PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information.PDF generated at: Thu, 06 Sep 2012 07:40:31 UTC

Magnetism

ContentsArticles

Antiferromagnetism 1Biot–Savart law 3Classical electromagnetism and special relativity 7Coercivity 13Diamagnetism 16Electromagnet 20Ferrimagnetism 30Ferromagnetism 31History of electromagnetic theory 37Lorentz force 74Magnet 85Magnetic bearing 97Magnetic circuit 102Magnetic dipole 107Magnetic domain 110Magnetic field 116Magnetic monopole 137Magnetic refrigeration 150Magnetic stirrer 158Magnetic structure 160Magnetism 161Metamagnetism 171Micromagnetics 172Molecule-based magnets 175Neodymium magnet 177Paramagnetism 182Plastic magnet 188Rare-earth magnet 189Single-molecule magnet 192Spin glass 197Spin wave 202Spontaneous magnetization 205Superparamagnetism 206Vibrating sample magnetometer 210

ReferencesArticle Sources and Contributors 211Image Sources, Licenses and Contributors 215

Article LicensesLicense 218

Antiferromagnetism 1

Antiferromagnetism

Antiferromagnetic ordering

In materials that exhibit antiferromagnetism, the magnetic momentsof atoms or molecules, usually related to the spins of electrons, align ina regular pattern with neighboring spins (on different sublattices)pointing in opposite directions. This is, like ferromagnetism andferrimagnetism, a manifestation of ordered magnetism. Generally,antiferromagnetic order may exist at sufficiently low temperatures,vanishing at and above a certain temperature, the Néel temperature(named after Louis Néel, who had first identified this type of magneticordering).[1] Above the Néel temperature, the material is typically paramagnetic.

MeasurementWhen no external field is applied, the antiferromagnetic structure corresponds to a vanishing total magnetization. Inan external magnetic field, a kind of ferrimagnetic behavior may be displayed in the antiferromagnetic phase, withthe absolute value of one of the sublattice magnetizations differing from that of the other sublattice, resulting in anonzero net magnetization.The magnetic susceptibility of an antiferromagnetic material typically shows a maximum at the Néel temperature. Incontrast, at the transition between the ferromagnetic to the paramagnetic phases the susceptibility will diverge. In theantiferromagnetic case, a divergence is observed in the staggered susceptibility.Various microscopic (exchange) interactions between the magnetic moments or spins may lead to antiferromagneticstructures. In the simplest case, one may consider an Ising model on an bipartite lattice, e.g. the simple cubic lattice,with couplings between spins at nearest neighbor sites. Depending on the sign of that interaction, ferromagnetic orantiferromagnetic order will result. Geometrical frustration or competing ferro- and antiferromagnetic interactionsmay lead to different and, perhaps, more complicated magnetic structures.

Antiferromagnetic materialsAntiferromagnetic materials occur commonly among transition metal compounds, especially oxides. An example isthe heavy-fermion superconductor URu2Si2. Better known examples include hematite, metals such as chromium,alloys such as iron manganese (FeMn), and oxides such as nickel oxide (NiO). There are also numerous examplesamong high nuclearity metal clusters. Organic molecules can also exhibit antiferromagnetic coupling under rarecircumstances, as seen in radicals such as 5-dehydro-m-xylylene.Antiferromagnets can couple to ferromagnets, for instance, through a mechanism known as exchange bias, in whichthe ferromagnetic film is either grown upon the antiferromagnet or annealed in an aligning magnetic field, causingthe surface atoms of the ferromagnet to align with the surface atoms of the antiferromagnet. This provides the abilityto "pin" the orientation of a ferromagnetic film, which provides one of the main uses in so-called spin valves, whichare the basis of magnetic sensors including modern hard drive read heads. The temperature at or above which anantiferromagnetic layer loses its ability to "pin" the magnetization direction of an adjacent ferromagnetic layer iscalled the blocking temperature of that layer and is usually lower than the Néel temperature.

Antiferromagnetism 2

Geometric frustrationUnlike ferromagnetism, anti-ferromagnetic interactions can lead to multiple optimal states (ground states—states ofminimal energy). In one dimension, the anti-ferromagnetic ground state is an alternating series of spins: up, down,up, down, etc. Yet in two dimensions, multiple ground states can occur.Consider an equilateral triangle with three spins, one on each vertex. If each spin can take on only two values (up ordown), there are 23 = 8 possible states of the system, six of which are ground states. The two situations which are notground states are when all three spins are up or are all down. In any of the other six states, there will be twofavorable interactions and one unfavorable one. This illustrates frustration: the inability of the system to find a singleground state. This type of magnetic behavior has been found in minerals that have a crystal stacking structure such asa Kagome lattice or hexagonal lattice.

Other propertiesAntiferromagnetism plays a crucial role in giant magnetoresistance, as had been discovered in 1988 by the Nobelprize winners Albert Fert and Peter Grünberg (awarded in 2007).There are also examples of disordered materials (such as iron phosphate glasses) that become antiferromagneticbelow their Néel temperature. These disordered networks 'frustrate' the antiparallelism of adjacent spins; i.e. it is notpossible to construct a network where each spin is surrounded by opposite neighbour spins. It can only bedetermined that the average correlation of neighbour spins is antiferromagnetic. This type of magnetism issometimes called speromagnetism.

References[1] L. Néel, Propriétées magnétiques des ferrites; Férrimagnétisme et antiferromagnétisme, Annales de Physique (Paris) 3, 137–198 (1948).

BiotSavart law 3

Biot–Savart lawIn physics, particularly electromagnetism, the Biot–Savart law (  /ˈbiːoʊsəˈvɑr/ or /ˈbjoʊsəˈvɑr/)[1] is anequation that describes the magnetic field generated by an electric current. It relates the magnetic field to themagnitude, direction, length, and proximity of the electric current. The law is valid in the magnetostaticapproximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism.[2]

Equation

Electric currents (along closed curve)The Biot–Savart law is used to compute the resultant magnetic field B at position r generated by a steady current I(for example due to a wire): a continual flow of charges which is constant in time and the charge neither accumulatesnor depletes at any point. The law is a physical example of a line integral: evaluated over the path C the electriccurrents flow. The equation in SI units is[3]

where r is the full displacement vector from the wire element to the point at which the field is being computed and r̂is the unit vector of r. Using this the equation can be equivalently written

where dl is a vector whose magnitude is the length of the differential element of the wire, in the direction ofconventional current, and μ0 is the magnetic constant. The symbols in boldface denote vector quantities.The integral is usually around a closed curve, since electric currents can only flow around closed paths. An infinitelylong wire (as used in the definition of the SI unit of electric current - the Ampere) is a counter-example.To apply the equation, the point in space where the magnetic field is to be calculated is chosen. Holding that pointfixed, the line integral over the path of the electric currents is calculated to find the total magnetic field at that point.The application of this law implicitly relies on the superposition principle for magnetic fields, i.e. the fact that themagnetic field is a vector sum of the field created by each infinitesimal section of the wire individually.[4]

Electric currents (throughout conductor volume)The formulations given above work well when the current can be approximated as running through aninfinitely-narrow wire. If the current has some thickness, the proper formulation of the Biot–Savart law (again in SIunits) is:

or equivalently

where dV is the differential element of volume and J is the current density vector in that volume.In this case the integral is over the volume of the conductor.The Biot–Savart law is fundamental to magnetostatics, playing a similar role to Coulomb's law in electrostatics.When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations.

BiotSavart law 4

Constant uniform currentIn the special case of a steady constant current I, the magnetic field B is

i.e. the current can be taken out the integral.

Point charge at constant velocityIn the case of a point charged particle q moving at a constant velocity v, then Maxwell's equations give the followingexpression for the electric field and magnetic field:[5]

where r̂ is the vector pointing from the current (non-retarded) position of the particle to the point at which the field isbeing measured, and θ is the angle between v and r.When v2 ≪ c2, the electric field and magnetic field can be approximated as[5]

These equations are called the "Biot–Savart law for a point charge"[6] due to its closely analogous form to the"standard" Biot–Savart law given previously. These equations were first derived by Oliver Heaviside in 1888.

Magnetic responses applicationsThe Biot–Savart law can be used in the calculation of magnetic responses even at the atomic or molecular level, e.g.chemical shieldings or magnetic susceptibilities, provided that the current density can be obtained from a quantummechanical calculation or theory.

Aerodynamics applications

The figure shows the velocity ( ) induced at apoint P by an element of vortex filament ( ) of

strength .

The Biot–Savart law is also used in aerodynamic theory tocalculate the velocity induced by vortex lines.

In the aerodynamic application, the roles of vorticity and currentare reversed as when compared to the magnetic application.

In Maxwell's 1861 paper 'On Physical Lines of Force',[7] magneticfield strength H was directly equated with pure vorticity (spin),whereas B was a weighted vorticity that was weighted for thedensity of the vortex sea. Maxwell considered magneticpermeability μ to be a measure of the density of the vortex sea.Hence the relationship,

1.1. Magnetic induction current

was essentially a rotational analogy to the linear electric current relationship,2.2. Electric convection current

BiotSavart law 5

where ρ is electric charge density. B was seen as a kind of magnetic current of vortices aligned in their axialplanes, with H being the circumferential velocity of the vortices.

The electric current equation can be viewed as a convective current of electric charge that involves linear motion. Byanalogy, the magnetic equation is an inductive current involving spin. There is no linear motion in the inductivecurrent along the direction of the B vector. The magnetic inductive current represents lines of force. In particular, itrepresents lines of inverse square law force.In aerodynamics the induced air currents are forming solenoidal rings around a vortex axis that is playing the rolethat electric current plays in magnetism. This puts the air currents of aerodynamics into the equivalent role of themagnetic induction vector B in electromagnetism.In electromagnetism the B lines form solenoidal rings around the source electric current, whereas in aerodynamics,the air currents form solenoidal rings around the source vortex axis.Hence in electromagnetism, the vortex plays the role of 'effect' whereas in aerodynamics, the vortex plays the role of'cause'. Yet when we look at the B lines in isolation, we see exactly the aerodynamic scenario in so much as that B isthe vortex axis and H is the circumferential velocity as in Maxwell's 1861 paper.For a vortex line of infinite length, the induced velocity at a point is given by

where Γ is the strength of the vortex and r is the perpendicular distance between the point and the vortex line.This is a limiting case of the formula for vortex segments of finite length:

where A and B are the (signed) angles between the line and the two ends of the segment.

The Biot–Savart law, Ampère's circuital law, and Gauss's law for magnetismThe magnetic field B as calculated from the Biot–Savart law will always satisfy Ampère's circuital law and Gauss'slaw for magnetism.[8]

Outline of proof that a magnetic field calculated by the Biot–Savart law will always satisfy Gauss's law for magnetism and Ampère'slaw.

[8]

Starting with the Biot–Savart law:

Substituting the relation

and using the product rule for curls, as well as the fact that J does not depend on the unprimed coordinates, this equation can be rewritten as[8]

Since the divergence of a curl is always zero, this establishes Gauss's law for magnetism. Next, taking the curl of both sides, using the formula forthe curl of a curl, and again using the fact that J does not depend on the unprimed coordinates, we eventually get the result[8]

Finally, plugging in the relations[8]

BiotSavart law 6

(where δ is the Dirac delta function), using the fact that the divergence of J is zero (due to the assumption of magnetostatics), and performing anintegration by parts, the result turns out to be[8]

i.e. Ampère's law (without Maxwell's correction, the displacement current.).

Notes[1] (http:/ / dictionary. reference. com/ browse/ biot+ savart+ law?qsrc=2446)[2] Jackson, John David (1999). Classical Electrodynamics (3rd ed. ed.). New York: Wiley. Chapter 5. ISBN 0-471-30932-X.[3] Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9[4] The superposition principle holds for the electric and magnetic fields because they are the solution to a set of linear differential equations,

namely Maxwell's equations, where the current is one of the "source terms".[5] Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 222–224, 435–440. ISBN 0-13-805326-X.[6] http:/ / maxwell. ucdavis. edu/ ~electro/ magnetic_field/ pointcharge. html[7] Maxwell, J. C.. "On Physical Lines of Force" (http:/ / commons. wikimedia. org/ wiki/ File:On_Physical_Lines_of_Force. pdf). Wikimedia

commons. . Retrieved 25 December 2011.[8] See Jackson, page 178–79 or Griffiths p. 222–24. The presentation in Griffiths is particularly thorough, with all the details spelled out.

References• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed. ed.). Prentice Hall. ISBN 0-13-805326-X.• Feynman, Richard (1966). The Feynman Lectures on Physics (2nd ed. ed.). Addison-Wesley. ISBN 0-63-20717.

Further reading•• Electricity and Modern Physics (2nd Edition), G.A.G. Bennet, Edward Arnold (UK), 1974, ISBN 0-7131-2459-8•• Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN

0-7195-3382-1•• The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN

978-0-521-57507-2.•• Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008,

ISBN 0-7167-8964-7•• Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN

(Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3•• McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3

External links• Electromagnetism (http:/ / www. lightandmatter. com/ html_books/ 0sn/ ch11/ ch11. html), B. Crowell, Fullerton

College• MISN-0-125 The Ampère–Laplace–Biot–Savart Law (http:/ / physnet2. pa. msu. edu/ home/ modules/ pdf_modules/

m125. pdf) by Orilla McHarris and Peter Signell for Project PHYSNET (http:/ / www. physnet. org).

Classical electromagnetism and special relativity 7

Classical electromagnetism and special relativityThe theory of special relativity plays an important role in the modern theory of classical electromagnetism. First ofall, it gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered undera Lorentz transformation from one inertial frame of reference to another. Secondly, it sheds light on the relationshipbetween electricity and magnetism, showing that frame of reference determines if an observation followselectrostatic or magnetic laws. Third, it motivates a compact and convenient notation for the laws ofelectromagnetism, namely the "manifestly covariant" tensor form.Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatiblewith special relativity.[1] Moreover, the apparent coincidences in which the same effect was observed due to differentphysical phenomena by two different observers would be shown to be not coincidental in the least by specialrelativity. In fact, half of Einstein's 1905 first paper on special relativity, "On the Electrodynamics of MovingBodies," explains how to transform Maxwell's equations.

Transformation of the fields between inertial frames

The E and B fieldsThis equation, also called the Joules-Bernoulli equation, considers two inertial frames. As notation, the fieldvariables in one frame are unprimed, and in a frame moving relative to the unprimed frame at velocity v, the fieldsare denoted with primes. In addition, the fields parallel to the velocity v are denoted by while the fieldsperpendicular to v are denoted as . In these two frames moving at relative velocity v, the E-fields and B-fieldsare related by:[2]

where

is called the Lorentz factor and c is the speed of light in free space. The inverse transformations are the same exceptv → −v.An equivalent, alternative expression is:[3]

where v̂ is the velocity unit vector.If one of the fields is zero in one frame of reference, that doesn't necessarily mean it is zero in all other frames ofreference. This can be seen by, for instance, making the unprimed electric field zero in the transformation to theprimed electric field. In this case, depending on the orientation of the magnetic field, the primed system could see anelectric field, even though there is none in the unprimed system.This does not mean two completely different sets of events are seen in the two frames, but that the same sequence ofevents is described in two different ways (see Moving magnet and conductor problem below).

Classical electromagnetism and special relativity 8

If a particle of charge q moves with velocity u with respect to frame S, then the Lorentz force in frame S is:

In frame S', the Lorentz force is:

If S and S' have aligned axes then[4]:

A derivation for the transformation of the Lorentz force for the particular case u = 0 is given here.[5] A more generalone can be seen here.[6]

Component by component, for relative motion along the x-axis, this works out to be the following, in SI units:

and in Gaussian-cgs units, the transformation is given by:[7]

where .The transformations in this form can be made more compact by introducing the electromagnetic tensor (definedbelow), which is a covariant tensor.

The D and H fieldsFor the electric displacement D and magnetic intensity H, using the constitutive relations and the result for c2:

gives

Analogously for E and B, the D and H form the electromagnetic displacement tensor.

Classical electromagnetism and special relativity 9

The φ and A fieldsAn alternative simpler transformation of the EM field uses the electromagnetic potentials - the electric potential φand magnetic potential A:[8]

where is the parallel component of A to the direction of relative velocity between frames v, and is theperpendicular component. These transparently resemble the characteristic form of other Lorentz transformations(like time-position and energy-momentum), while the transformations of E and B above are slightly morecomplicated. The components can be collected together as:

The ρ and J fieldsAnalogously for the charge density ρ and current density J,[8]

Collecting components together:

Non-relativistic approximationsFor speeds v ≪ c, the relativistic factor γ ≈ 1, which yields:

so that there is no need to distinguish between the spatial and temporal coordinates in Maxwell's equations.

Classical electromagnetism and special relativity 10

Relationship between electricity and magnetism

“One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect.”—Richard Feynman[9]

Deriving magnetism from electrostaticsThe chosen reference frame determines if an electromagnetic phenomenon is viewed as an effect of electrostatics ormagnetism. Authors usually derive magnetism from electrostatics when special relativity and charge invariance aretaken into account. The Feynman Lectures on Physics (vol. 2, ch. 13-6) uses this method to derive the "magnetic"force on a moving charge next to a current-carrying wire. See also Haskell,[10] Landau,[11] and Field.[12]

Fields intermix in different framesThe above transformation rules show that the electric field in one frame contributes to the magnetic field in anotherframe, and vice versa.[13] This is often described by saying that the electric field and magnetic field are twointerrelated aspects of a single object, called the electromagnetic field. Indeed, the entire electromagnetic field can beencoded in a single rank-2 tensor called the electromagnetic tensor; see below.

Moving magnet and conductor problemA famous example of the intermixing of electric and magnetic phenomena in different frames of reference is calledthe "moving magnet and conductor problem", cited by Einstein in his 1905 paper on Special Relativity.If a conductor moves with a constant velocity through the field of a stationary magnet, eddy currents will beproduced due to a magnetic force on the electrons in the conductor. In the rest frame of the conductor, on the otherhand, the magnet will be moving and the conductor stationary. Classical electromagnetic theory predicts thatprecisely the same microscopic eddy currents will be produced, but they will be due to an electric force.[14]

Covariant formulation in vacuumThe laws and mathematical objects in classical electromagnetism can be written in a form which is manifestlycovariant. Here, this is only done so for vacuum (or for the microscopic Maxwell equations, not using macroscopicdescriptions of materials such as electric permittivity), and uses SI units.This section uses Einstein notation, including Einstein summation convention. See also Ricci calculus for a summaryof tensor index notations, and raising and lowering indices for definition of superscript and subscript indices, andhow to switch between them. The Minkowski metric tensor η here has metric signature (+−−−).

Field tensor and 4-currentThe above relativistic transformations suggest the electric and magnetic fields are coupled together, in amathematical object with 6 components: an antisymmetric second-rank tensor, or a bivector. This is called theelectromagnetic field tensor, usually written as Fμν. In matrix form:[15]

where c the speed of light - in natural units c = 1.

Classical electromagnetism and special relativity 11

There is another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacing E/c → Band B → − E/c, to get the dual tensor Gμν.

In the context of special relativity, both of these transform according to the Lorentz transformation according to

,where Λα

ν is the Lorentz transformation tensor for a change from one reference frame to another. The same tensor isused twice in the summation.The charge and current density, the sources of the fields, also combine into the four-vector

called the four-current.

Maxwell's equations in tensor formUsing these tensors, Maxwell's equations reduce to:[15]

Maxwell's equations (Covariant formulation)

where the partial derivatives may be written in various ways, see 4-gradient. The first equation listed abovecorresponds to both Gauss's Law (for α = 0) and the Ampère-Maxwell Law (for α = 1, 2, 3). The second equationcorresponds to the two remaining equations, Gauss's law for magnetism (for α = 0) and Faraday's Law ( for α = 1, 2,3).These tensor equations are manifestly-covariant, meaning the equations can be seen to be covariant by the indexpositions. This short form of writing Maxwell's equations illustrates an idea shared amongst some physicists, namelythat the laws of physics take on a simpler form when written using tensors.By lowering the indices on Fαβ to obtain Fαβ (see raising and lowering indices):

the second equation can be written in terms of Fαβ as:

where is the contravariant Levi-Civita symbol. Notice the cyclic permutation of indices in this equation:

.

Another covariant electromagnetic object is the electromagnetic stress-energy tensor, a covariant rank-2 tensor whichincludes the Poynting vector, Maxwell stress tensor, and electromagnetic energy density.

Classical electromagnetism and special relativity 12

4-potentialThe EM field tensor can also be written[16]

where

is the four-potential and

is the four-position.Using the 4-potential in the Lorenz gauge, an alternative manifestly-covariant formulation can be found in a singleequation (a generalization of an equation due to Bernhard Riemann by Arnold Sommerfeld, known as theRiemann–Sommerfeld equation,[17] or the covariant form of the Maxwell equations[18]):

Maxwell's equations (Covariant Lorenz gauge formulation)

where is the d'Alembertian operator, or four-Laplacian. For a more comprehensive presentation of these topics,see Covariant formulation of classical electromagnetism.

Footnotes[1] Questions remain about the treatment of accelerating charges: Haskell, " Special relativity and Maxwell's equations. (http:/ / www. cse. secs.

oakland. edu/ haskell/ SpecialRelativity. htm)"[2] Tai L. Chow (2006). Electromagnetic theory (http:/ / books. google. com/ books?id=dpnpMhw1zo8C& pg=PA153&

dq=isbn:0763738271#PPA368,M1). Sudbury MA: Jones and Bartlett. p. Chapter 10.21; p. 402–403 ff. ISBN 0-7637-3827-1. .[3] Daniel, Herbert (1997), "4.5.1" (http:/ / books. google. com/ books?id=8vAC8YG41goC), Physik: Elektrodynamik, relativistische Physik,

Walter de Gruyter, pp. 360–361, ISBN 3-11-015777-2, , Extract of pages 360-361 (http:/ / books. google. com/ books?id=8vAC8YG41goC&pg=PA360)

[4][4] R.C.Tolman "Relativity Thermodynamics and Cosmology" pp25[5] Force Laws and Maxwell's Equations http:/ / www. mathpages. com/ rr/ s2-02/ 2-02. htm at MathPages[6] http:/ / www. hep. princeton. edu/ ~mcdonald/ examples/ EM/ ganley_ajp_31_510_62. pdf[7][7] Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X[8][8] The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.[9] Feynman Lectures vol. 2, ch. 1-1[10] http:/ / www. cse. secs. oakland. edu/ haskell/ SpecialRelativity. htm[11] E M Lifshitz, L D Landau (1980). The classical theory of fields (http:/ / worldcat. org/ isbn/ 0750627689). Course of Theoretical Physics.

Vol. 2 (Fourth Edition ed.). Oxford UK: Butterworth-Heinemann. ISBN 0-7506-2768-9. .[12] J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its

relationship to quantum electrodynamics". Phys. Scr. 74 702-717[13] Tai L. Chow (2006). Electromagnetic theory (http:/ / books. google. com/ books?id=dpnpMhw1zo8C& pg=PA153&

dq=isbn:0763738271#PPR6,M1). Sudbury MA: Jones and Bartlett. p. 395. ISBN 0-7637-3827-1. .[14] David J Griffiths (1999). Introduction to electrodynamics (http:/ / worldcat. org/ isbn/ 013805326X) (Third Edition ed.). Prentice Hall.

pp. 478–9. ISBN 0-13-805326-X. .[15] Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. p. 557. ISBN 0-13-805326-X.[16] DJ Griffiths (1999). Introduction to electrodynamics. Saddle River NJ: Pearson/Addison-Wesley. p. 541. ISBN 0-13-805326-X.[17] Carver A. Mead (2002-08-07). Collective Electrodynamics: Quantum Foundations of Electromagnetism (http:/ / books. google. com/

?id=GkDR4e2lo2MC& pg=PA37& dq=Riemann+ Summerfeld). MIT Press. pp. 37–38. ISBN 978-0-262-63260-7. .[18] Frederic V. Hartemann (2002). High-field electrodynamics (http:/ / books. google. com/ ?id=tIkflVrfkG0C& pg=PA102&

dq=d'Alembertian+ covariant-form+ maxwell-lorentz). CRC Press. p. 102. ISBN 978-0-8493-2378-2. .

Coercivity 13

Coercivity

A family of hysteresis loops for grain-orientedelectrical steel (BR denotes remanence and HC is

the coercivity).

In materials science, the coercivity, also called the coercive field orcoercive force, of a ferromagnetic material is the intensity of theapplied magnetic field required to reduce the magnetization of thatmaterial to zero after the magnetization of the sample has been drivento saturation. Thus coercivity measures the resistance of aferromagnetic material to becoming demagnetized. Coercivity isusually measured in oersted or ampere/meter units and is denoted HC.It can be measured using a B-H Analyzer or magnetometer.

Ferromagnetic materials with high coercivity are called magneticallyhard materials, and are used to make permanent magnets. Permanentmagnets find application in electric motors, magnetic recording media (e.g. hard drives, floppy disks, or magnetictape) and magnetic separation.

Materials with low coercivity are said to be magnetically soft. They are used in transformer and inductor cores,recording heads, microwave devices, and magnetic shielding.

Experimental determination

The coercivity is a horizontal intercept of thehysteresis loop.

Typically the coercivity of a magnetic material is determined bymeasurement of the hysteresis loop, also called the magnetizationcurve, as illustrated in the figure. The apparatus used to acquire thedata is typically a vibrating-sample or alternating-gradientmagnetometer. The applied field where the data line crosses zero is thecoercivity. If an antiferromagnet is present in the sample, thecoercivities measured in increasing and decreasing fields may beunequal as a result of the exchange bias effect.

Coercivities of soft and hard magnets:hardness grows as crystal (domain) size and strew,

shrinks as smoothness or glassiness

Material Coercivity[Oe (A/m)]

[.1Mn:]6Fe:27Ni:Mo, Supermalloy 0.002 [1] (0.16)

Fe:4Ni, Permalloy 0.01 [2]–1 [3] (0.8-80)

.9995 iron–filings 0.05 [1]–470 [4] (4-37,000)

11Fe:Si, silicon iron 0.4–0.9 [5] (32-72)

Raw iron (1896) 2 [6] (160)

.99 Nickel 0.7 [4]–290 [7] (56-23,000)

Coercivity 14

ZnxFeNi1-xO3,ferrite for magnetron

15–200 [8] (1200-16,000)

2Fe:Co [9], Iron pole 240 [4] (19,000)

>.99 cobalt 10 [10]–900 [10] (800-72,000)

6Al:18Fe:8Co:Cu:6Ni–3Ti:8Al:20Fe:20Co:2Cu:8Ni,alnico 5–9, fridge magnet and stronger

640 [11]–2000 [12] (51,000-1.6*105)

Cr:Co:Pt,disk drive recording media

1700 [13] (1.4*105)

2Nd:14Fe:B, neodymium-iron-boron 10,000 [14]–12,000 [15] ((8-9.5)*105)

12Fe:13Pt, Fe48Pt52 12,300+[16] (9.8*105)

?(Dy,Nb,Ga,Co):2Nd:14Fe:B 25,600 [17]–26,300 [18] (2*106)

2Sm:17Fe:3N, samarium-iron-nitrogen (10 K) <500 [19]–35,000 [20] (40,000-2.8*106)

Sm:5Co, samarium-cobalt 40,000 [21] (3.2*106)

The coercivity of a material depends on the time scale over which a magnetization curve is measured. Themagnetization of a material measured at an applied reversed field which is nominally smaller than the coercivitymay, over a long time scale, slowly relax to zero. Relaxation occurs when reversal of magnetization by domain wallmotion is thermally activated and is dominated by magnetic viscosity.[22] The increasing value of coercivity at highfrequencies is a serious obstacle to the increase of data rates in high-bandwidth magnetic recording, compounded bythe fact that increased storage density typically requires a higher coercivity in the media.

TheoryAt the coercive field, the vector component of the magnetization of a ferromagnet measured along the applied fielddirection is zero. There are two primary modes of magnetization reversal: single-domain rotation and domain wallmotion. When the magnetization of a material reverses by rotation, the magnetization component along the appliedfield is zero because the vector points in a direction orthogonal to the applied field. When the magnetization reversesby domain wall motion, the net magnetization is small in every vector direction because the moments of all theindividual domains sum to zero. Magnetization curves dominated by rotation and magnetocrystalline anisotropy arefound in relatively perfect magnetic materials used in fundamental research.[23] Domain wall motion is a moreimportant reversal mechanism in real engineering materials since defects like grain boundaries and impurities serveas nucleation sites for reversed-magnetization domains. The role of domain walls in determining coercivity iscomplex since defects may pin domain walls in addition to nucleating them. The dynamics of domain walls inferromagnets is similar to that of grain boundaries and plasticity in metallurgy since both domain walls and grainboundaries are planar defects.

SignificanceAs with any hysteretic process, the area inside the magnetization curve during one cycle represents the work that isperformed on the material by the external field in reversing the magnetization, and is dissipated as heat. Commondissipative processes in magnetic materials include magnetostriction and domain wall motion. The coercivity is ameasure of the degree of magnetic hysteresis and therefore characterizes the lossiness of soft magnetic materials fortheir common applications.

Coercivity 15

The squareness (saturation remanence divided by saturation magnetization) and coercivity are figures of merit forhard magnets although energy product (saturation magnetization times coercivity) is most commonly quoted. The1980s saw the development of rare earth magnets with high energy products but undesirably low Curie temperatures.Since the 1990s new exchange spring hard magnets with high coercivities have been developed.[24]

References[1] http:/ / mysite. du. edu/ ~jcalvert/ phys/ iron. htm#Magn[2] http:/ / www. science. upd. edu. ph/ nip/ images/ pdfs/

magnetism%20and%20magneto-impedance%20of%20electroplated%20ni-fe%20permalloy%20thin%20films. pdf[3] http:/ / dx. doi. org/ 10. 1063/ 1. 365100[4] http:/ / hyperphysics. phy-astr. gsu. edu/ Hbase/ tables/ magprop. html[5] http:/ / cartech. ides. com/ datasheet. aspx?E=193~192~191~190~189& CK=1967748[6] http:/ / books. google. com/ books?id=G0cOAAAAYAAJ& pg=PA133[7] http:/ / dx. doi. org/ 10. 1063/ 1. 355560[8] http:/ / dx. doi. org/ 10. 1109/ 20. 619559[9] http:/ / books. google. com/ books?id=y0FF19lud5YC& pg=PA142[10] http:/ / pubs. acs. org/ doi/ abs/ 10. 1021/ jp045554t[11] http:/ / www. dextermag. com/ uploadedFiles/ Alnico_Data_Sheet. pdf[12] http:/ / ieeexplore. ieee. org/ xpl/ abs_free. jsp?arNumber=1066731[13] http:/ / dx. doi. org/ 10. 1109/ 20. 278737[14] http:/ / dx. doi. org/ 10. 1063/ 1. 353563[15] http:/ / wondermagnet. com/ magfaq. html[16] Chen & Nikles 2002[17] http:/ / dx. doi. org/ 10. 1016/ j. jmmm. 2006. 04. 029[18] http:/ / dx. doi. org/ 10. 1016/ S0304-8853(01)00017-8[19] http:/ / dx. doi. org/ 10. 1109/ TJMJ. 1992. 4565502[20] http:/ / cat. inist. fr?aModele=afficheN& cpsidt=4841321[21] http:/ / dx. doi. org/ 10. 1063/ 1. 368075[22][22] Gaunt 1986[23][23] Genish et al. 2004[24] Kneller & Hawig 1991

• Chen, Min; Nikles, David E. (2002). "Synthesis, self-assembly, and magnetic properties of FexCoyPt100-x-ynanoparticles". Nano Letters 2 (3): 211–214. doi:10.1021/nl015649w.

• Gaunt, P. (1986). "Magnetic viscosity and thermal activation energy". Journal of Applied Physics 59 (12):4129–4132. Bibcode 1986JAP....59.4129G. doi:10.1063/1.336671.

• Genish, Isaschar; Kats, Yevgeny; Klein, Lior; Reiner, James W.; Beasley, M. R. (2004). "Local measurements ofmagnetization reversal in thin films of SrRbO3". physica status solidi (c) 1 (12): 3440–3442.doi:10.1002/pssc.200405476.

• Kneller, E. F.; Hawig, R. (1991). "The exchange-spring magnet: a new material principle for permanentmagnets". IEEE Transactions on Magnetics 27 (4): 3588–3560. Bibcode 1991ITM....27.3588K.doi:10.1109/20.102931.

• Livingston, J. D. (1981). "A review of coercivity mechanisms". Journal of Applied Physics 52 (3): 2541–2545.Bibcode 1981JAP....52.2544L. doi:10.1063/1.328996.

Coercivity 16

External links• Magnetization reversal applet (coherent rotation) (http:/ / www. bama. ua. edu/ ~tmewes/ Java/ Reversal/ reversal.

shtml)• For a table of coercivities of various magnetic recording media, see " Degaussing Data Storage Tape Magnetic

Media (http:/ / www. fujifilmusa. com/ shared/ bin/ Degauss_Data_Tape. pdf)" (PDF), at fujifilmusa.com.ml-Coercivity

Diamagnetism

Levitating pyrolytic carbon

Diamagnetism is the property of an object or material whichcauses it to create a magnetic field in opposition to an externallyapplied magnetic field. Unlike a ferromagnet, a diamagnet is not apermanent magnet. Diamagnetism is believed to be due toquantum mechanics (and is understood in terms of Landaulevels[1]) and occurs because the external field alters the orbitalvelocity of electrons around their nuclei, thus changing themagnetic dipole moment. According to Lenz's law, the field ofthese electrons will oppose the magnetic field changes provided bythe applied field. The magnetic permeability of diamagnets is lessthan (a relative permeability less than 1). In most materialsdiamagnetism is a weak effect, but in a superconductor a strongquantum effect repels the magnetic field entirely, apart from a thinlayer at the surface.

Diamagnets were first discovered when Sebald Justinus Brugmansobserved in 1778 that bismuth and antimony were repelled bymagnetic fields. The term diamagnetism was coined by MichaelFaraday in September 1845, when he realized that every material responded (in either a diamagnetic or paramagneticway) to an applied magnetic field.

Diamagnetic materials

Notable diamagnetic materials[2]

Material χv

(10−5)

Superconductor −105

Pyrolytic carbon −40.0

Bismuth −16.6

Mercury −2.9

Silver −2.6

Carbon (diamond) −2.1

Lead −1.8

Carbon (graphite) −1.6

Diamagnetism 17

Copper −1.0

Water −0.91

Diamagnetism, to a greater or lesser degree, is a property of all materials and will always make a weak contributionto the material's response to a magnetic field. However, for materials that show some other form of magnetism (suchas ferromagnetism or paramagnetism), the diamagnetic contribution becomes negligible. Substances that mostlydisplay diamagnetic behaviour are termed diamagnetic materials, or diamagnets. Materials that are said to bediamagnetic are those that are usually considered by non-physicists to be non-magnetic, and include water, wood,most organic compounds such as petroleum and some plastics, and many metals including copper, particularly theheavy ones with many core electrons, such as mercury, gold and bismuth. The magnetic susceptibility of variousmolecular fragments are called Pascal's constants.Diamagnetic materials have a relative magnetic permeability that is less than or equal to 1, and therefore a magneticsusceptibility which is less than 0 since susceptibility is defined as χv = μv − 1. This means that diamagneticmaterials are repelled by magnetic fields. However, since diamagnetism is such a weak property its effects are notobservable in everyday life. For example, the magnetic susceptibility of diamagnets such as water is χv =−9.05×10−6. The most strongly diamagnetic material is bismuth, χv = −1.66×10−4, although pyrolytic carbon mayhave a susceptibility of χv = −4.00×10−4 in one plane. Nevertheless, these values are orders of magnitudes smallerthan the magnetism exhibited by paramagnets and ferromagnets. Note that because χv is derived from the ratio of theinternal magnetic field to the applied field, it is a dimensionless value.All conductors exhibit an effective diamagnetism when they experience a changing magnetic field. The Lorentzforce on electrons causes them to circulate around forming eddy currents. The eddy currents then produce an inducedmagnetic field which opposes the applied field, resisting the conductor's motion.

A superconductor acts as an essentially perfectdiamagnetic material when placed in a magneticfield and it excludes the field, and the flux lines

avoid the region

Superconductors may be considered to be perfect diamagnets (χv =−1), since they expel all fields (except in a thin surface layer) due tothe Meissner effect. However this effect is not due to eddy currents, asin ordinary diamagnetic materials (see the article onsuperconductivity).

Demonstrations of diamagnetism

Curving water surfaces

If a powerful magnet (such as a supermagnet) is covered with a layerof water (that is thin compared to the diameter of the magnet) then the field of the magnet significantly repels thewater. This causes a slight dimple in the water's surface that may be seen by its reflection.[3][4]

Diamagnetism 18

Diamagnetic levitation

A live frog levitates inside a 32 mm diameter verticalbore of a Bitter solenoid in a magnetic field of about 16

teslas at the Nijmegen High Field MagnetLaboratory.[5]

Diamagnets may be levitated in stable equilibrium in a magneticfield, with no power consumption. Earnshaw's theorem seems topreclude the possibility of static magnetic levitation. However,Earnshaw's theorem only applies to objects with positive moments,such as ferromagnets (which have a permanent positive moment)and paramagnets (which induce a positive moment). These areattracted to field maxima, which do not exist in free space.Diamagnets (which induce a negative moment) are attracted tofield minima, and there can be a field minimum in free space.

A thin slice of pyrolytic graphite, which is an unusually strongdiamagnetic material, can be stably floated in a magnetic field,such as that from rare earth permanent magnets. This can be donewith all components at room temperature, making a visuallyeffective demonstration of diamagnetism.

The Radboud University Nijmegen, the Netherlands, hasconducted experiments where water and other substances were

successfully levitated. Most spectacularly, a live frog (see figure) was levitated.[6]

In September 2009, NASA's Jet Propulsion Laboratory in Pasadena, California announced they had successfullylevitated mice using a superconducting magnet,[7] an important step forward since mice are closer biologically tohumans than frogs.[8] They hope to perform experiments regarding the effects of microgravity on bone and musclemass.

Recent experiments studying the growth of protein crystals has led to a technique using powerful magnets to allowgrowth in ways that counteract Earth's gravity.[9]

A simple homemade device for demonstration can be constructed out of bismuth plates and a few permanentmagnets that will levitate a permanent magnet.[10]

Theory of diamagnetismThe Bohr–van Leeuwen theorem proves that there cannot be any diamagnetism or paramagnetism in a purelyclassical system. Yet the classical theory for Langevin diamagnetism gives the same prediction as the quantumtheory.[11] The classical theory is given below.

Langevin diamagnetismThe Langevin theory of diamagnetism applies to materials containing atoms with closed shells (see dielectrics). Afield with intensity B, applied to an electron with charge e and mass m, gives rise to Larmor precession withfrequency ω = eB / 2m. The number of revolutions per unit time is ω / 2π, so the current for an atom with Zelectrons is (in SI units)[11]

The magnetic moment of a current loop is equal to the current times the area of the loop. Suppose the field is alignedwith the z axis. The average loop area can be given as , where is the mean square distance of theelectrons perpendicular to the z axis. The magnetic moment is therefore

Diamagnetism 19

If the distribution of charge is spherically symmetric, we can suppose that the distribution of x,y,z coordinates areindependent and identically distributed. Then , where is the mean square distanceof the electrons from the nucleus. Therefore . If is the number of atoms per unitvolume, the diamagnetic susceptibility is

Diamagnetism in metalsThe Langevin theory does not apply to metals because they have non-localized electrons. The theory for thediamagnetism of a free electron gas is called Landau diamagnetism, and instead considers the weak counter-actingfield that forms when their trajectories are curved due to the Lorentz force. Landau diamagnetism, however, shouldbe contrasted with Pauli paramagnetism, an effect associated with the polarization of delocalized electrons' spins.[12]

References[1] http:/ / physics. ucsc. edu/ ~peter/ 231/ magnetic_field/ node5. html[2] Nave, Carl L.. "Magnetic Properties of Solids" (http:/ / hyperphysics. phy-astr. gsu. edu/ Hbase/ tables/ magprop. html). Hyper Physics. .

Retrieved 2008-11-09.[3] Beatty, Bill (2005). "Neodymium supermagnets: Some demonstrations—Diamagnetic water" (http:/ / amasci. com/ amateur/ neodymium.

html#water). Science Hobbyist. . Retrieved September 2011.[4] Quit007 (2011). "Diamagnetism Gallery" (http:/ / quit007. deviantart. com/ gallery/ 23787987). DeviantART. . Retrieved September 2011.[5] "The Frog That Learned to Fly" (http:/ / www. ru. nl/ hfml/ research/ levitation/ diamagnetic/ ). High Field Laboratory. Radboud University

Nijmegen. 2011. . Retrieved September 2011.[6] "The Real Levitation" (http:/ / www. ru. nl/ hfml/ research/ levitation/ diamagnetic/ ). High Field Laboratory. Radboud University Nijmegen.

