electrical mo-tors. More generally, Faraday’s law statesthat any change in the magnetic field linkinga circuit will generate an electromotive forceor EMF, a force tending to drive a current.

Mathematically, Faraday’s law is com-monly represented as:

where is the electromotive force (thevoltage generated around a closed loop) andΦm is the magnetic flux (the product of thearea times the magnetic field normal to thatarea). This law includes both the case whenthe flux changes because of the magneticfield generated by a time varying E-field (so-called transformer EMF) and the case whenthe flux changes because of movementthrough a magnetic field (so-called motionalEMF). The appearance of the magnetic fluxin this law is why engineers often refer to theB-field as the "magnetic flux density". Cer-tain calculations involving magnetic fieldsare easier when formulated in terms of fluxdensity, for example, in magnetic circuits.

A limited form of Faraday’s law of induc-tion that does not include motional electro-motive force is the Maxwell-Faradayequation:

which is one of Maxwell’s equations. Thisequation is valid even in the presence ofmagnetic material.[13]

Mathematical propertiesof BThe magnitude of B is defined (in SI units) interms of the voltage induced per unit area ona current carrying loop in a uniform magneticfield normal to the loop when the magneticfield is reduced to zero in a unit amount oftime.

The magnetic field vector is apseudovector (also called an axial vector).(This is a technical statement about how themagnetic field behaves when you reflect theworld in a mirror.) This fact is apparent frommany of the definitions and properties of thefield; for example, the magnitude of the field

is proportional to the torque on a dipole, andtorque is a well-known pseudovector.

Maxwell’s equationsAs a vector field, the B-field has two import-ant mathematical properties that relates thismagnetic field to its sources. These two prop-erties, along with the two correspondingproperties of the electric field, make up Max-well’s Equations. Maxwell’s Equations to-gether with the Lorentz force law form acomplete description of classical electro-dynamics including both electricity andmagnetism.

The first property is that a B-field line nev-er starts nor ends at a point but insteadforms a complete loop. This is mathematicallyequivalent to saying that the divergence of Bis zero. (Such vector fields are called solen-oidal vector fields.) This property is calledGauss’ law for magnetism and is equivalentto the statement that there are no magneticcharges or magnetic monopoles:

where ∇ · represents the divergenceoperation.

The second mathematical property of themagnetic field is that it always loops aroundthe source that creates it. This source couldbe a current, a magnet, or a changing elec-tric field, but it is always within the loops ofmagnetic field they create. Mathematically,this fact is described by the combination ofthe above Gauss’s law with the Ampère-Max-well equation:

where ∇ × represents the curl operation, J =complete microscopic current density and E= electric field.

Measuring the B-fieldDevices used to measure the local magneticfield are called magnetometers. Importantclasses of magnetometers include using a ro-tating coil, Hall effect magnetometers, NMRmagnetometer, SQUID magnetometer, and afluxgate magnetometer. The magnetic fieldsof distant astronomical objects can be de-termined by noting their effects on localcharged particles. For instance, electronsspiraling around a field line will produce

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synchotron radiation which is detectable inradio waves.

Hall effectWhen a current carrying conductor is placedin a transverse magnetic field the sidewaysLorentz force on the charge carriers resultsin a charge separation in a direction perpen-dicular to both the current and the magneticfield. The resultant voltage, due to thatcharge separation, is proportional to the ap-plied magnetic field. This is known as theHall effect. The Hall effect is often used tomeasure the magnitude of a magnetic field aswell as to find the sign of the dominantcharge carriers in semiconductors (negativeelectrons or positive holes).

SQUID magnetometerSee also: superconductivitySuperconductors are materials with both dis-tinctive electric properties (perfect conduct-ivity) and magnetic properties (such as theMeissner effect, in which many supercon-ductors can perfectly expel magnetic fields).Due to these properties, loops of supercon-ducting material broken up by Josephsonjunctions can function as very sensitive mag-netometers, called SQUIDs. SQUID magneto-meters are used in a Scanning SQUID micro-scope to create a 2D map of the magneticfield.

The H-fieldIn the formulation of Maxwell’s equations ata microscopic level where all charges andcurrents are treated explicitly, only the E-and B-fields occur. On the other hand, whencharges and currents are divided into "free"and "bound" categories, D- and H-fields areused, with the H-field determined by the"free" current and time rate of change ofD.[14] Thus, when the "free" and "bound" di-vision of currents and charges is introduced,the H-field appears and simplifies the equa-tions for the magnetic field because micro-scopic details of the B- and E-fields insidematerials can be treated separately as prob-lems of condensed-matter physics. The H-field is defined as:

(definition of H in SIunits)

(definition of H incgs units)

where M is magnetization density of anymagnetic material. H is measured in amperesper meter (A/m) in SI and in oersteds (Oe) forcgs. In SI units, μ0 is a defined constantcalled the magnetic constant (μ0 = 4π × 10−7Tm/A).

MagnetizationSee also: Magnetization

Hierarchy of types of magnetism. See My-ers.[15]

Materials placed in a magnetic field can be-come magnetized. Magnetization is due tothe accumulated effect of many tiny magneticdipole moments that occur on the atomiclevel. In non-magnetized materials, the mag-netic dipoles align randomly such that thenet magnetic moment cancels producing nonet magnetic field. But, if the magnetic di-poles of the material becomes aligned a netmagnetization and magnetic field is pro-duced. The magnetization field M representshow strongly a region is magnetized and isdefined as the volume density of the net mag-netic dipole moment in that region ofmaterial.

An equivalent way to represent magnetiza-tion is to add all of the currents of the dipolemoments that produce the magnetization.The resultant current is called bound currentand is the source of the magnetic field due tothe magnet. Mathematically, the curl of Mequals the bound current. Unlike B, though,magnetization must begin and end near thepoles. (There is no magnetization outside ofthe material.) Therefore, the divergence of Mmust be non-zero near the poles of a magnet.

Most materials produce a magnetization inresponse to an applied B-field. Typically, theresponse is very weak and exists only whenthe magnetic field is applied. Materials are

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divided into groups based upon their magnet-ic behavior:• Diamagnetic materials[16] produce a

magnetization that opposes the magneticfield.

• Paramagnetic materials[16] produce amagnetization in the same direction as theapplied magnetic field.

• Ferromagnetic materials and the closelyrelated ferrimagnetic materials andantferromagnetic materials[17][18] canhave a magnetization independent of anapplied B-field with a complexrelationship between the two fields.

• Superconductors (and ferromagneticsuperconductors)[19][20] are materials thatare characterized by perfect conductivitybelow a critical temperature and magneticfield. They also are highly magnetic andcan be perfect diamagnets below a lowercritical magnetic field. Superconductorsoften have a broad range of temperaturesand magnetic fields (the so named mixedstate) for which they exhibit a complexhysteretic dependence of M on B.

In the case of paramagnetism, and diamag-netism the B-field often is proportional to theH-field such that:

,

where μ is a material dependent parametercalled the permeability (see constitutiveequations). In some cases the permeabilitymay be a second rank tensor so that H maynot point in the same direction as B. Theserelations between B and H are examples ofconstitutive equations. However, supercon-ductors and ferromagnets have a more com-plex B to H relation, see hysteresis. In allcases the original definitions of H in terms ofB and M still are valid.

The advantage of the H-field is that itsbound sources are treated so differently thatthey can often be isolated from the freesources. For example, a line integral of theH-field in a closed loop will yield the totalfree current in the loop (not including thebound current). Similarly, a surface integralof H over any closed surface will pick out the’magnetic charges’ within that closedsurface.

Magnetic dipolesSee also: Spin magnetic moment andMicromagnetism

Magnetic field lines around a ”magnetostaticdipole” the magnetic dipole itself is in thecenter and is seen from the side.

The magnetic field of an ideal magnetic di-pole is depicted on the right. As discussed be-low, however, due to the inherent connectionbetween angular momentum and magnetism,magnetic dipoles in actual materials are notideal magnetic dipoles. The connectionbetween angular momentum and magnetismis the basis of the Einstein-de Haas effect "ro-tation by magnetization" and its inverse, theBarnett effect or "magnetization by rota-tion".[21]

The magnetic field of permanent magnetsand of all magnetic material originate at theatomic level. Orbiting electrons along withthe nucleus form tiny magnets.[22] The orbit-al component of these tiny magnets can bemodeled as tiny loops of current with associ-ated magnetic dipoles.[23] The dipole momentof that dipole is defined as the current timesthe area of the loop and represents thestrength of that magnet (magnetic dipole).However, in magnetic materials such as al-loys of iron, cobalt and nickel, the magnetismis almost entirely spin magnetism, not orbitalmagnetism.[24][25]

The magnetic dipole originating in anatom, electron, or nucleus is not a true di-pole, as is an electric dipole. Viewing a mag-netic dipole as a rotating charged spherebrings out the close connection betweenmagnetic moment and angular momentum.Both the magnetic moment and the angularmomentum increase with the rate of rotation

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of the sphere. The ratio of the two is calledthe gyromagnetic ratio, usually denoted bythe symbol γ.[24][26]

For an atom, individual electron spins areadded to get a total spin and individual orbit-al angular momenta are added to get a totalorbital angular momentum. These two thenare added using angular momentum couplingto get a total angular momentum. The mag-nitude of the atomic dipole moment isthen:[27]

where J is the total angular momentumquantum number, gJ is the Landé g-factor,and μB is the Bohr magneton. The componentof this magnetic moment along the directionof the magnetic field is then:[28]

where m is called the magnetic quantumnumber or the equatorial quantum number,which can take on any of 2J+1 values: -J,−(J-1), … , (J−1), J.[29] The negative sign oc-curs because electrons have negative charge.