2011. . Retrieved September 2011.[7] Liu, Yuanming; Zhu, Da-Ming; Strayer, Donald M.; Israelsson, Ulf E. (2010). "Magnetic levitation of large water droplets and mice".

Advances in Space Research 45 (1): 208–213. Bibcode 2010AdSpR..45..208L. doi:10.1016/j.asr.2009.08.033.[8] Choi, Charles Q. (09-09-2009). "Mice levitated in lab" (http:/ / www. livescience. com/ animals/ 090909-mouse-levitation. html). Live

Science. . Retrieved September 2011.[9] Kleiner, Kurt (08-10-2007). "Magnetic gravity trick grows perfect crystals" (http:/ / www. newscientist. com/ article/

dn12467-magnetic-gravity-trick-grows-perfect-crystals. html). New Scientist. . Retrieved September 2011.[10] "Fun with diamagnetic levitation" (http:/ / web. archive. org/ web/ 20080212011654/ http:/ / www. fieldlines. com/ other/ diamag1. html).

ForceField. 02-12-2008. . Retrieved September 2011.[11] Kittel, Charles (1986). Introduction to Solid State Physics (6th ed.). John Wiley & Sons. pp. 299–302. ISBN 0-471-87474-4.[12] Chang, M. C.. "Diamagnetism and paramagnetism" (http:/ / phy. ntnu. edu. tw/ ~changmc/ Teach/ SS/ SS_note/ chap11. pdf). NTNU lecture

notes. . Retrieved 2011-02-24.

External links• Video of a museum-style magnetic elevation train model which makes use of diamagnetism (http:/ / www.

youtube. com/ watch?v=8tFsrGRwOOM)• Videos of frogs and other diamagnets levitated in a strong magnetic field (http:/ / www. ru. nl/ hfml/ research/

levitation/ diamagnetic/ )• Video of levitating pyrolytic graphite (http:/ / www. grand-illusions. com/ images/ articles/ toyshop/

diamagnetic_levitation_2/ diamagnetic_levitation_2. wmv)• Video of Meissner-Ochsenfeld effect involving liquid nitrogen (http:/ / www. science. tv/ watch/

e257e44aa9d5bade97ba/ liquid-nitrogen-and-superconductor)• Video of a piece of neodymium magnet levitating between blocks of bismuth. (http:/ / netti. nic. fi/ ~054028/

images/ LevitorMK1. 0-1. mpg)• Website about this device, with images (in Finnish). (http:/ / netti. nic. fi/ ~054028/ )

Electromagnet 20

ElectromagnetAn electromagnet is a type of magnet in which the magnetic field is produced by the flow of electric current. Themagnetic field disappears when the current is turned off. Electromagnets are widely used as components of otherelectrical devices, such as motors, generators, relays, loudspeakers, hard disks, MRI machines, scientific instruments,and magnetic separation equipment, as well as being employed as industrial lifting electromagnets for picking up andmoving heavy iron objects like scrap iron.

A simple electromagnet consisting of a coil ofinsulated wire wrapped around an iron core. The

strength of magnetic field generated isproportional to the amount of current.

Current (I) through a wire produces a magneticfield (B). The field is oriented according to the

right-hand rule.

An electric current flowing in a wire creates a magnetic field aroundthe wire (see drawing below). To concentrate the magnetic field, in anelectromagnet the wire is wound into a coil with many turns of wirelying side by side. The magnetic field of all the turns of wire passesthrough the center of the coil, creating a strong magnetic field there. Acoil forming the shape of a straight tube (a helix) is called a solenoid.Much stronger magnetic fields can be produced if a "core" offerromagnetic material, such as soft iron, is placed inside the coil. Theferromagnetic core increases the magnetic field to thousands of timesthe strength of the field of the coil alone, due to the high magneticpermeability μ of the ferromagnetic material. This is called aferromagnetic-core or iron-core electromagnet.

Magnetic field produced by a solenoid (coil ofwire). This drawing shows a cross section

through the center of the coil. The crosses arewires in which current is moving into the page;the dots are wires in which current is moving up

out of the page.

The direction of the magnetic field through a coil of wire can be foundfrom a form of the right-hand rule.[1][2][3][4][5][6] If the fingers of theright hand are curled around the coil in the direction of current flow(conventional current, flow of positive charge) through the windings,the thumb points in the direction of the field inside the coil. The side ofthe magnet that the field lines emerge from is defined to be the northpole.

The main advantage of an electromagnet over a permanent magnet isthat the magnetic field can be rapidly manipulated over a wide rangeby controlling the amount of electric current. However, a continuoussupply of electrical energy is required to maintain the field.

Electromagnet 21

How the iron core worksThe material of the core of the magnet (usually iron) is composed of small regions called magnetic domains that actlike tiny magnets (see ferromagnetism). Before the current in the electromagnet is turned on, the domains in the ironcore point in random directions, so their tiny magnetic fields cancel each other out, and the iron has no large scalemagnetic field. When a current is passed through the wire wrapped around the iron, its magnetic field penetrates theiron, and causes the domains to turn, aligning parallel to the magnetic field, so their tiny magnetic fields add to thewire's field, creating a large magnetic field that extends into the space around the magnet. The larger the currentpassed through the wire coil, the more the domains align, and the stronger the magnetic field is. Finally all thedomains are lined up, and further increases in current only cause slight increases in the magnetic field: thisphenomenon is called saturation.When the current in the coil is turned off, most of the domains lose alignment and return to a random state and thefield disappears. However some of the alignment persists, because the domains have difficulty turning their directionof magnetization, leaving the core a weak permanent magnet. This phenomenon is called hysteresis and theremaining magnetic field is called remanent magnetism. The residual magnetization of the core can be removed bydegaussing.

Electromagnet used in the Tevatron particle accelerator, Fermilab, USA

Laboratory electromagnet used in physics experiments, around 1910

Magnet in a mass spectrometer

AC electromagnet on the stator of an electric motor

Magnets in an electric bell

Electromagnet 22

History

Sturgeon's electromagnet, 1824

Danish scientist Hans Christian Ørsted discovered in 1820 that electric currentscreate magnetic fields. British scientist William Sturgeon invented theelectromagnet in 1824.[7][8] His first electromagnet was a horseshoe-shaped pieceof iron that was wrapped with about 18 turns of bare copper wire (insulated wiredidn't exist yet). The iron was varnished to insulate it from the windings. When acurrent was passed through the coil, the iron became magnetized and attractedother pieces of iron; when the current was stopped, it lost magnetization.Sturgeon displayed its power by showing that although it only weighed sevenounces (roughly 200 grams), it could lift nine pounds (roughly 4 kilos) when thecurrent of a single-cell battery was applied. However, Sturgeon's magnets wereweak because the uninsulated wire he used could only be wrapped in a singlespaced out layer around the core, limiting the number of turns. Beginning in1827, US scientist Joseph Henry systematically improved and popularized theelectromagnet.[9] By using wire insulated by silk thread he was able to windmultiple layers of wire on cores, creating powerful magnets with thousands of turns of wire, including one that couldsupport 2063 lb (unknown operator: u'strong' kg). The first major use for electromagnets was in telegraphsounders.

The magnetic domain theory of how ferromagnetic cores work was first proposed in 1906 by French physicistPierre-Ernest Weiss, and the detailed modern quantum mechanical theory of ferromagnetism was worked out in the1920s by Werner Heisenberg, Lev Landau, Felix Bloch and others.

Uses of electromagnets

Industrial electromagnet lifting scrapiron, 1914

Electromagnets are very widely used in electric and electromechanical devices,including:

• Motors and generators• Transformers• Relays, including reed relays originally used in telephone exchanges• Electric bells• Loudspeakers• Magnetic recording and data storage equipment: tape recorders, VCRs, hard

disks• Scientific instruments such as MRI machines and mass spectrometers• Particle accelerators• Magnetic locks• Magnetic separation of material•• Industrial lifting magnets• Electromagnetic suspension used for MAGLEV trains

Electromagnet 23

Analysis of ferromagnetic electromagnetsFor definitions of the variables below, see box at end of article.

The magnetic field of electromagnets in the general case is given by Ampere's Law:

which says that the integral of the magnetizing field H around any closed loop of the field is equal to the sum of thecurrent flowing through the loop. Another equation used, that gives the magnetic field due to each small segment ofcurrent, is the Biot-Savart law. Computing the magnetic field and force exerted by ferromagnetic materials isdifficult for two reasons. First, because the strength of the field varies from point to point in a complicated way,particularly outside the core and in air gaps, where fringing fields and leakage flux must be considered. Second,because the magnetic field B and force are nonlinear functions of the current, depending on the nonlinear relationbetween B and H for the particular core material used. For precise calculations, computer programs that can producea model of the magnetic field using the finite element method are employed.

Magnetic circuit – the constant B field approximation

Magnetic field (green) of a typical electromagnet, with the iron core C forming aclosed loop with two air gaps G in it. Most of the magnetic field B is concentrated inthe core. However some of the field lines BL, called the "leakage flux", do not follow

the full core circuit and so do not contribute to the force exerted by theelectromagnet. In the gaps G the field lines spread out beyond the boundaries of the

core in "fringing fields" BF. This increases the "resistance" (reluctance) of themagnetic circuit, decreasing the total magnetic flux in the core. Both the leakage fluxand the fringing fields get larger as the gaps are increased, reducing the force exerted

by the magnet. Line L shows the average length of the magnetic circuit, used inequation (1) below. It is the sum of the length Lcore in the iron core and the length

Lgap in the air gaps

In many practical applications ofelectromagnets, such as motors, generators,transformers, lifting magnets, andloudspeakers, the iron core is in the formof a loop or magnetic circuit, possiblybroken by a few narrow air gaps. This isbecause iron presents much less"resistance" (reluctance) to the magneticfield than air, so a stronger field can beobtained if most of the magnetic field'spath is within the core.

Since most of the magnetic field isconfined within the outlines of the coreloop, this allows a simplification of themathematical analysis. See the drawing atright. A common simplifying assumptionsatisfied by many electromagnets, whichwill be used in this section, is that themagnetic field strength B is constantaround the magnetic circuit and zerooutside it. Most of the magnetic field willbe concentrated in the core material (C).Within the core the magnetic field (B) willbe approximately uniform across any crosssection, so if in addition the core hasroughly constant area throughout itslength, the field in the core will beconstant. This just leaves the air gaps (G),

Electromagnet 24

if any, between core sections. In the gaps the magnetic field lines are no longer confined by the core, so they 'bulge'out beyond the outlines of the core before curving back to enter the next piece of core material, reducing the fieldstrength in the gap. The bulges (BF) are called fringing fields. However, as long as the length of the gap is smallerthan the cross section dimensions of the core, the field in the gap will be approximately the same as in the core. Inaddition, some of the magnetic field lines (BL) will take 'short cuts' and not pass through the entire core circuit, andthus will not contribute to the force exerted by the magnet. This also includes field lines that encircle the wirewindings but do not enter the core. This is called leakage flux. Therefore the equations in this section are valid forelectromagnets for which:1.1. the magnetic circuit is a single loop of core material, possibly broken by a few air gaps2.2. the core has roughly the same cross sectional area throughout its length.3.3. any air gaps between sections of core material are not large compared with the cross sectional dimensions of the

core.4.4. there is negligible leakage fluxThe main nonlinear feature of ferromagnetic materials is that the B field saturates at a certain value, which is around1.6 teslas (T) for most high permeability core steels. The B field increases quickly with increasing current up to thatvalue, but above that value the field levels off and becomes almost constant, regardless of how much current is sentthrough the windings. So the strength of the magnetic field possible from an iron core electromagnet is limited toaround 1.6 to 2 T.

Magnetic field created by a currentThe magnetic field created by an electromagnet is proportional to both the number of turns in the winding, N, and thecurrent in the wire, I, hence this product, NI, in ampere-turns, is given the name magnetomotive force. For anelectromagnet with a single magnetic circuit, of which length Lcore is in the core material and length Lgap is in airgaps, Ampere's Law reduces to:[10][11]

where is the permeability of free space (or air); note that in this definition is

amperes.This is a nonlinear equation, because the permeability of the core, μ, varies with the magnetic field B. For an exactsolution, the value of μ at the B value used must be obtained from the core material hysteresis curve. If B isunknown, the equation must be solved by numerical methods. However, if the magnetomotive force is well abovesaturation, so the core material is in saturation, the magnetic field will be approximately the saturation value Bsat forthe material, and won't vary much with changes in NI. For a closed magnetic circuit (no air gap) most core materialssaturate at a magnetomotive force of roughly 800 ampere-turns per meter of flux path.

For most core materials, .[11] So in equation (1) above, the second term dominates.Therefore, in magnetic circuits with an air gap, the strength of the magnetic field B depends strongly on the length ofthe air gap, and the length of the flux path in the core doesn't matter much.

Electromagnet 25

Force exerted by magnetic fieldThe force exerted by an electromagnet on a section of core material is:

The 1.6 T limit on the field mentioned above sets a limit on the maximum force per unit core area, or pressure, aniron-core electromagnet can exert; roughly:

In more intuitive units it's useful to remember that at 1T the magnetic pressure is approximately 4 atmospheres, orkg/cm2.Given a core geometry, the B field needed for a given force can be calculated from (2); if it comes out to much morethan 1.6 T, a larger core must be used.

Closed magnetic circuit

Cross section of lifting electromagnet like that inabove photo, showing cylindrical construction.

The windings (C) are flat copper strips towithstand the Lorentz force of the magnetic field.The core is formed by the thick iron housing (D)

that wraps around the windings.

For a closed magnetic circuit (no air gap), such as would be found inan electromagnet lifting a piece of iron bridged across its poles,equation (1) becomes:

Substituting into (2), the force is:

It can be seen that to maximize the force, a core with a short flux pathL and a wide cross sectional area A is preferred. To achieve this, inapplications like lifting magnets (see photo above) and loudspeakers aflat cylindrical design is often used. The winding is wrapped around a short wide cylindrical core that forms onepole, and a thick metal housing that wraps around the outside of the windings forms the other part of the magneticcircuit, bringing the magnetic field to the front to form the other pole.

Force between electromagnetsThe above methods are inapplicable when most of the magnetic field path is outside the core. For electromagnets (orpermanent magnets) with well defined 'poles' where the field lines emerge from the core, the force between twoelectromagnets can be found using the 'Gilbert model' which assumes the magnetic field is produced by fictitious'magnetic charges' on the surface of the poles, with pole strength m and units of Ampere-turn meter. Magnetic polestrength of electromagnets can be found from:

The force between two poles is:

This model doesn't give the correct magnetic field inside the core, and thus gives incorrect results if the pole of onemagnet gets too close to another magnet.

Electromagnet 26

Side effects in large electromagnetsThere are several side effects which become important in large electromagnets and must be provided for in theirdesign:

Ohmic heating

Large aluminum busbars carrying current into theelectromagnets at the LNCMI (Laboratoire

National des Champs Magnétiques Intenses) highfield laboratory.

The only power consumed in a DC electromagnet is due to theresistance of the windings, and is dissipated as heat. Some largeelectromagnets require cooling water circulating through pipes in thewindings to carry off the waste heat.

Since the magnetic field is proportional to the product NI, the numberof turns in the windings N and the current I can be chosen to minimizeheat losses, as long as their product is constant. Since the powerdissipation, P = I2R, increases with the square of the current but onlyincreases approximately linearly with the number of windings, thepower lost in the windings can be minimized by reducing I andincreasing the number of turns N proportionally. For example halving Iand doubling N halves the power loss. This is one reason mostelectromagnets have windings with many turns of wire.

However, the limit to increasing N is that the larger number ofwindings takes up more room between the magnet's core pieces. If thearea available for the windings is filled up, more turns require going toa smaller diameter of wire, which has higher resistance, which cancelsthe advantage of using more turns. So in large magnets there is aminimum amount of heat loss that can't be reduced. This increases withthe square of the magnetic flux B2.

Inductive voltage spikesAn electromagnet is a large inductor, and resists changes in the current through its windings. Any sudden changes inthe winding current cause large voltage spikes across the windings. This is because when the current through themagnet is increased, such as when it is turned on, energy from the circuit must be stored in the magnetic field. Whenit is turned off the energy in the field is returned to the circuit.If an ordinary switch is used to control the winding current, this can cause sparks at the terminals of the switch. Thisdoesn't occur when the magnet is switched on, because the voltage is limited to the power supply voltage. But whenit is switched off, the energy in the magnetic field is suddenly returned to the circuit, causing a large voltage spikeand an arc across the switch contacts, which can damage them. With small electromagnets a capacitor is often usedacross the contacts, which reduces arcing by temporarily storing the current. More often a diode is used to preventvoltage spikes by providing a path for the current to recirculate through the winding until the energy is dissipated asheat. The diode is connected across the winding, oriented so it is reverse-biased during steady state operation anddoesn't conduct. When the supply voltage is removed, the voltage spike forward-biases the diode and the reactivecurrent continues to flow through the winding, through the diode and back into the winding. A diode used in thisway is called a flyback diode.Large electromagnets are usually powered by variable current electronic power supplies, controlled by amicroprocessor, which prevent voltage spikes by accomplishing current changes slowly, in gentle ramps. It may takeseveral minutes to energize or deenergize a large magnet.

Electromagnet 27

Lorentz forcesIn powerful electromagnets, the magnetic field exerts a force on each turn of the windings, due to the Lorentz force

acting on the moving charges within the wire. The Lorentz force is perpendicular to both the axis of thewire and the magnetic field. It can be visualized as a pressure between the magnetic field lines, pushing them apart.It has two effects on an electromagnet's windings:• The field lines within the axis of the coil exert a radial force on each turn of the windings, tending to push them

outward in all directions. This causes a tensile stress in the wire.•• The leakage field lines between each turn of the coil exert a repulsive force between adjacent turns, tending to

push them apart.The Lorentz forces increase with B2. In large electromagnets the windings must be firmly clamped in place, toprevent motion on power-up and power-down from causing metal fatigue in the windings. In the Bitter design,below, used in very high field research magnets, the windings are constructed as flat disks to resist the radial forces,and clamped in an axial direction to resist the axial ones.

Core lossesIn alternating current (AC) electromagnets, used in transformers, inductors, and AC motors and generators, themagnetic field is constantly changing. This causes energy losses in their magnetic cores that are dissipated as heat inthe core. The losses stem from two processes:• Eddy currents: From Faraday's law of induction, the changing magnetic field induces circulating electric currents

inside nearby conductors, called eddy currents. The energy in these currents is dissipated as heat in the electricalresistance of the conductor, so they are a cause of energy loss. Since the magnet's iron core is conductive, andmost of the magnetic field is concentrated there, eddy currents in the core are the major problem. Eddy currentsare closed loops of current that flow in planes perpendicular to the magnetic field. The energy dissipated isproportional to the area enclosed by the loop. To prevent them, the cores of AC electromagnets are made of stacksof thin steel sheets, or laminations, oriented parallel to the magnetic field, with an insulating coating on thesurface. The insulation layers prevent eddy current from flowing between the sheets. Any remaining eddycurrents must flow within the cross section of each individual lamination, which reduces losses greatly. Anotheralternative is to use a ferrite core, which is a nonconductor.

• Hysteresis losses: Reversing the direction of magnetization of the magnetic domains in the core material eachcycle causes energy loss, because of the coercivity of the material. These losses are called hysteresis. The energylost per cycle is proportional to the area of the hysteresis loop in the BH graph. To minimize this loss, magneticcores used in transformers and other AC electromagnets are made of "soft" low coercivity materials, such assilicon steel or soft ferrite.

The energy loss per cycle of the AC current is constant for each of these processes, so the power loss increaseslinearly with frequency.

Electromagnet 28

High field electromagnets

Superconducting electromagnets

The most powerful electromagnet in the world,the 45 T hybrid Bitter-superconducting magnet atthe US National High Magnetic Field Laboratory,

Tallahassee, Florida, USA

When a magnetic field higher than the ferromagnetic limit of 1.6 T isneeded, superconducting electromagnets can be used. Instead of usingferromagnetic materials, these use superconducting windings cooledwith liquid helium, which conduct current without electrical resistance.These allow enormous currents to flow, which generate intensemagnetic fields. Superconducting magnets are limited by the fieldstrength at which the winding material ceases to be superconducting.Current designs are limited to 10–20 T, with the current (2009) recordof 33.8 T.[12] The necessary refrigeration equipment and cryostat makethem much more expensive than ordinary electromagnets. However, inhigh power applications this can be offset by lower operating costs,since after startup no power is required for the windings, since noenergy is lost to ohmic heating. They are used in particle accelerators,MRI machines, and research.

Bitter electromagnets

Both iron-core and superconducting electromagnets have limits to thefield they can produce. Therefore the most powerful man-mademagnetic fields have been generated by air-core nonsuperconductingelectromagnets of a design invented by Francis Bitter in 1933, called Bitter electromagnets.[13] Instead of wirewindings, a Bitter magnet consists of a solenoid made of a stack of conducting disks, arranged so that the currentmoves in a helical path through them. This design has the mechanical strength to withstand the extreme Lorentzforces of the field, which increase with B2. The disks are pierced with holes through which cooling water passes tocarry away the heat caused by the high current. The strongest continuous field achieved with a resistive magnet iscurrently (2008) 35 T, produced by a Bitter electromagnet.[12] The strongest continuous magnetic field, 45 T,[13] wasachieved with a hybrid device consisting of a Bitter magnet inside a superconducting magnet.

Exploding electromagnetsThe factor limiting the strength of electromagnets is the inability to dissipate the enormous waste heat, so morepowerful fields, up to 100 T,[12] have been obtained from resistive magnets by sending brief pulses of currentthrough them. The most powerful manmade magnetic fields have been created by using explosives to compress themagnetic field inside an electromagnet as it is pulsed. The implosion compresses the magnetic field to values ofaround 1000 T[13] for a few microseconds. While this method may seem very destructive there are methods tocontrol the blast so that neither the experiment nor the magnetic structure are harmed, by redirecting the brunt of theforce radially outwards. These devices are known as destructive pulsed electromagnets. They are used in physics andmaterials science research to study the properties of materials at high magnetic fields.

Electromagnet 29

Definition of terms

square meter cross sectional area of core

tesla Magnetic field (Magnetic flux density)

newton Force exerted by magnetic field

ampere per meter Magnetizing field

ampere Current in the winding wire

meter Total length of the magnetic field path

meter Length of the magnetic field path in the core material

meter Length of the magnetic field path in air gaps

ampere meter Pole strength of the electromagnet

newton per square ampere Permeability of the electromagnet core material

newton per square ampere Permeability of free space (or air) = 4π(10−7)

- Relative permeability of the electromagnet core material

- Number of turns of wire on the electromagnet

meter Distance between the poles of two electromagnets

References[1] Olson, Andrew (2008). "Right hand rules" (http:/ / www. ece. unb. ca/ Courses/ EE2683/ AW/ hand_rules. pdf). Science fair project

resources. Science Buddies. . Retrieved 2008-08-11.[2] Wilson, Adam (2008). "Hand Rules" (http:/ / www. ece. unb. ca/ Courses/ EE2683/ AW/ hand_rules. pdf). Course outline, EE2683 Electric

Circuits and Machines. Faculty of Engineering, Univ. of New Brunswick. . Retrieved 2008-08-11.[3] Gussow, Milton (1983). Schaum's Outline of Theory and Problems of Basic Electricity (http:/ / books. google. com/ ?id=T8t4MwtiLioC&

pg=PA166). New York: McGraw-Hill. pp. 166. ISBN 978-0-07-025240-0. .[4] Millikin, Robert; Edwin Bishop (1917). Elements of Electricity (http:/ / books. google. com/ ?id=dZM3AAAAMAAJ& pg=PA125). Chicago:

American Technical Society. pp. 125. .[5] Fleming, John Ambrose (1892). Short Lectures to Electrical Artisans, 4th Ed. (http:/ / books. google. com/ ?id=wzdHAAAAIAAJ&

pg=PA38). London: E.& F. N. Spon. pp. 38–40. .[6] Fleming, John Ambrose (1902). Magnets and Electric Currents, 2nd Edition (http:/ / books. google. com/ ?id=ASUYAAAAYAAJ&

pg=PA173). London: E.& F. N. Spon. pp. 173–174. .[7] Sturgeon, W. (1825). "Improved Electro Magnetic Apparatus". Trans. Royal Society of Arts, Manufactures, & Commerce (London) 43:

37–52. cited in Miller, T.J.E (2001). Electronic Control of Switched Reluctance Machines (http:/ / books. google. com/ ?id=E8VroIWyjB8C&pg=PA7). Newnes. pp. 7. ISBN 0-7506-5073-7. .

[8] Windelspecht, Michael. Groundbreaking Scientific Experiments, Inventions, and Discoveries of the 19th Century (http:/ / books. google.com/ books?id=hX1jPbJVSu4C& pg=PR22& lpg=PR22& dq="William+ Sturgeon"+ electromagnet+ 1825& source=web& ots=BhXj3j9j4t&sig=6gI6QNC-Yc5YMCY5RpEE43eIfgU& hl=en& sa=X& oi=book_result& resnum=9& ct=result#PPR22,M1), xxii, Greenwood PublishingGroup, 2003, ISBN 0-313-31969-3.

[9] Sherman, Roger (2007). "Joseph Henry's contributions to the electromagnet and the electric motor" (http:/ / siarchives. si. edu/ history/ jhp/joseph21. htm). The Joseph Henry Papers. The Smithsonian Institution. . Retrieved 2008-08-27.

[10] Feynmann, Richard P. (1963). Lectures on Physics, Vol. 2. New York: Addison-Wesley. pp. 36–9 to 36–11. ISBN 0-201-02117-XP., eq.36-26

[11] Fitzgerald, A.; Charles Kingsley, Alexander Kusko (1971). Electric Machinery, 3rd Ed.. USA: McGraw-Hill. pp. 3–5. ISBN 07021140X.[12] "Mag Lab World Records" (http:/ / www. magnet. fsu. edu/ mediacenter/ factsheets/ records. html). Media Center. National High Magnetic

Field Laboratory, USA. 2008. . Retrieved 2008-08-31.[13] Coyne, Kristin (2008). "Magnets: from Mini to Mighty" (http:/ / www. magnet. fsu. edu/ education/ tutorials/ magnetacademy/ magnets/

fullarticle. html). Magnet Lab U. National High Magnetic Field Laboratory. . Retrieved 2008-08-31.

Electromagnet 30

External links• Magnets from Mini to Mighty: Primer on electromagnets and other magnets (http:/ / www. magnet. fsu. edu/

education/ tutorials/ magnetacademy/ magnets/ ) National High Magnetic Field Laboratory• Magnetic Fields and Forces (http:/ / instruct. tri-c. edu/ fgram/ web/ mdipole. htm) Cuyahoga Community College• Fundamental Relationships (http:/ / geophysics. ou. edu/ solid_earth/ notes/ mag_basic/ mag_basic. html) School

of Geology and Geophysics, University of Oklahoma

FerrimagnetismNot to be confused with Ferromagnetism; for an overview see Magnetism

Ferrimagnetic ordering

In physics, a ferrimagnetic material is one in which the magneticmoments of the atoms on different sublattices are opposed, as inantiferromagnetism; however, in ferrimagnetic materials, the opposingmoments are unequal and a spontaneous magnetization remains. Thishappens when the sublattices consist of different materials or ions(such as Fe2+ and Fe3+).

Ferrimagnetism is exhibited by ferrites and magnetic garnets. Theoldest-known magnetic substance, magnetite (iron(II,III) oxide;Fe3O4), is a ferrimagnet; it was originally classified as a ferromagnet before Néel's discovery of ferrimagnetism andantiferromagnetism in 1948.[1]

Some ferrimagnetic materials are YIG (yttrium iron garnet) and ferrites composed of iron oxides and other elementssuch as aluminum, cobalt, nickel, manganese and zinc.

Effects of temperature

➀ Below the magnetization compensation point,ferrimagnetic material is magnetic. ➁ At the

compensation point, the magnetic componentscancel each other and the total magnetic moment is

zero. ➂ Above the Curie point, material losesmagnetism.

Ferrimagnetic materials are like ferromagnets in that they hold aspontaneous magnetization below the Curie temperature, and show nomagnetic order (are paramagnetic) above this temperature. However,there is sometimes a temperature below the Curie temperature atwhich the two sublattices have equal moments, resulting in a netmagnetic moment of zero; this is called the magnetizationcompensation point. This compensation point is observed easily ingarnets and rare earth - transition metal alloys (RE-TM). Furthermore,ferrimagnets may also exhibit an angular momentum compensationpoint at which the angular momentum of the magnetic sublattices iscompensated. This compensation point is a crucial point for achievinghigh speed magnetization reversal in magnetic memory devices.[2]

Properties

Ferrimagnetic materials have high resistivity and have anisotropic properties. The anisotropy is actually induced byan external applied field. When this applied field aligns with the magnetic dipoles it causes a net magnetic dipolemoment and causes the magnetic dipoles to precess at a frequency controlled by the applied field, called Larmor or

precession frequency. As a particular example, a microwave signal circularly polarized in the same direction as this precession strongly interacts with the magnetic dipole moments; when it is polarized in the opposite direction the

Ferrimagnetism 31

interaction is very low. When the interaction is strong, the microwave signal can pass through the material. Thisdirectional property is used in the construction of microwave devices like isolators, circulators and gyrators.Ferrimagnetic materials are also used to produce optical isolators and circulators.

Molecular ferrimagnetsFerrimagnetism can also occur in molecular magnets. A classic example is a dodecanuclear Manganese moleculewith an effective spin of S = 10 derived from antiferromagnetic interaction on Mn(IV) metal centres with Mn(III)and Mn(II) metal centres.[3]

References[1] L. Néel, Propriétées magnétiques des ferrites; Férrimagnétisme et antiferromagnétisme, Annales de Physique (Paris) 3, 137-198 (1948).[2] C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and Th. Rasing, Ultrafast spin dynamics across compensation

points in ferrimagnetic GdFeCo: The role of angular momentum compensation, Phys. Rev. B 73, 220402(R) (2006).[3] Sessoli, Roberta; Tsai, Hui Lien ; Schake, Ann R. ; Wang, Sheyi; Vincent, John B.; Folting, Kirsten; Gatteschi, Dante; Christou, George;

Hendrickson, David N. (1993). "High-spin molecules: [Mn12O12(O2CR)16(H2O)4]". J. Am. Chem. Soc., 115 (5): 1804–1816.doi:10.1021/ja00058a027.

FerromagnetismNot to be confused with Ferrimagnetism; for an overview see Magnetism

A magnet made of alnico, an iron alloy.Ferromagnetism is the physical theory which

explains how materials become magnets.

Ferromagnetism is the basic mechanism by which certain materials(such as iron) form permanent magnets, or are attracted to magnets. Inphysics, several different types of magnetism are distinguished.Ferromagnetism (including ferrimagnetism)[1] is the strongest type; itis the only type that creates forces strong enough to be felt, and isresponsible for the common phenomena of magnetism encountered ineveryday life. Other substances respond weakly to magnetic fields withtwo other types of magnetism, paramagnetism and diamagnetism, butthe forces are so weak that they can only be detected by sensitiveinstruments in a laboratory. An everyday example of ferromagnetism isa refrigerator magnet used to hold notes on a refrigerator door. Theattraction between a magnet and ferromagnetic material is "the qualityof magnetism first apparent to the ancient world, and to us today".[2]

Permanent magnets (materials that can be magnetized by an externalmagnetic field and remain magnetized after the external field isremoved) are either ferromagnetic or ferrimagnetic, as are other materials that are noticeably attracted to them. Onlya few substances are ferromagnetic. The common ones are iron, nickel, cobalt and most of their alloys, somecompounds of rare earth metals, and a few naturally-occurring minerals such as lodestone.

Ferromagnetism is very important in industry and modern technology, and is the basis for many electrical andelectromechanical devices such as electromagnets, electric motors, generators, transformers, and magnetic storagesuch as tape recorders, and hard disks.

Ferromagnetism 32

History and distinction from ferrimagnetismHistorically, the term ferromagnet was used for any material that could exhibit spontaneous magnetization: a netmagnetic moment in the absence of an external magnetic field. This general definition is still in common use. Morerecently, however, different classes of spontaneous magnetization have been identified when there is more than onemagnetic ion per primitive cell of the material, leading to a stricter definition of "ferromagnetism" that is often usedto distinguish it from ferrimagnetism. In particular, a material is "ferromagnetic" in this narrower sense only if all ofits magnetic ions add a positive contribution to the net magnetization. If some of the magnetic ions subtract from thenet magnetization (if they are partially anti-aligned), then the material is "ferrimagnetic".[3] If the moments of thealigned and anti-aligned ions balance completely so as to have zero net magnetization, despite the magnetic ordering,then it is an antiferromagnet. These alignment effects only occur at temperatures below a certain critical temperature,called the Curie temperature (for ferromagnets and ferrimagnets) or the Néel temperature (for antiferromagnets).Among the first investigations of ferromagnetism are the pioneering works of Aleksandr Stoletov on measurement ofthe magnetic permeability of ferromagnetics, known as the Stoletov curve.

Ferromagnetic materials

Curie temperatures for some crystalline ferromagnetic (* = ferrimagnetic) materials[4]

Material Curietemp. (K)

Co 1388

Fe 1043

Fe2O3* 948

FeOFe2O3* 858

NiOFe2O3* 858

CuOFe2O3* 728

MgOFe2O3* 713

MnBi 630

Ni 627

MnSb 587

MnOFe2O3* 573

Y3Fe5O12* 560

CrO2 386

MnAs 318

Gd 292

Dy 88

EuO 69

The table on the right lists a selection of ferromagnetic and ferrimagnetic compounds, along with the temperatureabove which they cease to exhibit spontaneous magnetization (see Curie temperature).Ferromagnetism is a property not just of the chemical make-up of a material, but of its crystalline structure and microscopic organization. There are ferromagnetic metal alloys whose constituents are not themselves

Ferromagnetism 33

ferromagnetic, called Heusler alloys, named after Fritz Heusler. Conversely there are non-magnetic alloys, such astypes of stainless steel, composed almost exclusively of ferromagnetic metals.One can also make amorphous (non-crystalline) ferromagnetic metallic alloys by very rapid quenching (cooling) of aliquid alloy. These have the advantage that their properties are nearly isotropic (not aligned along a crystal axis); thisresults in low coercivity, low hysteresis loss, high permeability, and high electrical resistivity. One such typicalmaterial is a transition metal-metalloid alloy, made from about 80% transition metal (usually Fe, Co, or Ni) and ametalloid component (B, C, Si, P, or Al) that lowers the melting point.A relatively new class of exceptionally strong ferromagnetic materials are the rare-earth magnets. They containlanthanide elements that are known for their ability to carry large magnetic moments in well-localized f-orbitals.

Actinide ferromagnetsA number of actinide compounds are ferromagnets at room temperature or become ferromagnets below the Curietemperature (TC). PuP is one actinide pnictide that is a paramagnet and has cubic symmetry at room temperature, butupon cooling undergoes a lattice distortion to tetragonal when cooled to below its Tc = 125 K. PuP has an easy axisof <100>,[5] so that

at 5 K.[6] The lattice distortion is presumably a consequence of strain induced by the magnetoelastic interactions asthe magnetic moments aligned parallel within magnetic domains.In NpFe2 the easy axis is <111>.[7] Above TC ~500 K NpFe2 is also paramagnetic and cubic. Cooling below theCurie temperature produces a rhombohedral distortion wherein the rhombohedral angle changes from 60° (cubicphase) to 60.53°. An alternate description of this distortion is to consider the length c along the unique trigonal axis(after the distortion has begun) and a as the distance in the plane perpendicular to c. In the cubic phase this reduces to

= 1.00. Below the Curie temperature

which is the largest strain in any actinide compound.[6] NpNi2 undergoes a similar lattice distortion below TC = 32 K,with a strain of (43 ± 5) × 10−4.[6] NpCo2 is a ferrimagnet below 15 K.

Lithium gasIn 2009, a team of MIT physicists demonstrated that a lithium gas cooled to less than one Kelvin can exhibitferromagnetism.[8] The team cooled fermionic lithium-6 to less than 150 billionths of one Kelvin above absolutezero using infrared laser cooling. This demonstration is the first time that ferromagnetism has been demonstrated in agas.

ExplanationThe Bohr–van Leeuwen theorem shows that magnetism cannot occur in purely classical solids. Without quantummechanics, there would be no diamagnetism, paramagnetism or ferromagnetism. The property of ferromagnetism isdue to the direct influence of two effects from quantum mechanics: spin and the Pauli exclusion principle.[9]

Ferromagnetism 34

Origin of magnetismOne of the fundamental properties of an electron (besides that it carries charge) is that it has a dipole moment, i.e. itbehaves itself as a tiny magnet. This dipole moment comes from the more fundamental property of the electron thatit has quantum mechanical spin. The quantum mechanical nature of this spin causes the electron to only be able to bein two states, with the magnetic field either pointing "up" or "down" (for any choice of up and down). The spin of theelectrons in atoms is the main source of ferromagnetism, although there is also a contribution from the orbitalangular momentum of the electron about the nucleus. When these tiny magnetic dipoles are aligned in the samedirection, their individual magnetic fields add together to create a measurable macroscopic field.However in materials with a filled electron shell, the total dipole moment of the electrons is zero because the spinsare in up/down pairs. Only atoms with partially filled shells (i.e., unpaired spins) can have a net magnetic moment,so ferromagnetism only occurs in materials with partially filled shells. Because of Hund's rules, the first fewelectrons in a shell tend to have the same spin, thereby increasing the total dipole moment.These unpaired dipoles (often called simply "spins" even though they also generally include angular momentum)tend to align in parallel to an external magnetic field, an effect called paramagnetism. Ferromagnetism involves anadditional phenomenon, however: the dipoles tend to align spontaneously, giving rise to a spontaneousmagnetization, even when there is no applied field.

Exchange interactionAccording to classical electromagnetism, two nearby magnetic dipoles will tend to align in opposite directions, sotheir magnetic fields will oppose one another and cancel out. However, this effect is very weak, because themagnetic fields generated by individual spins are small and the resulting alignment is easily destroyed by thermalfluctuations. In a few materials, a much stronger interaction between spins arises because the change in the directionof the spin leads to a change in electrostatic repulsion between neighboring electrons, due to a particular quantummechanical effect called the exchange interaction. At short distances, the exchange interaction is much stronger thanthe dipole-dipole magnetic interaction. As a result, in a few materials, the ferromagnetic ones, nearby spins tend toalign in the same direction.The exchange interaction is related to the Pauli exclusion principle, which says that two electrons with the same spincannot also have the same "position". Therefore, under certain conditions, when the orbitals of the unpaired outervalence electrons from adjacent atoms overlap, the distributions of their electric charge in space are further apartwhen the electrons have parallel spins than when they have opposite spins. This reduces the electrostatic energy ofthe electrons when their spins are parallel compared to their energy when the spins are anti-parallel, so theparallel-spin state is more stable. In simple terms, the electrons, which repel one another, can move "further apart" byaligning their spins, so the spins of these electrons tend to line up. This difference in energy is called the exchangeenergy.The materials in which the exchange interaction is much stronger than the competing dipole-dipole interaction arefrequently called magnetic materials. For instance, in iron (Fe) the exchange force is about 1000 times stronger thanthe dipole interaction. Therefore below the Curie temperature virtually all of the dipoles in a ferromagnetic materialwill be aligned. The exchange interaction is also responsible for the other types of spontaneous ordering of atomicmagnetic moments occurring in magnetic solids, antiferromagnetism and ferrimagnetism. There are differentexchange interaction mechanisms which create the magnetism in different ferromagnetic, ferrimagnetic, andantiferromagnetic substances. These mechanisms include direct exchange, RKKY exchange, double exchange, andsuperexchange.

Ferromagnetism 35

Magnetic anisotropyAlthough the exchange interaction keeps spins aligned, it does not align them in a particular direction. Withoutmagnetic anisotropy, the spins in a magnet randomly change direction in response to thermal fluctuations and themagnet is superparamagnetic. There are several kinds of magnetic anisotropy, the most common of which ismagnetocrystalline anisotropy. This is a dependence of the energy on the direction of magnetization relative to thecrystallographic lattice. Another common source of anisotropy, inverse magnetostriction, is induced by internalstrains. Single-domain magnets also can have a shape anisotropy due to the magnetostatic effects of the particleshape. As the temperature of a magnet increases, the anisotropy tends to decrease, and there is often a blockingtemperature at which a transition to superparamagnetism occurs.[10]

Magnetic domainsThe above would seem to suggest that every piece of ferromagnetic material should have a strong magnetic field,since all the spins are aligned, yet iron and other ferromagnets are often found in an "unmagnetized" state.