Because of the angular momentum, thedynamics of a magnetic dipole in a magneticfield differs from that of an electric dipole inan electric field. The field does exert a torqueon the magnetic dipole tending to align itwith the field. However, torque is proportion-al to rate of change of angular momentum, soprecession occurs: the direction of spinchanges. This behavior is described by theLandau-Lifshitz-Gilbert equation:[30][31]

where γ =gyromagnetic ratio, m = magneticmoment, λ = damping coefficient and Heff =effective magnetic field (the external fieldplus any self-field), and ’×’ = vector crossproduct. The first term describes precessionof the moment about the effective field, whilethe second is a damping term related to dis-sipation of energy caused by interaction withsurroundings.

Uses of the H-fieldEnergy stored in magnetic fieldsIn asking how much energy does it take tocreate a specific magnetic field using a par-ticular current it is important to distinguish

between free and bound currents. It is thefree current that we directly ’push’ on to cre-ate the magnetic field. The bound currentscreate a magnetic field that the free currenthas to work against without doing any of thework.

It is not surprising, therefore, that the H-field is important in magnetic energy calcula-tions since it treats the two sources differ-ently. In general the incremental amount ofwork per unit volume δW needed to cause asmall change of magnetic field δB is:

If there are no magnetic materials aroundthen we can replace H with B ⁄ μ0,

For linear materials (such that B = μH ), theenergy density can be expressed as:

(Valid onlyfor linear materials)

Nonlinear materials cannot use the aboveequation but must return to the first equationwhich is always valid. In particular, the en-ergy density stored in the fields of hystereticmaterials such as ferromagnets and super-conductors will depend on how the magneticfield was created.

Magnetic circuitsA second use for H is in magnetic circuitswhere inside a linear material B = μ H. Here,μ is the magnetic permeability of the materi-al. This result is similar in form to Ohm’s LawJ = σ E, where J is the current density, σ isthe conductance and E is the electric field.Extending this analogy we derive the coun-terpart to the macroscopic Ohm’s law ( I = V ⁄R ) as:

where is the magnetic flux

in the circuit, is the mag-netomotive force applied to the circuit, andRm is the reluctance of the circuit. Here the

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reluctance Rm is a quantity similar in natureto resistance for the flux.

Using this analogy it is straight-forward tocalculate the magnetic flux of complicatedmagnetic field geometries, by using all theavailable techniques of circuit theory.

History of B and HThe modern understanding that the B-field isthe more fundamental field with the H-fieldbeing an auxiliary field was not easy to arriveat. Indeed, largely because of mathematicalsimilarities to the electric field, the H-fieldwas developed first and was thought at firstto be the more fundamental of the two. Abrief history of this important transition inthought is instructional in giving insight intothe nature of both H and B.

Perhaps the earliest description of a mag-netic field was performed by Petrus Pereg-rinus and published in his “Epistola Petri Per-egrini de Maricourt ad Sygerum de Foucauc-ourt Militem de Magnete” and is dated 1269A.D. Petrus Peregrinus mapped out the mag-netic field on the surface of a spherical mag-net. Noting that the resulting field linescrossed at two points he named those points’poles’ in analogy to Earth’s poles. Almostthree centuries later, near the end of the six-teenth century, William Gilbert of Colchesterreplicated Petrus Peregrinus work and wasthe first to state explicitly that Earth itselfwas a magnet. William Gilbert’s great workDe Magnete was published in 1600 A.D. andhelped to establish the study of magnetism asa science.

The modern distinction between the B-and H- fields does not become important un-til Siméon-Denis Poisson (1781–1840) de-veloped one of the first mathematical theor-ies of magnetism. Poisson’s model, developedin 1824, assumed that magnetism was due tomagnetic charges. In analogy to electriccharges, these magnetic charges produce aH-field. In modern notation, Poisson’s modelwas exactly analogous to electrostatics withthe H-field replacing the electric field E-fieldand the B-field replacing the auxiliary D-field.

Poisson’s model was, unfortunately, incor-rect. Magnetism is not due to magneticcharges. Nor is magnetism created by the H-field polarizing magnetic charge in a materi-al. The model, however, was remarkably suc-cessful for being fundamentally wrong. It

predicts the correct relationship between theH-field and the B-field, even though itwrongly places H as the fundamental fieldwith B as the auxiliary field. It predicts thecorrect forces between magnets.

It even predicts the correct energy storedin the magnetic fields. By the definition ofmagnetization, in this model, and in analogyto the physics of springs, the work done perunit volume, in stretching and twisting thebonds between magnetic charge to incrementthe magnetization by μ0δM is W = H · μ0δM.In this model, B = μ0 (H + M ) is an effectivemagnetization which includes the H-fieldterm to account for the energy of setting upthe magnetic field in a vacuum. Therefore thetotal energy density increment needed to in-crement the magnetic field is W = H · δB.This is the correct result, but it is derivedfrom an incorrect model.

In retrospect the success of this model isdue largely to the remarkable coincidencethat from the ’outside’ the field of an electricdipole has the exact same form as that of amagnetic dipole. It is therefore only for thephysics of magnetism ’inside’ of magneticmaterial where the simpler model of magnet-ic charges fails. It is also important to notethat this model is still useful in many situ-ations dealing with magnetic material. Oneexample of its utility is the concept of mag-netic circuits.

The formation of the correct theory ofmagnetism begins with a series of revolution-ary discoveries in 1820, four years beforePoisson’s model was developed. (The firstclue that something was amiss, though, wasthat unlike electrical charges magnetic polescannot be separated from each other or formmagnetic currents.) The revolution beganwhen Hans Christian Oersted discovered thatan electrical current generates a magneticfield that encircles the wire. In a quick suc-cession that discovery was followed by AndreMarie Ampere showing that parallel wireshaving currents in the same direction attract,and by Jean-Baptiste Biot and Felix Savartdeveloping the correct equation, the Biot-Savart Law, for the magnetic field of a cur-rent carrying wire. In 1825, Ampere exten-ded this revolution by publishing hisAmpere’s Law which provided a more math-ematically subtle and correct description ofthe magnetic field generated by a currentthan the Biot-Savart Law.

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Subsequent development in the nine-teenth century interlinked magnetic and elec-tric phenomena even tighter, until theconcept of magnetic charge was not needed.Magnetism became an electric phenomenonwith even the magnetism of permanent mag-nets being due to small loops of current intheir interior. This development was aidedgreatly by Michael Faraday, who in 1831showed that a changing magnetic field gener-ates an encircling electric field.

In 1861, James Clerk-Maxwell wrote a pa-per entitled ’On Physical Lines of Force’ [1]in which he attempted to explain Faraday’smagnetic lines of force in terms of a sea oftiny molecular vortices. These molecular vor-tices occupied all space and they werealigned in a solenoidal fashion such that theirrotation axes traced out the magnetic lines offorce. When two like magnetic poles repeleach other, the magnetic lines of forcespread outwards from each other in thespace between the two poles. Maxwell con-sidered that magnetic repulsion was the con-sequence of a lateral pressure between adja-cent lines of force, due to centrifugal force inthe equatorial plane of the molecular vor-tices. When deriving the equation for mag-netic force in part I of his 1861 paper, Max-well used a quantity which was closely re-lated to the circumferential speed of the vor-tices. This quantity was therefore a measureof the vorticity in the magnetic lines of force,and Maxwell referred to it as the intensity ofthe magnetic force. In the 1861 paper, themagnetic intensity which we will denote as v,was always multiplied by the term μ as aweighting for the cross sectional density ofthe lines of force. The quantity v correspondsreasonably closely to the modern magneticfield vector H, and the product μv corres-ponds very closely to the modern magneticflux density B, where μ is referred to as themagnetic permeability.

Although the classical theory of electro-dynamics was essentially complete with Max-well’s equations, the twentieth century saw anumber of improvements and extensions tothe theory. Albert Einstein, in his great paperof 1905 that established relativity, showedthat both the electric and magnetic fieldswere part of the same phenomena viewedfrom different reference frames. Finally, theemergent field of quantum mechanics wasmerged with electrodynamics to formquantum electrodynamics or QED.

Special relativity andelectromagnetismMagnetic fields played an important role inhelping to develop the theory of specialrelativity.

Moving magnet and conductorproblemImagine a moving conducting loop that ispassing by a stationary magnet, as seen byan observer on the magnet, and contrast thiswith an observer on the loop, who sees a sta-tionary loop near a moving magnet. The ob-servable phenomenon here depends only onthe relative motion of the conductor and themagnet, whereas the customary view draws asharp distinction between the two cases inwhich either the one or the other of thesebodies is in motion: in the stationary magnetcase, carriers moving in a magnetic field aresubject to a magnetic force that gives rise tothe current (the so-called motional electro-motive force), while if the magnet is in mo-tion and the conductor at rest, a changingmagnetic field induces an electric field thatdrives the current (the so-called transformerelectromotive force). Bringing these two de-scriptions together was one factor that ledAlbert Einstein to develop his theory of spe-cial relativity.

In more detail, an observer for whom themagnet is stationary would see an unchan-ging magnetic field and a moving conductingloop. Because the loop is moving, all of thecharges that make up the loop also are mov-ing. Each of these charges will have a side-ways, Lorentz force, acting on it due to theB-field, and this force generates the current.Contrariwise, an observer on the moving loopwould see a changing magnetic field becausethe loop is not moving in this observer’s ref-erence frame, but the magnet is. This chan-ging magnetic field generates an electricfield that generates the current.

The observer for whom the magnet is sta-tionary claims there is only a magnetic fieldthat creates a magnetic force on a movingcharge. The observer for whom the loop isstationary claims that there is both a magnet-ic and an electric field but all of the force isdue to the electric field. Which is true? Doesthe electric field exist or not? The answer, ac-cording to special relativity, is that both

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observers are right from their referenceframe. A pure magnetic field in one referencecan be a mixture of magnetic and electricfield in another reference frame.

Electric and magnetic fields dif-ferent aspects of the samephenomenonAccording to special relativity, electric andmagnetic forces are part of a single physicalphenomenon, electromagnetism; an electricforce perceived by one observer will be per-ceived by another observer in a differentframe of reference as a mixture of electricand magnetic forces. Magnetic and electricforces are facets of the underlying electro-magnetic force, and the partition of the elec-tromagnetic force into separate electric andmagnetic components is not fundamental, butvaries with the observational frame ofreference.