Weiss domains microstructure

The reason for this is that a bulk piece of ferromagnetic material isdivided into tiny magnetic domains[11] (also known as Weiss domains).Within each domain, the spins are aligned, but (if the bulk material isin its lowest energy configuration, i.e. unmagnetized), the spins ofseparate domains point in different directions and their magnetic fieldscancel out, so the object has no net large scale magnetic field.

Ferromagnetic materials spontaneously divide into magnetic domainsbecause the exchange interaction is a short-range force, so over longdistances of many atoms the tendency of the magnetic dipoles toreduce their energy by orienting in opposite directions wins out. If allthe dipoles in a piece of ferromagnetic material are aligned parallel, itcreates a large magnetic field extending into the space around it. Thiscontains a lot of magnetostatic energy. The material can reduce thisenergy by splitting into many domains pointing in different directions, so the magnetic field is confined to smalllocal fields in the material, reducing the volume of the field. The domains are separated by thin domain walls anumber of molecules thick, in which the direction of magnetization of the dipoles rotates smoothly from onedomain's direction to the other.

Thus, a piece of iron in its lowest energy state ("unmagnetized") generally has little or no net magnetic field.However, if it is placed in a strong enough external magnetic field, the domain walls will move, reorienting thedomains so more of the dipoles are aligned with the external field. The domains will remain aligned when theexternal field is removed, creating a magnetic field of their own extending into the space around the material, thuscreating a "permanent" magnet. The domains do not go back to their original minimum energy configuration whenthe field is removed because the domain walls tend to become 'pinned' or 'snagged' on defects in the crystal lattice,preserving their parallel orientation. This is shown by the Barkhausen effect: as the magnetizing field is changed, themagnetization changes in thousands of tiny discontinuous jumps as the domain walls suddenly "snap" past defects.This magnetization as a function of the external field is described by a hysteresis curve. Although this state ofaligned domains found in a piece of magnetized ferromagnetic material is not a minimal-energy configuration, it ismetastable, and can persist for long periods, as shown by samples of magnetite from the sea floor which havemaintained their magnetization for millions of years.Alloys used for the strongest permanent magnets are "hard" alloys made with many defects in their crystal structure where the domain walls "catch" and stabilize. The net magnetization can be destroyed by heating and then cooling (annealing) the material without an external field, however. The thermal motion allows the domain boundaries to

Ferromagnetism 36

move, releasing them from any defects, to return to their low-energy unaligned state.

Curie temperatureAs the temperature increases, thermal motion, or entropy, competes with the ferromagnetic tendency for dipoles toalign. When the temperature rises beyond a certain point, called the Curie temperature, there is a second-orderphase transition and the system can no longer maintain a spontaneous magnetization, although it still respondsparamagnetically to an external field. Below that temperature, there is a spontaneous symmetry breaking and randomdomains form (in the absence of an external field). The Curie temperature itself is a critical point, where themagnetic susceptibility is theoretically infinite and, although there is no net magnetization, domain-like spincorrelations fluctuate at all length scales.The study of ferromagnetic phase transitions, especially via the simplified Ising spin model, had an important impacton the development of statistical physics. There, it was first clearly shown that mean field theory approaches failedto predict the correct behavior at the critical point (which was found to fall under a universality class that includesmany other systems, such as liquid-gas transitions), and had to be replaced by renormalization group theory.

References[1][1] Chikazumi 2009, p. 118[2] Richard M. Bozorth, Ferromagnetism, first published 1951, reprinted 1993 by IEEE Press, New York as a "Classic Reissue." ISBN

0-7803-1032-2.[3] Herrera, J. M.; Bachschmidt, A, Villain, F, Bleuzen, A, Marvaud, V, Wernsdorfer, W, Verdaguer, M (13 January 2008). "Mixed valency and

magnetism in cyanometallates and Prussian blue analogues". Philosophical Transactions of the Royal Society A: Mathematical, Physical andEngineering Sciences 366 (1862): 127–138. doi:10.1098/rsta.2007.2145.

[4] Kittel, Charles (1986). Introduction to Solid State Physics (sixth ed.). John Wiley and Sons. ISBN 0-471-87474-4.[5] Lander GH, Lam DJ (1976). "Neutron diffraction study of PuP: The electronic ground state". Phys Rev B. 14 (9): 4064–7.

Bibcode 1976PhRvB..14.4064L. doi:10.1103/PhysRevB.14.4064.[6] Mueller MH, Lander GH, Hoff HA, Knott HW, Reddy JF (Apr 1979). "Lattice distortions measured in actinide ferromagnets PuP, NpFe2, and

NpNi2" (http:/ / hal. archives-ouvertes. fr/ docs/ 00/ 21/ 88/ 17/ PDF/ ajp-jphyscol197940C421. pdf). J Phys Colloque C4, supplement 40 (4):C4–68–C4–69. .

[7] Aldred AT, Dunlap BD, Lam DJ, Lander GH, Mueller MH, Nowik I (1975). "Magnetic properties of neptunium Laves phases: NpMn2,NpFe2, NpCo2, and NpNi2". Phys Rev B. 11 (1): 530–44. Bibcode 1975PhRvB..11..530A. doi:10.1103/PhysRevB.11.530.

[8] G-B Jo, Y-R Lee, J-H Choi, C. A. Christensen, T. H. Kim, J. H. Thywissen, D. E. Pritchard, and W. Ketterle (2009). "ItinerantFerromagnetism in a Fermi Gas of Ultracold Atoms". Science 325 (5947): 1521–1524. Bibcode 2009Sci...325.1521J.doi:10.1126/science.1177112. PMID 19762638.

[9] Feynman, Richard P.; Robert Leighton, Matthew Sands (1963). The Feynman Lectures on Physics, Vol.2. USA: Addison-Wesley. pp. Ch. 37.ISBN 0-201-02011-4H.

[10] Aharoni, Amikam (1996). Introduction to the Theory of Ferromagnetism (http:/ / www. oup. com/ us/ catalog/ general/ subject/ Physics/ElectricityMagnetism/ ?view=usa& ci=9780198508090). Clarendon Press. ISBN 0-19-851791-2. .

[11] Feynman, Richard P.; Robert B. Leighton, Matthew Sands (1963). The Feynman Lectures on Physics, Vol. I (http:/ / books. google. com/books?id=bDF-uoUmttUC& pg=SA4-PA4& dq="inclined+ plane"+ + "conservation+ of+ energy"& hl=en& sa=X&ei=gQtdT6iLCanSiAK22tCsCw& ved=0CGwQ6AEwBg#v=onepage& q="inclined plane" "conservation of energy"& f=false). USA:California Inst. of Technology. pp. 37.5-37.6. ISBN 0-201-02117-XP. .

Ferromagnetism 37

Bibliography• Ashcroft, Neil W.; Mermin, N. David (1977). Solid state physics (27. repr. ed.). New York: Holt, Rinehart and

Winston. ISBN 978-0-03-083993-1.• Chikazumi, Sōshin (2009). Physics of ferromagnetism. English edition prepared with the assistance of C.D.

Graham, Jr (2nd ed.). Oxford: Oxford University Press. ISBN 9780199564811.• Jackson, John David (1998). Classical electrodynamics (3rd ed.). New York: Wiley. ISBN 978-0-471-30932-1.• E. P. Wohlfarth, ed., Ferromagnetic Materials (North-Holland, 1980).• "Heusler alloy," Encyclopædia Britannica Online, retrieved Jan. 23, 2005.• F. Heusler, W. Stark, and E. Haupt, Verh. der Phys. Ges. 5, 219 (1903).• S. Vonsovsky Magnetism of elementary particles (Mir Publishers, Moscow, 1975).• Tyablikov S. V. (1995): Methods in the Quantum Theory of Magnetism. Springer; 1st edition. ISBN

0-306-30263-2.

External links• Electromagnetism (http:/ / www. lightandmatter. com/ html_books/ 0sn/ ch11/ ch11. html) - a chapter from an

online textbook• Sandeman, Karl (January 2008). "Ferromagnetic Materials" (http:/ / www. msm. cam. ac. uk/ doitpoms/ tlplib/

ferromagnetic/ printall. php). DoITPoMS. Dept. of Materials Sci. and Metallurgy, Univ. of Cambridge. Retrieved2008-08-27. Detailed nonmathematical description of ferromagnetic materials with animated illustrations

History of electromagnetic theoryThe history of electromagnetism (including its use) dates back over several thousand years. In the history ofelectromagnetic theory, the ancients would have been acquainted with the effects of atmospheric electricity, inparticular lightning[1] as thunderstorms in most southern latitudes are common, and they also knew of St. Elmo's fire.They however had little understanding of electricity, and were unable to scientifically explain those phenomena.[2]

Electricity is treated jointly with magnetism, because both generally appear together; wherever electricity is inmotion, magnetism is also present.[3] The phenomenon of magnetism was observed early in the history ofmagnetism, but was not explained in contemporary understanding until the idea of magnetic induction wasdeveloped.[4] The phenomenon of electricity was observed early in the history of electricity, but was not fullyexplained in contemporary understanding until the idea of electric charge was fully developed.

History of electromagnetic theory 38

Ancient and classical historyThe knowledge of static electricity dates back to the earliest civilizations, but for millennia it remained merely aninteresting and mystifying phenomenon, without a theory to explain its behavior and often confused with magnetism.The ancients were acquainted with rather curious properties possessed by two minerals, amber (ἤλεκτρον) andmagnetic iron ore. Amber, when rubbed, attracts light bodies; magnetic iron ore has the power of attracting iron.[5]

The discovery of the property ofmagnets.

Magnets were first found in a naturalstate; certain iron oxides were discoveredin various parts of the world, notably inMagnesia in Asia Minor, that had theproperty of attracting small pieces of

iron, which is shown here.

Based on his find of an Olmec hematite artifact in Central America, theAmerican astronomer John Carlson has suggested that "the Olmec may havediscovered and used the geomagnetic lodestone compass earlier than 1000BC". If true, this "predates the Chinese discovery of the geomagneticlodestone compass by more than a millennium".[6][7] Carlson speculates thatthe Olmecs may have used similar artifacts as a directional device forastrological or geomantic purposes, or to orient their temples, the dwellings ofthe living or the interments of the dead. The earliest Chinese literaturereference to magnetism lies in a 4th century BC book called Book of the DevilValley Master (鬼 谷 子): "The lodestone makes iron come or it attractsit."[8]

The discovery of amber and other similar substances[9] in the ancient timessuggests the possible perception of it by pre-historic man.[10][11] Theaccidental rubbing against the skins with which he clothed himself may havecaused an attraction by the resin, thus electrified, of the light fur insufficiently marked degree to arrest his attention.[12] Between such a mereobservation of the fact, however and the making of any deduction from it,vast periods may have elapsed; but there came a time at last, when the amberwas looked upon as a strange inanimate substance which could influence oreven draw to itself other things; and this by its own apparent capacity and not through any mechanical bond orconnection extending from it to them; when it was recognized, in brief, that nature held a lifeless thing showing anattribute of life.[12]

Electric catfish are found in tropicalAfrica and the Nile River.

Long before any knowledge of electromagnetism existed, people wereindirectly aware of the effects of electricity. Lightning and certain othermanifestations of electricity were known in ancient times, but it was notunderstood that these phenomena had a common origin.[13] Ancient Egyptianswere aware of shocks when interacting with electric fish (such as the electriccatfish) or other animals (such as electric eels).[14] The shocks from animalswere apparent to observers since pre-history by a variety of peoples that cameinto contact with them. Texts from 2750 BC by the ancient Egyptians referredto these fish as "thunderer of the Nile" and saw them as the "protectors" of allthe other fish.[5] Another possible approach to the discovery of the identity of lightning and electricity from any othersource, is to be attributed to the Arabs, who before the 15th century used the same Arabic word for lightning (barq)and the electric ray.[13]

Thales of Miletus, writing at around 600 BC, noted that rubbing fur on various substances, such as amber wouldcause them to attract specks of dust and other light objects.[15] Thales wrote on the effect now known as staticelectricity. The Greeks noted that if they rubbed the amber for long enough they could even get an electric spark tojump.The electrostatic phenomena was again reported millennia later by Roman and Arabic naturalists and physicians.[16]

Several ancient writers, such as Pliny the Elder and Scribonius Largus, attested to the numbing effect of electric

History of electromagnetic theory 39

shocks delivered by catfish and torpedo rays. Pliny in his books writes: "The ancient Tuscans by their learning holdthat there are nine gods that send forth lightning and those of eleven sorts." This was in general the early pagan ideaof lightning.[13] The ancients held some concept that shocks could travel along conducting objects.[17] Patientssuffering from ailments such as gout or headache were directed to touch electric fish in the hope that the powerfuljolt might cure them.[18]

A number of objects found in Iraq in 1938 dated to the early centuries AD (Sassanid Mesopotamia), called theBaghdad Battery, resembles a galvanic cell and is believed by some to have been used for electroplating.[19] Theclaims are controversial because of supporting evidence and theories for the uses of the artifacts,[20][21] physicalevidence on the objects conducive for electrical functions,[22] and if they were electrical in nature. As a result thenature of these objects is based on speculation, and the function of these artifacts remains in doubt.[23]

Middle Ages and the RenaissanceThe attempt to account for magnetic attraction as the working of a soul in the stone led to the first attack of humanreason upon superstition and the foundation of philosophy. After the lapse of centuries, a new capacity of thelodestone became revealed in its polarity, or the appearance of opposite effects at opposite ends; then came the firstutilization of the knowledge thus far gained, in the mariner's compass, leading to the discovery of the New World,and the throwing wide of all the portals of the Old to trade and civilization.[12]

Shen Kua wrote Dream Pool Essays (夢溪 筆 談); Shen also first described the

magnetic needle.

In the 11th century, the Chinese scientist Shen Kuo (1031–1095) was the firstperson to write of the magnetic needle compass and that it improved theaccuracy of navigation by employing the astronomical concept of true north(Dream Pool Essays, AD 1088 ), and by the 12th century the Chinese wereknown to use the lodestone compass for navigation. In 1187, AlexanderNeckam was the first in Europe to describe the compass and its use fornavigation.

Magnetism was one of the few sciences which progressed in medievalEurope; for in the thirteenth century Peter Peregrinus, a native of Maricourt inPicardy, made a discovery of fundamental importance.[24] The French 13thcentury scholar conducted experiments on magnetism and wrote the firstextant treatise describing the properties of magnets and pivoting compassneedles.[5] The dry compass was invented around 1300 by Italian inventor Flavio Gioja.[25]

Archbishop Eustathius of Thessalonica, Greek scholar and writer of the 12th century, records that Woliver, king ofthe Goths, was able to draw sparks from his body. The same writer states that a certain philosopher was able whiledressing to draw sparks from his clothes, a result seemingly akin to that obtained by Robert Symmer in his silkstocking experiments, a careful account of which may be found in the 'Philosophical Transactions,' 1759.[13]

Italian physician Gerolamo Cardano wrote about electricity in De Subtilitate (1550) distinguishing, perhaps for thefirst time, between electrical and magnetic forces.Toward the late 16th century, a physician of Queen Elizabeth's time, Dr. William Gilbert, in De Magnete, expandedon Cardano's work and invented the New Latin word electricus from ἤλεκτρον (elektron), the Greek word for"amber". Gilbert, a native of Colchester, Fellow of St John's College, Cambridge, and sometime President of theCollege of Physicians, was one of the earliest and most distinguished English men of science — a man whose workGalileo thought enviably great. He was appointed Court physician, and a pension was settled on him to set him freeto continue his researches in Physics and Chemistry.[26]

Gilbert undertook a number of careful electrical experiments, in the course of which he discovered that many substances other than amber, such as sulphur, wax, glass, etc.,[27] were capable of manifesting electrical properties. Gilbert also discovered that a heated body lost its electricity and that moisture prevented the electrification of all

History of electromagnetic theory 40

bodies, due to the now well-known fact that moisture impaired the insulation of such bodies. He also noticed thatelectrified substances attracted all other substances indiscriminately, whereas a magnet only attracted iron. The manydiscoveries of this nature earned for Gilbert the title of founder of the electrical science.[13] By investigating theforces on a light metallic needle, balanced on a point, he extended the list of electric bodies, and found also thatmany substances, including metals and natural magnets, showed no attractive forces when rubbed. He noticed thatdry weather with north or east wind was the most favourable atmospheric condition for exhibiting electricphenomena—an observation liable to misconception till the difference between conductor and insulator wasunderstood.[26]

Robert Boyle.

Gilbert's work was followed up by Robert Boyle (1627—1691), the famousnatural philosopher who was once described as "father of Chemistry, anduncle of the Earl of Cork." Boyle was one of the founders of the RoyalSociety when it met privately in Oxford, and became a member of theCouncil after the Society was incorporated by Charles II. in 1663. He workedfrequently at the new science of electricity, and added several substances toGilbert's list of electrics. He left a detailed account of his researches under thetitle of Experiments on the Origin of Electricity.[26] Boyle, in 1675, stated thatelectric attraction and repulsion can act across a vacuum. One of hisimportant discoveries was that electrified bodies in a vacuum would attractlight substances, this indicating that the electrical effect did not depend uponthe air as a medium. He also added resin to the then known list ofelectrics.[13][28][29][30]

This was followed in 1660 by Otto von Guericke, who invented an earlyelectrostatic generator. By the end of the 17th Century, researchers had developed practical means of generatingelectricity by friction with an electrostatic generator, but the development of electrostatic machines did not begin inearnest until the 18th century, when they became fundamental instruments in the studies about the new science ofelectricity.

The first usage of the word electricity is ascribed to Sir Thomas Browne in his 1646 work, Pseudodoxia Epidemica.

History of electromagnetic theory 41

18th century

Improving the electric machine

Generator built by Francis Hauksbee.[31]

The electric machine was subsequently improved by Francis Hauksbee,Litzendorf, and by Prof. Georg Matthias Bose, about 1750. Litzendorfsubstituted a glass ball for the sulphur ball of Guericke. Boze was the first toemploy the "prime conductor" in such machines, this consisting of an iron rodheld in the hand of a person whose body was insulated by standing on a blockof resin. Ingenhousz, during 1746, invented electric machines made of plateglass.[32] Experiments with the electric machine were largely aided by thediscovery of the property of a glass plate, when coated on both sides withtinfoil, of accumulating a charge of electricity when connected with a sourceof electromotive force. The electric machine was soon further improved byAndrew Gordon, a Scotsman, Professor at Erfurt, who substituted a glasscylinder in place of a glass globe; and by Giessing of Leipzig who added a"rubber" consisting of a cushion of woollen material. The collector, consistingof a series of metal points, was added to the machine by Benjamin Wilsonabout 1746, and in 1762, John Canton of England (also the inventor of thefirst pith-ball electroscope) improved the efficiency of electric machines bysprinkling an amalgam of tin over the surface of the rubber.[13]

Electrics and non-electricsIn 1729, Stephen Gray conducted a series of experiments that demonstrated the difference between conductors andnon-conductors (insulators), showing amongst other things that a metal wire and even pack thread conductedelectricity, whereas silk did not. In one of his experiments he sent an electric current through 800 feet of hempenthread which was suspended at intervals by loops of silk thread. When he tried to conduct the same experimentsubstituting the silk for finely spun brass wire, he found that the electric current was no longer carried throughout thehemp cord, but instead seemed to vanish into the brass wire. From this experiment he classified substances into twocategories: "electrics" like glass, resin and silk and "non-electrics" like metal and water. "Electrics" conductedcharges while "non-electrics" held the charge.[13][33]

Vitreous and resinousIntrigued by Gray's results, in 1732, C. F. du Fay began to conduct several experiments. In his first experiment, DuFay concluded that all objects except metals, animals, and liquids could be electrified by rubbing and that metals,animals and liquids could be electrified by means of an electric machine, thus discrediting Gray's "electrics" and"non-electrics" classification of substances.In 1737 Du Fay and Hauksbee independently discovered what they believed to be two kinds of frictional electricity;one generated from rubbing glass, the other from rubbing resin. From this, Du Fay theorized that electricity consistsof two electrical fluids, "vitreous" and "resinous", that are separated by friction and that neutralize each other whencombined.[34] This two-fluid theory would later give rise to the concept of positive and negative electrical chargesdevised by Benjamin Franklin.[13]

History of electromagnetic theory 42

Leyden jar

Pieter van Musschenbroek

The Leyden jar, a type of capacitor for electrical energy in large quantities,was invented independently by Ewald Georg von Kleist on 11 October 1744and by Pieter van Musschenbroek in 1745—1746 at Leiden University (thelatter location giving the device its name).[35] William Watson, whenexperimenting with the Leyden jar, discovered in 1747 that a discharge ofstatic electricity was equivalent to an electric current. Capacitance was firstobserved by Von Kleist of Leyden in 1754.[36] Von Kleist happened to hold,near his electric machine, a small bottle, in the neck of which there was aniron nail. Touching the iron nail accidentally with his other hand he received asevere electric shock. In much the same way Musschenbroeck assisted byCunaens received a more severe shock from a somewhat similar glass bottle.Sir William Watson of England greatly improved this device, by covering thebottle, or jar, outside and in with tinfoil. This piece of electrical apparatus willbe easily recognized as the well-known Leyden jar, so called by the Abbot Nollet of Paris, after the place of itsdiscovery.[13]

In 1741, John Ellicott "proposed to measure the strength of electrification by its power to raise a weight in one scaleof a balance while the other was held over the electrified body and pulled to it by its attractive power". The SirWilliam Watson already mentioned conducted numerous experiments, about 1749, to ascertain the velocity ofelectricity in a wire. These experiments, although perhaps not so intended, also demonstrated the possibility oftransmitting signals to a distance by electricity. In these experiments, the signal appeared to travel the 12,276-footlength of the insulated wire instantaneously. Le Monnier in France had previously made somewhat similarexperiments, sending shocks through an iron wire 1,319 feet long.[13]

About 1750, first experiments in electrotherapeutics were made. Various experimenters made tests to ascertain thephysiological and therapeutical effects of electricity. Demainbray in Edinburgh examined the effects of electricityupon plants and concluded that the growth of two myrtle trees was quickened by electrification. These myrtles wereelectrified "during the whole month of October, 1746, and they put forth branches and blossoms sooner than othershrubs of the same kind not electrified.".[37] Abbé Ménon in France tried the effects of a continued application ofelectricity upon men and birds and found that the subjects experimented on lost weight, thus apparently showing thatelectricity quickened the excretions. The efficacy of electric shocks in cases of paralysis was tested in the countyhospital at Shrewsbury, England, with rather poor success.[38]

History of electromagnetic theory 43

Late 18th century

Benjamin Franklin

In 1752, Benjamin Franklin is frequently confused as the key luminary behindelectricity. William Watson and Benjamin Franklin share the discovery ofelectrical potentials. Benjamin Franklin promoted his investigations ofelectricity and theories through the famous, though extremely dangerous,experiment of flying a kite through a storm-threatened sky. A key attached tothe kite string sparked and charged a Leyden jar, thus establishing the linkbetween lightning and electricity.[39] Following these experiments heinvented a lightning rod. It is either Franklin (more frequently) or EbenezerKinnersley of Philadelphia (less frequently) who is considered as theestablisher of the convention of positive and negative electricity.

Theories regarding the nature of electricity were quite vague at this period,and those prevalent were more or less conflicting. Franklin considered thatelectricity was an imponderable fluid pervading everything, and which, in itsnormal condition, was uniformly distributed in all substances. He assumedthat the electrical manifestations obtained by rubbing glass were due to the production of an excess of the electricfluid in that substance and that the manifestations produced by rubbing wax were due to a deficit of the fluid. Thistheory was opposed by the "Two-fluid" theory due to Robert Symmer, 1759. By Symmer's theory the vitreous andresinous electricities were regarded as imponderable fluids, each fluid being composed of mutually repellentparticles while the particles of the opposite electricities arc mutually attractive. When the two fluids unite by reasonof their attraction for one another, their effect upon external objects is neutralized. The act of rubbing a bodydecomposes the fluids one of which remains in excess on the body and manifests itself as vitreous or resinouselectricity.[13]

Up to the time of Franklin's historic kite experiment[40] the identity of the electricity developed by rubbing and byelectrostatic machines (frictional electricity), with lightning had not been generally established Dr. Wall,[41] AbbotNollet, Hauksbee,[42] Stephen Gray[43] and John Henry Winkler[44] had indeed suggested the resemblance betweenthe phenomena of "electricity" and "lightning," Gray having intimated that they only differed in degree. It wasdoubtless Franklin, however, who first proposed tests to determine the sameness of the phenomena. In a letter toPeter Comlinson, London, 19 October 1752. Franklin, referring to his kite experiment, wrote,

"At this key the phial (Leyden jar) may be charged; and from the electric fire thus obtained spirits may bekindled, and all the other electric experiments be formed which are usually done by the help of a rubbed glassglobe or tube, and thereby the sameness of the electric matter with that of lightning be completelydemonstrated."[45]

Thomas-François Dalibard, at Marley, near Paris, on 10 May 1742, by means of a vertical iron rod 40 feet long,obtained results corresponding to those recorded by Franklin and somewhat prior to the date of Franklin'sexperiment. Franklin's important demonstration of the sameness of frictional electricity and lightning doubtlessadded zest to the efforts of the many experimenters in this field in the last half of the 18th century, to advance theprogress of the science.[13]

Franklin's observations aided later scientists such as Michael Faraday, Luigi Galvani, Alessandro Volta, André-Marie Ampère, and Georg Simon Ohm whose work provided the basis for modern electrical technology. The work of Faraday, Volta, Ampere, and Ohm is honored by society, in that fundamental units of electrical measurement are named after them. Others would also advance the field of knowledge including those workers William Watson, Boze, Smeaton, Louis Guillaume Le Monnier, Jacques de Romas, Jean Jallabert, Beccaria, Tiberius Cavallo, John Canton, Robert Symmer, Abbot Nollet, John Henry Winkler, Richman, Dr. Wilson, Kinnersley, Joseph Priestley, Franz Aepinus, Edward Hussey Délavai, Henry Cavendish, and Charles-Augustin de Coulomb. A

History of electromagnetic theory 44

description of many of the experiments and discoveries of these early workers in the fields of electrical science andart will be found in the scientific publications of the time; notably the 'Philosophical Transactions', 'PhilosophicalMagazine', Cambridge Mathematical Journal, Young's Natural Philosophy,' Priestley's 'History of Electricity,' 'Franklin's 'Experiments and Observations on Electricity,' Cavalli's 'Treatise on Electricity,' De la Rive's 'Treatise onElectricity.'[13]

Henry Elles was one of the first people to suggest links between electricity and magnetism. In 1757 he claimed thathe had written to the Royal Society in 1755 about the links between electricity and magnetism, asserting that "thereare some things in the power of magnetism very similar to those of electricity" but he did "not by any means thinkthem the same". In 1760 he similarly claimed that in 1750 he had been the first "to think how the electric fire may bethe cause of thunder".[46] Among the more important of the electrical experiments and researches at this period werethose of Franz Aepinus, a noted German scholar (1724–1802) and Henry Cavendish of London, England.[13]

To Aepinus is accorded the credit of having been the first to conceive the view of the reciprocal relationship ofelectricity and magnetism. In his work 'Tentamen Theoria Electricitatis et Magnetism,'[47] published in SaintPetersburg, 1759. he gives the following amplification of Franklin's theory, which in some of its features ismeasurably in accord with present day views: "The particles of the electric fluid repel each other, attract and areattracted by the particles of all bodies with a force that decreases in proportion as the distance increases; theelectric fluid exists in the pores of bodies; it moves unobstructedly through non-electric (conductors), but moves withdifficulty in insulators; the manifestations of electricity are due to the unequal distribution of the fluid in a body, orto the approach of bodies unequally charged with the fluid." Aepinus formulated a corresponding theory ofmagnetism excepting that in the case of magnetic phenomena the fluids only acted on the particles of iron. He alsomade numerous electrical experiments, amongst others those apparently showing that in order to manifest electricaleffects tourmaline requires to be heated to a temperature between 37.5 °С and 100 °C. In fact, tourmaline remainsunelectrified when its temperature is uniform, but manifests electrical properties when its temperature is rising orfalling. Crystals which manifest electrical properties in this way are termed pyro-electrics, amongst which, besidestourmaline, are sulphate of quinine and quartz.[13]

Cavendish independently conceived a theory of electricity nearly akin to that of Aepinus.[48] He also (1784) wasperhaps the first to utilize the electric spark to produce the explosion of hydrogen and oxygen in the properproportions to produce pure water. The same philosopher also discovered the inductive capacity of dielectrics(insulators) and as early as 1778 measured the specific inductive capacity for beeswax and other substances bycomparison with an air condenser.

Drawing of Coulomb's torsion balance.From Plate 13 of his 1785 memoir.

About 1784 C. A. Coulomb, after whom is named the electrical unit ofquantity, devised the torsion balance, by means of which he discovered whatis known as Coulomb's law; — The force exerted between two smallelectrified bodies varies inversely as the square of the distance; not asAepinus in his theory of electricity had assumed, merely inversely as thedistance. According to the theory advanced by Cavendish "the particlesattract and are attracted inversely as some less power of the distance than thecube."[13] A large part of the domain of electricity became virtually annexedby Coulomb's discovery of the law of inverse squares.

With the discovery, by the experiments of Watson and others, that electricitycould be transmitted to a distance, the idea of making practical use of thisphenomenon began, about 1753, to engross the minds of "inquisitive"persons, and to this end suggestions looking to the employment of electricityin the transmission of intelligence were made. The first of the methods

History of electromagnetic theory 45

devised for this purpose was probably that, due to Georges Lesage (1774).[49][50][51] This method consisted in theemployment of 24 wires, insulated from one another and each of which had a pith ball connected to its distant end.Each wire represented a letter of the alphabet. To send a message, a desired wire was charged momentarily withelectricity from an electric machine, whereupon the pith ball connected to that wire would fly out; and in this waymessages were transmitted. Other methods of telegraphing in which frictional electricity was employed were alsotried, some of which are described in the history on the telegraph.[13]

Hitherto the only electricity known was that developed by friction or rubbing, which was therefore termed frictionalelectricity. We now come to the era of galvanic or voltaic electricity. Volta discovered that chemical reactions couldbe used to create positively charged anodes and negatively charged cathodes. When a conductor was attachedbetween these, the difference in the electrical potential (also known as voltage) drove a current between themthrough the conductor. The potential difference between two points is measured in units of volts in recognition ofVolta's work.[13]

The first mention of voltaic electricity, although not recognized as such at the time, was probably made by Sulzer in1767, who on placing a small disc of zinc under his tongue and a small disc of copper over it, observed a peculiartaste when the respective metals touched at their edges. Sulzer assumed that when the metals came together theywere set into vibration, this acting upon the nerves of the tongue, producing the effects noticed. In 1790 Prof. LuigiAlyisio Galvani of Bologna on one occasion, while conducting experiments on "animal electricity," as he termed it,to which his attention had been turned by the twitching of a frog's legs in the presence of an electric machine,observed that the muscles of a frog which was suspended on an iron balustrade by a copper hook that passed throughits dorsal column underwent lively convulsions without any extraneous cause; the electric machine being at this timeabsent.[13]

To account for this phenomenon Galvani assumed that electricity of opposite kinds existed in the nerves and musclesof the frog; the muscles and nerves constituting the charged coatings of a Leyden jar. Galvani published the resultsof his discoveries, together with his hypothesis, which at once engrossed the attention of the physicists of that time;the most prominent of whom, Alexander Volta, professor of physics at Pavia, contended that the results observed byGalvani were due to the two metals, copper and iron, acting as "electromotors," and that the muscles of the frogplayed the part of a conductor, completing the circuit. This precipitated a long discussion between the adherents ofthe conflicting views; one set of adherents holding with Volta that the electric current was the result of anelectromotive force of contact at the two metals; the other set adopting a modification of Galvani's view andasserting that the current was due to a chemical affinity between the metals and the acids in the pile. MichaelFaraday wrote in the preface to his Experimental Researches, relative to the question whether metallic contact is or isnot productive of a part of the electricity of the voltaic pile: I see no reason as yet to alter the opinion I have given;... but the point itself is of such great importance that I intend at the first opportunity renewing the inquiry, and, if Ican, rendering the proofs either on the one side or the other, undeniable to all."[13]

Even Faraday himself, however, did not settle the controversy, and while the views of the advocates on both sides ofthe question have undergone modifications, as subsequent investigations and discoveries demanded, up to 1918diversity of opinion on these points continued to crop out. Volta made numerous experiments in support of his theoryand ultimately developed the pile or battery,[52] which was the precursor of all subsequent chemical batteries, andpossessed the distinguishing merit of being the first means by which a prolonged continuous current of electricitywas obtainable. Volta communicated a description of his pile to the Royal Society of London and shortly thereafterNicholson and Cavendish (1780) produced the decomposition of water by means of the electric current, using Volta'spile as the source of electromotive force.[13]

History of electromagnetic theory 46

19th century

Early 19th century

Alessandro Volta

In 1800 Alessandro Volta constructed the first device to produce a largeelectric current, later known as the electric battery. Napoleon, informed of hisworks, summoned him in 1801 for a command performance of hisexperiments. He received many medals and decorations, including the Légiond'honneur.

Davy in 1806, employing a voltaic pile of approximately 250 cells, orcouples, decomposed potash and soda, showing that these substances wererespectively the oxides of potassium and sodium, which metals previouslyhad been unknown. These experiments were the beginning ofelectrochemistry, the investigation of which Faraday took up, and concerningwhich in 1833 he announced his important law of electrochemicalequivalents, viz.: "The same quantity of electricity — that is, the same electric current — decomposes chemicallyequivalent quantities of all the bodies which it traverses; hence the weights of elements separated in theseelectrolytes are to each other as their chemical equivalents." Employing a battery of 2,000 elements of a voltaic pileHumphry Davy in 1809 gave the first public demonstration of the electric arc light, using for the purpose charcoalenclosed in a vacuum.[13]

Somewhat important to note, it was not until many years after the discovery of the voltaic pile that the sameness ofannual and frictional electricity with voltaic electricity was clearly recognized and demonstrated. Thus as late asJanuary 1833 we find Faraday writing[53] in a paper on the electricity of the electric ray. "After an examination of theexperiments of Walsh,[54][55] Ingenhousz, Henry Cavendish, Sir H. Davy, and Dr. Davy, no doubt remains on mymind as to the identity of the electricity of the torpedo with common (frictional) and voltaic electricity; and I presumethat so little will remain on the mind of others as to justify my refraining from entering at length into thephilosophical proof of that identity. The doubts raised by Sir Humphry Davy have been removed by his brother, Dr.Davy; the results of the latter being the reverse of those of the former. ... The general conclusion which must, I think,be drawn from this collection of facts (a table showing the similarity, of properties of the diversely namedelectricities) is, that electricity, whatever may be its source, is identical in its nature."[13]

It is proper to state, however, that prior to Faraday's time the similarity of electricity derived from different sourceswas more than suspected. Thus, William Hyde Wollaston,[56] wrote in 1801:[57] "This similarity in the means bywhich both electricity and galvanism (voltaic electricity) appear to be excited in addition to the resemblance that hasbeen traced between their effects shows that they are both essentially the same and confirm an opinion that hasalready been advanced by others, that all the differences discoverable in the effects of the latter may be owing to itsbeing less intense, but produced in much larger quantity." In the same paper Wollaston describes certainexperiments in which he uses very fine wire in a solution of sulphate of copper through which he passed electriccurrents from an electric machine. This is interesting in connection with the later day use of almost similarlyarranged fine wires in electrolytic receivers in wireless, or radio-telegraphy.[13]

History of electromagnetic theory 47

Hans Christian Ørsted

In the first half of the 19th century many very important additions were madeto the world's knowledge concerning electricity and magnetism. For example,in 1819 Hans Christian Ørsted of Copenhagen discovered the deflecting effectof an electric current traversing a wire upon- a suspended magnetic needle.[13]

This discovery gave a clue to the subsequently proved intimate relationshipbetween electricity and magnetism which was promptly followed up byAmpère who shortly thereafter (1821) announced his celebrated theory ofelectrodynamics, relating to the force that one current exerts upon another, byits electro-magnetic effects, namely[13]

1.1. Two parallel portions of a circuit attract one another if the currents in themare flowing in the same direction, and repel one another if the currentsflow in the opposite direction.

2.2. Two portions of circuits crossing one another obliquely attract one anotherif both the currents flow either towards or from the point of crossing, and repel one another if one flows to and theother from that point.

3.3. When an element of a circuit exerts a force on another element of a circuit, that force always tends to urge thesecond one in a direction at right angles to its own direction.

Ampere brought a multitude of phenomena into theory by his investigations of the mechanical forces betweenconductors supporting currents and magnets.Professor Seebeck, of Berlin, in 1821 discovered that when heat is applied to the junction of two metals that hadbeen soldered together an electric current is set up. This is termed Thermo-Electricity. Seebeck's device consists of astrip of copper bent at each end and soldered to a plate of bismuth. A magnetic needle is placed parallel with thecopper strip. When the heat of a lamp is applied to the junction of the copper and bismuth an electric current is set upwhich deflects the needle.[13]

Around this time, Siméon Denis Poisson attacked the difficult problem of induced magnetization, and his results,though differently expressed, are still the theory, as a most important first approximation. It was in the application ofmathematics to physics that his services to science were performed. Perhaps the most original, and certainly the mostpermanent in their influence, were his memoirs on the theory of electricity and magnetism, which virtually created anew branch of mathematical physics.George Green wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity andMagnetism in 1828. The essay introduced several important concepts, among them a theorem similar to the modernGreen's theorem, the idea of potential functions as currently used in physics, and the concept of what are now calledGreen's functions. George Green was the first person to create a mathematical theory of electricity and magnetismand his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, WilliamThomson, and others.Peltier in 1834 discovered an effect opposite to Thermo-Electricity, namely, that when a current is passed through acouple of dissimilar metals the temperature is lowered or raised at the junction of the metals, depending on thedirection of the current. This is termed the Peltier "effect". The variations of temperature are found to be proportionalto the strength of the current and not to the square of the strength of the current as in the case of heat due to theordinary resistance of a conductor. This second law is the C^2R law,[58] discovered experimentally in 1841 by theEnglish physicist Joule. In other words, this important law is that the heat generated in any part of an electric circuitis directly proportional to the product of the resistance of this part of the circuit and to the square of the strength ofcurrent flowing in the circuit.[13]

In 1822 Johann Schweigger devised the first galvanometer. This instrument was subsequently much improved by Wilhelm Weber (1833). In 1825 William Sturgeon of Woolwich, England, invented the horseshoe and straight bar

History of electromagnetic theory 48

electromagnet, receiving therefor the silver medal of the Society of Arts.[59] In 1837 Gauss and Weber (both notedworkers of this period) jointly invented a reflecting galvanometer for telegraph purposes. This was the forerunner ofthe Thomson reflecting and other exceedingly sensitive galvanometers once used in submarine signaling and stillwidely employed in electrical measurements. Arago in 1824 made the important discovery that when a copper disc isrotated in its own plane, and if a magnetic needle be freely suspended on a pivot over the disc, the needle will rotatewith the disc. If on the other hand the needle is fixed it will tend to retard the motion of the disc. This effect wastermed Arago's rotations.[13][60][61]

Georg Simon Ohm

Futile attempts were made by Charles Babbage, Peter Barlow, John Herscheland others to explain this phenomenon. The true explanation was reserved forFaraday, namely, that electric currents are induced in the copper disc by thecutting of the magnetic lines of force of the needle, which currents in turnreact on the needle. Georg Simon Ohm did his work on resistance in the years1825 and 1826, and published his results in 1827 as the book Die galvanischeKette, mathematisch bearbeitet.[62][63] He drew considerable inspiration fromFourier's work on heat conduction in the theoretical explanation of his work.For experiments, he initially used voltaic piles, but later used a thermocoupleas this provided a more stable voltage source in terms of internal resistanceand constant potential difference. He used a galvanometer to measure current,and knew that the voltage between the thermocouple terminals wasproportional to the junction temperature. He then added test wires of varyinglength, diameter, and material to complete the circuit. He found that his datacould be modeled through a simple equation with variable composed of the reading from a galvanometer, the lengthof the test conductor, thermocouple junction temperature, and a constant of the entire setup. From this, Ohmdetermined his law of proportionality and published his results. In 1827, he announced the now famous law thatbears his name, that is:

Electromotive force = Current × Resistance[64]

Ohm brought into order a host of puzzling facts connecting electromotive force and electric current in conductors,which all previous electricians had only succeeded in loosely binding together qualitatively under some rather vaguestatements. Ohm found that the results could be summed up in such a simple law and by Ohm's discovery a largepart of the domain of electricity became annexed to theory.

Faraday and Henry

History of electromagnetic theory 49

Joseph Henry

Michael Faraday

The discovery of electromagnetic induction was made almost simultaneously,although independently, by Michael Faraday and Joseph Henry. WhileFaraday's early results preceded those of Henry, Henry was first in his use ofthe transformer principle. Henry's discovery of self-induction and his work onspiral conductors using a copper coil were made public in 1835, just beforethose of Faraday.[65][66][67]

In 1831 began the epoch-making researches of Michael Faraday, the famouspupil and successor of Humphry Davy at the head of the Royal Institution,London, relating to electric and electromagnetic induction. The remarkableresearches of Faraday, the prince of experimentalists, on electrostatics andelectrodynamics and the induction of currents. These were rather long inbeing brought from the crude experimental state to a compact system,expressing the real essence. Faraday was not a competentmathematician,[68][69][70] but had he been one, he would have been greatlyassisted in his researches, have saved himself much useless speculation, andwould have anticipated much later work. He would, for instance, knowingAmpere's theory, by his own results have readily been led to Neumann'stheory, and the connected work of Helmholtz and Thomson. Faraday's studiesand researches extended from 1831 to 1855 and a detailed description of hisexperiments, deductions and speculations are to be found in his compiledpapers, entitled Experimental Researches in Electricity.' Faraday was byprofession a chemist. He was not in the remotest degree a mathematician inthe ordinary sense — indeed it is a question if in all his writings there is asingle mathematical formula.[13]

The experiment which led Faraday to the discovery of electric induction wasmade as follows: He constructed what is now and was then termed aninduction coil, the primary and secondary wires of which were wound on awooden bobbin, side by side, and insulated from one another. In the circuit ofthe primary wire he placed a battery of approximately 100 cells. In the

secondary wire he inserted a galvanometer. On making his first test he observed no results, the galvanometerremaining quiescent, but on increasing the length of the wires he noticed a deflection of the galvanometer in thesecondary wire when the circuit of the primary wire was made and broken. This was the first observed instance ofthe development of electromotive force by electromagnetic induction.[13]

He also discovered that induced currents are established in a second closed circuit when the current strength is variedin the first wire, and that the direction of the current in the secondary circuit is opposite to that in the first circuit.Also that a current is induced in a secondary circuit when another circuit carrying a current is moved to and from thefirst circuit, and that the approach or withdrawal of a magnet to or from a closed circuit induces momentary currentsin the latter. In short, within the space of a few months Faraday discovered by experiment virtually all the laws andfacts now known concerning electro-magnetic induction and magneto-electric induction. Upon these discoveries,with scarcely an exception, depends the operation of the telephone, the dynamo machine, and incidental to thedynamo electric machine practically all the gigantic electrical industries of the world, including electric lighting,electric traction, the operation of electric motors for power purposes, and electro-plating, electrotyping, etc.[13]

In his investigations of the peculiar manner in which iron filings arrange themselves on a cardboard or glass in proximity to the poles of a magnet, Faraday conceived the idea of magnetic "lines of force" extending from pole to pole of the magnet and along which the filings tend to place themselves. On the discovery being made that magnetic

History of electromagnetic theory 50

effects accompany the passage of an electric current in a wire, it was also assumed that similar magnetic lines offorce whirled around the wire. For convenience and to account for induced electricity it was then assumed that whenthese lines of force are "cut" by a wire in passing across them or when the lines of force in rising and falling cut thewire, a current of electricity is developed, or to be more exact, an electromotive force is developed in the wire thatsets up a current in a closed circuit. Faraday advanced what has been termed the molecular theory of electricity[71]

which assumes that electricity is the manifestation of a peculiar condition of the molecule of the body rubbed or theether surrounding the body. Faraday also, by experiment, discovered paramagnetism and diamagnetism, namely, thatall solids and liquids are either attracted or repelled by a magnet. For example, iron, nickel, cobalt, manganese,chromium, etc., are paramagnetic (attracted by magnetism), whilst other substances, such as bismuth, phosphorus,antimony, zinc, etc., are repelled by magnetism or are diamagnetic.[13][72]

Brugans of Leyden in 1778 and Le Baillif and Becquerel in 1827[73] had previously discovered diamagnetism in thecase of bismuth and antimony. Faraday also rediscovered specific inductive capacity in 1837, the results of theexperiments by Cavendish not having been published at that time. He also predicted[74] the retardation of signals onlong submarine cables due to the inductive effect of the insulation of the cable, in other words, the static capacity ofthe cable.[13]

The 25 years immediately following Faraday's discoveries of electric induction were fruitful in the promulgation oflaws and facts relating to induced currents and to magnetism. In 1834 Heinrich Lenz and Moritz von Jacobiindependently demonstrated the now familiar fact that the currents induced in a coil are proportional to the numberof turns in the coil. Lenz also announced at that time his important law that, in all cases of electromagnetic inductionthe induced currents have such a direction that their reaction tends to stop the motion that produces them, a law thatwas perhaps deducible from Faraday's explanation of Arago's rotations.[13][75]

The induction coil was first designed by Nicholas Callan in 1836. In 1845 Joseph Henry, the American physicist,published an account of his valuable and interesting experiments with induced currents of a high order, showing thatcurrents could be induced from the secondary of an induction coil to the primary of a second coil, thence to itssecondary wire, and so on to the primary of a third coil, etc.[76] Heinrich Daniel Ruhmkorff further developes theinduction coil, the Ruhmkorff coil was patented in 1851,[77] and he utilized long windings of copper wire to achievea spark of approximately 2 inches (50 mm) in length. In 1857, after examining a greatly improved version made byan American inventor, Edward Samuel Ritchie,[78][79] Ruhmkorff improved his design (as did other engineers), usingglass insulation and other innovations to allow the production of sparks more than 300 millimetres (unknownoperator: u'strong' in) long.[80]

Middle 19th century

“The electromagnetic theory of light adds to the old undulatory theory an enormous province of transcendent interest and importance; itdemands of us not merely an explanation of all the phenomena of light and radiant heat by transverse vibrations of an elastic solid called ether,but also the inclusion of electric currents, of the permanent magnetism of steel and lodestone, of magnetic force, and of electrostatic force, in acomprehensive ethereal dynamics." ”

—Lord Kelvin[81]

Up to the middle of the 19th century, indeed up to about 1870, electrical science was, it may be said, a sealed book to the majority of electrical workers. Prior to this time a number of handbooks had been published on electricity and magnetism, notably Auguste de La Rive's exhaustive ' Treatise on Electricity,'[82] in 1851 (French) and 1853 (English); August Beer's Einleitung in die Elektrostatik, die Lehre vom Magnetismus und die Elektrodynamik,[83] Wiedemann's ' Galvanismus,' and Reiss'[84] 'Reibungsal-elektricitat.' But these works consisted in the main in details of experiments with electricity and magnetism, and but little with the laws and facts of those phenomena. Henry d'Abria[85][86] published the results of some researches into the laws of induced currents, but owing to their complexity of the investigation it was not productive of very notable results.[87] Around the mid-19th century,

History of electromagnetic theory 51

Fleeming Jenkin's work on ' Electricity and Magnetism[88] ' and Clerk Maxwell's ' Treatise on Electricity andMagnetism ' were published.[13]

These books were departures from the beaten path. As Jenkin states in the preface to his work the science of theschools was so dissimilar from that of the practical electrician that it was quite impossible to give students sufficient,or even approximately sufficient, textbooks. A student he said might have mastered de la Rive's large and valuabletreatise and yet feel as if in an unknown country and listening to an unknown tongue in the company of practicalmen. As another writer has said, with the coming of Jenkin's and Maxwell's books all impediments in the way ofelectrical students were removed, "the full meaning of Ohm's law becomes clear; electromotive force, difference ofpotential, resistance, current, capacity, lines of force, magnetization and chemical affinity were measurable, andcould be reasoned about, and calculations could be made about them with as much certainty as calculations indynamics".[13][89]

About 1850 Kirchhoff published his laws relating to branched or divided circuits. He also showed mathematicallythat according to the then prevailing electrodynamic theory, electricity would be propagated along a perfectlyconducting wire with the velocity of light. Helmholtz investigated mathematically the effects of induction upon thestrength of a current and deduced therefrom equations, which experiment confirmed, showing amongst otherimportant points the retarding effect of self-induction under certain conditions of the circuit.[13][90]

Sir William Thomson

In 1853 Sir William Thomson (later Lord Kelvin) predicted as a result ofmathematical calculations the oscillatory nature of the electric discharge of acondenser circuit. To Henry, however, belongs the credit of discerning as aresult of his experiments in 1842 the oscillatory nature of the Leyden jardischarge. He wrote:[91] The phenomena require us to admit the existence of aprincipal discharge in one direction, and then several reflex actionsbackward and forward, each more feeble than the preceding, until theequilibrium is obtained. These oscillations were subsequently observed by B.W. Feddersen (1857)[92][93] who using a rotating concave mirror projected animage of the electric spark upon a sensitive plate, thereby obtaining aphotograph of the spark which plainly indicated the alternating nature of thedischarge. Sir William Thomson was also the discoverer of the electricconvection of heat (the "Thomson" effect). He designed for electricalmeasurements of precision his quadrant and absolute electrometers. Thereflecting galvanometer and siphon recorder, as applied to submarine cable signaling, are also due to him.[13]

About 1876 Prof. H. A. Rowland of Baltimore demonstrated the important fact that a static charge carried aroundproduces the same magnetic effects as an electric current.[94][95] The Importance of this discovery consists in that itmay afford a plausible theory of magnetism, namely, that magnetism may be the result of directed motion of rows ofmolecules carrying static charges.[13]

After Faraday's discovery that electric currents could be developed in a wire by causing it to cut across the lines offorce of a magnet, it was to be expected that attempts would be made to construct machines to avail of this fact in thedevelopment of voltaic currents.[96] The first machine of this kind was due to Hippolyte Pixii, 1832. It consisted oftwo bobbins of iron wire, opposite which the poles of a horseshoe magnet were caused to rotate. As this produced inthe coils of the wire an alternating current, Pixii arranged a commutating device (commutator) that converted thealternating current of the coils or armature into a direct current in the external circuit. This machine was followed byimproved forms of magneto-electric machines due to RItchie, Saxton, Clarke 1834, Stohrer 1843, Nollet 1849,Shepperd 1856, Van Maldern, Siemens, Wilde and others.[13]

A notable advance in the art of dynamo construction was made by Mr. S. A. Varley in 1866[97] and by Dr. Charles William Siemens and Mr. Charles Wheatstone,[98] who independently discovered that when a coil of wire, or armature, of the dynamo machine is rotated between the poles (or in the "field") of an electromagnet, a weak current

History of electromagnetic theory 52

is set up in the coil due to residual magnetism in the iron of the electromagnet, and that if the circuit of the armaturebe connected with the circuit of the electromagnet, the weak current developed in the armature increases themagnetism in the field. This further increases the magnetic lines of force in which the armature rotates, which stillfurther increases the current in the electromagnet, thereby producing a corresponding increase in the fieldmagnetism, and so on, until the maximum electromotive force which the machine is capable of developing isreached. By means of this principle the dynamo machine develops its own magnetic field, thereby much increasingits efficiency and economical operation. Not by any means, however, was the dynamo electric machine perfected atthe time mentioned.[13]

In 1860 an important improvement had been made by Dr. Antonio Pacinotti of Pisa who devised the first electricmachine with a ring armature. This machine was first used as an electric motor, but afterward as a generator ofelectricity. The discovery of the principle of the reversibility of the dynamo electric machine (variously attributed toWalenn 1860; Pacinotti 1864 ; Fontaine, Gramme 1873; Deprez 1881, and others) whereby it may be used as anelectric motor or as a generator of electricity has been termed one of the greatest discoveries of the 19th century.[13]

In 1872 the drum armature was devised by Hefner-Alteneck. This machine in a modified form was subsequentlyknown as the Siemens dynamo. These machines were presently followed by the Schuckert, Gulcher,[99]

Fein,[100][101] Brush, Hochhausen, Edison and the dynamo machines of numerous other inventors. In the early daysof dynamo machine construction the machines were mainly arranged as direct current generators, and perhaps themost important application of such machines at that time was in electro-plating, for which purpose machines of lowvoltage and large current strength were employed.[13][102]

Beginning about 1887 alternating current generators came into extensive operation and the commercial developmentof the transformer, by means of which currents of low voltage and high current strength are transformed to currentsof high voltage and low current strength, and vice-versa, in time revolutionized the transmission of electric power tolong distances. Likewise the introduction of the rotary converter (in connection with the "step-down" transformer)which converts alternating currents into direct currents (and vice-versa) has effected large economies in the operationof electric power systems.[13][103]

Before the introduction of dynamo electric machines, voltaic, or primary, batteries were extensively used forelectro-plating and in telegraphy. There are two distinct types of voltaic cells, namely, the "open" and the "closed,"or "constant," type. The open type in brief is that type which operated on closed circuit becomes, after a short time,polarized; that is, gases are liberated in the cell which settle on the negative plate and establish a resistance thatreduces the current strength. After a brief interval of open circuit these gases are eliminated or absorbed and the cellis again ready for operation. Closed circuit cells are those in which the gases in the cells are absorbed as quickly asliberated and hence the output of the cell is practically uniform. The Leclanché and Daniell cells, respectively, arefamiliar examples of the "open" and "closed" type of voltaic cell. The "open" cells are used very extensively atpresent, especially in the dry cell form, and in annunciator and other open circuit signal systems. Batteries of theDaniell or "gravity" type were employed almost generally in the United States and Canada as the source ofelectromotive force in telegraphy before the dynamo machine became available, and still are largely used for thisservice or as "local" cells. Batteries of the "gravity" and the Edison-Lalande types are still much used in "closedcircuit" systems.[13]

In the late 19th century, the term luminiferous aether, meaning light-bearing aether, was the term used to describe amedium for the propagation of light.[104] The word aether stems via Latin from the Greek αιθήρ, from a rootmeaning to kindle, burn, or shine. It signifies the substance which was thought in ancient times to fill the upperregions of space, beyond the clouds.

History of electromagnetic theory 53

Maxwell, Hertz, and Tesla

James Clerk Maxwell

In 1864 James Clerk Maxwell of Edinburgh announced his electromagnetictheory of light, which was perhaps the greatest single step in the world'sknowledge of electricity.[105] Maxwell had studied and commented on thefield of electricity and magnetism as early as 1855/6 when On Faraday's linesof force[106] was read to the Cambridge Philosophical Society. The paperpresented a simplified model of Faraday's work, and how the two phenomenawere related. He reduced all of the current knowledge into a linked set ofdifferential equations with 20 equations in 20 variables. This work was laterpublished as On Physical Lines of Force in March 1861.[107] In order todetermine the force which is acting on any part of the machine we must findits momentum, and then calculate the rate at which this momentum is beingchanged. This rate of change will give us the force. The method of calculationwhich it is necessary to employ was first given by Lagrange, and afterwardsdeveloped, with some modifications, by Hamilton's equations. It is usuallyreferred to as Hamilton's principle; when the equations in the original form are used they are known as Lagrange'sequations. Now Maxwell logically showed how these methods of calculation could be applied to theelectro-magnetic field.[108] The energy of a dynamical system is partly kinetic, partly potential. Maxwell supposesthat the magnetic energy of the field is kinetic energy, the electric energy potential.[109]

Around 1862, while lecturing at King's College, Maxwell calculated that the speed of propagation of anelectromagnetic field is approximately that of the speed of light. He considered this to be more than just acoincidence, and commented "We can scarcely avoid the conclusion that light consists in the transverse undulationsof the same medium which is the cause of electric and magnetic phenomena."[110]

Working on the problem further, Maxwell showed that the equations predict the existence of waves of oscillatingelectric and magnetic fields that travel through empty space at a speed that could be predicted from simple electricalexperiments; using the data available at the time, Maxwell obtained a velocity of 310,740,000 m/s. In his 1864 paperA Dynamical Theory of the Electromagnetic Field, Maxwell wrote, The agreement of the results seems to show thatlight and magnetism are affections of the same substance, and that light is an electromagnetic disturbancepropagated through the field according to electromagnetic laws.[111]

As already noted herein Faraday, and before him, Ampère and others, had inklings that the luminiferous ether ofspace was also the medium for electric action. It was known by calculation and experiment that the velocity ofelectricity was approximately 186,000 miles per second; that is, equal to the velocity of light, which in itself suggeststhe idea of a relationship between -electricity and "light." A number of the earlier philosophers or mathematicians, asMaxwell terms them, of the 19th century, held the view that electromagnetic phenomena were explainable by actionat a distance. Maxwell, following Faraday, contended that the seat of the phenomena was in the medium. Themethods of the mathematicians in arriving at their results were synthetical while Faraday's methods were analytical.Faraday in his mind's eye saw lines of force traversing all space where the mathematicians saw centres of forceattracting at a distance. Faraday sought the seat of the phenomena in real actions going on in the medium; they weresatisfied that they had found it in a power of action at a distance on the electric fluids.[112]

Both of these methods, as Maxwell points out, had succeeded in explaining the propagation of light as anelectromagnetic phenomenon while at the same time the fundamental conceptions of what the quantities concernedare, radically differed. The mathematicians assumed that insulators were barriers to electric currents; that, forinstance, in a Leyden jar or electric condenser the electricity was accumulated at one plate and that by some occultaction at a distance electricity of an opposite kind was attracted to the other plate.

History of electromagnetic theory 54

Maxwell, looking further than Faraday, reasoned that if light is an electromagnetic phenomenon and is transmissiblethrough dielectrics such as glass, the phenomenon must be in the nature of electromagnetic currents in the dielectrics.He therefore contended that in the charging of a condenser, for instance, the action did not stop at the insulator, butthat some "displacement" currents are set up in the insulating medium, which currents continue until the resistingforce of the medium equals that of the charging force. In a closed conductor circuit, an electric current is also adisplacement of electricity.The conductor offers a certain resistance, akin to friction, to the displacement of electricity, and heat is developed inthe conductor, proportional to the square of the current(as already stated herein), which current flows as long as theimpelling electric force continues. This resistance may be likened to that met with by a ship as it displaces in thewater in its progress. The resistance of the dielectric is of a different nature and has been compared to thecompression of multitudes of springs, which, under compression, yield with an increasing back pressure, up to apoint where the total back pressure equals the initial pressure. When the initial pressure is withdrawn the energyexpended in compressing the "springs" is returned to the circuit, concurrently with the return of the springs to theiroriginal condition, this producing a reaction in the opposite direction. Consequently the current due to thedisplacement of electricity in a conductor may be continuous, while the displacement currents in a dielectric aremomentary and, in a circuit or medium which contains but little resistance compared with capacity or inductancereaction, the currents of discharge are of an oscillatory or alternating nature.[113]

Maxwell extended this view of displacement currents in dielectrics to the ether of free space. Assuming light to bethe manifestation of alterations of electric currents in the ether, and vibrating at the rate of light vibrations, thesevibrations by induction set up corresponding vibrations in adjoining portions of the ether, and in this way theundulations corresponding to those of light are propagated as an electromagnetic effect in the ether. Maxwell'selectromagnetic theory of light obviously involved the existence of electric waves in free space, and his followers setthemselves the task of experimentally demonstrating the truth of the theory. By 1871, he presented the Remarks onthe mathematical classification of physical quantities.[114]

In 1887, Prof. Heinrich Hertz in a series of experiments proved the actual existence of such waves. The discovery ofelectric waves in space naturally led to the discovery and introduction in the closing years of the 19th century ofwireless telegraphy, various systems of which are now in successful use on shipboard, lighthouses and shore andinland stations throughout the world, by means of which intelligence is transmitted across the widest oceans andlarge parts of continents.

Nikola Tesla, c. 1896

In 1891, notable additions to our knowledge of the phenomena ofelectromagnetic frequency and high potential current were contributed byNikola Tesla.[115] Amongst the novel experiments performed by Tesla was totake in his hand a glass tube from which the air had been exhausted, thenbringing his body into contact with a wire carrying currents of high potential,the tube was suffused with a pleasing bright glow. Another experiment was tograsp a bulb that was suspended from a single wire attached to a highpotential, high frequency current circuit, when a platinum button within thebulb was brought to vivid incandescence, the experimenter at this timestanding on an insulating platform. The frequency and potential involved inthe experiments made by Tesla at this time were of the order of one or moremillion cycles and volts. For further information relative to these experimentsthe reader may be referred to Tesla's Experiments with Alternate Currents ofHigh Potential and High Frequency.[13]

History of electromagnetic theory 55

End of the 19th century

William Crookes

Oliver Heaviside

The electron as a unit of charge in electrochemistry was posited by G.Johnstone Stoney in 1874, who also coined the term electron in 1894. Plasmawas first identified in a Crookes tube, and so described by Sir WilliamCrookes in 1879 (he called it "radiant matter").[116] The place of electricity inleading up to the discovery of those beautiful phenomena of the CrookesTube (due to Sir William Crookes), viz., Cathode rays,[117] and later to thediscovery of Roentgen or X-rays, must not be overlooked, since withoutelectricity as the excitant of the tube the discovery of the rays might havebeen postponed indefinitely. It has been noted herein that Dr. William Gilbertwas termed the founder of electrical science. This must, however, be regardedas a comparative statement.[13]

Oliver Heaviside was a self-taught scholar who reformulated Maxwell's fieldequations in terms of electric and magnetic forces and energy flux, andindependently co-formulated vector analysis. His series of articles continuedthe work entitled "Electromagnetic Induction and its Propagation,"commenced in The Electrician in 1885 to dearly 1887 (ed., the latter part ofthe work dealing with the propagation of electromagnetic waves along wiresthrough the dielectric surrounding them), when the great pressure on spaceand the want of readers appeared to necessitate its abrupt discontinuance.[118]

(A straggler piece appeared December 31, 1887.) He wrote an interpretationof the transcendental formulae of electromagnetism. Following the real objectof true naturalists[119] when they employ mathematics to assist them, he wroteto find out the connections of known phenomena, and by deductive reasoning,to obtain a knowledge of electromagnetic phenomena. Although at odds withthe scientific establishment for most of his life, Heaviside changed the face ofmathematics and science for years to come.

Of the changes in the field of electromagnetic theory, certain conclusionsfrom Electro-Magnetic Theory[120] by Heaviside are, if not drawn, at leastindicated in this book. Two of them may be stated as follows:

1.1. That magnetism is a phenomenon of motion and not a staticalphenomenon; also that this motion is more likely to be translational than vortical.

2.2. That all electric currents are phenomena consequent upon the emission of electro-magnetic wave disturbances inthe aether, and that the proper treatment of all the phenomena of currents and magnetic flux should be consideredas the consequence, and not as the cause, of electro-magnetic waves.

The ultimate results of his work are twofold. (1) The first ultimate result is purely mathematical, which is importantonly to those who study mathematical physics. The system of vectorial algebra[121] as developed by Mr. Heavisidewas used because of ease for physical investigations to the methods of quaternions. (2) The second ultimate result isphysical. It consists in more closely uniting the more recondite problems of telegraphy, telephony, Teslaicphenomena and Hertzian phenomena with the fundamental properties of the aether. In elucidating this connection,the merit of the book appears most prominently as a stepping-stone to the goal in the full view of all physicalanalysis, namely, the resolution of all physical phenomena to the activities of the aether, and of matter in the aether,under the laws of dynamics.[122]

During the late 1890s a number of physicists proposed that electricity, as observed in studies of electrical conduction in conductors, electrolytes, and cathode ray tubes, consisted of discrete units, which were given a variety of names,

History of electromagnetic theory 56

but the reality of these units had not been confirmed in a compelling way. However, there were also indications thatthe cathode rays had wavelike properties.[13]

Faraday, Weber, Helmholtz, Clifford and others had glimpses of this view; and the experimental works of Zeeman,Goldstein, Crookes, J. J. Thomson and others had greatly strengthened this view. Weber predicted that electricalphenomena were due to the existence of electrical atoms, the influence of which on one another depended on theirposition and relative accelerations and velocities. Helmholtz and others also contended that the existence of electricalatoms followed from Faraday's laws of electrolysis, and Johnstone Stoney, to whom is due the term "electron,"showed that each chemical ion of the decomposed electrolyte carries a definite and constant quantity of electricity,and inasmuch as these charged ions are separated on the electrodes as neutral substances there must be an instant,however brief, when the charges must be capable of existing separately as electrical atoms; while in 1887, Cliffordwrote: "There is great reason to believe that every material atom carries upon it a small electric current, if it does notwholly consist of this current."[13]

J.J. Thomson

In 1896 J.J. Thomson performed experiments indicating that cathode raysreally were particles, found an accurate value for their charge-to-mass ratioe/m, and found that e/m was independent of cathode material. He made goodestimates of both the charge e and the mass m, finding that cathode rayparticles, which he called "corpuscles", had perhaps one thousandth of themass of the least massive ion known (hydrogen). He further showed that thenegatively charged particles produced by radioactive materials, by heatedmaterials, and by illuminated materials, were universal. The nature of theCrookes tube "cathode ray" matter was identified by Thomson in 1897.[123]

In the late 19th century, the Michelson-Morley experiment was performed byAlbert Michelson and Edward Morley at what is now Case Western ReserveUniversity. It is generally considered to be the evidence against the theory ofa luminiferous aether. The experiment has also been referred to as "thekicking-off point for the theoretical aspects of the Second ScientificRevolution."[124] Primarily for this work, Albert Michelson was awarded the Nobel Prize in 1907. Dayton Millercontinued with experiments, conducting thousands of measurements and eventually developing the most accurateinterferometer in the world at that time. Miller and others, such as Morley, continue observations and experimentsdealing with the concepts.[125] A range of proposed aether-dragging theories could explain the null result but thesewere more complex, and tended to use arbitrary-looking coefficients and physical assumptions.[13]

By the end of the 19th century electrical engineers had become a distinct profession, separate from physicists andinventors. They created companies that investigated, developed and perfected the techniques of electricitytransmission, and gained support from governments all over the world for starting the first worldwide electricaltelecommunication network, the telegraph network. Pioneers in this field included Werner von Siemens, founder ofSiemens AG in 1847, and John Pender, founder of Cable & Wireless.The late 19th century produced such giants of electrical engineering as Nikola Tesla, inventor of the polyphaseinduction motor. The first public demonstration of a "alternator system" took place in 1886.[126][127] Largetwo-phase alternating current generators were built by a British electrician, J.E.H. Gordon, in 1882. Lord Kelvin andSebastian Ferranti also developed early alternators, producing frequencies between 100 and 300 hertz. In 1891,Nikola Tesla patented a practical "high-frequency" alternator (which operated around 15,000 hertz).[128] After 1891,polyphase alternators were introduced to supply currents of multiple differing phases.[129] Later alternators weredesigned for varying alternating-current frequencies between sixteen and about one hundred hertz, for use with arclighting, incandescent lighting and electric motors.[130]

The possibility of obtaining the electric current in large quantities, and economically, by means of dynamo electric machines gave impetus to the development of incandescent and arc lighting. Until these machines had attained a

History of electromagnetic theory 57

commercial basis voltaic batteries were the only available source of current for electric lighting and power. The costof these batteries, however, and the difficulties of maintaining them in reliable operation were prohibitory of theiruse for practical lighting purposes. The date of the employment of arc and incandescent lamps may be set at about1877.[13]

Even in 1880, however, but little headway had been made toward the general use of these illuminants; the rapidsubsequent growth of this industry is a matter of general knowledge.[131] The employment of storage batteries, whichwere originally termed secondary batteries or accumulators, began about 1879. Such batteries are now utilized on alarge scale as auxiliaries to the dynamo machine in electric power-houses and substations, in electric automobilesand in immense numbers in automobile ignition and starting systems, also in fire alarm telegraphy and other signalsystems.[13]

World's Fair Tesla presentation

In 1893, the World's Columbian International Exposition washeld in a building which was devoted to electrical exhibits.General Electric Company (backed by Edison and J.P.Morgan) had proposed to power the electric exhibits withdirect current at the cost of one million dollars. However,Westinghouse, armed with Tesla's alternating current system,proposed to illuminate the Columbian Exposition in Chicagofor half that price, and Westinghouse won the bid. It was anhistorical moment and the beginning of a revolution, asNikola Tesla and George Westinghouse introduced the public to electrical power by illuminating the Exposition.

Second Industrial Revolution

Thomas Edison

The AC motor helped usher in the Second Industrial Revolution. The rapidadvance of electrical technology in the latter 19th and early 20th centuries ledto commercial rivalries. In the War of Currents in the late 1880s, GeorgeWestinghouse and Thomas Edison became adversaries due to Edison'spromotion of direct current (DC) for electric power distribution overalternating current (AC) advocated by Westinghouse and Nikola Tesla.Tesla's patents and theoretical work formed the basis of modern alternatingcurrent electric power (AC) systems, including the polyphase powerdistribution systems.[132][133]

Several inventors helped develop commercial systems. Samuel Morse,inventor of a long-range telegraph; Thomas Edison, inventor of the firstcommercial electrical energy distribution network; George Westinghouse,inventor of the electric locomotive; Alexander Graham Bell, the inventor ofthe telephone and founder of a successful telephone business.

In 1871 the electric telegraph had grown to large proportions and was in usein every civilized country in the world, its lines forming a network in alldirections over the surface of the land. The system most generally in use was the electromagnetic telegraph due to S.F. B. Morse of New York, or modifications of his system.[134] Submarine cables[135] connecting the Eastern andWestern hemispheres were also in successful operation at that time.[13]

When, however, in 1918 one views the vast applications of electricity to electric light, electric railways, electric power and other purposes (all it may be repeated made possible and practicable by the perfection of the dynamo machine), it is difficult to believe that no longer ago than 1871 the author of a book published in that year, in referring to the state of the art of applied electricity at that time, could have truthfully written: "The most important

History of electromagnetic theory 58

and remarkable of the uses which have been made of electricity consists in its application to telegraph purposes".[136]

The statement was, however, quite accurate and perhaps the time could have been carried forward to the year 1876without material modification of the remarks. In that year the telephone, due to Alexander Graham Bell, wasinvented, but it was not until several years thereafter that its commercial employment began in earnest. Since thattime also the sister branches of electricity just mentioned have advanced and are advancing with such gigantic stridesin every direction that it is difficult to place a limit upon their progress. Electrical devices account of the use ofelectricity in the arts and industries.[13]

Charles Proteus Steinmetz, theoreticianof alternating current.

AC replaced DC for central station power generation and power distribution,enormously extending the range and improving the safety and efficiency ofpower distribution. Edison's low-voltage distribution system using DCultimately lost to AC devices proposed by others: primarily Tesla's polyphasesystems, and also other contributors, such as Charles Proteus Steinmetz (in1888, he was working in Pittsburgh for Westinghouse[137]). The successfulNiagara Falls system was a turning point in the acceptance of alternatingcurrent. Eventually, the General Electric company (formed by a mergerbetween Edison's companies and the AC-based rival Thomson-Houston)began manufacture of AC machines. Centralized power generation becamepossible when it was recognized that alternating current electric power linescan transport electricity at low costs across great distances by takingadvantage of the ability to change voltage across the distribution path usingpower transformers. The voltage is raised at the point of generation (a representative number is a generator voltage inthe low kilovolt range) to a much higher voltage (tens of thousands to several hundred thousand volts) for primarytransmission, followed to several downward transformations, to as low as that used in residential domestic use.[13]

The International Electro-Technical Exhibition of 1891 featuring the long distance transmission of high-power,three-phase electric current. It was held between 16 May and 19 October on the disused site of the three former"Westbahnhöfe" (Western Railway Stations) in Frankfurt am Main. The exhibition featured the first long distancetransmission of high-power, three-phase electric current, which was generated 175 km away at Lauffen am Neckar.As a result of this successful field trial, three-phase current became established for electrical transmission networksthroughout the world.[13]

Much was done in the direction in the improvement of railroad terminal facilities, and it is difficult to find one steamrailroad engineer who would have denied that all the important steam railroads of this country were not to beoperated electrically. In other directions the progress of events as to the utilization of electric power was expected tobe equally rapid. In every part of the world the power of falling water, nature's perpetual motion machine, which hasbeen going to waste since the world began, is now being converted into electricity and transmitted by wire hundredsof miles to points where it is usefully and economically employed.[13][138]

The first windmill for electricity production was built in Scotland in July 1887 by Prof James Blyth of Anderson'sCollege, Glasgow (the precursor of Strathclyde University.[139] Across the Atlantic, in Cleveland, Ohio a larger andheavily engineered machine was designed and constructed in 1887-1888 by Charles F. Brush,[140] this was built byhis engineering company at his home and operated from 1886 until 1900.[141] The Brush wind turbine had a rotor 56feet (unknown operator: u'strong' m) in diameter and was mounted on a 60-foot (18 m) tower. Although large bytoday's standards, the machine was only rated at 12 kW; it turned relatively slowly since it had 144 blades. Theconnected dynamo was used either to charge a bank of batteries or to operate up to 100 incandescent light bulbs,three arc lamps, and various motors in Brush's laboratory. The machine fell into disuse after 1900 when electricitybecame available from Cleveland's central stations, and was abandoned in 1908.[142]

History of electromagnetic theory 59

20th centuryVarious units of electricity and magnetism have been adopted and named by representatives of the electricalengineering institutes of the world, which units and names have been confirmed and legalized by the governments ofthe United States and other countries. Thus the volt, from the Italian Volta, has been adopted as the practical unit ofelectromotive force, the ohm, from the enunciator of Ohm's law, as the practical unit of resistance; the ampere, afterthe eminent French scientist of that name, as the practical unit of current strength, the henry as the practical unit ofinductance, after Joseph Henry and in recognition of his early and important experimental work in mutualinduction.[143]

Dewar and John Ambrose Fleming predicted that at absolute zero, pure metals would become perfectelectromagnetic conductors (though, later, Dewar altered his opinion on the disappearance of resistance believingthat there would always be some resistance). Walther Hermann Nernst developed the third law of thermodynamicsand stated that absolute zero was unattainable. Carl von Linde and William Hampson, both commercial researchers,nearly at the same time filed for patents on the Joule-Thomson effect. Linde's patent was the climax of 20 years ofsystematic investigation of established facts, using a regenerative counterflow method. Hampson's design was also ofa regenerative method. The combined process became known as the Linde-Hampson liquefaction process. HeikeKamerlingh Onnes purchased a Linde machine for his research. On March 21, 1900, Tesla was granted a US patentfor the means for increasing the intensity of electrical oscillations by lowering temperature, which was caused bylowered resistance, a phenomenon previously observed by Olszewski and Wroblewski. Within this patent itdescribes the increase intensity and duration of electric oscillations of a low temperature resonating circuit. It isbelieved that Tesla had intended that Linde's machine would be used to attain the cooling agents. A milestone wasachieved on 10 July 1908 when Onnes at the Leiden University in Leiden produced, for the first time, liquifiedhelium and achieved superconductivity.In 1900, William Du Bois Duddell develops the Singing Arc and produced melodic sounds, from a low to ahigh-tones, from this arc lamp.

Lorentz and Poincaré

Hendrik Lorentz

Between 1900 and 1910, many scientists like Wilhelm Wien, Max Abraham,Hermann Minkowski, or Gustav Mie believed that all forces of nature are ofelectromagnetic origin (the so called "electromagnetic world view"). This wasconnected with the electron theory developed between 1892 and 1904 byHendrik Lorentz. Lorentz introduced a strict separation between matter(electrons) and ether, whereby in his model the ether is completelymotionless, and it won't be set in motion in the neighborhood of ponderablematter. Contrary to other electron models before, the electromagnetic field ofthe ether appears as a mediator between the electrons, and changes in thisfield can propagate not faster than the speed of light.

In 1896, three years after submitting his thesis on the Kerr effect, PieterZeeman disobeyed the direct orders of his supervisor and used laboratoryequipment to measure the splitting of spectral lines by a strong magneticfield. Lorentz theoretically explained the Zeeman effect on the basis of histheory, for which both received the Nobel Prize in Physics in 1902. Afundamental concept of Lorentz's theory in 1895 was the "theorem of corresponding states" for terms of order v/c.This theorem states that a moving observer (relative to the ether) in his "fictitious" field makes the same observations

as a resting observers in his "real" field. This theorem was extended for terms of all orders by Lorentz in 1904. Lorentz noticed, that it was necessary to change the space-time variables when changing frames and introduced

History of electromagnetic theory 60

concepts like physical length contraction (1892) to explain the Michelson-Morley experiment, and the mathematicalconcept of local time (1895) to explain the aberration of light and the Fizeau experiment. That resulted in theformulation of the so called Lorentz transformation by Joseph Larmor (1897, 1900) and Lorentz (1899,1904).[144][145][146]

Henri Poincaré

Continuing the work of Lorentz, Henri Poincaré between 1895 and 1905formulated on many occasions the Principle of Relativity and tried toharmonize it with electrodynamics. He declared simultaneity only aconvenient convention which depends on the speed of light, whereby theconstancy of the speed of light would be a useful postulate for making thelaws of nature as simple as possible. In 1900 he interpreted Lorentz's localtime as the result of clock synchronization by light signals, and introduced theelectromagnetic momentum by ascribing to electromagnetic energy the"fictitious" mass . And finally in June and July 1905 he declaredthe relativity principle a general law of nature, including gravitation. Hecorrected some mistakes of Lorentz and proved the Lorentz covariance of theelectromagnetic equations. Poincaré also found out that there existnon-electrical forces to stabilize the electron configuration and asserted thatgravitation is a non-electrical force as well. So the electromagnetic worldview was shown by Poincaré to be invalid. However, he remained the notionof an ether and still distinguished between "apparent" and "real" time and therefore failed to invent what is nowcalled special relativity.[146][147][148][149][150][151]

Einstein's Annus Mirabilis

Albert Einstein, 1905

In 1905, while he was working in the patent office, Albert Einstein had fourpapers published in the Annalen der Physik, the leading German physicsjournal. These are the papers that history has come to call the Annus MirabilisPapers:

• His paper on the particulate nature of light put forward the idea that certainexperimental results, notably the photoelectric effect, could be simplyunderstood from the postulate that light interacts with matter as discrete"packets" (quanta) of energy, an idea that had been introduced by MaxPlanck in 1900 as a purely mathematical manipulation, and which seemedto contradict contemporary wave theories of light (Einstein 1905a). Thiswas the only work of Einstein's that he himself called "revolutionary."

• His paper on Brownian motion explained the random movement of verysmall objects as direct evidence of molecular action, thus supporting theatomic theory. (Einstein 1905b)

• His paper on the electrodynamics of moving bodies introduced the radical theory of special relativity, whichshowed that the observed independence of the speed of light on the observer's state of motion requiredfundamental changes to the notion of simultaneity. Consequences of this include the time-space frame of amoving body slowing down and contracting (in the direction of motion) relative to the frame of the observer. Thispaper also argued that the idea of a luminiferous aether—one of the leading theoretical entities in physics at thetime—was superfluous. (Einstein 1905c)

• In his paper on mass–energy equivalence (previously considered to be distinct concepts), Einstein deduced from his equations of special relativity what later became the well-known expression: , suggesting that tiny

History of electromagnetic theory 61

amounts of mass could be converted into huge amounts of energy. (Einstein 1905d)All four papers are today recognized as tremendous achievements—and hence 1905 is known as Einstein's"Wonderful Year". At the time, however, they were not noticed by most physicists as being important, and many ofthose who did notice them rejected them outright. Some of this work—such as the theory of light quanta—remainedcontroversial for years.[152][153] Einstein establishes a new concept of the aether,[154] through relativation, and wasthe outcome of the Lorentzian aether.[155]

Latter half of the 20th Century

Paul Adrien Maurice Dirac

The first formulation of a quantum theory describing radiation and matterinteraction is due to Paul Adrien Maurice Dirac, who, during 1920, was firstable to compute the coefficient of spontaneous emission of an atom.[156] PaulDirac described the quantization of the electromagnetic field as an ensembleof harmonic oscillators with the introduction of the concept of creation andannihilation operators of particles. In the following years, with contributionsfrom Wolfgang Pauli, Eugene Wigner, Pascual Jordan, Werner Heisenbergand an elegant formulation of quantum electrodynamics due to EnricoFermi,[157] physicists came to believe that, in principle, it would be possibleto perform any computation for any physical process involving photons andcharged particles. However, further studies by Felix Bloch with ArnoldNordsieck,[158] and Victor Weisskopf,[159] in 1937 and 1939, revealed thatsuch computations were reliable only at a first order of perturbation theory, aproblem already pointed out by Robert Oppenheimer.[160] At higher orders inthe series infinities emerged, making such computations meaningless andcasting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at thetime, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics.