More specifically, rather than treating theelectric and magnetic fields as separatefields, special relativity shows that they nat-urally mix together into a rank-2 tensor,called the electromagnetic tensor. This isanalogous to the way that special relativity"mixes" space and time into spacetime, andmass, momentum and energy into four-momentum.

Magnetic field shapedescriptions• An magnetic field is one that runs east-

west.• A magnetic field is one that runs north-

south. In the solar dynamo model of theSun, differential rotation of the solarplasma causes the meridional magneticfield to stretch into an azimuthal magneticfield, a process called the omega-effect.The reverse process is called the alpha-effect.[32]

• A dipole magnetic field is one seenaround a bar magnet or around a chargedelementary particle with nonzero spin.

• A quadrupole magnetic field is one seen,for example, between the poles of four barmagnets. The field strength grows linearlywith the radial distance from itslongitudinal axis.

• A magnetic field is similar to a dipolemagnetic field, except that a solid bar

Schematic quadrupole magnet ("four-pole")magnetic field. There are four steel pole tips,two opposing magnetic north poles and twoopposing magnetic south poles.

magnet is replaced by a hollowelectromagnetic coil magnet.

• A magnetic field occurs in a doughnut-shaped coil, the electric current spiralingaround the tube-like surface, and is found,for example, in a tokamak.

• A magnetic field is generated by a currentflowing in a ring, and is found, forexample, in a tokamak.

• A magnetic field is one in which the fieldlines are directed from the centeroutwards, similar to the spokes in abicycle wheel. An example can be found ina loudspeaker transducers (driver).[33]

• A magnetic field is corkscrew-shaped, andsometimes seen in space plasmas such asthe Orion Molecular Cloud.[34]

Important uses and ex-amples of magnetic fieldEarth’s magnetic fieldSee also: North Magnetic Pole and SouthMagnetic PoleBecause of Earth’s magnetic field, a compassplaced anywhere on Earth will turn so thatthe "north pole" of the magnet inside thecompass points roughly north, toward Earth’snorth magnetic pole in northern Canada. Thisis the traditional definition of the "north pole"of a magnet, although other equivalent

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A sketch of Earth’s magnetic field repres-enting the source of Earth’s magnetic field asa magnet. The north pole of earth is near thetop of the diagram, the south pole near thebottom. Notice that the south pole of thatmagnet is deep in Earth’s interior belowEarth’s North Magnetic Pole. Earth’s mag-netic field is produced in the outer liquid partof its core due to a dynamo that produceelectrical currents there.

definitions are also possible. One confusionthat arises from this definition is that if Earthitself is considered as a magnet, the southpole of that magnet would be the one nearerthe north magnetic pole, and vice-versa. (Op-posite poles attract and the north pole of thecompass magnet is attracted to the northmagnetic pole.) The north magnetic pole is sonamed not because of the polarity of the fieldthere but because of its geographicallocation.

The figure to the right is a sketch ofEarth’s magnetic field represented by fieldlines. The magnetic field at any given pointdoes not point straight toward (or away) fromthe poles and has a significant up/down com-ponent for most locations. (In addition, thereis an East/West component as Earth’s mag-netic poles do not coincide exactly withEarth’s geological pole.) The magnetic field isas if there were a magnet deep in Earth’sinterior.

Earth’s magnetic field is probably due to adynamo that produces electric currents inthe outer liquid part of its core. Earth’s mag-netic field is not constant: Its strength andthe location of its poles vary. The poles even

periodically reverse direction, in a processcalled geomagnetic reversal.

Rotating magnetic fieldsThe rotating magnetic field is a key principlein the operation of alternating-current mo-tors. A permanent magnet in such a field willrotate so as to maintain its alignment withthe external field. This effect was conceptual-ized by Nikola Tesla, and later utilized in his,and others’, early AC (alternating-current)electric motors. A rotating magnetic field canbe constructed using two orthogonal coilswith 90 degrees phase difference in their ACcurrents. However, in practice such a systemwould be supplied through a three-wire ar-rangement with unequal currents. This in-equality would cause serious problems instandardization of the conductor size and so,in order to overcome it, three-phase systemsare used where the three currents are equalin magnitude and have 120 degrees phasedifference. Three similar coils having mutualgeometrical angles of 120 degrees will createthe rotating magnetic field in this case. Theability of the three-phase system to create arotating field, utilized in electric motors, isone of the main reasons why three-phase sys-tems dominate the world’s electrical powersupply systems.

Because magnets degrade with time, syn-chronous motors and induction motors useshort-circuited rotors (instead of a magnet)following the rotating magnetic field of amulticoiled stator. The short-circuited turnsof the rotor develop eddy currents in the ro-tating field of the stator, and these currentsin turn move the rotor by the Lorentz force.

In 1882, Nikola Tesla identified theconcept of the rotating magnetic field. In1885, Galileo Ferraris independently re-searched the concept. In 1888, Tesla gainedU.S. Patent 381,968 for his work. Also in1888, Ferraris published his research in a pa-per to the Royal Academy of Sciences inTurin.