In December 1938, the German chemists Otto Hahn and Fritz Strassmann sent a manuscript to Naturwissenschaftenreporting they had detected the element barium after bombarding uranium with neutrons;[161] simultaneously, theycommunicated these results to Lise Meitner. Meitner, and her nephew Otto Robert Frisch, correctly interpreted theseresults as being nuclear fission.[162] Frisch confirmed this experimentally on 13 January 1939.[163] In 1944, Hahnreceived the Nobel Prize for Chemistry for the discovery of nuclear fission. Some historians who have documentedthe history of the discovery of nuclear fission believe Meitner should have been awarded the Nobel Prize withHahn.[164][165][166]

Difficulties with the Quantum theory increased through the end of 1940. Improvements in microwave technologymade it possible to take more precise measurements of the shift of the levels of a hydrogen atom,[167] now known asthe Lamb shift and magnetic moment of the electron.[168] These experiments unequivocally exposed discrepancieswhich the theory was unable to explain. With the invention of bubble chambers and spark chambers in the 1950s,experimental particle physics discovered a large and ever-growing number of particles called hadrons. It seemed thatsuch a large number of particles could not all be fundamental.Shortly after the end of the war in 1945, Bell Labs formed a Solid State Physics Group, led by William Shockley and chemist Stanley Morgan; other personnel including John Bardeen and Walter Brattain, physicist Gerald Pearson, chemist Robert Gibney, electronics expert Hilbert Moore and several technicians. Their assignment was to seek a solid-state alternative to fragile glass vacuum tube amplifiers. Their first attempts were based on Shockley's ideas about using an external electrical field on a semiconductor to affect its conductivity. These experiments failed every time in all sorts of configurations and materials. The group was at a standstill until Bardeen suggested a theory that invoked surface states that prevented the field from penetrating the semiconductor. The group changed its focus to study these surface states and they met almost daily to discuss the work. The rapport of the group was excellent, and

History of electromagnetic theory 62

ideas were freely exchanged.[169]

As to the problems in the electron experiments, a path to a solution was given by Hans Bethe. In 1947, while he wastraveling by train to reach Schenectady from New York,[170] after giving a talk at the conference at Shelter Island onthe subject, Bethe completed the first non-relativistic computation of the shift of the lines of the hydrogen atom asmeasured by Lamb and Retherford.[171] Despite the limitations of the computation, agreement was excellent. Theidea was simply to attach infinities to corrections at mass and charge that were actually fixed to a finite value byexperiments. In this way, the infinities get absorbed in those constants and yield a finite result in good agreementwith experiments. This procedure was named renormalization.

Richard Feynman

Based on Bethe's intuition and fundamental papers on the subject by Sin-ItiroTomonaga,[172] Julian Schwinger,[173][174] Richard Feynman[175][176][177] andFreeman Dyson,[178][179] it was finally possible to get fully covariantformulations that were finite at any order in a perturbation series of quantumelectrodynamics. Sin-Itiro Tomonaga, Julian Schwinger and RichardFeynman were jointly awarded with a Nobel prize in physics in 1965 for theirwork in this area.[180] Their contributions, and those of Freeman Dyson, wereabout covariant and gauge invariant formulations of quantum electrodynamicsthat allow computations of observables at any order of perturbation theory.Feynman's mathematical technique, based on his diagrams, initially seemedvery different from the field-theoretic, operator-based approach of Schwingerand Tomonaga, but Freeman Dyson later showed that the two approacheswere equivalent.[178] Renormalization, the need to attach a physical meaning at certain divergences appearing in thetheory through integrals, has subsequently become one of the fundamental aspects of quantum field theory and hascome to be seen as a criterion for a theory's general acceptability. Even though renormalization works very well inpractice, Feynman was never entirely comfortable with its mathematical validity, even referring to renormalizationas a "shell game" and "hocus pocus".[181] QED has served as the model and template for all subsequent quantumfield theories. Peter Higgs, Jeffrey Goldstone, and others, Sheldon Glashow, Steven Weinberg and Abdus Salamindependently showed how the weak nuclear force and quantum electrodynamics could be merged into a singleelectroweak force.

Robert Noyce credited Kurt Lehovec for the principle of p-n junction isolation caused by the action of a biased p-njunction (the diode) as a key concept behind the integrated circuit.[182] Jack Kilby recorded his initial ideasconcerning the integrated circuit in July 1958 and successfully demonstrated the first working integrated circuit onSeptember 12, 1958.[183] In his patent application of February 6, 1959, Kilby described his new device as "a body ofsemiconductor material ... wherein all the components of the electronic circuit are completely integrated."[184] Kilbywon the 2000 Nobel Prize in Physics for his part of the invention of the integrated circuit.[185] Robert Noyce alsocame up with his own idea of an integrated circuit half a year later than Kilby. Noyce's chip solved many practicalproblems that Kilby's had not. Noyce's chip, made at Fairchild Semiconductor, was made of silicon, whereas Kilby'schip was made of germanium.Philo Farnsworth developed the Farnsworth–Hirsch Fusor, or simply fusor, is an apparatus designed by Farnsworthto create nuclear fusion. Unlike most controlled fusion systems, which slowly heat a magnetically confined plasma,the fusor injects high temperature ions directly into a reaction chamber, thereby avoiding a considerable amount ofcomplexity. When the Farnsworth-Hirsch Fusor was first introduced to the fusion research world in the late 1960s,the Fusor was the first device that could clearly demonstrate it was producing fusion reactions at all. Hopes at thetime were high that it could be quickly developed into a practical power source. However, as with other fusionexperiments, development into a power source has proven difficult. Nevertheless, the fusor has since become apractical neutron source and is produced commercially for this role.[186]

History of electromagnetic theory 63

The first step towards the Standard Model was Sheldon Glashow's discovery, in 1960, of a way to combine theelectromagnetic and weak interactions.[187] In 1967, Steven Weinberg[188] and Abdus Salam[189] incorporated theHiggs mechanism[190][191][192] into Glashow's electroweak theory, giving it is's modern form. The Higgs mechanismis believed to give rise to the masses of all the elementary particles in the Standard Model. This includes the massesof the W and Z bosons, and the masses of the fermions - i.e. the quarks and leptons. After the neutral weak currentscaused by Z boson exchange were discovered at CERN in 1973,[193][194][195][196] the electroweak theory becamewidely accepted and Glashow, Salam, and Weinberg shared the 1979 Nobel Prize in Physics for discovering it. TheW and Z bosons were discovered experimentally in 1981, and their masses were found to be as the Standard Modelpredicted. The theory of the strong interaction, to which many contributed, acquired its modern form around1973–74, when experiments confirmed that the hadrons were composed of fractionally charged quarks. With theestablishment of quantum chromodynamics in the 1970s finalized a set of fundamental and exchange particles,which allowed for the establishment of a "standard model" based on the mathematics of gauge invariance, whichsuccessfully described all forces except for gravity, and which remains generally accepted within the domain towhich it is designed to be applied.The 'standard model' groups the electroweak interaction theory and quantum chromodynamics into a structuredenoted by the gauge group SU(3)×SU(2)×U(1). The formulation of the unification of the electromagnetic and weakinteractions in the standard model is due to Abdus Salam, Steven Weinberg and, subsequently, Sheldon Glashow.After the discovery, made at CERN, of the existence of neutral weak currents,[197][198][199][200] mediated by the Zboson foreseen in the standard model, the physicists Salam, Glashow and Weinberg received the 1979 Nobel Prize inPhysics for their electroweak theory.[201] Since then, discoveries of the bottom quark (1977), the top quark (1995)and the tau neutrino (2000) have given credence to the standard model. Because of its success in explaining a widevariety of experimental results.The first superstring theory revolution lead to important discoveries roughly between 1984 and 1986. It was realisedthat string theory was capable of describing all elementary particles as well as the interactions between them.Hundreds of physicists started to work on string theory as the most promising idea to unify physical theories. Therevolution was started by a discovery of anomaly cancellation in type I string theory via the Green-Schwarzmechanism in 1984. Several other ground-breaking discoveries, such as the heterotic string, were made in 1985. Itwas also realised in 1985 that to obtain supersymmetry, the six small extra dimensions need to becompactified on a Calabi-Yau manifold.

Electrodynamic tethersBefore the turn of the 20th to 21st century, the electrodynamic tether[202] being oriented at an angle to the localvertical between the object and a planet with a magnetic field cut the Earth's magnetic field and generated a current;thereby it converted some of the orbiting body's kinetic energy to electrical energy. The tether's far end can be leftbare, making electrical contact with the ionosphere, creating a generator. As part of a tether propulsion system, craftscan use long, strong conductors[203] to change the orbits of spacecraft. It has the potential to make space travelsignificantly cheaper. It is a simplified, very low-budget magnetic sail. It can be used either to accelerate or brake anorbiting spacecraft. When direct current is pumped through the tether, it exerts a force against the magnetic field, andthe tether accelerates the spacecraft.

History of electromagnetic theory 64

21st century

Electromagnetic technologiesThere are a range of emerging energy technologies. By 2007, solid state micrometer-scale electric double-layercapacitors based on advanced superionic conductors had been for low-voltage electronics such as deep-sub-voltagenanoelectronics and related technologies (the 22 nm technological node of CMOS and beyond). Also, the nanowirebattery, a lithium-ion battery, was invented by a team led by Dr. Yi Cui in 2007.

Magnetic resonance

Reflecting the fundamental importance and applicability of Magnetic resonance imaging[204] in medicine, PaulLauterbur of the University of Illinois at Urbana-Champaign and Sir Peter Mansfield of the University ofNottingham were awarded the 2003 Nobel Prize in Physiology or Medicine for their "discoveries concerningmagnetic resonance imaging". The Nobel citation acknowledged Lauterbur's insight of using magnetic fieldgradients to determine spatial localization, a discovery that allowed rapid acquisition of 2D images.

Wireless electricity

"Wireless electricity" describes a form of wireless energy transfer,[205] the ability to provide electrical energy toremote objects without wires. The term WiTricity was coined in 2005 by Dave Gerding and later used for a projectled by Prof. Marin Soljačić in 2007.[206][207] The MIT researchers successfully demonstrated the ability to power a60 watt light bulb wirelessly, using two 5-turn copper coils of 60 cm (24 in) diameter, that were 2 m (7 ft) away, atroughly 45% efficiency.[208] This technology can potentially be used in a large variety of applications, includingconsumer, industrial, medical and military. Its aim is to reduce the dependence on batteries. Further applications forthis technology include transmission of information—it would not interfere with radio waves and thus could be usedas a cheap and efficient communication device without requiring a license or a government permit.

Unified TheoriesAs of 2010, there is still no hard evidence that nature is described by a Grand Unified Theory. Moreover, since theHiggs particle has not yet been verified, the smaller electroweak unification is still pending.[209] The discovery ofneutrino oscillations indicates that the Standard Model is incomplete and has led to renewed interest toward certainGUT such as . One of the few possible experimental tests of certain GUT is proton decay and also fermionmasses. There are a few more special tests for supersymmetric GUT. The gauge coupling strengths of QCD, theweak interaction and hypercharge seem to meet at a common length scale called the GUT scale and equalapproximately to GeV, which is slightly suggestive. This interesting numerical observation is called the gaugecoupling unification, and it works particularly well if one assumes the existence of superpartners of the StandardModel particles. Still it is possible to achieve the same by postulating, for instance, that ordinary (nonsupersymmetric) models break with an intermediate gauge scale, such as the one of Pati-Salam group.The Theory of Everything (TOE) is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and, ideally, has predictive power for the outcome of any experiment that could be carried out in principle. M-Theory is not yet complete, but the underlying structure of the mathematics has been established and is in agreement with not only all the string theories, but with all of our scientific observations of the universe. Furthermore, it has passed many tests of internal mathematical consistency that many other attempts to combine quantum mechanics and gravity had failed. Unfortunately, until we can find some way to observe higher dimensions (impossible with our current level of technology) M-Theory has a very difficult time making predictions which can be tested in a laboratory. Technologically, it may never be possible for it to be "proven". Physicist and author Michio Kaku has remarked that M-Theory may present us with a "Theory of Everything" which is so concise that its underlying formula would fit on a t-shirt.[210] Stephen Hawking originally believed that M-Theory may be

History of electromagnetic theory 65

the ultimate theory but later suggested that the search for understanding of mathematics and physics will never becomplete.[211]

Open problemsThe magnetic monopole[212] in the quantum theory of magnetic charge started with a paper by the physicist PaulA.M. Dirac in 1931.[213] The detection of magnetic monopoles is an open problem in experimental physics. In sometheoretical models, magnetic monopoles are unlikely to be observed, because they are too massive to be created inparticle accelerators, and also too rare in the Universe to enter a particle detector with much probability.After more than twenty years of intensive research the origin of high-temperature superconductivity is still not clear,but it seems that instead of electron-phonon attraction mechanisms, as in conventional superconductivity, one isdealing with genuine electronic mechanisms (e.g. by antiferromagnetic correlations), and instead of s-wave pairing,d-wave pairings[214] are substantial.[215] One goal of all this research is room-temperature superconductivity.[216]

ReferencesCitations and notes[1] Bruno Kolbe, Francis ed Legge, Joseph Skellon, tr., " An Introduction to Electricity (http:/ / books. google. com/

books?vid=0o90G64Z2FDIyKUsLs9& id=150IAAAAIAAJ)". Kegan Paul, Trench, Trübner, 1908. 429 pages. Page 391 (http:/ / books.google. com/ books?id=150IAAAAIAAJ& printsec=titlepage& cad=0#PPA391,M1). (cf., "[...] high poles covered with copper plates andwith gilded tops were erected 'to break the stones coming from on high'. J. Dümichen, Baugeschichte des Dendera-Tempels, Strassburg,1877")

[2] Urbanitzky, A. v., & Wormell, R. (1886). Electricity in the service of man: a popular and practical treatise on the applications of electricity inmodern life (http:/ / books. google. com/ books?id=rkgOAAAAYAAJ). London: Cassell &.

[3] Lyons, T. A. (1901). A treatise on electromagnetic phenomena, and on the compass and its deviations aboard ship. Mathematical, theoretical,and practical. New York: J. Wiley & Sons.

[4][4] The Encyclopaedia Britannica; a dictionary of arts, sciences and general literature. (1890). New York: The Henry G. Allen Company.[5][5] Whittaker, E. T. (1910). A history of the theories of aether and electricity from the age of Descartes to the close of the nineteenth century.

Dublin University Press series. London: Longmans, Green and Co.; [etc.].[6] Carlson, John B. (1975) "Lodestone Compass: Chinese or Olmec Primacy?: Multidisciplinary analysis of an Olmec hematite artifact from San

Lorenzo, Veracruz, Mexico", Science, 189 (4205 : 5 September), p. 753-760, doi:10.1126/science.189.4205.753. p. 753–760[7] Lodestone Compass: Chinese or Olmec Primacy?: Multidisciplinary analysis of an Olmec hematite artifact from San Lorenzo, Veracruz,

Mexico - Carlson 189 (4205): 753 - Science (http:/ / www. sciencemag. org/ cgi/ content/ abstract/ 189/ 4205/ 753)[8][8] Li Shu-hua, p. 175[9] If there was another substance, having the same attractive quality as the amber, was known to the ancients, it was probably jet — a species of

lignite resembling cannel coal, but harder and susceptible of a high polish. It does not seem possible, however, to resolve that doubt, owing tothe many kinds of coal and other fossil deposits which not only old writers but even modern commentators constantly confuse. Theophrastusspeaks of a material which is plainly anthracite coal and Pliny (xxxvi. 18), of the Gagates, his description of which answers generally to that ofjet; but neither author mentions any phenomenon similar to that of the amber as pertaining to it. Later writers apply the word "gagates" toalmost any black bituminous material, though they commonly mean "jet" by the term. Leonardus regards the gagate as another species ofamber — "black amber" — in contradistinction to yellow and he describes it as "black, light, dry and lucid, not transparent and if put into firehas, as it were, the smell of pitch. Being heated with rubbing it attracts straws and chaff." Marbodeus gives almost the same account and statesthat it is found in Britain, where it is still obtained in the tertiary clays along the Yorkshire coast. This unfortunate confusion of yellow amberand jet, probably first due to Leonardus, has rendered it impossible to tell, from the references to amber attraction by the writers of the 16thand even of the 17th century, which substance is meant. It appears not at all unlikely that the English were then much more familiar with theattraction of jet than they were with that of amber.

[10][10] The Phoenicians have transmitted to us in their romantic language the story that the pieces of Amber sometimes washed up by the waves ofthe ocean were the petrified tears of maidens, who, disappointed in love, had cast themselves into the arms of Mother Ocean and had afteryears returned like Galatea to their original source.

[11] Barrett, J. P. (1894). Electricity at the Columbian Exposition, including an account of the exhibits in the Electricity Building, the power plantin Machinery Hall, the arc and incandescent lighting of the grounds and buildings (http:/ / books. google. com/ books?id=lF5KAAAAMAAJ)... etc. Chicago: R.R. Donnelley. Page 4

[12] Benjamin, P. (1898). A history of electricity (The intellectual rise in electricity) from antiquity to the days of Benjamin Franklin. New York:J. Wiley & Sons.

[13] Maver, William Jr.: "Electricity, its History and Progress", The Encyclopedia Americana; a library of universal knowledge, vol. X, pp. 172ff (http:/ / www. archive. org/ stream/ encyclopediaame21unkngoog#page/ n210/ mode/ 1up). (1918). New York: Encyclopedia Americana

History of electromagnetic theory 66

Corp.[14] Heinrich Karl Brugsch-Bey and Henry Danby Seymour, " A History of Egypt Under the Pharaohs (http:/ / books. google. com/

books?vid=0CJl3KVQupibKmzuADNu17& id=LoiTizgRo9kC)". J. Murray, 1881. Page 422. (cf., [... the symbol of a] 'serpent' is rather afish, which still serves, in the Coptic language, to designate the electric fish [...])

[15] Seeman, Bernard and Barry, James E. The Story of Electricity and Magnetism, Harvey House 1967, p. 19[16] Moller, Peter; Kramer, Bernd (December 1991), "Review: Electric Fish", BioScience (American Institute of Biological Sciences) 41 (11):

794–6 [794], doi:10.2307/1311732, JSTOR 1311732[17] Bullock, Theodore H. (2005), Electroreception, Springer, pp. 5–7, ISBN 0-387-23192-7[18] Morris, Simon C. (2003), Life's Solution: Inevitable Humans in a Lonely Universe, Cambridge University Press, pp. 182–185,

ISBN 0-521-82704-3[19] Riddle of 'Baghdad's batteries' (http:/ / news. bbc. co. uk/ 1/ hi/ sci/ tech/ 2804257. stm). BBC News.[20] After the Second World War, Willard Gray demonstrated current production by a reconstruction of the inferred battery design when filled

with grape juice. W. Jansen experimented with 1,4-Benzoquinone (some beetles produce quinones) and vinegar in a cell and got satisfactoryperformance.

[21][21] An alternative, but still electrical explanation was offered by Paul Keyser. It was suggested that a priest or healer, using an iron spatula tocompound a vinegar based potion in a copper vessel, may have felt an electrical tingle and used the phenomenon either forelectro-acupuncture, or to amaze supplicants by electrifying a metal statue.

[22] Copper and iron form an electrochemical couple, so that in the presence of any electrolyte, an electric potential (voltage) will be produced.König had observed a number of very fine silver objects from ancient Iraq which were plated with very thin layers of gold, and speculated thatthey were electroplated using batteries of these "cells".

[23][23] Corder, Gregory, "Using an Unconventional History of the Battery to engage students and explore the importance of evidence", VirginiaJournal of Science Education 1

[24] His Epistola was written in 1269.[25] Lane, Frederic C. (1963) "The Economic Meaning of the Invention of the Compass", The American Historical Review, 68 (3: April), p.

605–617[26][26] Dampier, W. C. D. (1905). The theory of experimental electricity. Cambridge physical series. Cambridge [Eng.: University Press.[27][27] consult ' Priestley's 'History of Electricity,' London 1757[28][28] Robert Boyle (1676). Experiments and notes about the mechanical origin or production of particular qualities.[29] Benjamin, P. (1895). A history of electricity (http:/ / books. google. com/ books?id=hkMPAAAAMAAJ): (The intellectual rise in

electricity) from antiquity to the days of Benjamin Franklin. New York: J. Wiley & Sons.[30][30] Consult Boyle's 'Experiments on the Origin of Electricity,'" and Priestley's 'History of Electricity'.[31] From Physico-Mechanical Experiments, 2nd Ed., London 1719[32] Consult Dr. Carpue's 'Introduction to Electricity and Galvanism,' London 1803.[33] Krebs, Robert E. (2003), Groundbreaking Scientific Experiments, Inventions, and Discoveries of the 18th Century, Greenwood Publishing

Group, p. 82, ISBN 0-313-32015-2[34] Keithley, Joseph F. (1999), The Story of Electrical and Magnetic Measurements: From 500 B.C. to the 1940s, Wiley, ISBN 0-7803-1193-0[35] Biography, Pieter (Petrus) van Musschenbroek (http:/ / chem. ch. huji. ac. il/ history/ musschenbroek. htm)[36][36] According to Priestley ('History of Electricity,' 3d ed., Vol. I, p. 102)[37][37] Priestley's 'History of Electricity,' p. 138[38] Cheney Hart: " Part of a letter from Cheney Hart, M.D. to William Watson, F.R.S. giving Account of the Effects of Electricity in the County

Hospital at Shrewsbury (http:/ / rstl. royalsocietypublishing. org/ content/ 48/ 786. full. pdf)", Phil. Trans. 1753:48 (http:/ / rstl.royalsocietypublishing. org/ content/ 48. toc), pp. 786–788. Read on November 14, 1754.

[39] Kite Experiment (http:/ / www. ieeeghn. org/ wiki/ index. php/ Kite_Experiment) (2011). IEEE Global History Network.[40] see atmospheric electricity[41] Dr. Wall Experiments of the Luminous Qualities of Amber, Diamonds, and Gum Lac (http:/ / rstl. royalsocietypublishing. org/ content/ 26/

313-324/ 69. full. pdf+ html), by Dr. Wall, in a Letter to Dr. Sloane, R. S. Secr. Phil. Trans. 1708 26:69-76; doi:10.1098/rstl.1708.0011[42][42] Physico-mechanical experiments, on various subjects; with, explanations of all the machines engraved on copper[43] Vail, A. (1845). The American electro magnetic telegraph: With the reports of Congress, and a description of all telegraphs known,

employing electricity or galvanism. Philadelphia: Lea & Blanchard[44] Hutton, C., Shaw, G., Pearson, R., & Royal Society (Great Britain). (1665). Philosophical transactions of the Royal Society of London:

From their commencement, in 1665 to the year 1800. London: C. and R. Baldwin. PaGE 345 (http:/ / books. google. com/books?id=QkNKAAAAYAAJ& pg=PR345).

[45][45] Franklin, 'Experiments and Observations on Electricity'[46][46] Royal Society Papers, vol. IX (BL. Add MS 4440): Henry Elles, from Lismore, Ireland, to the Royal Society, London, 9 August 1757, f.12b;

9 August 1757, f.166.[47] Tr., Test Theory of Electricity and Magnetism[48][48] Philosophical Transactions 1771[49] Electric Telegraph, apparatus by wh. signals may be transmitted to a distance by voltaic currents propagated on metallic wires; fnded. on

experimts. of Gray 1729, Nollet, Watson 1745, Lesage 1774, Lamond 1787, Reusserl794, Cavallo 1795, Betancourt 1795, Soemmering 1811,

History of electromagnetic theory 67

Gauss & Weber 1834, &c. Telegraphs constructed by Wheatstone & Independently by Steinheil 1837, improved by Morse, Cooke,Woolaston, &c.

[50][50] Cassell's miniature cyclopaedia By Sir William Laird Clowes. Page 288.[51] Die Geschichte Der Physik in Grundzügen: th. In den letzten hundert jahren (1780–1880) 1887-90 (tr. The history of physics in broad terms:

th. In the last hundred years (1780–1880) 1887-90) By Ferdinand Rosenberger. F. Vieweg und sohn, 1890. Page 288.[52] See Voltaic pile[53][53] 'Philosophical Transactions,' 1833[54] Of Torpedos Found on the Coast of England. In a Letter from John Walsh, Esq; F. R. S. to Thomas Pennant, Esq; F. R. S. John Walsh

Philosophical Transactions (1683–1775) Vol. 64, (1774), pp. 464-473[55] The works of Benjamin Franklin: containing several political and historical tracts not included in any former ed., and many letters official

and private, not hitherto published; with notes and a life of the author, Volume 6 Page 348 (http:/ / books. google. com/books?id=dvQ_AAAAYAAJ& pg=PA348).

[56][56] another noted and careful experimenter in electricity and the discoverer of palladium and rhodium[57][57] Philosophical Magazine, Vol. Ill, p. 211[58] (coulomb^2) * the molar gas constant = 8.314472 m2 kg A2 K-1 mol-1[59][59] 'Trans. Society of Arts,1 1825[60] Meteorological essays By François Arago, Sir Edward Sabine. Page 290. " On Rotation Magnetism (http:/ / books. google. com/

books?id=j0wlAAAAMAAJ& pg=PA290). Proces verbal, Academy of Sciences, 22 November 1824."[61] For more, see Rotating magnetic field.[62] Tr., "The galvanic Circuit investigated mathematically".[63] G. S. Ohm (1827). Die galvanische Kette, mathematisch bearbeitet (http:/ / www. ohm-hochschule. de/ bib/ textarchiv/ Ohm.

Die_galvanische_Kette. pdf). Berlin: T. H. Riemann. .[64][64] The Encyclopedia Americana: a library of universal knowledge, 1918.[65][65] Tsverava, G. K. 1981. "FARADEI, GENRI, I OTKRYTIE INDUKTIROVANNYKH TOKOV." Voprosy Istorii Estestvoznaniia i Tekhniki

no. 3: 99-106. Historical Abstracts, EBSCOhost . Retrieved October 17, 2009.[66] Bowers, Brian. 2004. "Barking Up the Wrong (Electric Motor) Tree." Proceedings of the IEEE 92, no. 2: 388-392. Computers & Applied

Sciences Complete, EBSCOhost . Retrieved October 17, 2009.[67] 1998. "Joseph Henry." Issues in Science & Technology 14, no. 3: 96. Associates Programs Source, EBSCOhost . Retrieved October 17,

2009.[68] According to Oliver Heaviside[69][69] Oliver Heaviside, Electromagnetic theory: Complete and unabridged ed. of v.1, no.2, and: Volume 3. 1950.[70][70] Oliver Heaviside, Electromagnetic theory, v.1. "The Electrician" printing and publishing company, limited, 1893.[71] A treatise on electricity, in theory and practice, Volume 1 By Auguste de La Rive. Page 139 (http:/ / books. google. com/

books?id=IvQEAAAAYAAJ& pg=PA139).[72][72] 'Phil. Trans.,' 1845.[73] Elementary Lessons in Electricity and Magnetism By Silvanus Phillips Thompson. Page 363 (http:/ / books. google. com/

books?id=LZzB2UhRw94C& pg=PA363).[74][74] Phil. Mag-., March 1854[75] For more, see Counter-electromotive force.[76][76] Philosophical Magazine, 1849.[77] Ruhmkorff's version coil was such a success that in 1858 he was awarded a 50,000-franc prize by Napoleon III for the most important

discovery in the application of electricity.[78] American Academy of Arts and Sciences, Proceedings of the American Academy of Arts and Sciences, Vol. XXIII, May 1895 - May 1896,

Boston: University Press, John Wilson and Son (1896), pp. 359-360: Ritchie's most powerful version of his induction coil, using stagedwindings, achieved electrical bolts 2 inches (unknown operator: u'strong' cm) or longer in length.

[79] Page, Charles G., History of Induction: The American Claim to the Induction Coil and Its Electrostatic Developments, Boston: HarvardUniversity, Intelligencer Printing house (1867), pp. 104-106

[80][80] American Academy, pp. 359-360[81] Lyons, T. A. (1901). A treatise on electromagnetic phenomena, and on the compass and its deviations aboard ship. Mathematical,

theoretical, and practical. New York: J. Wiley & Sons. Page 500.[82] La, R. A. (1853). A treatise on electricity: In theory and practice (http:/ / books. google. com/ books?id=IvQEAAAAYAAJ). London:

Longman, Brown, Green, and Longmans.[83][83] tr., Introduction to electrostatics, the study of magnetism and electrodynamics[84] May be Johann Philipp Reis, of Friedrichsdorf, Germany[85] "On a permanent Deflection of the Galvanometer-needle under the influence of a rapid series of equal and opposite induced Currents". By

Lord Rayleigh, F.R.S.. Philosophical magazine, 1877. Page 44 (http:/ / books. google. com/ books?id=wVIwAAAAIAAJ& pg=PA44).[86] Annales de chimie et de physique, Page 385 (http:/ / books. google. com/ books?id=KikFAAAAQAAJ& pg=PA385). "Sur l'aimantation par

les courants" (tr. "On the magnetization by currents").[87][87] 'Ann. de Chimie III,' i, 385.

History of electromagnetic theory 68

[88] Jenkin, F. (1873). Electricity and magnetism (http:/ / books. google. com/ books?id=9OkDAAAAQAAJ). Text-books of science. London:Longmans, Green, and Co

[89][89] Introduction to 'Electricity in the Service of Man'.[90][90] 'Poggendorf Ann.1 1851.[91][91] Proc. Am. Phil. Soc.,Vol. II, pp. 193[92] Annalen der Physik, Volume 103 (http:/ / books. google. com/ books?id=j2UEAAAAYAAJ). Contributions to the acquaintance with the

electric spark, B. W. Feddersen. Page 69+.[93][93] Special information on method and apparatus can be found in Feddersen's Inaugural Dissertation, Kiel 1857th (In the Commission der

Schwers'sehen Buchhandl Handl. In Kiel.)[94] Rowland, H. A. (1902). The physical papers of Henry Augustus Rowland: Johns Hopkins University, 1876-1901 (http:/ / books. google.

com/ books?id=3plMAAAAYAAJ). Baltimore: The Johns Hopkins Press.[95] LII. On the electromagnetic effect of convection-currents Henry A. Rowland; Cary T. Hutchinson Philosophical Magazine Series 5,

1941-5990, Volume 27, Issue 169, Pages 445 – 460[96] See electric machinery, electric direct current, electrical generators.[97][97] consult his British patent of that year[98] consult 'Royal Society Proceedings, 1867 VOL. 10—12[99][99] RJ Gulcher, of Biala, near Bielitz, Austria.[100] The Electrical journal, Volume 7 (http:/ / books. google. com/ books?id=MrbmAAAAMAAJ). 1881. Page117+ (http:/ / books. google.

com/ books?id=MrbmAAAAMAAJ& pg=PA117)[101] ETA: Electrical magazine: A. Ed, Volume 1 (http:/ / books. google. com/ books?id=SCrOAAAAMAAJ)[102] See electric direct current.[103][103] See Electric alternating current machinery.[104] The 19th century science book A Guide to the Scientific Knowledge of Things Familiar provides a brief summary of scientific thinking in

this field at the time.[105][105] Consult Maxwell's 'Electricity and Magnetism,1 Vol. II, Chap. xx[106] On Faraday’s Lines of Force’ byJames Clerk Maxwell 1855 (http:/ / www. blazelabs. com/ On Faraday's Lines of Force. pdf)[107] James Clerk Maxwell, On Physical Lines of Force, Philosophical Magazine, 1861[108][108] In November 1847, Clerk Maxwell entered the University of Edinburgh, learning mathematics from Kelland, natural philosophy from J. D.

Forbes, and logic from Sir W. R. Hamilton.[109] Glazebrook, R. (1896). James Clerk Maxwell and modern physics. New York: Macmillan. Pg. 190 (http:/ / books. google. com/

books?id=rX9LAAAAMAAJ& pg=PA190)[110] J J O'Connor and E F Robertson, James Clerk Maxwell (http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ Biographies/ Maxwell. html),

School of Mathematics and Statistics, University of St Andrews, Scotland, November 1997[111] James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of the Royal Society of London 155,

459-512 (1865).[112][112] Maxwell's 'Electricity and Magnetism,' preface[113] See oscillating current, telegraphy, wireless.[114] Proceedings of the London Mathematical Society, Volume 3. London Mathematical Society, 1871. Pg. 224 (http:/ / books. google. com/

books?id=lekKAAAAYAAJ& pg=PA224)[115][115] Consult 'Proc. Am. Inst. El. Engrs.,' 1901[116] Crookes presented a lecture to the British Association for the Advancement of Science, in Sheffield, on Friday, 22 August 1879 (http:/ /

www. worldcatlibraries. org/ wcpa/ top3mset/ 5dcb9349d366f8ec. html) (http:/ / www. tfcbooks. com/ mall/ more/ 315rm. htm)[117][117] consult 'Proc. British Association,' 1879[118] Perhaps there were other reasons than those mentioned for the discontinuance. We do not dwell in the Palace of Truth.[119] in Sir W. Thomson's meaning of the word[120][120] Electro-Magnetic Theory. By Oliver HeaviBide. Vol. I. Electrician Printing: and Publishing Company, Ltd. London, 1893[121] In mathematics, vectorial algebra may mean a linear algebra, specifically the basic algebraic operations of vector addition and scalar

multiplication; see vector space. The algebraic operations in vector calculus, namely the specific additional structure of vectors in3-dimensional Euclidean space of dot product and especially cross product. In this sense, vector algebra is contrasted with geometricalgebra, which provides an alternative generalization to higher dimensions. Original vector algebras of the 19th century like quaternions,tessarines, or coquaternions, each of which has its own product. The vector algebras biquaternions and hyperbolic quaternions enabled therevolution in physics called special relativity by providing mathematical models.

[122] Electrical engineer, Volume 18. Page299 (http:/ / books. google. com/ books?id=WbrmAAAAMAAJ& pg=PA299)[123] Announced in his evening lecture to the Royal Institution on Friday, 30 April 1897, and published in Philosophical Magazine, 44, 293

(http:/ / web. lemoyne. edu/ ~GIUNTA/ thomson1897. html)[124] Earl R. Hoover, Cradle of Greatness: National and World Achievements of Ohio’s Western Reserve (Cleveland: Shaker Savings

Association, 1977).[125] Dayton C. Miller, "Ether-drift Experiments at Mount Wilson Solar Observatory," Physical Review (http:/ / prola. aps. org/ abstract/ PR/

v19/ i4/ p407_1), S2, V19, N4, pp. 407-408 (April 1922).

History of electromagnetic theory 69

[126] Alternating current generating systems were known in simple forms from the discovery of the magnetic induction of electric current. Theearly machines were developed by pioneers such as Michael Faraday and Hippolyte Pixii. Faraday developed the "rotating rectangle", whoseoperation was heteropolar - each active conductor passed successively through regions where the magnetic field was in opposite directions.

[127] Blalock, Thomas J., " Alternating Current Electrification, 1886 (http:/ / www. ieee. org/ organizations/ history_center/ stanley. html)".IEEE History Center, IEEE Milestone. (ed. first practical demonstration of a dc generator - ac transformer system.)

[128] US 447921 (http:/ / worldwide. espacenet. com/ textdoc?DB=EPODOC& IDX=US447921), Tesla, Nikola, "Alternating Electric CurrentGenerator".

[129] Thompson, Silvanus P., Dynamo-Electric Machinery. pp. 17[130] Thompson, Silvanus P., Dynamo-Electric Machinery. pp. 16[131] See electric lighting[132][132] Lomas, Robert (1999). The Man who Invented the 20th century. London: Headline. ISBN 0-7472-7588-2.[133] See War of Currents and International Electro-Technical Exhibition - 1891[134] See telegraph[135] see transatlantic telegraph cable[136][136] Miller's 'Magnetism and Electricity,' p. 460[137] Thomas Hughes, Networks of Power, page 120[138] See Electric transmission of energy.[139][139] 'James Blyth - Britain's first modern wind power pioneer', by Trevor Price, 2003, Wind Engineering, vol 29 no. 3, pp 191-200][140][140] [Anon, 1890, 'Mr. Brush's Windmill Dynamo', Scientific American, vol 63 no. 25, 20 December, p. 54][141] A Wind Energy Pioneer: Charles F. Brush (http:/ / www. windpower. org/ en/ pictures/ brush. htm), Danish Wind Industry Association.

Retrieved 2007-05-02.[142] History of Wind Energy in Cutler J. Cleveland,(ed) Encyclopedia of Energy Vol.6, Elsevier, ISBN 978-1-60119-433-6, 2007, pp. 421-422[143] See electrical units, electrical terms.[144][144] Miller 1981, Ch. 1[145][145] Pais 1982, Ch. 6b[146][146] Janssen, 2007[147][147] Galison 2002[148][148] Darrigol 2005[149][149] Katzir 2005[150] Miller 1981, Ch. 1.7 & 1.14[151] Pais 1982, Ch. 6 & 8[152] On the reception of relativity theory around the world, and the different controversies it encountered, see the articles in Thomas F. Glick,

ed., The Comparative Reception of Relativity (Kluwer Academic Publishers, 1987), ISBN 90-277-2498-9.[153] Pais, Abraham (1982), Subtle is the Lord. The Science and the Life of Albert Einstein, Oxford University Press, pp. 382–386,

ISBN 0-19-520438-7[154][154] The state of the aether is at every place determined by connections with the matter and the state of the ether in neighbouring places, which

are amenable to law in the form of differential equations.[155] Sidelights On Relativity (http:/ / books. google. com/ books?id=cDtk_23SsXgC), Albert Einstein.[156] P.A.M. Dirac (1927). "The Quantum Theory of the Emission and Absorption of Radiation". Proceedings of the Royal Society of London A

114: 243–265. Bibcode 1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039.[157] E. Fermi (1932). "Quantum Theory of Radiation". Reviews of Modern Physics 4: 87–132. Bibcode 1932RvMP....4...87F.

doi:10.1103/RevModPhys.4.87.[158] F. Bloch; A. Nordsieck (1937). "Note on the Radiation Field of the Electron". Physical Review 52: 54–59. Bibcode 1937PhRv...52...54B.

doi:10.1103/PhysRev.52.54.[159] V. F. Weisskopf (1939). "On the Self-Energy and the Electromagnetic Field of the Electron". Physical Review 56: 72–85.

Bibcode 1939PhRv...56...72W. doi:10.1103/PhysRev.56.72.[160] R. Oppenheimer (1930). "Note on the Theory of the Interaction of Field and Matter". Physical Review 35: 461–477.

Bibcode 1930PhRv...35..461O. doi:10.1103/PhysRev.35.461.[161] O. Hahn and F. Strassmann. Über den Nachweis und das Verhalten der bei der Bestrahlung des Urans mittels Neutronen entstehenden

Erdalkalimetalle ("On the detection and characteristics of the alkaline earth metals formed by irradiation of uranium with neutrons"),Naturwissenschaften Volume 27, Number 1, 11–15 (1939). The authors were identified as being at the Kaiser-Wilhelm-Institut für Chemie,Berlin-Dahlem. Received 22 December 1938.

[162] Lise Meitner and O. R. Frisch. "Disintegration of Uranium by Neutrons: a New Type of Nuclear Reaction", Nature, Volume 143,Number 3615, 239–240 (11 February 1939) (http:/ / www. nature. com/ physics/ looking-back/ meitner/ index. html). The paper is dated 16January 1939. Meitner is identified as being at the Physical Institute, Academy of Sciences, Stockholm. Frisch is identified as being at theInstitute of Theoretical Physics, University of Copenhagen.

[163] O. R. Frisch. "Physical Evidence for the Division of Heavy Nuclei under Neutron Bombardment", Nature, Volume 143, Number 3616, 276–276 (18 February 1939) (http:/ / dbhs. wvusd. k12. ca. us/ webdocs/ Chem-History/ Frisch-Fission-1939. html). The paper is dated 17 January 1939. [The experiment for this letter to the editor was conducted on 13 January 1939; see Richard Rhodes The Making of the

History of electromagnetic theory 70

Atomic Bomb. 263 and 268 (Simon and Schuster, 1986).][164] Ruth Lewin Sime. From Exceptional Prominence to Prominent Exception: Lise Meitner at the Kaiser Wilhelm Institute for Chemistry

Ergebnisse 24 (http:/ / www. mpiwg-berlin. mpg. de/ KWG/ Ergebnisse/ Ergebnisse24. pdf) Forschungsprogramm Geschichte derKaiser-Wilhelm-Gesellschaft im Nationalsozialismus (2005).

[165] Ruth Lewin Sime. Lise Meitner: A Life in Physics (University of California, 1997).[166] Elisabeth Crawford, Ruth Lewin Sime, and Mark Walker. "A Nobel Tale of Postwar Injustice", Physics Today Volume 50, Issue 9, 26–32

(1997).[167] W. E. Lamb; R. C. Retherford (1947). "Fine Structure of the Hydrogen Atom by a Microwave Method,". Physical Review 72: 241–243.

Bibcode 1947PhRv...72..241L. doi:10.1103/PhysRev.72.241.[168] P. Kusch; H. M. Foley (1948). "On the Intrinsic Moment of the Electron". Physical Review 73: 412. Bibcode 1948PhRv...73..412F.

doi:10.1103/PhysRev.73.412.[169] Brattain quoted in Michael Riordan and Lillian Hoddeson; Crystal Fire: The Invention of the Transistor and the Birth of the Information

Age. New York: Norton (1997) ISBN 0-393-31851-6 pbk. p. 127[170] Schweber, Silvan (1994). "Chapter 5". QED and the Men Who Did it: Dyson, Feynman, Schwinger, and Tomonaga. Princeton University

Press. p. 230. ISBN 978-0-691-03327-3.[171] H. Bethe (1947). "The Electromagnetic Shift of Energy Levels". Physical Review 72: 339–341. Bibcode 1947PhRv...72..339B.

doi:10.1103/PhysRev.72.339.[172] S. Tomonaga (1946). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields". Progress of Theoretical

Physics 1: 27–42. doi:10.1143/PTP.1.27.[173] J. Schwinger (1948). "On Quantum-Electrodynamics and the Magnetic Moment of the Electron". Physical Review 73: 416–417.

Bibcode 1948PhRv...73..416S. doi:10.1103/PhysRev.73.416.[174] J. Schwinger (1948). "Quantum Electrodynamics. I. A Covariant Formulation". Physical Review 74: 1439–1461.

Bibcode 1948PhRv...74.1439S. doi:10.1103/PhysRev.74.1439.[175] R. P. Feynman (1949). "Space-Time Approach to Quantum Electrodynamics". Physical Review 76: 769–789.

Bibcode 1949PhRv...76..769F. doi:10.1103/PhysRev.76.769.[176] R. P. Feynman (1949). "The Theory of Positrons". Physical Review 76: 749–759. Bibcode 1949PhRv...76..749F.

doi:10.1103/PhysRev.76.749.[177] R. P. Feynman (1950). "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction". Physical Review 80: 440–457.

Bibcode 1950PhRv...80..440F. doi:10.1103/PhysRev.80.440.[178] F. Dyson (1949). "The Radiation Theories of Tomonaga, Schwinger, and Feynman". Physical Review 75: 486–502.

Bibcode 1949PhRv...75..486D. doi:10.1103/PhysRev.75.486.[179] F. Dyson (1949). "The S Matrix in Quantum Electrodynamics". Physical Review 75: 1736–1755. Bibcode 1949PhRv...75.1736D.

doi:10.1103/PhysRev.75.1736.[180] "The Nobel Prize in Physics 1965" (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1965/ index. html). Nobel Foundation. .

Retrieved 2008-10-09.[181] Feynman, Richard (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 128. ISBN 978-0-691-12575-6.[182] Kurt Lehovec's patent on the isolation p-n junction: U.S. Patent 3029366 (http:/ / www. google. com/ patents?vid=3029366) granted on

April 10, 1962, filed April 22, 1959. Robert Noyce credits Lehovec in his article – "Microelectronics", Scientific American, September 1977,Volume 23, Number 3, pp. 63–9.

[183] The Chip that Jack Built (http:/ / www. ti. com/ corp/ docs/ kilbyctr/ jackbuilt. shtml), (c. 2008), (HTML), Texas Instruments, accessedMay 29, 2008.

[184] Winston, Brian. Media technology and society: a history: from the telegraph to the Internet (http:/ / books. google. com/books?id=gfeCXlElJTwC& pg=RA2-PA221& dq="wherein+ all+ the+ components+ of+ the+ electronic+ circuit"#v=onepage& q="whereinall the components of the electronic circuit"& f=false), (1998), Routeledge, London, ISBN 0-415-14230-X ISBN 978-0-415-14230-4, p. 221

[185] Nobel Web AB, (October 10, 2000),( The Nobel Prize in Physics 2000 (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 2000/press. html), Retrieved on May 29, 2008

[186] Cartlidge, Edwin. The Secret World of Amateur Fusion. Physics World, March 2007: IOP Publishing Ltd, pp. 10-11. ISSN: 0953-8585.[187] S.L. Glashow (1961). "Partial-symmetries of weak interactions". Nuclear Physics 22: 579–588. Bibcode 1961NucPh..22..579G.

doi:10.1016/0029-5582(61)90469-2.[188] S. Weinberg (1967). "A Model of Leptons". Physical Review Letters 19: 1264–1266. Bibcode 1967PhRvL..19.1264W.

doi:10.1103/PhysRevLett.19.1264.[189] A. Salam (1968). N. Svartholm. ed. Elementary Particle Physics: Relativistic Groups and Analyticity. Eighth Nobel Symposium.

Stockholm: Almquvist and Wiksell. pp. 367.[190] F. Englert, R. Brout (1964). "Broken Symmetry and the Mass of Gauge Vector Mesons". Physical Review Letters 13: 321–323.

Bibcode 1964PhRvL..13..321E. doi:10.1103/PhysRevLett.13.321.[191] P.W. Higgs (1964). "Broken Symmetries and the Masses of Gauge Bosons". Physical Review Letters 13: 508–509.

Bibcode 1964PhRvL..13..508H. doi:10.1103/PhysRevLett.13.508.[192] G.S. Guralnik, C.R. Hagen, T.W.B. Kibble (1964). "Global Conservation Laws and Massless Particles". Physical Review Letters 13:

585–587. Bibcode 1964PhRvL..13..585G. doi:10.1103/PhysRevLett.13.585.

History of electromagnetic theory 71

[193] F.J. Hasert et al. (1973). "Search for elastic muon-neutrino electron scattering". Physics Letters B 46: 121. Bibcode 1973PhLB...46..121H.doi:10.1016/0370-2693(73)90494-2.

[194] F.J. Hasert et al. (1973). "Observation of neutrino-like interactions without muon or electron in the gargamelle neutrino experiment".Physics Letters B 46: 138. Bibcode 1973PhLB...46..138H. doi:10.1016/0370-2693(73)90499-1.

[195] F.J. Hasert et al. (1974). "Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment".Nuclear Physics B 73: 1. Bibcode 1974NuPhB..73....1H. doi:10.1016/0550-3213(74)90038-8.

[196] D. Haidt (4 October 2004). "The discovery of the weak neutral currents" (http:/ / cerncourier. com/ cws/ article/ cern/ 29168). CERNCourier. . Retrieved 2008-05-08.

[197] F. J. Hasert et al. Phys. Lett. 46B 121 (1973).[198] F. J. Hasert et al. Phys. Lett. 46B 138 (1973).[199] F. J. Hasert et al. Nucl. Phys. B73 1(1974).[200] The discovery of the weak neutral currents (http:/ / cerncourier. com/ cws/ article/ cern/ 29168), CERN courier, 2004-10-04, , retrieved

2008-05-08[201] The Nobel Prize in Physics 1979 (http:/ / www. nobel. se/ physics/ laureates/ 1979), Nobel Foundation, , retrieved 2008-09-10[202][202] A long conductor attached to an object.[203] It is noted that though not all space tethers are conductive.[204][204] A medical imaging technique used in radiology to visualize detailed internal structures. The good contrast it provides between the different

soft tissues of the body make it especially useful in brain, muscles, heart, and cancer compared with other medical imaging techniques such ascomputed tomography (CT) or X-rays.

[205][205] Wireless power is the transmission of electrical energy from a power source to an electrical load without interconnecting wires. Wirelesstransmission is useful in cases where interconnecting wires are inconvenient, hazardous, or impossible.

[206] "Wireless electricity could power consumer, industrial electronics" (http:/ / web. mit. edu/ newsoffice/ 2006/ wireless. html). MIT News.2006-11-14. .

[207] "Goodbye wires…" (http:/ / web. mit. edu/ newsoffice/ 2007/ wireless-0607. html). MIT News. 2007-06-07. .[208] "Wireless Power Demonstrated" (http:/ / thefutureofthings. com/ pod/ 250/ wireless-power-demonstrated. html). . Retrieved 2008-12-09.[209] Hawking, S.W. (1996). A Brief History of Time: The Updated and Expanded Edition. (2nd ed.). Bantam Books. p. XXX.

ISBN 0-553-38016-8.[210] M-Theory: The Mother of all SuperStrings (http:/ / www. mkaku. org/ articles/ m_theory. php)[211] Hawking, Stephen. Gödel and the end of physics (http:/ / www. damtp. cam. ac. uk/ strings02/ dirac/ hawking/ ), July 20, 2002.[212] A hypothetical particle in particle physics that is a magnet with only one magnetic pole. In more technical terms, a magnetic monopole

would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unification and superstringtheories, which predict their existence. See Particle Data Group summary of magnetic monopole search (http:/ / pdg. lbl. gov/ 2004/ listings/s028. pdf); Wen, Xiao-Gang; Witten, Edward, Electric and magnetic charges in superstring models,Nuclear Physics B, Volume 261, p.651-677; and Coleman, The Magnetic Monopole 50 years Later, reprinted in Aspects of Symmetry for more

[213] Paul Dirac, "Quantised Singularities in the Electromagnetic Field". Proc. Roy. Soc. (London) A 133, 60 (1931). Free web link (http:/ /users. physik. fu-berlin. de/ ~kleinert/ files/ dirac1931. pdf).

[214] d-Wave Pairing (http:/ / musr. ca/ theses/ Sonier/ MSc/ node17. html). musr.ca.[215] The Motivation for an Alternative Pairing Mechanism (http:/ / musr. ca/ theses/ Sonier/ MSc/ node16. html). musr.ca.[216] A. Mourachkine (2004). Room-Temperature Superconductivity. Cambridge International Science Publishing (Cambridge, UK) (also http:/ /

xxx. lanl. gov/ abs/ cond-mat/ 0606187). & #32;ISBN& nbsp;1-904602-27-4.

Bibliography• Bakewell, F. C. (1853). Electric science; its history, phenomena, and applications (http:/ / books. google. com/

books?id=Lks1AAAAMAAJ). London: Ingram, Cooke.• Benjamin, P. (1898). A history of electricity (The intellectual rise in electricity) from antiquity to the days of

Benjamin Franklin (http:/ / books. google. com/ books?id=VLsKAAAAIAAJ). New York: J. Wiley & Sons.• Darrigol, Olivier (2005), "The Genesis of the theory of relativity" (http:/ / www. bourbaphy. fr/ darrigol2. pdf)

(PDF), Séminaire Poincaré 1: 1–22, retrieved 2009-06-21• Durgin, W. A. (1912). Electricity, its history and development (http:/ / books. google. com/

books?id=hQtJAAAAIAAJ). Chicago: A.C. McClurg.• Einstein, Albert: "Ether and the Theory of Relativity" (1920), republished in Sidelights on Relativity (Dover, NY,

1922).• Einstein, Albert, The Investigation of the State of Aether in Magnetic Fields (http:/ / www. worldscibooks. com/

phy_etextbook/ 4454/ 4454_chap1. pdf), 1895. (PDF format)

History of electromagnetic theory 72

• Einstein, Albert (1905a), "On a Heuristic Viewpoint Concerning the Production and Transformation of Light",Annalen der Physik 17: 132–148, Bibcode 1905AnP...322..132E, doi:10.1002/andp.19053220607. This annusmirabilis paper on the photoelectric effect was received by Annalen der Physik March 18.

• Einstein, Albert (1905b), "On the Motion—Required by the Molecular Kinetic Theory of Heat—of SmallParticles Suspended in a Stationary Liquid", Annalen der Physik 17: 549–560, Bibcode 1905AnP...322..549E,doi:10.1002/andp.19053220806. This annus mirabilis paper on Brownian motion was received May 11.

• Einstein, Albert (1905c), "On the Electrodynamics of Moving Bodies", Annalen der Physik 17: 891–921,Bibcode 1905AnP...322..891E, doi:10.1002/andp.19053221004. This annus mirabilis paper on special relativitywas received June 30.

• Einstein, Albert (1905d), "Does the Inertia of a Body Depend Upon Its Energy Content?", Annalen der Physik 18:639–641, Bibcode 1905AnP...323..639E, doi:10.1002/andp.19053231314. This annus mirabilis paper onmass-energy equivalence was received September 27.

• " Aether (http:/ / encyclopedia. jrank. org/ ADA_AIZ/ AETHER_or_ETHER_Gr_deli_p_proba. html)",Encyclopædia Britannica, Eleventh Edition (1910–1911). Volume Vol. 1, Page 297.

• The Encyclopedia Americana; a library of universal knowledge (http:/ / books. google. com/books?id=62UMAAAAYAAJ); "Electricity, its history and Progress". (1918). New York: EncyclopediaAmericana Corp. Page 171 (http:/ / books. google. com/ books?id=62UMAAAAYAAJ&pg=PA171#PPA171,M1)

• Galison, Peter (2003), Einstein's Clocks, Poincaré's Maps: Empires of Time, New York: W.W. Norton,ISBN 0-393-32604-7

• Gibson, C. R. (1907). Electricity of to-day, its work & mysteries described in non-technical language (http:/ /books. google. com/ books?id=lwpVAAAAMAAJ). London: Seeley and co., limited

• Heaviside, O. (1894). Electromagnetic theory (http:/ / books. google. com/ books?id=9ukEAAAAYAAJ).London: "The Electrician" Print. and Pub.

• Ireland commissioners of nat. educ., (1861). Electricity, galvanism, magnetism, electro-magnetism, heat, and thesteam engine (http:/ / books. google. com/ books?id=1AoFAAAAQAAJ). Oxford University.

• Janssen, Michel & Mecklenburg, Matthew (2007), From classical to relativistic mechanics: Electromagneticmodels of the electron (http:/ / www. tc. umn. edu/ ~janss011/ ), in V. F. Hendricks, et al., , Interactions:Mathematics, Physics and Philosophy (Dordrecht: Springer): 65–134

• Jeans, J. H. (1908). The mathematical theory of electricity and magnetism (http:/ / books. google. com/books?id=jKYTAAAAYAAJ). Cambridge: University Press.

• Katzir, Shaul (2005), "Poincaré’s Relativistic Physics: Its Origins and Nature", Phys. Perspect. 7: 268–292,Bibcode 2005PhP.....7..268K, doi:10.1007/s00016-004-0234-y

• Lord Kelvin (Sir William Thomson), "On Vortex Atoms". Proceedings of the Royal Society of Edinburgh, Vol.VI, 1867, pp. 94–105. (ed., Reprinted in Phil. Mag. Vol. XXXIV, 1867, pp. 15–24.)

• Kolbe, Bruno; Francis ed Legge, Joseph Skellon, tr., " An Introduction to Electricity (http:/ / books. google. com/books?vid=0o90G64Z2FDIyKUsLs9& id=150IAAAAIAAJ)". Kegan Paul, Trench, Trübner, 1908.

• Lodge, Oliver, "Ether", Encyclopædia Britannica, Thirteenth Edition (1926).•• Lodge, Oliver, "The Ether of Space". ISBN 1-4021-8302-X (paperback) ISBN 1-4021-1766-3 (hardcover)•• Lodge, Oliver, "Ether and Reality". ISBN 0-7661-7865-X• Lyons, T. A. (1901). A treatise on electromagnetic phenomena, and on the compass and its deviations aboard ship

(http:/ / books. google. com/ books?id=JXkpAAAAYAAJ). Mathematical, theoretical, and practical. New York:J. Wiley & Sons.

• Maxwell, James Clerk, "Ether", Encyclopædia Britannica, Ninth Edition (1875–89).• Maxwell, J. C., & Thompson, J. J. (1892). A treatise on electricity and magnetism (http:/ / books. google. com/

books?id=qdYcAAAAMAAJ). Clarendon Press series. Oxford: Clarendon.

History of electromagnetic theory 73

• Miller, Arthur I. (1981), Albert Einstein’s special theory of relativity. Emergence (1905) and early interpretation(1905–1911), Reading: Addison–Wesley, ISBN 0-201-04679-2

• Pais, Abraham (1982), Subtle is the Lord: The Science and the Life of Albert Einstein, New York: OxfordUniversity Press, ISBN 0-19-520438-7

• Priestley, J., & Mynde, J. (1775). The history and present state of electricity, with original experiments (http:/ /books. google. com/ books?id=RkpkAAAAMAAJ). London: Printed for C. Bathurst, and T. Lowndes; J.Rivington, and J. Johnson; S. Crowder [and 4 others in London].

• Schaffner, Kenneth F. : 19th-century aether theories, Oxford: Pergamon Press, 1972. (contains several reprints oforiginal papers of famous physicists)

• Slingo, M., Brooker, A., Urbanitzky, A., Perry, J., & Dibner, B. (1895). The cyclopædia of electrical engineering:containing a history of the discovery and application of electricity with its practice and achievements from theearliest period to the present time: the whole being a practical guide to artisans, engineers and students interestedin the practice and development of electricity, electric lighting, motors, thermo-piles, the telegraph, the telephone,magnets and every other branch of electrical application (http:/ / books. google. com/books?id=EexMAAAAMAAJ). Philadelphia: The Gebbie Pub. Co., Limited.

• Steinmetz, C. P., " Transient Electric Phenomena (http:/ / books. google. com/ books?id=PBsAAAAAMAAJ&pg=RA1-PA40#PRA1-PA38,M1)". Page 38 (http:/ / books. google. com/ books?id=PBsAAAAAMAAJ&pg=RA1-PA40#PRA1-PA38,M1). (ed., contained in: General Electric Company. General Electric review.Schenectady: General Electric Co. (http:/ / books. google. com/ books?id=PBsAAAAAMAAJ).)

• A New System of Alternating Current Motors and Transformers, by Nichola Tesla, 1888• Thompson, S. P. (1891). The electromagnet, and electromagnetic mechanism (http:/ / books. google. com/

books?id=CLmFTg_j0pwC). London: E. & F.N. Spon.• Whittaker, E. T., " A History of the Theories of Aether and Electricity, from the Age of Descartes to the Close of

the 19th century (http:/ / www. archive. org/ details/ historyoftheorie00whitrich)". Dublin University Press series.London: Longmans, Green and Co.;

• Urbanitzky, A. v., & Wormell, R. (1886). Electricity in the service of man: a popular and practical treatise on theapplications of electricity in modern life (http:/ / books. google. com/ books?id=rkgOAAAAYAAJ). London:Cassell &.

Lorentz force 74

Lorentz force

Trajectory of a particle with a positive or negative charge q under theinfluence of a magnetic field B, which is directed perpendicularly out

of the screen.

Beam of electrons moving in a circle, due to the presence of amagnetic field. Purple light is emitted along the electron path, due to

the electrons colliding with gas molecules in the bulb. Using aTeltron tube.

In physics, particularly electromagnetism, the Lorentzforce is the force on a point charge due toelectromagnetic fields. If a particle of charge q moveswith velocity v in the presence of an electric field Eand a magnetic field B, then it will experience a force

Variations on this basic formula describe the magneticforce on a current-carrying wire (sometimes calledLaplace force), the electromotive force in a wire loopmoving through a magnetic field (an aspect ofFaraday's law of induction), and the force on a particlewhich might be traveling near the speed of light(relativistic form of the Lorentz force).

The first derivation of the Lorentz force is commonlyattributed to Oliver Heaviside in 1889,[1] although otherhistorians suggest an earlier origin in an 1865 paper byJames Clerk Maxwell.[2] Lorentz derived it a few yearsafter Heaviside.

Equation (SI units)

One charged particle

The force F acting on a particle of electric charge qwith instantaneous velocity v, due to an externalelectric field E and magnetic field B, is given by:[3]

where × is the vector cross product. All boldface quantities are vectors. More explicitly stated:

in which r is the position vector of the charged particle, t is time, and the overdot is a time derivative.A positively charged particle will be accelerated in the same linear orientation as the E field, but will curveperpendicularly to both the instantaneous velocity vector v and the B field according to the right-hand rule (in detail,if the thumb of the right hand points along v and the index finger along B, then the middle finger points along F).

The term qE is called the electric force, while the term qv × B is called the magnetic force.[4] According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,[5] with the total

Lorentz force 75

electromagnetic force (including the electric force) given some other (nonstandard) name. This article will not followthis nomenclature: In what follows, the term "Lorentz force" will refer only to the expression for the total force.The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wirein a magnetic field. In that context, it is also called the Laplace force.

Continuous charge distributionFor a continuous charge distribution in motion, the Lorentz force equation becomes:

where dF is the force on a small piece of the charge distribution with charge dq. If both sides of this equation aredivided by the volume of this small piece of the charge distribution dV, the result is:

where f is the force density (force per unit volume) and ρ is the charge density (charge per unit volume). Next, thecurrent density corresponding to the motion of the charge continuum is

so the continuous analogue to the equation is[6]

The total force is the volume integral over the charge distribution:

By eliminating ρ and J, using Maxwell's equations, and manipulating using the theorems of vector calculus, thisform of the equation can be used to derive the Maxwell stress tensor T, used in General relativity.[6]

In terms of the tensor field T and the Poynting vector S, another way to write the Lorentz force (per unit volume)is[6]

where c is the speed of light and ∇• denotes the divergence of a tensor field. Rather than the amount of charge and itsvelocity in electric and magnetic fields, this equation relates the energy flux (flow of energy per unit time per unitdistance) in the fields to the force exerted on a charge distribution.

HistoryEarly attempts to quantitatively describe the electromagnetic force were made in the mid-18th century. It wasproposed that the force on magnetic poles, by Johann Tobias Mayer and others in 1760, and electrically chargedobjects, by Henry Cavendish in 1762, obeyed an inverse-square law. However, in both cases the experimental proofwas neither complete nor conclusive. It was not until 1784 when Charles-Augustin de Coulomb, using a torsionbalance, was able to definitively show through experiment that this was true.[7] Soon after the discovery in 1820 byH. C. Ørsted that a magnetic needle is acted on by a voltaic current, André-Marie Ampère that same year was able todevise through experimentation the formula for the angular dependence of the force between two currentelements.[8][9] In all these descriptions, the force was always given in terms of the properties of the objects involvedand the distances between them rather than in terms of electric and magnetic fields.[10]

The modern concept of electric and magnetic fields first arose in the theories of Michael Faraday, particularly his idea of lines of force, later to be given full mathematical description by Lord Kelvin and James Clerk Maxwell.[11]

From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents,[2] however, in the time of Maxwell it was not evident how

Lorentz force 76

his equations related to the forces on moving charged objects. J. J. Thomson was the first to attempt to derive fromMaxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's propertiesand external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays,Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as[1]

Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incompletedescription of the displacement current, included an incorrect scale-factor of a half in front of the formula. It wasOliver Heaviside, who had invented the modern vector notation and applied them to Maxwell's field equations, thatin 1885 and 1889 fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force ona moving charged object.[1][12][13] Finally, in 1892, Hendrik Lorentz derived the modern day form of the formula forthe electromagnetic force which includes the contributions to the total force from both the electric and the magneticfields. Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction. Instead, Lorentz madea distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopicscale. Using the Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangianmechanics (see below), Lorentz arrived at the correct and complete form of the force law that now bears hisname.[14][15]

Trajectories of particles due to the Lorentz force

Charged particle drifts in a homogeneous magnetic field. (A) Nodisturbing force (B) With an electric field, E (C) With an independentforce, F (e.g. gravity) (D) In an inhomogeneous magnetic field, grad H

In many cases of practical interest, the motion in amagnetic field of an electrically charged particle(such as an electron or ion in a plasma) can be treatedas the superposition of a relatively fast circularmotion around a point called the guiding center anda relatively slow drift of this point. The drift speedsmay differ for various species depending on theircharge states, masses, or temperatures, possiblyresulting in electric currents or chemical separation.

Significance of the Lorentz force

While the modern Maxwell's equations describe howelectrically charged particles and currents or movingcharged particles give rise to electric and magneticfields, the Lorentz force law completes that pictureby describing the force acting on a moving pointcharge q in the presence of electromagneticfields.[3][16] The Lorentz force law describes theeffect of E and B upon a point charge, but suchelectromagnetic forces are not the entire picture.Charged particles are possibly coupled to otherforces, notably gravity and nuclear forces. Thus,Maxwell's equations do not stand separate from otherphysical laws, but are coupled to them via the chargeand current densities. The response of a point chargeto the Lorentz law is one aspect; the generation of E and B by currents and charges is another.

Lorentz force 77

In real materials the Lorentz force is inadequate to describe the behavior of charged particles, both in principle and asa matter of computation. The charged particles in a material medium both respond to the E and B fields and generatethese fields. Complex transport equations must be solved to determine the time and spatial response of charges, forexample, the Boltzmann equation or the Fokker–Planck equation or the Navier–Stokes equations. For example, seemagnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, stellar evolution. An entirephysical apparatus for dealing with these matters has developed. See for example, Green–Kubo relations and Green'sfunction (many-body theory).

Lorentz force law as the definition of E and BIn many textbook treatments of classical electromagnetism, the Lorentz Force Law is used as the definition of theelectric and magnetic fields E and B.[17] To be specific, the Lorentz Force is understood to be the followingempirical statement:

The electromagnetic force F on a test charge at a given point and time is a certain function of its charge q andvelocity v, which can be parameterized by exactly two vectors E and B, in the functional form:

If this empirical statement is valid (countless experiments have shown that it is), then two vector fields E and B arethereby defined throughout space and time, and these are called the "electric field" and "magnetic field". Note thatthe fields are defined everywhere in space and time with respect to what force a test charge would receive regardlessof whether a charge is present to experience the force.Note also that as a definition of E and B, the Lorentz force is only a definition in principle because a real particle (asopposed to the hypothetical "test charge" of infinitesimally-small mass and charge) would generate its own finite Eand B fields, which would alter the electromagnetic force that it experiences. In addition, if the charge experiencesacceleration, as if forced into a curved trajectory by some external agency, it emits radiation that causes braking ofits motion. See for example Bremsstrahlung and synchrotron light. These effects occur through both a direct effect(called the radiation reaction force) and indirectly (by affecting the motion of nearby charges and currents).Moreover, net force must include gravity, electroweak, and any other forces aside from electromagnetic force.

Force on a current-carrying wire

Right-hand rule for a current-carrying wire in a magnetic field B

When a wire carrying an electrical current is placed in amagnetic field, each of the moving charges, whichcomprise the current, experiences the Lorentz force,and together they can create a macroscopic force on thewire (sometimes called the Laplace force). Bycombining the Lorentz force law above with thedefinition of electrical current, the following equationresults, in the case of a straight, stationary wire:

where ℓ is a vector whose magnitude is the length ofwire, and whose direction is along the wire, aligned with the direction of conventional current flow I.

If the wire is not straight but curved, the force on it can be computed by applying this formula to each infinitesimalsegment of wire dℓ, then adding up all these forces by integration. Formally, the net force on a stationary, rigid wirecarrying a steady current I is

Lorentz force 78

This is the net force. In addition, there will usually be torque, plus other effects if the wire is not perfectly rigid.One application of this is Ampère's force law, which describes how two current-carrying wires can attract or repeleach other, since each experiences a Lorentz force from the other's magnetic field. For more information, see thearticle: Ampère's force law.

EMFThe magnetic force (q v × B) component of the Lorentz force is responsible for motional electromotive force (ormotional EMF), the phenomenon underlying many electrical generators. When a conductor is moved through amagnetic field, the magnetic force tries to push electrons through the wire, and this creates the EMF. The term"motional EMF" is applied to this phenomenon, since the EMF is due to the motion of the wire.In other electrical generators, the magnets move, while the conductors do not. In this case, the EMF is due to theelectric force (qE) term in the Lorentz Force equation. The electric field in question is created by the changingmagnetic field, resulting in an induced EMF, as described by the Maxwell-Faraday equation (one of the four modernMaxwell's equations).[18]

Both of these EMF's, despite their different origins, can be described by the same equation, namely, the EMF is therate of change of magnetic flux through the wire. (This is Faraday's law of induction, see above.) Einstein's theory ofspecial relativity was partially motivated by the desire to better understand this link between the two effects.[18] Infact, the electric and magnetic fields are different faces of the same electromagnetic field, and in moving from oneinertial frame to another, the solenoidal vector field portion of the E-field can change in whole or in part to a B-fieldor vice versa.[19]

Lorentz force and Faraday's law of inductionGiven a loop of wire in a magnetic field, Faraday's law of induction states the induced electromotive force (EMF) inthe wire is:

where

is the magnetic flux through the loop, B is the magnetic field, Σ(t) is a surface bounded by the closed contour ∂Σ(t),at all at time t, dA is an infinitesimal vector area element of Σ(t) (magnitude is the area of an infinitesimal patch ofsurface, direction is orthogonal to that surface patch).The sign of the EMF is determined by Lenz's law. Note that this is valid for not only a stationary wire — but also fora moving wire.From Faraday's law of induction (that is valid for a moving wire, for instance in a motor) and the MaxwellEquations, the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the Maxwell Equationscan be used to derive the Faraday Law.Let Σ(t) be the moving wire, moving together without rotation and with constant velocity v and Σ(t) be the internalsurface of the wire. The EMF around the closed path ∂Σ(t) is given by:[20]

where

is the electric field and dℓ is an infinitesimal vector element of the contour ∂Σ(t).

Lorentz force 79

NB: Both dℓ and dA have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in thearticle Kelvin-Stokes theorem.The above result can be compared with the version of Faraday's law of induction that appears in the modernMaxwell's equations, called here the Maxwell-Faraday equation:

The Maxwell-Faraday equation also can be written in an integral form using the Kelvin-Stokes theorem:.[21]

So we have, the Maxwell Faraday equation:

and the Faraday Law,

The two are equivalent if the wire is not moving. Using the Leibniz integral rule and that div B = 0, results in,

and using the Maxwell Faraday equation,

since this is valid for any wire position it implies that,

Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process ofdeformation, and it holds whether the magnetic field is constant in time or changing. However, there are cases whereFaraday's law is either inadequate or difficult to use, and application of the underlying Lorentz force law isnecessary. See inapplicability of Faraday's law.If the magnetic field is fixed in time and the conducting loop moves through the field, the magnetic flux ΦB linkingthe loop can change in several ways. For example, if the B-field varies with position, and the loop moves to alocation with different B-field, ΦB will change. Alternatively, if the loop changes orientation with respect to theB-field, the B • dA differential element will change because of the different angle between B and dA, also changingΦB. As a third example, if a portion of the circuit is swept through a uniform, time-independent B-field, and anotherportion of the circuit is held stationary, the flux linking the entire closed circuit can change due to the shift in relativeposition of the circuit's component parts with time (surface ∂Σ(t) time-dependent). In all three cases, Faraday's lawof induction then predicts the EMF generated by the change in ΦB.Note that the Maxwell Faraday's equation implies that the Electric Field E is non conservative when the MagneticField B varies in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theoremsince its rotational is not zero. See also.[20][22]

Lorentz force 80

Lorentz force in terms of potentialsThe E and B fields can be replaced by the magnetic vector potential A and (scalar) electrostatic potential ϕ by

where ∇ is the gradient, ∇• is the divergence, ∇ × is the curl.The force becomes

and using an identity for the triple product simplifies to

Lorentz force and Lagrangian mechanicsThe Lagrangian for a charged particle of mass m and charge q in an electromagnetic field equivalently describes thedynamics of the particle in terms of its energy, rather than the force exerted on it. The classical expression is givenby:[23]

where A and ϕ are the potential fields as above. Using Lagrange's equations, the equation for the Lorentz force canbe obtained.

Derivation of Lorentz force (SI units)

For an A field, a particle moving with velocity v = ṙ has potential momentum , so its potential energy is . For a ϕ field, theparticle's potential energy is .The total potential energy is then:

and the kinetic energy is:

hence the Lagrangian:

Lagrange's equations are

(same for y and z). So calculating the partial derivatives:

Lorentz force 81

equating and simplifying:

and similarly for the y and z directions. Hence the force equation is:

The potential energy depends on the velocity of the particle, so the force is velocity dependent, so it is notconservative.

Equation (cgs units)The above-mentioned formulae use SI units which are the most common among experimentalists, technicians, andengineers. In cgs-Gaussian units, which are somewhat more common among theoretical physicists, one has instead

where c is the speed of light. Although this equation looks slightly different, it is completely equivalent, since onehas the following relations:

where ε0 is the vacuum permittivity and μ0 the vacuum permeability. In practice, the subscripts "cgs" and "SI" arealways omitted, and the unit system has to be assessed from context.

Relativistic form of the Lorentz forceBecause the electric and magnetic fields are dependent on the velocity of an observer, the relativistic form of theLorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagneticand magnetic fields,[24] , and an arbitrary time-direction, , where

and

is a space-time plane (bivector), which has six degrees of freedom corresponding to boosts (rotations inspace-time planes) and rotations (rotations in space-space planes). The dot product with the vector pulls a vectorfrom the translational part, while the wedge-product creates a space-time trivector, whose dot product with thevolume element (the dual above) creates the magnetic field vector from the spatial rotation part. Only the parts of theabove two formulas perpendicular to gamma are relevant. The relativistic velocity is given by the (time-like) changesin a time-position vector , where

Lorentz force 82

(which shows our choice for the metric) and the velocity is

Then the Lorentz force law is simply (note that the order is important)

Covariant form of the Lorentz force

Field tensor

Using the metric signature (-1,1,1,1), The Lorentz force for a charge q can be written in covariant form:

where pα is the four-momentum, defined as:

the proper time of the particle, Fαβ the contravariant electromagnetic tensor

and U is the covariant 4-velocity of the particle, defined as:

where is the Lorentz factor defined above.The fields are transformed to a frame moving with constant relative velocity by:

where Λμα is the Lorentz transformation tensor.

Translation to vector notationThe α = 1 component (x-component) of the force is

Substituting the components of the covariant electromagnetic tensor F yields

Using the components of covariant four-velocity yields

The calculation for α = 2, 3 (force components in the y and z directions) yields similar results, so collecting the 3equations into one:

Lorentz force 83

which is the Lorentz force.

ApplicationsThe Lorentz force occurs in many devices, including:• Cyclotrons and other circular path particle accelerators• Mass spectrometers•• Velocity Filters• MagnetronsIn its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devicesincluding:

• Electric motors• Railguns• Linear motors• Loudspeakers

• Magnetoplasmadynamic thrusters• Electrical generators• Homopolar generators• Linear alternators

Footnotes[1] Oliver Heaviside By Paul J. Nahin, p120 (http:/ / books. google. com/ books?id=e9wEntQmA0IC& pg=PA120)[2] Huray, Paul G. (2009). Maxwell's Equations (http:/ / books. google. com/ books?id=0QsDgdd0MhMC& pg=PA22#v=onepage& q& f=false).

Wiley-IEEE. p. 22. ISBN 0-470-54276-4. .[3] See Jackson page 2. The book lists the four modern Maxwell's equations, and then states, "Also essential for consideration of charged particle

motion is the Lorentz force equation, F = q ( E+ v × B ), which gives the force acting on a point charge q in the presence of electromagneticfields."

[4][4] See Griffiths page 204.[5] For example, see the website of the "Lorentz Institute": \ (http:/ / ilorentz. org/ history/ lorentz/ lorentz. html), or Griffiths.[6] Griffiths, David J. (1999). Introduction to electrodynamics. reprint. with corr. (3rd ed.). Upper Saddle River, NJ [u.a.]: Prentice Hall.

ISBN 9780138053260.[7] Meyer, Herbert W. (1972). A History of Electricity and Magnetism. Norwalk, CT: Burndy Library. pp. 30–31. ISBN 0-262-13070-X.[8] Verschuur, Gerrit L. (1993). Hidden Attraction : The History And Mystery Of Magnetism. New York: Oxford University Press. pp. 78–79.

ISBN 0-19-506488-7.[9] Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 9, 25.

ISBN 0-19-850593-0[10] Verschuur, Gerrit L. (1993). Hidden Attraction : The History And Mystery Of Magnetism. New York: Oxford University Press. p. 76.

ISBN 0-19-506488-7.[11] Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 126–131, 139–144.

ISBN 0-19-850593-0[12] Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. pp. 200, 429–430.

ISBN 0-19-850593-0[13] Heaviside, Oliver. "On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric" (http:/ / en. wikisource. org/

wiki/ Motion_of_Electrification_through_a_Dielectric). Philosophical Magazine, April 1889, p. 324. .[14] Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford, [England]: Oxford University Press. p. 327.

ISBN 0-19-850593-0[15] Whittaker, E. T. (1910). A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth

Century (http:/ / books. google. com/ books?id=CGJDAAAAIAAJ& printsec=frontcover#v=onepage& q& f=false). Longmans, Green andCo.. pp. 420–423. ISBN 1-143-01208-9. .

[16][16] See Griffiths page 326, which states that Maxwell's equations, "together with the [Lorentz] force law...summarize the entire theoreticalcontent of classical electrodynamics".

[17][17] See, for example, Jackson p777-8.[18] See Griffiths pages 301–3.[19] Tai L. Chow (2006). Electromagnetic theory (http:/ / books. google. com/ ?id=dpnpMhw1zo8C& pg=PA153& dq=isbn=0763738271).

Sudbury MA: Jones and Bartlett. p. 395. ISBN 0-7637-3827-1. .[20] Landau, L. D., Lifshit︠s︡, E. M., & Pitaevskiĭ, L. P. (1984). Electrodynamics of continuous media; Volume 8 [[Course of Theoretical

Physics (http:/ / worldcat. org/ search?q=0750626348& qt=owc_search)]] (Second ed.). Oxford: Butterworth-Heinemann. p. §63 (§49 pp.205–207 in 1960 edition). ISBN 0-7506-2634-8. .

Lorentz force 84

[21] Roger F Harrington (2003). Introduction to electromagnetic engineering (http:/ / books. google. com/ ?id=ZlC2EV8zvX8C& pg=PA57&dq="faraday's+ law+ of+ induction"). Mineola, NY: Dover Publications. p. 56. ISBN 0-486-43241-6. .

[22] M N O Sadiku (2007). Elements of elctromagnetics (http:/ / books. google. com/ ?id=w2ITHQAACAAJ& dq=isbn:0-19-530048-3) (Fourthed.). NY/Oxford: Oxford University Press. p. 391. ISBN 0-19-530048-3. .

[23][23] Classical Mechanics (2nd Edition), T.W.B. Kibble, European Physics Series, Mc Graw Hill (UK), 1973, ISBN 07-084018-0.[24] Hestenes, David. "SpaceTime Calculus" (http:/ / geocalc. clas. asu. edu/ html/ STC. html). .

ReferencesThe numbered references refer in part to the list immediately below.• Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics (3

vol.). Pearson / Addison-Wesley. ISBN 0-8053-9047-2: volume 2.• Griffiths, David J. (1999). Introduction to electrodynamics (3rd ed.). Upper Saddle River, [NJ.]: Prentice-Hall.

ISBN 0-13-805326-X• Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 0-471-30932-X• Serway, Raymond A.; Jewett, John W., Jr. (2004). Physics for scientists and engineers, with modern physics.

Belmont, [CA.]: Thomson Brooks/Cole. ISBN 0-534-40846-X• Srednicki, Mark A. (2007). Quantum field theory (http:/ / books. google. com/ ?id=5OepxIG42B4C&

pg=PA315& dq=isbn=9780521864497). Cambridge, [England] ; New York [NY.]: Cambridge University Press.ISBN 978-0-521-86449-7

External links• Interactive Java tutorial on the Lorentz force (http:/ / www. magnet. fsu. edu/ education/ tutorials/ java/

lorentzforce/ index. html) National High Magnetic Field Laboratory• Lorentz force (demonstration) (http:/ / www. youtube. com/ watch?v=mxMMqNrm598)• Faraday's law: Tankersley and Mosca (http:/ / www. nadn. navy. mil/ Users/ physics/ tank/ Public/ FaradaysLaw.

pdf)• Notes from Physics and Astronomy HyperPhysics at Georgia State University (http:/ / hyperphysics. phy-astr.

gsu. edu/ HBASE/ hframe. html); see also home page (http:/ / hyperphysics. phy-astr. gsu. edu/ HBASE/ hframe.html)

• Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field (http:/ /chair. pa. msu. edu/ applets/ Lorentz/ a. htm) by Wolfgang Bauer

Magnet 85

Magnet

A "horseshoe magnet" made of alnico, an ironalloy. The magnet is made in the shape of a

horseshoe to bring the two magnetic poles closeto each other, to create a strong magnetic field

there that can pick up heavy pieces of iron

Iron filings that have oriented in the magneticfield produced by a bar magnet

Magnetic field lines of a solenoid electromagnet,which are similar to a bar magnet as illustrated

above with the iron filings

A magnet (from Greek μαγνήτις λίθος magnḗtis líthos, "Magnesianstone") is a material or object that produces a magnetic field. Thismagnetic field is invisible but is responsible for the most notableproperty of a magnet: a force that pulls on other ferromagneticmaterials, such as iron, and attracts or repels other magnets.