See alsoGeneral• Electric field — field produced by electric

charges and changing magnetic fields thataffects charged particles.

• Electromagnetic field — a field composedof the electric field and the magnetic field.

From Wikipedia, the free encyclopedia Magnetic field

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• Electromagnetism — the physics of theelectromagnetic field.

• Faraday’s law of induction — theconnection between electric and magneticfields as found in motors and generators

• Lorentz force — the connection betweenfields and forces

• Magnetism — phenomenon by whichmaterials exert a magnetic force on othermaterials.

• Magnetohydrodynamics — the study of thedynamics of electrically conducting fluids.

• Magnetic flux — amount of ’magneticfield’ through a given loop.

• Magnetic monopole — hypotheticalparticle which causes nonzero divergenceof magnetic field.

• Magnetic nanoparticles — extremely smallmagnetic particles that are tens of atomswide

• Magnetic reconnection — an effect whichcauses solar flares and auroras.

• Magnetic potential — the vector andscalar potential representation ofmagnetism.

• SI electromagnetism units — commonunits used in electromagnetism.

• Orders of magnitude (magnetic field) —list of magnetic field sources andmeasurement devices from smallestmagnetic fields to largest detected.

Mathematics• Ampère’s law — law describing how

currents act as circulation sources formagnetic fields.

• Biot-Savart law — the magnetic field setup by a steadily flowing line current.

• Magnetic helicity — extent to which amagnetic field "wraps around itself".

• Maxwell’s equations — four equationsdescribing the behavior of electric andmagnetic fields and their interaction withmatter.

Applications• Dynamo theory — a proposed mechanism

for the creation of the Earth’s magneticfield.

• Earth’s magnetic field — a discussion ofthe magnetic field of the Earth.

• Electric motor — AC motors usedmagnetic fields.

• Helmholtz coil — a device for producing aregion of nearly uniform magnetic field.

• Magnetic field viewing film — Film used toview the magnetic field of an area.

• Maxwell coil — a device for producing alarge volume of almost constant magneticfield.

• Stellar magnetic field — a discussion ofthe magnetic field of stars.

• Teltron Tube — device used to display anelectron beam and demonstrates effect ofelectric and magnetic fields on movingcharges.

Further readingWeb• Nave, R.. "Magnetic Field Strength H".

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfield.html.

• Oppelt, Arnulf (2006-11-02). "magneticfield strength".http://searchsmb.techtarget.com/sDefinition/0,290660,sid44_gci763586,00.html.

• "magnetic field strength converter".http://www.unitconversion.org/unit_converter/magnetic-field-strength.html.

Books• Durney, Carl H. and Johnson, Curtis C.

(1969). Introduction to modernelectromagnetics. McGraw-Hill. ISBN0-07-018388-0.

• Rao, Nannapaneni N. (1994). Elements ofengineering electromagnetics (4th ed.).Prentice Hall. ISBN 0-13-948746-8. OCLC221993786.

• Griffiths, David J. (1999). Introduction toElectrodynamics (3rd ed.). Prentice Hall.ISBN 0-13-805326-X. OCLC 40251748.

• Jackson, John D. (1999). ClassicalElectrodynamics (3rd ed.). Wiley. ISBN0-471-30932-X. OCLC 224523909.

• Tipler, Paul (2004). Physics for Scientistsand Engineers: Electricity, Magnetism,Light, and Elementary Modern Physics(5th ed.). W. H. Freeman. ISBN0-7167-0810-8. OCLC 51095685.

• Furlani, Edward P. (2001). PermanentMagnet and Electromechanical Devices:Materials, Analysis and Applications.Academic Press Series inElectromagnetism. ISBN 0-12-269951-3.OCLC 162129430.

Notes and references[1] The standard graduate textbook by J. D.

Jackson "Classical Electrodynamics"

From Wikipedia, the free encyclopedia Magnetic field

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specifically follows the historicaltradition, specifically, "In the presence ofmagnetic materials the dipole tends toalign itself in a certain direction. Thatdirection is by definition the direction ofthe magnetic flux density, denoted by B,provided the dipole is sufficiently smalland weak that it does not perturb theexisting field". Similarly, in Section 5 ofJackson, H is referred to as the magneticfield. Hence, Edward Purcell, inElectricity and Magnetism, McGraw-Hill,1963, writes, Even some modern writerswho treat B as the primary field feelobliged to call it the magnetic inductionbecause the name magnetic field washistorically preempted by H. This seemsclumsy and pedantic. If you go into thelaboratory and ask a physicist whatcauses the pion trajectories in his bubblechamber to curve, he’ll probably answer"magnetic field", not "magneticinduction." You will seldom hear ageophysicist refer to the Earth’smagnetic induction, or an astrophysicisttalk about the magnetic induction of thegalaxy. We propose to keep on calling Bthe magnetic field. As for H, althoughother names have been invented for it,we shall call it "the field H" or even "themagnetic field H." In a similar vein, MGerloch (1983). Magnetism and Ligand-field Analysis. Cambridge UniversityPress. p. 110. ISBN 0521249392.http://books.google.com/books?id=Ovo8AAAAIAAJ&pg=PA110.says: “So we may think of both B and Has magnetic fields, but drop the word’magnetic’ from H so as to maintain thedistinction … As Purcell points out, ’it isonly the names that give trouble, not thesymbols’.”

[2] Magnetic Field Strength H[3] Magnetic Field Strength Converter[4] H. P. Myers (1997). Introductory solid

state physics (2 ed.). Taylor & Francis.p. 366. ISBN 074840659X.http://books.google.com/books?id=QhqyWH7DDQ0C&pg=PA366.

[5] See Eq. 11.42 in E. Richard Cohen, DavidR. Lide, George L. Trigg (2003). AIPphysics desk reference (3 ed.).Birkhäuser. p. 381. ISBN 0387989730.http://books.google.com/books?id=JStYf6WlXpgC&pg=PA381.

[6] Yuval Ne ̕eman, Y. Kirsh (1996). TheParticle Hunters (2 ed.). CambridgeUniversity Press. p. 56. ISBN0521476860. http://books.google.com/books?id=K4jcfCguj8YC&pg=PA56.

[7] John S Townsend (2000). "Stern-Gerlachexperiments". A Modern Approach toQuantum Mechanics (2 ed.). UniversityScience Books. pp. 1–23. ISBN1891389130. http://books.google.com/books?id=3_7uriPX028C&pg=PA3.

[8] Note that when a magnetic field isdepicted with field lines, it is not meantto imply that the field is only nonzeroalong the drawn-in field lines. The use ofiron filings to display a field presentssomething of an exception to thispicture: the magnetic field is in factmuch larger along the "lines" of iron, dueto the large permeability of iron relativeto air.

[9] To see that this must be true imagineplacing a compass inside of the magnet.The north pole of the compass will pointtoward the north pole of the magnetsince magnets stacked on each otherpoint in the same direction.

[10]Two experiments produced candidateevents that were initially interpreted asmonopoles, but these are now regardedto be inconclusive. For details andreferences, see magnetic monopole.

[11] In special relativity this means that theelectrical field and the magnetic fieldmust be two parts of the samephenomenon. For a moving single chargeor charges moving together we canalways shift to a reference system inwhich they are not moving. In thatreference system there is no magneticfield. Yet, the physics has to be the samein all reference systems. It turns out theelectric field changes as well whichproduces the same force in the originalreference frame. It is probably a mistake,though, to say that the electric fieldcauses the magnetic field when relativityis accounted for, since relativity favorsno particular reference frame. (Onecould just as easily say that the magneticfield caused an electric field). Moreimportantly it is not always possible tomove into a coordinate system in whichall of the charges are stationary. Seeclassical electromagnetism and specialrelativity for more information.

From Wikipedia, the free encyclopedia Magnetic field

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[12] In practice the Biot-Savart law and otherlaws of magnetostatics can often be usedeven when the charge is changing intime as long as it is not changing tooquickly. This situation is known as beingquasistatic.

[13]A complete expression for Faraday’s lawof induction in terms of the electric Eand magnetic fields can be written as:

where ∂Σ(t) is the moving closed pathbounding the moving surface Σ(t), anddA is an element of surface area of Σ(t).The first integral calculates the workdone moving a charge a distance dℓbased upon the Lorentz force law. In thecase where the bounding surface isstationary, the Kelvin-Stokes theoremcan be used to show this equation isequivalent to the Maxwell-Faradayequation.

[14] John Clarke Slater, Nathaniel HermanFrank (1969). Electromagnetism (firstpublished in 1947 ed.). Courier DoverPublications. p. 69. ISBN 0486622630.http://books.google.com/books?id=GYsphnFwUuUC&pg=PA69.

[15]HP Meyers (1997). Introductory solidstate physics (2 ed.). CRC Press. p. 322;Figure 11.1. ISBN 0748406603.http://books.google.com/books?id=Uc1pCo5TrYUC&pg=PA322.

[16]^ RJD Tilley (2004). UnderstandingSolids. Wiley. p. 368. ISBN 0470852755.http://books.google.com/books?id=ZVgOLCXNoMoC&pg=PA368.

[17]Sōshin Chikazumi, Chad D. Graham(1997). Physics of ferromagnetism (2ed.). Oxford University Press. p. 118.ISBN 0198517769.http://books.google.com/books?id=AZVfuxXF2GsC&printsec=frontcover.

[18]Amikam Aharoni (2000). Introduction tothe theory of ferromagnetism (2 ed.).Oxford University Press. p. 27. ISBN0198508085. http://books.google.com/books?id=9RvNuIDh0qMC&pg=PA27.

[19]M Brian Maple et al. (2008)."Unconventional superconductivity innovel materials". in K. H. Bennemann,John B. Ketterson. Superconductivity.Springer. p. 640. ISBN 3540732527.http://books.google.com/books?id=PguAgEQTiQwC&pg=PA640.

[20]Naoum Karchev (2003). "Itinerantferromagnetism and superconductivity".in Paul S. Lewis, D. Di (CON) Castro.Superconductivity research at theleading edge. Nova Publishers. p. 169.ISBN 1590338618.http://books.google.com/books?id=3AFo_yxBkD0C&pg=PA169.