A permanent magnet is an object made from a material that ismagnetized and creates its own persistent magnetic field. An everydayexample is a refrigerator magnet used to hold notes on a refrigeratordoor. Materials that can be magnetized, which are also the ones that arestrongly attracted to a magnet, are called ferromagnetic (orferrimagnetic). These include iron, nickel, cobalt, some alloys of rareearth metals, and some naturally occurring minerals such as lodestone.Although ferromagnetic (and ferrimagnetic) materials are the only onesattracted to a magnet strongly enough to be commonly consideredmagnetic, all other substances respond weakly to a magnetic field, byone of several other types of magnetism.

Ferromagnetic materials can be divided into magnetically "soft"materials like annealed iron, which can be magnetized but do not tendto stay magnetized, and magnetically "hard" materials, which do.Permanent magnets are made from "hard" ferromagnetic materials suchas alnico and ferrite that are subjected to special processing in apowerful magnetic field during manufacture, to align their internalmicrocrystalline structure, making them very hard to demagnetize. Todemagnetize a saturated magnet, a certain magnetic field must beapplied, and this threshold depends on coercivity of the respectivematerial. "Hard" materials have high coercivity, whereas "soft"materials have low coercivity.

An electromagnet is made from a coil of wire that acts as a magnetwhen an electric current passes through it but stops being a magnetwhen the current stops. Often, the coil is wrapped around a core offerromagnetic material like steel, which enhances the magnetic fieldproduced by the coil.

The overall strength of a magnet is measured by its magnetic momentor, alternatively, the total magnetic flux it produces. The local strengthof magnetism in a material is measured by its magnetization.

Discovery and development

Ancient people learned about magnetism from lodestones, naturally magnetized pieces of iron ore. They arenaturally created magnets, which attract pieces of iron. The word magnet in Greek meant "stone from Magnesia", apart of ancient Greece where lodestones were found. Lodestones suspended so they could turn were the first

magnetic compasses. The earliest known surviving descriptions of magnets and their properties are from Greece, India, and China around 2500 years ago.[1][2][3] The properties of lodestones and their affinity for iron were written

Magnet 86

of by Pliny the Elder in his encyclopedia Naturalis Historia.[4]

By the 12th to 13th centuries AD, magnetic compasses were used in navigation in China, Europe, and elsewhere.[5]

Background on the physics of magnetism and magnets

An ovoid-shaped rare-earth magnet hanging fromanother

Magnetic field

The magnetic flux density (also called magnetic B field or justmagnetic field, usually denoted B) is a vector field. The magnetic Bfield vector at a given point in space is specified by two properties:

1. Its direction, which is along the orientation of a compass needle.2. Its magnitude (also called strength), which is proportional to how

strongly the compass needle orients along that direction.

In SI units, the strength of the magnetic B field is given in teslas.[6]

Magnetic moment

A magnet's magnetic moment (also called magnetic dipole moment andusually denoted μ) is a vector that characterizes the magnet's overallmagnetic properties. For a bar magnet, the direction of the magneticmoment points from the magnet's south pole to its north pole,[7] and themagnitude relates to how strong and how far apart these poles are. InSI units, the magnetic moment is specified in terms of A•m2.

A magnet both produces its own magnetic field and responds to magnetic fields. The strength of the magnetic field itproduces is at any given point proportional to the magnitude of its magnetic moment. In addition, when the magnet isput into an external magnetic field, produced by a different source, it is subject to a torque tending to orient themagnetic moment parallel to the field.[8] The amount of this torque is proportional both to the magnetic moment andthe external field. A magnet may also be subject to a force driving it in one direction or another, according to thepositions and orientations of the magnet and source. If the field is uniform in space, the magnet is subject to no netforce, although it is subject to a torque.[9]

A wire in the shape of a circle with area A and carrying current I is a magnet, with a magnetic moment of magnitudeequal to IA.

MagnetizationThe magnetization of a magnetized material is the local value of its magnetic moment per unit volume, usuallydenoted M, with units A/m.[10] It is a vector field, rather than just a vector (like the magnetic moment), becausedifferent areas in a magnet can be magnetized with different directions and strengths (for example, because ofdomains, see below). A good bar magnet may have a magnetic moment of magnitude 0.1 A•m2 and a volume of1 cm3, or 1×10−6 m3, and therefore an average magnetization magnitude is 100,000 A/m. Iron can have amagnetization of around a million amperes per meter. Such a large value explains why iron magnets are so effectiveat producing magnetic fields.

Magnet 87

Modelling magnets

Field of a cylindrical bar magnet calculated withAmpère's model

Two different models exist for magnets: magnetic poles and atomiccurrents.Although for many purposes it is convenient to think of a magnet ashaving distinct north and south magnetic poles, the concept of polesshould not be taken literally: it is merely a way of referring to the twodifferent ends of a magnet. The magnet does not have distinct north orsouth particles on opposing sides. If a bar magnet is broken into twopieces, in an attempt to separate the north and south poles, the resultwill be two bar magnets, each of which has both a north and southpole. However, a version of the magnetic-pole approach is used byprofessional magneticians to design permanent magnets. In thisapproach, the divergence of the magnetization ∇•M inside a magnet and the surface normal component M•n aretreated as a distribution of magnetic monopoles. This is a mathematical convenience and does not imply that thereare actually monopoles in the magnet. If the magnetic-pole distribution is known, then the pole model gives themagnetic field H. Outside the magnet, the field B is proportional to H, while inside the magnetization must be addedto H. An extension of this method that allows for internal magnetic charges is used in theories of ferromagnetism.

Another model is the Ampère model, where all magnetization is due to the effect of microscopic, or atomic, circularbound currents, also called Ampèrian currents, throughout the material. For a uniformly magnetized cylindrical barmagnet, the net effect of the microscopic bound currents is to make the magnet behave as if there is a macroscopicsheet of electric current flowing around the surface, with local flow direction normal to the cylinder axis.[11]

Microscopic currents in atoms inside the material are generally canceled by currents in neighboring atoms, so onlythe surface makes a net contribution; shaving off the outer layer of a magnet will not destroy its magnetic field, butwill leave a new surface of uncancelled currents from the circular currents throughout the material.[12] Theright-hand rule tells which direction the current flows.

Pole naming conventionsThe north pole of a magnet is the pole that, when the magnet is freely suspended, points towards the Earth's NorthMagnetic Pole which is located in northern Canada. Since opposite poles (north and south) attract, the Earth's "NorthMagnetic Pole" is thus actually the south pole of the Earth's magnetic field.[13][14][15][16] As a practical matter, inorder to tell which pole of a magnet is north and which is south, it is not necessary to use the Earth's magnetic field atall. For example, one method would be to compare it to an electromagnet, whose poles can be identified by theright-hand rule. The magnetic field lines of a magnet are considered by convention to emerge from the magnet'snorth pole and reenter at the south pole.[16]

Magnetic materialsThe term magnet is typically reserved for objects that produce their own persistent magnetic field even in theabsence of an applied magnetic field. Only certain classes of materials can do this. Most materials, however, producea magnetic field in response to an applied magnetic field; a phenomenon known as magnetism. There are severaltypes of magnetism, and all materials exhibit at least one of them.The overall magnetic behavior of a material can vary widely, depending on the structure of the material, particularlyon its electron configuration. Several forms of magnetic behavior have been observed in different materials,including:• Ferromagnetic and ferrimagnetic materials are the ones normally thought of as magnetic; they are attracted to a

magnet strongly enough that the attraction can be felt. These materials are the only ones that can retain

Magnet 88

magnetization and become magnets; a common example is a traditional refrigerator magnet. Ferrimagneticmaterials, which include ferrites and the oldest magnetic materials magnetite and lodestone, are similar to butweaker than ferromagnetics. The difference between ferro- and ferrimagnetic materials is related to theirmicroscopic structure, as explained in Magnetism.

• Paramagnetic substances, such as platinum, aluminum, and oxygen, are weakly attracted to either pole of amagnet. This attraction is hundreds of thousands of times weaker than that of ferromagnetic materials, so it canonly be detected by using sensitive instruments or using extremely strong magnets. Magnetic ferrofluids, althoughthey are made of tiny ferromagnetic particles suspended in liquid, are sometimes considered paramagnetic sincethey cannot be magnetized.

• Diamagnetic means repelled by both poles. Compared to paramagnetic and ferromagnetic substances,diamagnetic substances, such as carbon, copper, water, and plastic, are even more weakly repelled by a magnet.The permeability of diamagnetic materials is less than the permeability of a vacuum. All substances notpossessing one of the other types of magnetism are diamagnetic; this includes most substances. Although force ona diamagnetic object from an ordinary magnet is far too weak to be felt, using extremely strong superconductingmagnets, diamagnetic objects such as pieces of lead and even mice[17] can be levitated, so they float in mid-air.Superconductors repel magnetic fields from their interior and are strongly diamagnetic.

There are various other types of magnetism, such as spin glass, superparamagnetism, superdiamagnetism, andmetamagnetism.

Common uses of magnets

Hard disk drives record data on a thin magneticcoating

Magnetic hand separator for heavy minerals

• Magnetic recording media: VHS tapes contain a reel of magnetictape. The information that makes up the video and sound is encodedon the magnetic coating on the tape. Common audio cassettes alsorely on magnetic tape. Similarly, in computers, floppy disks andhard disks record data on a thin magnetic coating.[18]

• Credit, debit, and ATM cards: All of these cards have a magneticstrip on one side. This strip encodes the information to contact anindividual's financial institution and connect with theiraccount(s).[19]

• Common televisions and computer monitors: TV and computerscreens containing a cathode ray tube employ an electromagnet toguide electrons to the screen.[20] Plasma screens and LCDs usedifferent technologies.

• Speakers and microphones: Most speakers employ a permanentmagnet and a current-carrying coil to convert electric energy (thesignal) into mechanical energy (movement that creates the sound).The coil is wrapped around a bobbin attached to the speaker coneand carries the signal as changing current that interacts with thefield of the permanent magnet. The voice coil feels a magnetic forceand in response, moves the cone and pressurizes the neighboring air,thus generating sound. Dynamic microphones employ the sameconcept, but in reverse. A microphone has a diaphragm ormembrane attached to a coil of wire. The coil rests inside a speciallyshaped magnet. When sound vibrates the membrane, the coil is vibrated as well. As the coil moves through themagnetic field, a voltage is induced across the coil. This voltage drives a current in the wire that is characteristicof the original sound.

Magnet 89

• Electric guitars use magnetic pickups to transduce the vibration of guitar strings into electric current that can thenbe amplified. This is different from the principle behind the speaker and dynamic microphone because thevibrations are sensed directly by the magnet, and a diaphragm is not employed. The Hammond organ used asimilar principle, with rotating tonewheels instead of strings.

• Electric motors and generators: Some electric motors rely upon a combination of an electromagnet and apermanent magnet, and, much like loudspeakers, they convert electric energy into mechanical energy. A generatoris the reverse: it converts mechanical energy into electric energy by moving a conductor through a magnetic field.

• Medicine: Hospitals use magnetic resonance imaging to spot problems in a patient's organs without invasivesurgery.

• Chucks are used in the metalworking field to hold objects. Magnets are also used in other types of fasteningdevices, such as the magnetic base, the magnetic clamp and the refrigerator magnet.

• Compasses: A compass (or mariner's compass) is a magnetized pointer free to align itself with a magnetic field,most commonly Earth's magnetic field.

• Art: Vinyl magnet sheets may be attached to paintings, photographs, and other ornamental articles, allowing themto be attached to refrigerators and other metal surfaces. Objects and paint can be applied directly to the magnetsurface to create collage pieces of art. Magnetic art is portable, inexpensive and easy to create. Vinyl magnetic artis not for the refrigerator anymore. Colorful metal magnetic boards, strips, doors, microwave ovens, dishwashers,cars, metal I beams, and any metal surface can be receptive of magnetic vinyl art. Being a relatively new mediafor art, the creative uses for this material is just beginning.

• Science projects: Many topic questions are based on magnets. For example: how is the strength of a magnetaffected by glass, plastic, and cardboard?

Magnets have many uses in toys. M-tic usesmagnetic rods connected to metal spheres for

construction. Note the geodesic pyramid

• Toys: Given their ability to counteract the force of gravity at closerange, magnets are often employed in children's toys, such as theMagnet Space Wheel and Levitron, to amusing effect.

•• Magnets can be used to make jewelry. Necklaces and bracelets canhave a magnetic clasp, or may be constructed entirely from a linkedseries of magnets and ferrous beads.

•• Magnets can pick up magnetic items (iron nails, staples, tacks,paper clips) that are either too small, too hard to reach, or too thinfor fingers to hold. Some screwdrivers are magnetized for thispurpose.

•• Magnets can be used in scrap and salvage operations to separatemagnetic metals (iron, cobalt, and nickel) from non-magnetic metals(aluminum, non-ferrous alloys, etc.). The same idea can be used inthe so-called "magnet test", in which an auto body is inspected witha magnet to detect areas repaired using fiberglass or plastic putty.

• Magnetic levitation transport, or maglev, is a form of transportation that suspends, guides and propels vehicles(especially trains) through electromagnetic force. The maximum recorded speed of a maglev train is 581kilometers per hour (unknown operator: u'strong' mph).

• Magnets may be used to serve as a fail-safe device for some cable connections. For example, the power cords ofsome laptops are magnetic to prevent accidental damage to the port when tripped over. The MagSafe powerconnection to the Apple MacBook is one such example.

Magnet 90

Medical issues and safetyBecause human tissues have a very low level of susceptibility to static magnetic fields, there is little mainstreamscientific evidence showing a health hazard associated with exposure to static fields. Dynamic magnetic fields maybe a different issue, however; correlations between electromagnetic radiation and cancer rates have been postulateddue to demographic correlations (see Electromagnetic radiation and health).If a ferromagnetic foreign body is present in human tissue, an external magnetic field interacting with it can pose aserious safety risk.[21]

A different type of indirect magnetic health risk exists involving pacemakers. If a pacemaker has been embedded in apatient's chest (usually for the purpose of monitoring and regulating the heart for steady electrically induced beats),care should be taken to keep it away from magnetic fields. It is for this reason that a patient with the device installedcannot be tested with the use of an MRI, which is a magnetic imaging device.Children sometimes swallow small magnets from toys, and this can be hazardous if two or more magnets areswallowed, as the magnets can pinch or puncture internal tissues; one death has been reported.[22]

Magnetizing ferromagnetsFerromagnetic materials can be magnetized in the following ways:• Heating the object above its Curie temperature, allowing it to cool in a magnetic field and hammering it as it

cools. This is the most effective method and is similar to the industrial processes used to create permanentmagnets.

• Placing the item in an external magnetic field will result in the item retaining some of the magnetism on removal.Vibration has been shown to increase the effect. Ferrous materials aligned with the Earth's magnetic field that aresubject to vibration (e.g., frame of a conveyor) have been shown to acquire significant residual magnetism.

•• Stroking: An existing magnet is moved from one end of the item to the other repeatedly in the same direction.

Demagnetizing ferromagnetsMagnetized ferromagnetic materials can be demagnetized (or degaussed) in the following ways:• Heating a magnet past its Curie temperature; the molecular motion destroys the alignment of the magnetic

domains. This always removes all magnetization.• Placing the magnet in an alternating magnetic field with intensity above the material's coercivity and then either

slowly drawing the magnet out or slowly decreasing the magnetic field to zero. This is the principle used incommercial demagnetizers to demagnetize tools and erase credit cards and hard disks and degaussing coils usedto demagnetize CRTs.

• Some demagnetization or reverse magnetization will occur if any part of the magnet is subjected to a reverse fieldabove the magnetic material's coercivity.

•• Demagnetisation progressively occurs if the magnet is subjected to cyclic fields sufficient to move the magnetaway from the linear part on the second quadrant of the B-H curve of the magnetic material (the demagnetisationcurve).

•• Hammering or jarring: the mechanical disturbance tends to randomize the magnetic domains. This will leavesome residual magnetization.

Magnet 91

Types of permanent magnets

A stack of ferrite magnets

Magnetic metallic elements

Many materials have unpaired electron spins, and the majority of thesematerials are paramagnetic. When the spins interact with each other insuch a way that the spins align spontaneously, the materials are calledferromagnetic (what is often loosely termed as magnetic). Because ofthe way their regular crystalline atomic structure causes their spins tointeract, some metals are ferromagnetic when found in their naturalstates, as ores. These include iron ore (magnetite or lodestone), cobaltand nickel, as well as the rare earth metals gadolinium and dysprosium(when at a very low temperature). Such naturally occurringferromagnets were used in the first experiments with magnetism.Technology has since expanded the availability of magnetic materialsto include various man-made products, all based, however, on naturally magnetic elements.

CompositesCeramic, or ferrite, magnets are made of a sintered composite of powdered iron oxide and barium/strontiumcarbonate ceramic. Given the low cost of the materials and manufacturing methods, inexpensive magnets (ornon-magnetized ferromagnetic cores, for use in electronic components such as radio antennas, for example) ofvarious shapes can be easily mass-produced. The resulting magnets are non-corroding but brittle and must be treatedlike other ceramics.Alnico magnets are made by casting or sintering a combination of aluminium, nickel and cobalt with iron and smallamounts of other elements added to enhance the properties of the magnet. Sintering offers superior mechanicalcharacteristics, whereas casting delivers higher magnetic fields and allows for the design of intricate shapes. Alnicomagnets resist corrosion and have physical properties more forgiving than ferrite, but not quite as desirable as ametal. Trade names for alloys in this family include: Alni, Alcomax, Hycomax, Columax, and Ticonal.[23]

Injection-molded magnets are a composite of various types of resin and magnetic powders, allowing parts ofcomplex shapes to be manufactured by injection molding. The physical and magnetic properties of the productdepend on the raw materials, but are generally lower in magnetic strength and resemble plastics in their physicalproperties.Flexible magnets are similar to injection-molded magnets, using a flexible resin or binder such as vinyl, andproduced in flat strips, shapes or sheets. These magnets are lower in magnetic strength but can be very flexible,depending on the binder used. Flexible magnets can be used in industrial printers.

Magnet 92

Rare-earth magnetsRare earth (lanthanoid) elements have a partially occupied f electron shell (which can accommodate up to 14electrons). The spin of these electrons can be aligned, resulting in very strong magnetic fields, and therefore, theseelements are used in compact high-strength magnets where their higher price is not a concern. The most commontypes of rare-earth magnets are samarium-cobalt and neodymium-iron-boron (NIB) magnets.

Single-molecule magnets (SMMs) and single-chain magnets (SCMs)In the 1990s, it was discovered that certain molecules containing paramagnetic metal ions are capable of storing amagnetic moment at very low temperatures. These are very different from conventional magnets that storeinformation at a magnetic domain level and theoretically could provide a far denser storage medium thanconventional magnets. In this direction, research on monolayers of SMMs is currently under way. Very briefly, thetwo main attributes of an SMM are:1. a large ground state spin value (S), which is provided by ferromagnetic or ferrimagnetic coupling between the

paramagnetic metal centres2. a negative value of the anisotropy of the zero field splitting (D)Most SMMs contain manganese but can also be found with vanadium, iron, nickel and cobalt clusters. Morerecently, it has been found that some chain systems can also display a magnetization that persists for long times athigher temperatures. These systems have been called single-chain magnets.

Nano-structured magnetsSome nano-structured materials exhibit energy waves, called magnons, that coalesce into a common ground state inthe manner of a Bose-Einstein condensate.[24][25]

CostsThe current cheapest permanent magnets, allowing for field strengths, are flexible and ceramic magnets, but theseare also among the weakest types. The ferrite magnets are mainly low-cost magnets since they are made from cheapraw materials- iron oxide and Ba- or Sr-carbonate. However, a new low cost magnet- Mn-Al alloy has beendeveloped and is now dominating the low-cost magnets field. It has a higher saturation magnetization than the ferritemagnets. It also has more favorable temperature coefficients, although it can be thermally unstable.Neodymium-iron-boron (NIB) magnets are among the strongest. These cost more per kilogram than most othermagnetic materials but, owing to their intense field, are smaller and cheaper in many applications.[26]

TemperatureTemperature sensitivity varies, but when a magnet is heated to a temperature known as the Curie point, it loses all ofits magnetism, even after cooling below that temperature. The magnets can often be remagnetized, however.Additionally, some magnets are brittle and can fracture at high temperatures.The maximum usable temperature is highest for alnico magnets at over 540 °C (unknown operator: u'strong' °F),around 300 °C (unknown operator: u'strong' °F) for ferrite and SmCo, about 140 °C (unknown operator:u'strong' °F) for NIB and lower for flexible ceramics, but the exact numbers depend on the grade of material.

Magnet 93

ElectromagnetsAn electromagnet, in its simplest form, is a wire that has been coiled into one or more loops, known as a solenoid.When electric current flows through the wire, a magnetic field is generated. It is concentrated near (and especiallyinside) the coil, and its field lines are very similar to those of a magnet. The orientation of this effective magnet isdetermined by the right hand rule. The magnetic moment and the magnetic field of the electromagnet areproportional to the number of loops of wire, to the cross-section of each loop, and to the current passing through thewire.[27]

If the coil of wire is wrapped around a material with no special magnetic properties (e.g., cardboard), it will tend togenerate a very weak field. However, if it is wrapped around a soft ferromagnetic material, such as an iron nail, thenthe net field produced can result in a several hundred- to thousandfold increase of field strength.Uses for electromagnets include particle accelerators, electric motors, junkyard cranes, and magnetic resonanceimaging machines. Some applications involve configurations more than a simple magnetic dipole; for example,quadrupole and sextupole magnets are used to focus particle beams.

Units and calculationsFor most engineering applications, MKS (rationalized) or SI (Système International) units are commonly used. Twoother sets of units, Gaussian and CGS-EMU, are the same for magnetic properties and are commonly used inphysics.In all units, it is convenient to employ two types of magnetic field, B and H, as well as the magnetization M, definedas the magnetic moment per unit volume.1. The magnetic induction field B is given in SI units of teslas (T). B is the magnetic field whose time variation

produces, by Faraday's Law, circulating electric fields (which the power companies sell). B also produces adeflection force on moving charged particles (as in TV tubes). The tesla is equivalent to the magnetic flux (inwebers) per unit area (in meters squared), thus giving B the unit of a flux density. In CGS, the unit of B is thegauss (G). One tesla equals 104 G.

2. The magnetic field H is given in SI units of ampere-turns per meter (A-turn/m). The turns appears because whenH is produced by a current-carrying wire, its value is proportional to the number of turns of that wire. In CGS, theunit of H is the oersted (Oe). One A-turn/m equals 4π×10−3 Oe.

3. The magnetization M is given in SI units of amperes per meter (A/m). In CGS, the unit of M is the oersted (Oe).One A/m equals 10−3 emu/cm3. A good permanent magnet can have a magnetization as large as a millionamperes per meter.

4. In SI units, the relation B = μ0(H + M) holds, where μ0 is the permeability of space, which equals4π×10−7 T•m/A. In CGS, it is written as B = H + 4πM. (The pole approach gives μ0H in SI units. A μ0M term inSI must then supplement this μ0H to give the correct field within B, the magnet. It will agree with the field Bcalculated using Ampèrian currents]

Materials that are not permanent magnets usually satisfy the relation M = χH in SI, where χ is the (dimensionless)magnetic susceptibility. Most non-magnetic materials have a relatively small χ (on the order of a millionth), but softmagnets can have χ on the order of hundreds or thousands. For materials satisfying M = χH, we can also write B =μ0(1 + χ)H = μ0μrH = μH, where μr = 1 + χ is the (dimensionless) relative permeability and μ =μ0μr is the magneticpermeability. Both hard and soft magnets have a more complex, history-dependent, behavior described by what arecalled hysteresis loops, which give either B vs. H or M vs. H. In CGS, M = χH, but χSI = 4πχCGS, and μ = μr.Caution: in part because there are not enough Roman and Greek symbols, there is no commonly agreed-upon symbol for magnetic pole strength and magnetic moment. The symbol m has been used for both pole strength (unit A•m, where here the upright m is for meter) and for magnetic moment (unit A•m2). The symbol μ has been used in some texts for magnetic permeability and in other texts for magnetic moment. We will use μ for magnetic permeability and

Magnet 94

m for magnetic moment. For pole strength, we will employ qm. For a bar magnet of cross-section A with uniformmagnetization M along its axis, the pole strength is given by qm = MA, so that M can be thought of as a pole strengthper unit area.

Fields of a magnetFar away from a magnet, the magnetic field created by that magnet is almost always described (to a goodapproximation) by a dipole field characterized by its total magnetic moment. This is true regardless of the shape ofthe magnet, so long as the magnetic moment is non-zero. One characteristic of a dipole field is that the strength ofthe field falls off inversely with the cube of the distance from the magnet's center.Closer to the magnet, the magnetic field becomes more complicated and more dependent on the detailed shape andmagnetization of the magnet. Formally, the field can be expressed as a multipole expansion: A dipole field, plus aquadrupole field, plus an octupole field, etc.At close range, many different fields are possible. For example, for a long, skinny bar magnet with its north pole atone end and south pole at the other, the magnetic field near either end falls off inversely with the square of thedistance from that pole.

Calculating the magnetic force

Force between two magnetic poles

Further information: Magnetic moment#Forces between two magnetic dipolesClassically, the force between two magnetic poles is given by:[28]

whereF is force (SI unit: newton)qm1 and qm2 are the magnitudes of magnetic poles (SI unit: ampere-meter)μ is the permeability of the intervening medium (SI unit: tesla meter per ampere, henry per meter or newtonper ampere squared)r is the separation (SI unit: meter).

The pole description is useful to the engineers designing real-world magnets, but real magnets have a poledistribution more complex than a single north and south. Therefore, implementation of the pole idea is not simple. Insome cases, one of the more complex formulae given below will be more useful.

Force between two nearby magnetized surfaces of area A

The mechanical force between two nearby magnetized surfaces can be calculated with the following equation. Theequation is valid only for cases in which the effect of fringing is negligible and the volume of the air gap is muchsmaller than that of the magnetized material:[29][30]

where:A is the area of each surface, in m2

H is their magnetizing field, in A/mμ0 is the permeability of space, which equals 4π×10−7 T•m/AB is the flux density, in T.

Magnet 95

Force between two bar magnets

The force between two identical cylindrical bar magnets placed end to end is given by:[29]

where:B0 is the magnetic flux density very close to each pole, in T,A is the area of each pole, in m2,L is the length of each magnet, in m,R is the radius of each magnet, in m, andx is the separation between the two magnets, in m.

relates the flux density at the pole to the magnetization of the magnet.

Note that all these formulations are based on Gilbert's model, which is usable in relatively great distances. In othermodels (e.g., Ampère's model), a more complicated formulation is used that sometimes cannot be solvedanalytically. In these cases, numerical methods must be used.

Force between two cylindrical magnets

For two cylindrical magnets with radius and height , with their magnetic dipole aligned, the force can be wellapproximated (even at distances of the order of ) by,[31]

where is the magnetization of the magnets and is the gap between the magnets. In disagreement to thestatement in the previous section, a measurement of the magnetic flux density very close to the magnet is relatedto by the formula

The effective magnetic dipole can be written as

Where is the volume of the magnet. For a cylinder, this is .When , the point dipole approximation is obtained,

which matches the expression of the force between two magnetic dipoles.

Notes[1] Fowler, Michael (1997). "Historical Beginnings of Theories of Electricity and Magnetism" (http:/ / galileoandeinstein. physics. virginia. edu/

more_stuff/ E& M_Hist. html). . Retrieved 2008-04-02.[2] Vowles, Hugh P. (1932). "Early Evolution of Power Engineering". Isis 17 (2): 412–420 [419–20]. doi:10.1086/346662.[3] Li Shu-hua (1954). "Origine de la Boussole II. Aimant et Boussole". Isis 45 (2): 175. JSTOR 227361.[4] Pliny the Elder, The Natural History, BOOK XXXIV. THE NATURAL HISTORY OF METALS., CHAP. 42.—THE METAL CALLED

LIVE IRON (http:/ / www. perseus. tufts. edu/ hopper/ text?doc=Perseus:text:1999. 02. 0137:book=34:chapter=42& highlight=magnet).Perseus.tufts.edu. Retrieved on 2011-05-17.

[5] Schmidl, Petra G. (1996–1997). "Two Early Arabic Sources On The Magnetic Compass" (http:/ / www. lancs. ac. uk/ jais/ volume/ docs/vol1/ 1_081-132schmidl2. pdf). Journal of Arabic and Islamic Studies 1: 81–132. .

[6] Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 255–8. ISBN 0-13-805326-X. OCLC 40251748.[7] Knight, Jones, & Field, "College Physics" (2007) p. 815

Magnet 96

[8] B. D. Cullity, C. D. Graham (2008). Introduction to Magnetic Materials (http:/ / books. google. com/ ?id=ixAe4qIGEmwC& pg=PA103) (2ed.). Wiley-IEEE Press. p. 103. ISBN 0-471-47741-9. .

[9] Boyer, Timothy H. (1988). "The Force on a Magnetic Dipole". American Journal of Physics 56 (8): 688–692.Bibcode 1988AmJPh..56..688B. doi:10.1119/1.15501.

[10] "Units for Magnetic Properties" (http:/ / www. magneticmicrosphere. com/ resources/ Units_for_Magnetic_Properties. pdf). Lake ShoreCryotronics, Inc.. . Retrieved 2009-10-24.

[11] Zachariah Allen (1852). Philosophy of the Mechanics of Nature, and the Source and Modes of Action of Natural Motive-Power (http:/ /books. google. com/ books?id=EpUIAAAAIAAJ& pg=PA252). D. Appleton and Company. p. 252. .

[12] Wayne M. Saslow (2002). Electricity, Magnetism, and Light (http:/ / books. google. com/ books?id=4liwlxqt9NIC& pg=PA426) (3rd ed.).Academic Press. p. 426. ISBN 978-0-12-619455-5. .

[13] Serway, Raymond A.; Chris Vuille (2006). Essentials of college physics (http:/ / books. google. com/ books?id=8n4NCyRgUMEC&pg=PA493). USA: Cengage Learning. p. 493. ISBN 0-495-10619-4. .

[14] Emiliani, Cesare (1992). Planet Earth: Cosmology, Geology, and the Evolution of Life and Environment (http:/ / books. google. com/books?id=MfAGpVq8gpQC& pg=PA228). UK: Cambridge University Press. p. 228. ISBN 0-521-40949-7. .

[15] Manners, Joy (2000). Static Fields and Potentials (http:/ / books. google. com/ books?id=vJyqbRPsXYQC& pg=PA148). USA: CRC Press.p. 148. ISBN 0-7503-0718-8. .

[16] Nave, Carl R. (2010). "Bar Magnet" (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ hframe. html). Hyperphysics. Dept. of Physics andAstronomy, Georgia State Univ.. . Retrieved 2011-04-10.

[17] Mice levitated in NASA lab (http:/ / www. livescience. com/ animals/ 090909-mouse-levitation. html). Livescience.com (2009-09-09).Retrieved on 2011-10-08.

[18] Mallinson, John C. (1987). The foundations of magnetic recording (2nd ed.). Academic Press. ISBN 0-12-466626-4.[19] "The stripe on a credit card" (http:/ / money. howstuffworks. com/ personal-finance/ debt-management/ credit-card2. htm). How Stuff Works.

. Retrieved July 2011.[20] "Electromagnetic deflection in a cathode ray tube, I" (http:/ / www. magnet. fsu. edu/ education/ tutorials/ java/ cathoderaytube/ index.

html). National High Magnetic Field Laboratory. . Retrieved July 2011.[21] Schenck JF (2000). "Safety of strong, static magnetic fields". J Magn Reson Imaging 12 (1): 2–19.

doi:10.1002/1522-2586(200007)12:1<2::AID-JMRI2>3.0.CO;2-V. PMID 10931560.[22] Oestreich AE (2008). "Worldwide survey of damage from swallowing multiple magnets". Pediatr Radiol 39 (2): 142.

doi:10.1007/s00247-008-1059-7. PMID 19020871.[23] Brady, George Stuart; Henry R. Clauser & John A. Vaccari (2002). Materials Handbook: An Encyclopedia for Managers (http:/ / books.

google. com/ books?id=vIhvSQLhhMEC& pg=PA577). McGraw-Hill Professional. p. 577. ISBN 0-07-136076-X. .[24] "Nanomagnets Bend The Rules" (http:/ / www. spacedaily. com/ news/ nanotech-05zm. html). . Retrieved November 14, 2005.[25] Della Torre, E.; Bennett, L.; Watson, R. (2005). "Extension of the Bloch T3/2 Law to Magnetic Nanostructures: Bose-Einstein

Condensation". Physical Review Letters 94 (14): 147210. Bibcode 2005PhRvL..94n7210D. doi:10.1103/PhysRevLett.94.147210.[26] Frequently Asked Questions (http:/ / www. magnetsales. com/ Design/ FAQs_frames/ FAQs_3. htm#howrated). Magnet sales. Retrieved on

2011-10-08.[27] Ruskell, Todd; Tipler, Paul A. ; Mosca, Gene (2007). Physics for Scientists and Engineers (6 ed.). Macmillan. ISBN 1-4292-0410-9.[28] "Basic Relationships" (http:/ / geophysics. ou. edu/ solid_earth/ notes/ mag_basic/ mag_basic. html). Geophysics.ou.edu. . Retrieved

2009-10-19.[29] "Magnetic Fields and Forces" (http:/ / instruct. tri-c. edu/ fgram/ web/ Mdipole. htm). . Retrieved 2009-12-24.[30] "The force produced by a magnetic field" (http:/ / info. ee. surrey. ac. uk/ Workshop/ advice/ coils/ force. html). . Retrieved 2010-03-09.[31] David Vokoun, Marco Beleggia, Ludek Heller, Petr Sittner (2009). "Magnetostatic interactions and forces between cylindrical permanent

magnets". Journal of Magnetism and Magnetic Materials 321 (22): 3758–3763. Bibcode 2009JMMM..321.3758V.doi:10.1016/j.jmmm.2009.07.030.

References• "positive pole n". The Concise Oxford English Dictionary. Catherine Soanes and Angus Stevenson. Oxford

University Press, 2004. Oxford Reference Online. Oxford University Press.• Wayne M. Saslow, Electricity, Magnetism, and Light, Academic (2002). ISBN 0-12-619455-6. Chapter 9

discusses magnets and their magnetic fields using the concept of magnetic poles, but it also gives evidence thatmagnetic poles do not really exist in ordinary matter. Chapters 10 and 11, following what appears to be a19th-century approach, use the pole concept to obtain the laws describing the magnetism of electric currents.

• Edward P. Furlani, Permanent Magnet and Electromechanical Devices:Materials, Analysis and Applications,Academic Press Series in Electromagnetism (2001). ISBN 0-12-269951-3.

Magnet 97

External links• HyperPhysics E/M (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ hframe. html), good complete tree diagram of

electromagnetic relationships with magnets• Maxwell's Equations and some history• Detailed Theory on Designing a Solenoid (http:/ / www. coilgun. info) or a coil gun• Video: The physicist Richard Feynman answers the question, Why do bar magnets attract or repel each other?

(http:/ / www. youtube. com/ watch?v=wMFPe-DwULM)• Articles, tutorials and other educational information about magnets (http:/ / www. magnet. fsu. edu/ education/

tutorials/ electricitymagnetism. html) National High Magnetic Field Laboratory• Answers to several questions from curious kids about magnets (http:/ / static. scribd. com/ docs/ ghnvi6g2fepvm.

swf)• Magnetic units discussed (http:/ / www. magnets. bham. ac. uk/ magneticmaterials/ units. shtml)• EU requires warning on toys containing magnets (http:/ / newsletter. sgs. com/ eNewsletterPro/ uploadedimages/

000006/ SafeGuardS_03608_EU_requires_warning_on_toys_containing_magnets_v2. pdf)• Information on Permanent Magnets (http:/ / www. stanfordmagnets. com/ magnet. html#ref)• About Magnets (http:/ / www. thomasnet. com/ about/ magnets-49490402. html)• International Magnetics Association (http:/ / www. intl-magnetics. org/ )• Online magnetic pull force calculator (http:/ / www. kjmagnetics. com/ calculator. asp)• Magnet (How Products Are Made Volume 2) (http:/ / www. madehow. com/ Volume-2/ Magnet. html)• Why are all metals not attracted to a magnet? (http:/ / wiki. answers. com/ Q/

Why_are_all_metals_not_attracted_to_a_magnet)

Magnetic bearing

A magnetic bearing

A magnetic bearing is a bearing which supports a load using magneticlevitation. Magnetic bearings support moving machinery withoutphysical contact; for example, they can levitate a rotating shaft andpermit relative motion with very low friction and no mechanical wear.Magnetic bearings are in service in such industrial applications aselectric power generation, petroleum refining, machine tool operation,and natural gas pipelines. They are also used in the Zippe-typecentrifuge[1] used for uranium enrichment. Magnetic bearings are usedin turbomolecular pumps, where oil-lubricated bearings would be asource of contamination. Magnetic bearings support the highest speedsof any kind of bearing; they have no known maximum relative speed.

DescriptionIt is difficult to build a magnetic bearing using permanent magnets due to the limitations described by Earnshaw'stheorem, and techniques using diamagnetic materials are relatively undeveloped. As a result, most magnetic bearingsrequire continuous power input and an active control system to hold the load stable. Many bearings can usepermanent magnets to carry the static load, and then only use power when the levitated object deviates from itsoptimum position. Magnetic bearings also typically require some kind of back-up bearing in case of power or controlsystem failure and during initial start-up conditions.Two sorts of instabilities are very typically present with magnetic bearings. Firstly, attractive magnets give an unstable static force that decreases with greater distance and increases at close distances. Secondly since magnetism

Magnetic bearing 98

is a conservative force, in and of itself it gives little if any damping, and oscillations may cause loss of successfulsuspension if any driving forces are present, which they very typically are.With the use of an induction-based levitation system present in maglev technologies such as the Inductrack system,magnetic bearings could do away with complex control systems by using Halbach Arrays and simple closed loopcoils. These systems gain in simplicity, but are less advantageous when it comes to eddy current losses. For rotatingsystems it is possible to use homopolar magnet designs instead of multipole Halbach structures, which reduces lossesconsiderably. An example of this - that has bypassed the Earnshaw's theorem issues - is the homopolarelectrodynamic bearings invented by Dr Torbjörn Lembke.[2][3][4]

Active magnetic bearing

Basic operation for a single axis

An active magnetic bearing (AMB) works on the principle ofelectromagnetic suspension and consists of an electromagnet assembly,a set of power amplifiers which supply current to the electromagnets, acontroller, and gap sensors with associated electronics to provide thefeedback required to control the position of the rotor within the gap.These elements are shown in the diagram. The power amplifiers supplyequal bias current to two pairs of electromagnets on opposite sides of arotor. This constant tug-of-war is mediated by the controller whichoffsets the bias current by equal but opposite perturbations of currentas the rotor deviates by a small amount from its center position.

The gap sensors are usually inductive in nature and sense in adifferential mode. The power amplifiers in a modern commercial application are solid state devices which operate ina pulse width modulation (PWM) configuration. The controller is usually a microprocessor or DSP.

Active bearings have several advantages, they do not suffer from wear, they have low friction, and they can oftenaccommodate irregularities in the mass distribution automatically, allowing it to spin around its centre of mass withvery low vibration.

HistoryThe evolution of active magnetic bearings may be traced through the patents issued in this field. The table belowlists several early patents for active magnetic bearings. Earlier patents for magnetic suspensions can be found but areexcluded here because they consist of assemblies of permanent magnets of problematic stability per Earnshaw'sTheorem.Early active magnetic bearing patents were assigned to Jesse Beams[5][6] at the University of Virginia during WorldWar II and are concerned with ultracentrifuges for purification of the isotopes of various elements for themanufacture of the first nuclear bombs, but the technology did not mature until the advances of solid-stateelectronics and modern computer-based control technology with the work of Habermann[7] and Schweitzer.[8]

Extensive modern work in magnetic bearings has continued at the University of Virginia in the Rotating Machineryand Controls Industrial Research Program. The first international symposium for active magnetic bearing technologywas held in 1988 with the founding of the International Society of Magnetic Bearings by Prof. Schweitzer (ETHZ),Prof. Allaire (University of Virginia), and Prof. Okada (Ibaraki University).In 1987 further improved AMB designs were created in Australia by E.Croot [9] (see reference below as well) butthese designs were not manufactured due to expensive costs of production. However, some of those designs havesince been used by Japanese electronics companies, they remain a specialty item: where extremely high RPM isrequired.