[21]B. D. Cullity, C. D. Graham (2008).Introduction to Magnetic Materials (2ed.). Wiley-IEEE. p. 103. ISBN0471477419. http://books.google.com/books?id=ixAe4qIGEmwC&pg=PA103.

[22]The total magnetic moment of an atom isdue to a combination of ’currents’ ofelectrons ’orbiting’ the nuclei of themagnetic material plus a spin componentof the magnetic moment of the electronsand the nucleus. (The true nature of theinternal magnetic field of the electronsand of the nucleons that make up thenucleus is relativistic in nature.) UweKrey, Anthony Owen (2007). BasicTheoretical Physics. Springer. p. 151.ISBN 3540368043.http://books.google.com/books?id=xZ_QelBmkxYC&pg=PA151.and H. Haken, Hans Christoph Wolf,William D Brewer (2000). The physics ofatoms and quanta (6 ed.). Springer. ISBN3540672745. http://books.google.com/books?id=SPrAMy8glocC&pg=PA187.

[23]A. E. Siegman (1986). Lasers. UniversityScience Books. pp. 1215–1216. ISBN0935702113. http://books.google.com/books?id=1BZVwUZLTkAC&pg=PA1234#PPA1215,M1.

[24]^ Uwe Krey & Anthony Owen (2007).Basic Theoretical Physics. Springer.pp. 151–152. ISBN 3540368043.http://books.google.com/books?id=xZ_QelBmkxYC&pg=PA151.

[25]Ferromagnetic materials contain manyatoms with unpaired electron spins.When these tiny atomic magnetic dipolesare aligned in the same direction, theycreate a measurable macroscopic field.

[26]Richard B. Buxton (2002). Introductionto functional magnetic resonanceimaging. Cambridge University Press.

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p. 136. ISBN 0521581133.http://books.google.com/books?id=6XVu0NKzgekC&pg=PA136.

[27]RJD Tilley (2004). Understanding Solids.John Wiley and Sons. p. 368. ISBN0470852755. http://books.google.com/books?id=ZVgOLCXNoMoC&pg=PA368.

[28]Paul Allen Tipler, Ralph A. Llewellyn(2002). Modern Physics (4 ed.).Macmillan. p. 310. ISBN 0716743450.http://books.google.com/books?id=tpU18JqcSNkC&pg=PA310.

[29] JA Crowther (2007). Ions, Electrons andIonizing Radiations (reprintedCambridge (1934) 6 ed.). Rene Press.p. 277. ISBN 1406720399.http://books.google.com/books?id=H_sft9-zm5AC&pg=PA277.

[30]Stuart Alan Rice (2004). Advances inchemical physics. Wiley. pp. 208 ff. ISBN0471445282. http://books.google.com/books?id=wK3Vhq-VnBQC&pg=PA208.

[31]Marcus Steiner (2004). Micromagnetismand Electrical Resistance ofFerromagnetic Electrodes for SpinInjection Devices. Cuvillier Verlag. p. 6.ISBN 3865371760.http://books.google.com/books?id=tnX1edkCB-wC&pg=PA6.

[32]The Solar Dynamo, retrieved Sep 15,2007.

[33] I. S. Falconer and M. I. Large (edited byI. M. Sefton), "Magnetism: Fields andForces" Lecture E6, The University ofSydney, retrieved 3 Oct 2008

[34]Robert Sanders, "Astronomers findmagnetic Slinky in Orion", 12 January2006 at UC Berkeley. Retrieved 3 Oct2008

External linksInformation• Crowell, B., "Electromagnetism".• Nave, R., "Magnetic Field". HyperPhysics.• "Magnetism", The Magnetic Field.

theory.uwinnipeg.ca.• Hoadley, Rick, "What do magnetic fields

look like?" 17 July 2005.Field density• Jiles, David (1994). Introduction to

Electronic Properties of Materials (1sted.). Springer. ISBN 0-412-49580-5.

Rotating magnetic fields• "Rotating magnetic fields". Integrated

Publishing.• "Introduction to Generators and Motors",

rotating magnetic field. IntegratedPublishing.

• "Induction Motor-Rotating Fields".Diagrams• McCulloch, Malcolm,"A2: Electrical Power

and Machines", Rotating magnetic field.eng.ox.ac.uk.

• "AC Motor Theory" Figure 2 RotatingMagnetic Field. Integrated Publishing.

Journal Articles• Yaakov Kraftmakher, "Two experiments

with rotating magnetic field". 2001 Eur. J.Phys. 22 477-482.

• Bogdan Mielnik and David J. FernándezC., "An electron trapped in a rotatingmagnetic field". Journal of MathematicalPhysics, February 1989, Volume 30, Issue2, pp. 537-549.

• Sonia Melle, Miguel A. Rubio and GeraldG. Fuller "Structure and dynamics ofmagnetorheological fluids in rotatingmagnetic fields". Phys. Rev. E 61, 4111 –4117 (2000).

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