Magnetic bearing 99

Since then there have been nine succeeding symposia. Kasarda[10] reviews the history of AMB in depth. She notesthat the first commercial application of AMB’s was with turbomachinery. The AMB allowed the elimination of oilreservoirs on compressors for the NOVA Gas Transmission Ltd. (NGTL) gas pipelines in Alberta, Canada. Thisreduced the fire hazard allowing a substantial reduction in insurance costs. The success of these magnetic bearinginstallations led NGTL to pioneer the research and development of a digital magnetic bearing control system as areplacement for the analog control systems supplied by the American company Magnetic Bearings Inc. (MBI). In1992, NGTL's magnetic bearing research group formed the company Revolve Technologies Inc [11]. tocommercialize the digital magnetic bearing technology. This firm was later purchased by SKF of Sweden. TheFrench company S2M, founded in 1976, was the first to commercially market AMB’s. Extensive research onmagnetic bearings continues at the University of Virginia in the Rotating Machinery and Controls IndustrialResearch Program [12].Starting from 1996 the Dutch oil and gas company NAM installed over a period of 10 years 20 large E-motor driven(with variable speed drive) gas compressors of 23 MW fully equipped with AMB's on both the E-motor and thecompressor. These compressors are used in the Groningen gas field to deplete the remaining gas from this large gasfield and to increase the field capacity. The motor - compressor design is done by Siemens and the AMB aredelivered by Waukesha (owned by Dover). (Originally these bearings were designed by Glacier, this company waslater on taken over by Federal Mogul and now part of Waukesha) By using AMB's and a direct drive between motorand compressor (so no the gearbox in between) and applying dry gas seals a full so called dry-dry system (=fully oilfree) has been installed. A few of the main advantages by applying AMB's in the driver as well as in the compressor(compared to the traditional configuration with a gearbox, plain bearings and a gasturbine-driver) is a relative simplesystem with a very wide operating envelope, high efficiencies (particularly at partial load) and also, as done in theGroningen field, to install the full installation outdoors (no large compressor building needed).

Early U.S. Patents in AMBInventor(s) Year Patent

No.Invention Title

Beams, Holmes 1941 2,256,937 Suspension of Rotatable Bodies

Beams 1954 2,691,306 Magnetically Supported Rotating Bodies

Beams 1962 3,041,482 Apparatus for Rotating Freely Suspended Bodies

Beams 1965 3,196,694 Magnetic Suspension System

Wolf 1967 3,316,032 Poly-Phase Magnetic Suspension Transformer

Lyman 1971 3,565,495 Magnetic Suspension Apparatus

Habermann 1973 3,731,984 Magnetic Bearing Block Device for Supporting a Vertical Shaft Adapted for Rotating at HighSpeed

Habermann, Loyen, Joli,Aubert

1974 3,787,100 Devices Including Rotating Members Supported by Magnetic Bearings

Habermann, Brunet 1977 4,012,083 Magnetic Bearings

Habermann, Brunet, LeClére 1978 4,114,960 Radial Displacement Detector Device for a Magnetic Bearings

Magnetic bearing 100

Electrodynamic bearing

An axial homopolar electrodynamic bearing

Electrodynamic bearings (EDB) are a novel type of bearing that is apassive magnetic technology. EDBs do not require any controlelectronics to operate. They work by the electrical currents generatedby motion causing a restoring force.

Applications

Magnetic bearing advantages include very low and predictable friction,ability to run without lubrication and in a vacuum. Magnetic bearingsare increasingly used in industrial machines such as compressors,turbines, pumps, motors and generators. Magnetic bearings arecommonly used in watt-hour meters by electric utilities to measurehome power consumption. Magnetic bearings are also used in high-precision instruments and to support equipmentin a vacuum, for example in flywheel energy storage systems. A flywheel in a vacuum has very low windage losses,but conventional bearings usually fail quickly in a vacuum due to poor lubrication. Magnetic bearings are also usedto support maglev trains in order to get low noise and smooth ride by eliminating physical contact surfaces.Disadvantages include high cost, and relatively large size.

A new application of magnetic bearings is their use in artificial hearts. The use of magnetic suspension in ventricularassist devices was pioneered by Prof. Paul Allaire and Prof. Houston Wood at the University of Virginia culminatingin the first magnetically suspended ventricular assist centrifugal pump (VAD) in 1999.

References[1] Charles, D., Spinning a Nuclear Comeback, Science, Vol. 315, (30 March 2007)[2] "Design and Analysis of a Novel Low Loss Homopolar Electrodynamic Bearing." (http:/ / www. magnetal. se/ Dokument/ PhDThesis. pdf)

Lembke, Torbjörn. PhD Thesis. Stockholm: Universitetsservice US AB, 2005. ISBN 91-7178-032-7[3] "3D-FEM Analysis of a Low Loss Homopolar Induction Bearing" (http:/ / www. kth. se/ ees/ forskning/ publikationer/ modules/

publications_polopoly/ reports/ 2004/ IR-EE-EME_2004_015. pdf?l=en_UK) Lembke, Torbjörn. 9th International Symposium on MagneticBearings (ISMB9). Aug. 2004.

[4] Seminar at KTH – the Royal Institute of Technology (http:/ / www. kth. se/ ees/ kalender/ seminarier/ 1. 54496) Stockholm. Feb 24. 2010[5] Beams, J. , Production and Use of High Centrifugal Fields, Science, Vol. 120, (1954)[6] Beams, J. , Magnetic Bearings, Paper 810A, Automotive Engineering Conference, Detroit, Michigan, USA, SAE (Jan. 1964)[7] Habermann,H. , Liard, G. Practical Magnetic Bearings , IEEE Spectrum, Vol. 16, No. 9, (September 1979)[8] Schweitzer, G. , Characteristics of a Magnetic Rotor Bearing for Active Vibration Control, Paper C239/76, First International Conference on

Vibrations in Rotating Machinery, (1976)[9] E. Croot, Australian Inventors Weekly, NSW Inventors Association, Vol. 3, (April 1987)[10] Kasarda, M. An Overview of Active Magnetic Bearing Technology and Applications, The Shock and Vibration Digest, Vol.32, No. 2: A

Publication of the Shock and Vibration Information Center, Naval Research Laboratory, (March 2000)[11] http:/ / www. skfmagneticbearings. com[12] http:/ / www. virginia. edu/ romac/

Magnetic bearing 101

Further reading• Schweitzer, G (2002). Active Magnetic Bearings – Chances and Limitations (http:/ / www. mcgs. ch/

web-content/ AMB-chances_and_limit. pdf).• Chiba, A., Fukao, T., Ichikawa, O., Oshima, M., Takemoto, M., Dorrel, D. (2005). Magnetic Bearings and

Bearingless Drives. Newnes.• Schweitzer, G., Maslen, H. (2009). Magnetic Bearings, Theory, Design,and Application to Rotating Machinery.

Springer.• Maslen, E. H. (1999). Course notes on Magnetic Bearings (http:/ / www. people. virginia. edu/ ~ehm7s/ courses/

magnetic_bearings/ home. html).• Jim Wilson (1999-September). "Beating Demon Friction" (http:/ / www. popularmechanics. com/ science/

research/ 1281766. html). Popular Mechanics.• E. Croot (1987 - 1995). Improved Magnetic Bearings (http:/ / pericles. ipaustralia. gov. au/ ols/ auspat/

quickSearch. do?queryString=Croot& resultsPerPage=). IPAustralia [Australian Patent Office database entries].• T. Lembke (2005). PhD Thesis "Design and Analysis of a Novel Low Loss Homopolar Electrodynamic Bearing"

(http:/ / www. magnetal. se/ Dokument/ PhDThesis. pdf). Stockholm: Universitetsservice US AB.ISBN 91-7178-032-7.

External links• Kinematic Models for Design Digital Library (KMODDL) (http:/ / kmoddl. library. cornell. edu/ index. php) -

Movies and photos of hundreds of working mechanical-systems models at Cornell University. Also includes ane-book library (http:/ / kmoddl. library. cornell. edu/ e-books. php) of classic texts on mechanical design andengineering.

• MADYN2000, Rotordynamics Software (http:/ / www. delta-js. ch/ english/ software/madyn-2000-for-rotordynamics/ magnetic-bearings/ ) supports computer-aided design of Magnetic Bearingcontrollers and provides multiple analytic reports of design quality.

Magnetic circuit 102

Magnetic circuitA magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usuallygenerated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting offerromagnetic materials like iron, although there may be air gaps or other materials in the path. Magnetic circuits areemployed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers,relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.The concept of a "magnetic circuit" exploits a one-to-one correspondence between the equations of the magneticfield in an unsaturated ferromagnetic material to that of an electrical circuit. Using this concept the magnetic fields ofcomplex devices such as transformers can be quickly solved using the methods and techniques developed forelectrical circuits.Some examples of magnetic circuits are:• horseshoe magnet with iron keeper (low-reluctance circuit)•• horseshoe magnet with no keeper (high-reluctance circuit)• electric motor (variable-reluctance circuit)

Magnetomotive force (MMF)Similar to the way that EMF drives a current of electrical charge in electrical circuits, magnetomotive force (MMF)'drives' magnetic flux through magnetic circuits. The term 'magnetomotive force', though, is a misnomer since it isnot a force nor is anything moving. It is perhaps better to call it simply MMF. In analogy to the definition of EMF,the magnetomotive force around a closed loop is defined as:

The MMF represents the potential that a hypothetical magnetic charge would gain by completing the loop. Themagnetic flux that is driven is not a current of magnetic charge; it merely has the same relationship to MMF thatelectric current has to EMF. (See microscopic origins of reluctance below for a further description.)The unit of magnetomotive force is the ampere-turn (At), represented by a steady, direct electric current of oneampere flowing in a single-turn loop of electrically conducting material in a vacuum. The gilbert (Gi), established bythe IEC in 1930 [1], is the CGS unit of magnetomotive force and is a slightly smaller unit than the ampere-turn. Theunit is named after William Gilbert (1544–1603) English physician and natural philosopher.

The magnetomotive force can often be quickly calculated using Ampère's law. For example, the magnetomotiveforce of long coil is:

,where N is the number of turns and I is the current in the coil. In practice this equation is used for the MMF of realinductors with N being the winding number of the inducting coil.

Magnetic fluxAn applied MMF 'drives' magnetic flux through the magnetic components of the system. The magnetic flux througha magnetic component is proportional to the number of magnetic field lines that pass through the cross sectional areaof that component. This is the net number, i.e. the number passing through in one direction, minus the numberpassing through in the other direction. The direction of the magnetic field vector B is by definition from the south tothe north pole of a magnet inside the magnet; outside the field lines go from north to south.

Magnetic circuit 103

The flux through an element of area perpendicular to the direction of magnetic field is given by the product of themagnetic field and the area element. More generally, magnetic flux Φ is defined by a scalar product of the magneticfield and the area element vector. Quantitatively, the magnetic flux through a surface S is defined as the integral ofthe magnetic field over the area of the surface

For a magnetic component the area S used to calculate the magnetic flux Φ is usually chosen to be thecross-sectional area of the component.The SI unit of magnetic flux is the weber (in derived units: volt-seconds), and the unit of magnetic field is the weberper square meter, or tesla.

Hopkinson's law: the magnetic analogy to Ohm's lawIn electronic circuits, Ohm's law is an empirical relation between the EMF applied across an element and thecurrent I it generates through that element. It is written as:

where R is the electrical resistance of that material. Hopkinson's law is a counterpart to Ohm's law used in magneticcircuits. The law is named after the British electrical engineer, John Hopkinson. It states that[2][3]

where is the magnetomotive force (MMF) across a magnetic element, is the magnetic flux through themagnetic element, and is the magnetic reluctance of that element. (It shall be shown later that this relationship isdue to the empirical relationship between the H-field and the magnetic field B, B=μH, where μ is the permeability ofthe material.) Like Ohm's law, Hopkinson's law can be interpreted either as an empirical equation that works forsome materials, or it may serve as a definition of reluctance.

ReluctanceMagnetic reluctance, or magnetic resistance, is analogous to resistance in an electrical circuit (although it does notdissipate magnetic energy). In likeness to the way an electric field causes an electric current to follow the path ofleast resistance, a magnetic field causes magnetic flux to follow the path of least magnetic reluctance. It is a scalar,extensive quantity, akin to electrical resistance.The total reluctance is equal to the ratio of the (MMF) in a passive magnetic circuit and the magnetic flux in thiscircuit. In an AC field, the reluctance is the ratio of the amplitude values for a sinusoidal MMF and magnetic flux.(see phasors)The definition can be expressed as:

where is the reluctance in ampere-turns per weber (a unit that is equivalent to turns per henry).Magnetic flux always forms a closed loop, as described by Maxwell's equations, but the path of the loop depends onthe reluctance of the surrounding materials. It is concentrated around the path of least reluctance. Air and vacuumhave high reluctance, while easily magnetized materials such as soft iron have low reluctance. The concentration offlux in low-reluctance materials forms strong temporary poles and causes mechanical forces that tend to move thematerials towards regions of higher flux so it is always an attractive force(pull).The inverse of reluctance is called permeance.

Magnetic circuit 104

Its SI derived unit is the henry (the same as the unit of inductance, although the two concepts are distinct).

Microscopic origins of reluctanceThe reluctance of a magnetically uniform magnetic circuit element can be calculated as:

wherel is the length of the element in metres

is the permeability of the material ( is the relative permeability of the material(dimensionless), and is the permeability of free space)A is the cross-sectional area of the circuit in square metres

This is similar to the equation for electrical resistance in materials, with permeability being analogous toconductivity; the reciprocal of the permeability is known as magnetic reluctivity and is analogous to resistivity.Longer, thinner geometries with low permeabilities lead to higher reluctance. Low reluctance, like low resistance inelectric circuits, is generally preferred.

Summary of analogy between magnetic circuits and electrical circuitsThe following table summarizes the mathematical analogy between electrical circuit theory and magnetic circuittheory. This is mathematical analogy and not a physical one. Objects in the same row have the same mathematicalrole; the physics of the two theories are very different. For example, current is the flow of electrical charge, whilemagnetic flux is not the flow of any quantity.

Analogy between 'magnetic circuits' and electrical circuits

Magnetic equivalent Symbol Units Electric equivalent Symbol

Magnetomotive force (MMF) ampere-turn Definition of EMF

Magnetic field H ampere/meter Electric field E

Magnetic flux φ weber Electric current I

Hopkinson's law or Rowland's law Ohm's law

Reluctance 1/henry Electrical resistance R

Permeance henry Electric conductance G = 1/R

relation between B and H Microscopic Ohm's law

Magnetic flux density B B tesla Current density J

permeability μ henry/meter Electrical conductivity σ

Magnetic circuit 105

Limitations of the analogyWhen using the analogy between magnetic circuits and electric circuits, the limitations of this analogy must be keptin mind. Electric and magnetic circuits are only superficially similar because of the similarity between Hopkinson'slaw and Ohm's law. Magnetic circuits have significant differences, which must be taken into account in theirconstruction:• Electric currents represent the flow of particles (electrons) and carry power, which is dissipated as heat in

resistances. Magnetic fields don't represent the "flow" of anything, and no power is dissipated in reluctances.• The current in typical electric circuits is confined to the circuit, with very little "leakage". In typical magnetic

circuits not all of the magnetic field is confined to the magnetic circuit; there is significant "leakage flux" in thespace outside the magnetic cores, which must be taken into account but is difficult to calculate.

• Most importantly, magnetic circuits are nonlinear; the reluctance in a magnetic circuit is not constant, asresistance is, but varies depending on the magnetic field. At high magnetic fluxes the ferromagnetic materialsused for the cores of magnetic circuits saturate, limiting the magnetic flux, so above this level the reluctanceincreases rapidly. The reluctance also increases at low fluxes. In addition, ferromagnetic materials suffer fromhysteresis so the flux in them depends not just on the instantaneous MMF but also on the history of MMF. Afterthe source of the magnetic flux is turned off, remanent magnetism is left in ferromagnetic circuits, creating a fluxwith no MMF.

Circuit Laws

Magnetic circuit

Magnetic circuits obey other laws that are similar to electrical circuitlaws. For example, the total reluctance of reluctances in series is:

This also follows from Ampère's law and is analogous to Kirchhoff'svoltage law for adding resistances in series. Also, the sum of magneticfluxes into any node is always zero:

This follows from Gauss's law and is analogous to Kirchhoff's currentlaw for analyzing electrical circuits.

Together, the three laws above form a complete system for analysing magnetic circuits, in a manner similar toelectric circuits. Comparing the two types of circuits shows that:• The equivalent to resistance R is the reluctance Rm• The equivalent to current I is the magnetic flux Φ• The equivalent to voltage V is the magnetomotive Force FMagnetic circuits can be solved for the flux in each branch by application of the magnetic equivalent of Kirchhoff'sVoltage Law (KVL) for pure source/resistance circuits. Specifically, whereas KVL states that the voltage excitationapplied to a loop is equal to the sum of the voltage drops (resistance times current) around the loop, the magneticanalogue states that the magnetomotive force (achieved from ampere-turn excitation) is equal to the sum of MMFdrops (product of flux and reluctance) across the rest of the loop. (If there are multiple loops, the current in eachbranch can be solved through a matrix equation—much as a matrix solution for mesh circuit branch currents isobtained in loop analysis—after which the individual branch currents are obtained by adding and/or subtracting theconstituent loop currents as indicated by the adopted sign convention and loop orientations.) Per Ampère's law, theexcitation is the product of the current and the number of complete loops made and is measured in ampere-turns.Stated more generally:

Magnetic circuit 106

(Note that, per Stokes's theorem, the closed line integral of H dot dl around a contour is equal to the open surfaceintegral of curl H dot dA across the surface bounded by the closed contour. Since, from Maxwell's equations, curl H= J, the closed line integral of H dot dl evaluates to the total current passing through the surface. This is equal to theexcitation, NI, which also measures current passing through the surface, thereby verifying that the net current flowthrough a surface is zero ampere-turns in a closed system that conserves energy.)More complex magnetic systems, where the flux is not confined to a simple loop, must be analysed from firstprinciples by using Maxwell's equations.

HistoryThe term reluctance was coined in May 1888 by Oliver Heaviside.[4] The notion of “magnetic resistance” was firstmentioned by James Joule [5] and the term "magnetomotive force” (MMF) was first named by Bosanquet.[6] The ideafor a magnetic flux law, similar to Ohm's law for closed electric circuits, is attributed to H. Rowland.[7]

Applications• Air gaps can be created in the cores of certain transformers to reduce the effects of saturation. This increases the

reluctance of the magnetic circuit, and enables it to store more energy before core saturation. This effect is alsoused in the flyback transformer.

• Variation of reluctance is the principle behind the reluctance motor (or the variable reluctance generator) and theAlexanderson alternator.

• Multimedia loudspeakers are typically shielded magnetically, in order to reduce magnetic interference caused totelevisions and other CRTs. The speaker magnet is covered with a material such as soft iron to minimize the straymagnetic field.

Reluctance can also be applied to:• Reluctance motors• Variable reluctance (magnetic) pickups

References[1] http:/ / www. iec. ch/ about/ history/[2] Magnetism (flash) (http:/ / www. ginerdelosrios. org/ pizarra/ electronica/ nemesio/ pizarra_neme/ simuladores/ parametros_magneticos. swf)[3] Tesche, Fredrick; Michel Ianoz, Torbjörn Karlsson (1997). EMC Analysis Methods and Computational Models. Wiley-IEEE. pp. 513.

ISBN 0-471-15573-X.[4] Heaviside O., Electrical Papers. Vol.2. – L.; N.Y.: Macmillan, 1892, p. 166.[5] Joule J., Scientific Papers, vol. 1. – 1884, p. 36.[6][6] Bosanquet, Phil. Mag., vol. 15, 1883, p. 205.[7][7] Rowland H., Phil. Mag. (4), vol. 46, 1873, p. 140.

External links• Magnetic-Electric Analogs (http:/ / www. analogzone. com/ col_0909. pdf) by Dennis L. Feucht, Innovatia

Laboratories (PDF)• Interactive Java Tutorial on Magnetic Shunts (http:/ / www. magnet. fsu. edu/ education/ tutorials/ java/

magneticshunt/ ) National High Magnetic Field Laboratory

Magnetic dipole 107

Magnetic dipole

Dipole moment m.

Electric current I.

A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of thesource are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electricdipole, but the analogy is not complete. In particular, a magnetic monopole, the magnetic analogue of an electriccharge, has never been observed. Moreover, one form of magnetic dipole moment is associated with a fundamentalquantum property, the spin of elementary particles.The magnetic field around any magnetic source looks increasingly like the field of a magnetic dipole as the distancefrom the source increases.

External magnetic field produced by a magnetic dipole moment

An electrostatic analogue for a magneticmoment: two opposing charges separated

by a finite distance. Each arrowrepresents the direction of the field vector

at that point.

In classical physics, the magnetic field of a dipole is calculated as the limit ofeither a current loop or a pair of charges as the source shrinks to a point whilekeeping the magnetic moment m constant. For the current loop, this limit ismost easily derived for the vector potential. Outside of the source region, thispotential is (in SI units) [1]

and the magnetic flux density (strength of the B-field) in teslas is

Magnetic dipole 108

The magnetic field of a current loop. The ringrepresents the current loop, which goes into the page at

the x and comes out at the dot.

Alternatively one can obtain the scalar potential first from the magnetic pole limit,

and hence the magnetic field strength (or strength of the H-field) in ampere-turns per meter is

The magnetic field is symmetric under rotations about the axis of the magnetic moment.

Internal magnetic field of a dipoleThe two models for a dipole (current loop and magnetic poles) give the same predictions for the magnetic field farfrom the source. However, inside the source region they give different predictions. The magnetic field between polesis in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge),while inside a current loop it is in the same direction (see the figure to the right). Clearly, the limits of these fieldsmust also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used tocalculate fields inside a magnetic material.If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current andarea constant, the limiting field is

.

where n=x/|x|. Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer togetherbut keeping the product of magnetic pole-charge and distance constant, the limiting field is

These fields are related by B = μ0(H+M), where

Magnetic dipole 109

is the magnetization.

Forces between two magnetic dipolesThe force F exerted by one dipole moment m1 on another m2 separated in space by a vector r can be calculated using

or [2]

where r is the distance between dipoles. The force acting on m1 is in the opposite direction.The torque can be obtained from the formula

Dipolar fields from finite sourcesThe magnetic scalar potential ψ produced by a finite source, but external to it, can be represented by a multipoleexpansion. Each term in the expansion is associated with a characteristic moment and a potential having acharacteristic rate of decrease with distance r from the source. Monopole moments have a 1/r rate of decrease,dipole moments have a 1/r2 rate, quadrupole moments have a 1/r3 rate, and so on. The higher the order, the fasterthe potential drops off. Since the lowest-order term observed in magnetic sources is the dipolar term, it dominates atlarge distances. Therefore, at large distances any magnetic source looks like a dipole with the same magneticmoment.

Notes[1] Chow 2006, pp. 146–150[2][2] Furlani 2001

References• Chow, Tai L. (2006). Introduction to electromagnetic theory: a modern perspective. Jones & Bartlett Learning.

ISBN 978-0-7637-3827-3.• Jackson, John D. (1999). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X. OCLC 224523909.• Furlani, Edward P. (2001). Permanent Magnet and Electromechanical Devices: Materials, Analysis, and

Applications (http:/ / books. google. com/ ?id=irsdLnC5SrsC& dq=permanent+ magnet+ and+electromechanical+ devices& printsec=frontcover& q=3. 130). Academic Press. p. 140. ISBN 0-12-269951-3.

• Schill, R. A. (2003). "General relation for the vector magnetic field of a circular current loop: A closer look".IEEE Transactions on Magnetics 39 (2): 961–967. Bibcode 2003ITM....39..961S.doi:10.1109/TMAG.2003.808597.

Magnetic domain 110

Magnetic domain

Several grains of NdFeB with magnetic domainsmade visible with a Kerr microscope. The

domains are the light and dark stripes visiblewithin each grain.

A magnetic domain is a region within a magnetic material which hasuniform magnetization. This means that the individual magneticmoments of the atoms are aligned with one another and they point inthe same direction. When cooled below a temperature called the Curietemperature, the magnetization of a piece of ferromagnetic materialspontaneously divides into many small regions called magneticdomains. The magnetization within each domain points in a uniformdirection, but the magnetization of different domains may point indifferent directions. Magnetic domain structure is responsible for themagnetic behavior of ferromagnetic materials like iron, nickel, cobaltand their alloys, ferrites etc. such as the formation of permanentmagnets. The regions separating magnetic domains are called domainwalls, where the magnetization rotates coherently from the direction inone domain to that in the next domain. The study of magnetic domainsis called micromagnetics.

Development of domain theoryMagnetic domain theory was developed by French physicist Pierre-Ernest Weiss[1] who in 1906 suggested existenceof magnetic domains in ferromagnets.[2] He suggested that large number of atomic magnetic moments (typically1012-1018) were aligned parallel. The direction of alignment varies from domain to domain in a more or less randommanner although certain crystallographic axis may be preferred by the magnetic moments, called easy axes. Weissstill had to explain the reason for the spontaneous alignment of atomic moments within a ferromagnetic material, andhe came up with the so-called Weiss mean field : he assumed that a given magnetic moment in a materialexperienced a very high effective magnetic field due to the magnetization of its neighbors. In the original Weisstheory the mean field was proportional to the bulk magnetization M, so that

where is the mean field constant. However this is not applicable to ferromagnets due to the variation ofmagnetization from domain to domain. In this case, the interaction field is

Where is the saturation magnetization at 0K.Later, the quantum theory made it possible to understand the microscopic origin of the Weiss field. The exchangeinteraction between localized spins favored a parallel (in ferromagnets) or an anti-parallel (in anti-ferromagnets) stateof neighboring magnetic moments.

Magnetic domain 111

Domain structure

How dividing a ferromagnetic material into magnetic domains reduces themagnetostatic energy

Why domains form

The reason a piece of magnetic material such asiron spontaneously divides into separate domains,rather than exist in a state with magnetization inthe same direction throughout the material, is tominimize its internal energy.[3] A large region offerromagnetic material with a constantmagnetization throughout will create a largemagnetic field extending into the space outsideitself (diagram a, right). This requires a lot ofmagnetostatic energy stored in the field. To reduce this energy, the sample can split into two domains, with themagnetization in opposite directions in each domain (diagram b right). The magnetic field lines pass in loops inopposite directions through each domain, reducing the field outside the material. To reduce the field energy further,each of these domains can split also, resulting in smaller parallel domains with magnetization in alternatingdirections, with smaller amounts of field outside the material.

The domain structure of actual magnetic materials does not usually form by the process of large domains splittinginto smaller ones as described here. When a sample is cooled below the Curie temperature, for example, theequilibrium domain configuration simply appears. But the description of domains splitting is often used to reveal theenergy tradeoffs in domain formation.

Size of domainsSo a domain which is too big is unstable, and will divide into smaller domains. But a small enough domain will bestable and will not split, and this determines the size of the domains created in a material. This size depends on thebalance of several energies within the material.[3] Each time a region of magnetization splits into two domains, itcreates a "domain wall" between the domains, where magnetic dipoles (molecules) with magnetization pointing indifferent directions are adjacent. The exchange interaction which creates the magnetization is a force which tends toalign nearby dipoles so they point in the same direction. So forcing adjacent dipoles to point in different directionsrequires energy. Therefore creating a domain wall requires extra energy, called the "exchange energy", which isproportional to the area of the wall.So the net amount that the energy is reduced when a domain splits is equal to the difference between the magneticfield energy saved, and the additional energy of the domain wall created. The field energy saved is proportional tothe cube of the domain size, while the domain wall energy is proportional to the square of the domain size. So as thedomains get smaller, the net energy saved by splitting decreases. The domains keep dividing into smaller domainsuntil the energy cost of creating an additional domain wall is just equal to the field energy saved. Then the domainsof this size are stable. In most materials the domains are microscopic in size, around 10-4 - 10-6 m.

Magnetic domain 112

Magnetic anisotropy

Micrograph of surface of ferromagnetic material,showing the crystal grains, each divided intoseveral domains parallel to its "easy" axis of

magnetization, with the magnetization inalternating directions (red and green areas).

Animation showing how magnetostriction works.An external magnetic field causes the magnetic

dipoles to rotate, changing the dimensions of thecrystal lattice.

An additional way for the material to further reduce its magnetostaticenergy is to form domains with magnetization at right angles to theother domains (diagram c, right), instead of just in opposing paralleldirections.[3] These domains, called flux closure domains, allow thefield lines to turn 180° within the material, forming closed loopsentirely within the material, reducing the magnetostatic energy to zero.However, forming these domains incurs two additional energy costs.First, the crystal lattice of most magnetic materials has magneticanisotropy, which means it has an "easy" direction of magnetization,parallel to one of the crystal axes. Changing the magnetization of thematerial to any other direction takes additional energy, called the"magnetocrystalline anisotropy energy".

Magnetostriction

The other energy cost to creating domains with magnetization at anangle to the "easy" direction is caused by the phenomenon calledmagnetostriction.[3] When the magnetization of a piece of magneticmaterial is changed to a different direction, it causes a slight change inits shape. The change in magnetic field causes the magnetic dipolemolecules to change shape slightly, making the crystal lattice longer inone dimension and shorter in other dimensions. However, since themagnetic domain is "squished in" with its boundaries held rigid by thesurrounding material, it cannot actually change shape. So instead,changing the direction of the magnetization induces tiny mechanicalstresses in the material, requiring more energy to create the domain. This is called "magnetoelastic anisotropyenergy".

To form these closure domains with "sideways" magnetization requires additional energy due to the aforementionedtwo factors. So flux closure domains will only form where the magnetostatic energy saved is greater than the sum ofthe "exchange energy" to create the domain wall, the magnetocrystalline anisotropy energy, and the magnetoelasticanisotropy energy. Therefore most of the volume of the material is occupied by domains with magnetization either"up" or "down" along the "easy" direction, and the flux closure domains only form in small areas at the edges of theother domains where they are needed to provide a path for magnetic field lines to change direction (diagram c,above).

Grain structureThe above describes magnetic domain structure in a perfect crystal lattice, such as would be found in a single crystalof iron. However most magnetic materials are polycrystalline, composed of microscopic crystalline grains. Thesegrains are not the same as domains. Each grain is a little crystal, with the crystal lattices of separate grains oriented inrandom directions. In most materials, each grain is big enough to contain several domains. Each crystal has an "easy"axis of magnetization, and is divided into domains with the axis of magnetization parallel to this axis, in alternatedirections.

Magnetic domain 113

"Magnetized" statesIt can be seen that, although on a microscopic scale almost all the magnetic dipoles in a piece of ferromagneticmaterial are lined up parallel to their neighbors in domains, creating strong local magnetic fields, energyminimization results in a domain structure that minimizes the large-scale magnetic field. The domains point indifferent directions, confining the field lines to microscopic loops between neighboring domains, so the combinedfields cancel at a distance. Therefore a bulk piece of ferromagnetic material in its lowest energy state has little or noexternal magnetic field. The material is said to be "unmagnetized". However, the domains can also exist in otherconfigurations in which their magnetization mostly points in the same direction, creating an external magnetic field.Although these are not minimum energy configurations, due to a phenomenon where the domain walls become"pinned" to defects in the crystal lattice they can be local minimums of the energy, and therefore can be very stable.This is what happens when a piece of magnetic material is "magnetized" and becomes a permanent magnet.

Landau-Lifshitz energy equation

Moving domain walls in a grain of silicon steelcaused by an increasing external magnetic field in

the "downward" direction, observed in a Kerrmicroscope. White areas are domains with

magnetization directed up, dark areas are domainswith magnetization directed down.

The contributions of the different internal energy factors describedabove is expressed by the free energy equation proposed by LevLandau and Evgeny Lifshitz in 1935 [1], which forms the basis of themodern theory of magnetic domains. The domain structure of amaterial is the one which minimizes the Gibbs free energy of thematerial. For a crystal of magnetic material, this is the Landau-Lifshitzfree energy, E, which is the sum of these energy terms:[4]

where• E

ex is exchange energy: This is the energy due to the exchange

interaction between magnetic dipole molecules in ferromagnetic,ferrimagnetic and antiferromagnetic materials. It is lowest when thedipoles are all pointed in the same direction, so it is responsible formagnetization of magnetic materials. When two domains with different directions of magnetization are next toeach other, at the domain wall between them magnetic dipoles pointed in different directions lie next to eachother, increasing this energy. This additional exchange energy is proportional to the total area of the domainwalls.

• ED

is magnetostatic energy: This is a self-energy, due to the interaction of the magnetic field created by themagnetization in some part of the sample on other parts of the same sample. It is dependent on the volumeoccupied by the magnetic field extending outside the domain. This energy is reduced by minimizing the length ofthe loops of magnetic field lines outside the domain. For example, this tends to encourage the magnetization to beparallel to the surfaces of the sample, so the field lines won't pass outside the sample. Reducing this energy is themain reason for the creation of magnetic domains.

• Eλ

is magnetoelastic anisotropy energy: This energy is due to the effect of magnetostriction, a slight change inthe dimensions of the crystal when magnetized. This causes elastic strains in the lattice, and the direction ofmagnetization that minimizes these strain energies will be favored. This energy tends to be minimized when theaxis of magnetization of the domains in a crystal are all parallel.

• Ek

is magnetocrystalline anisotropy energy: Due to its magnetic anisotropy, the crystal lattice is "easy" to magnetize in one direction, and "hard" to magnetize in others. This energy is minimized when the magnetization is along the "easy" crystal axis, so the magnetization of most of the domains in a crystal grain tend to be in either direction along the "easy" axis. Since the crystal lattice in separate grains of the material is usually oriented in different random directions, this causes the dominant domain magnetization in different grains to be pointed in

Magnetic domain 114

different directions.• E

H is Zeeman energy: This is energy which is added to or subtracted from the magnetostatic energy, due to the

interaction between the magnetic material and an externally applied magnetic field. It is proportional to thenegative of the cosine of the angle between the field and magnetization vectors. Domains with their magneticfield oriented parallel to the applied field reduce this energy, while domains with their magnetic field orientedopposite to the applied field increase this energy. So applying a magnetic field to a ferromagnetic materialgenerally causes the domain walls to move so as to increase the size of domains lying mostly parallel to the field,at the cost of decreasing the size of domains opposing the field. This is what happens when ferromagneticmaterials are "magnetized". With a strong enough external field, the domains opposing the field will beswallowed up and disappear; this is called saturation.

Rotation of orientation and increase in size of magnetic to an externally applied field(compare Zeeman energy).

Some sources define a wall energy EWequal to the sum of the exchangeenergy and the magnetocrystallineanisotropy energy, which replaces Eexand Ek in the above equation.

A stable domain structure is amagnetization function M(X),considered as a continuous vectorfield, which minimizes the total energy E throughout the material. To find the minimums a variational method isused, resulting in a set of nonlinear differential equations, called Brown's equations after William Fuller Brown Jr.Although in principle these equations can be solved for the stable domain configurations M(X), in practice only thesimplest examples can be solved. Analytic solutions do not exist, and numerical solutions calculated by the finiteelement method are computationally intractable because of the large difference in scale between the domain size andthe wall size. Therefore micromagnetics has evolved approximate methods which assume that the magnetization ofdipoles in the bulk of the domain, away from the wall, all point in the same direction, and numerical solutions areonly used near the domain wall, where the magnetization is changing rapidly.

Observing domainsThere are a number of microscopy methods which can make the magnetization at a surface of a magnetic materialvisible, revealing the magnetic domains. Each method has a different application because not all domains are thesame. In magnetic materials, domains can be circular, square, irregular, elongated, and striped, all of which havevaried sizes and dimensions. Large domains, within the range of 25-100 micrometers can be easily seen by Kerrmicroscopy, which uses the magneto-optic Kerr effect, which is the rotation of the polarization of light reflectedfrom a magnetized surface. Smaller domains, down to the scale of a few nanometers, can be viewed by the use ofmagnetic force microscopy.

Magnetic domain 115

Domain structure of a shape-memory alloy (recorded using CMOS-MagView)

Domain structure of an examplary meander domain (recorded using CMOS-MagView)

Domain structure of an examplary magnetic bubble domain (recorded using CMOS-MagView)

References[1] P. Weiss (1906) La variation du ferromagnetisme du temperature, Comptes Rendus, 143, p.1136-1149, cited in Cullity, 2008 (http:/ / books.

google. com/ books?id=ixAe4qIGEmwC& pg=PA116& lpg=PA116), p.116[2] Cullity; C. D. Graham. Introduction to Magnetic Materials, 2nd ed. (http:/ / books. google. com/ books?id=ixAe4qIGEmwC& pg=PA116&

lpg=PA116). New York: Wiley-IEte=2008. pp. 116. ISBN 0-471-47741-9. .[3] Feynman, Richard P.; Robert B. Leighton, Matthew Sands (1963). The Feynman Lectures on Physics, Vol. I (http:/ / books. google. com/

books?id=bDF-uoUmttUC& pg=SA4-PA4& dq="inclined+ plane"+ + "conservation+ of+ energy"& hl=en& sa=X&ei=gQtdT6iLCanSiAK22tCsCw& ved=0CGwQ6AEwBg#v=onepage& q="inclined plane" "conservation of energy"& f=false). USA:California Inst. of Technology. pp. 37.5-37.6. ISBN 0-201-02117-XP. .

[4][4] Carey R., Isaac E.D., Magnetic domains and techniques for their observation, The English University Press Ltd, London, (1966).

• Jiles, David (1998). Introduction to magnetism and magnetic materials. London: Chapman & Hall.ISBN 0-412-79860-3.

Magnetic domain 116

External links• Interactive Java tutorial on magnetic domains (http:/ / www. magnet. fsu. edu/ education/ tutorials/ java/ domains/

index. html) National High Magnetic Field Laboratory• Magnetismus und Magnetooptik (http:/ / schulzeundschultze. anphy. uni-duesseldorf. de/ veroff/ Diplome/

Volker. Solinus/ node3. phtml) a German text about magnetism and magneto-optics

Magnetic field

Magnetic field of an ideal cylindrical magnet with its axis ofsymmetry inside the image plane.

A magnetic field may be represented by amathematical description of the magnetic influence ofelectric currents and magnetic materials. The magneticfield at any given point is specified by both a directionand a magnitude (or strength); as such it is a vectorfield.[1] The magnetic field is most commonly definedin terms of the Lorentz force it exerts on movingelectric charges. There are two separate but closelyrelated fields to which the name "magnetic field" canrefer: a magnetic B field and a magnetic H field.

Magnetic fields are produced by moving electriccharges and the intrinsic magnetic moments ofelementary particles associated with a fundamentalquantum property, their spin. In special relativity,electric and magnetic fields are two interrelated aspects of a single object, called the electromagnetic tensor; the splitof this tensor into electric and magnetic fields depends on the relative velocity of the observer and charge. Inquantum physics, the electromagnetic field is quantized and electromagnetic interactions result from the exchange ofphotons.

Magnetic fields have had many uses in ancient and modern society. The Earth produces its own magnetic field,which is important in navigation. Rotating magnetic fields are utilized in both electric motors and generators.Magnetic forces give information about the charge carriers in a material through the Hall effect. The interaction ofmagnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits.

Magnetic field 117

History

One of the first drawings of a magnetic field, by René Descartes, 1644. Itillustrated his theory that magnetism was caused by the circulation of tiny helical

particles, "threaded parts", through threaded pores in magnets.

Although magnets and magnetism wereknown much earlier, the study of themagnetic field began in 1269 when Frenchscholar Petrus Peregrinus de Maricourtmapped out the magnetic field on thesurface of a spherical magnet using ironneedles.[2] Noting that the resulting fieldlines crossed at two points he named thosepoints 'poles' in analogy to Earth's poles.Almost three centuries later, William Gilbertof Colchester replicated Petrus Peregrinus'work and was the first to state explicitly thatEarth is a magnet.[3] Published in 1600,Gilbert's work, De Magnete, helped toestablish magnetism as a science.

In 1750, John Michell stated that magneticpoles attract and repel in accordance with aninverse square law.[4] Charles-Augustin deCoulomb experimentally verified this in1785 and stated explicitly that the North and South poles cannot be separated.[5] Building on this force betweenpoles, Siméon-Denis Poisson (1781–1840) created the first successful model of the magnetic field which hepresented in 1824.[6] In this model, a magnetic H-field is produced by 'magnetic poles' and magnetism is due to smallpairs of north/south magnetic poles.

Three discoveries challenged this foundation of magnetism, though. First, in 1819, Hans Christian Oersteddiscovered that an electric current generates a magnetic field encircling it. Then in 1820, André-Marie Ampèreshowed that parallel wires having currents in the same direction attract one another. Finally, Jean-Baptiste Biot andFélix Savart discovered the Biot–Savart law in 1820 which correctly predicts the magnetic field around anycurrent-carrying wire.Extending these experiments, Ampère published his own successful model of magnetism in 1825. In it, he showedthe equivalence of electrical currents to magnets[7] and proposed that magnetism is due to perpetually flowing loopsof current instead of the dipoles of magnetic charge in Poisson's model.[8] This has the additional benefit ofexplaining why magnetic charge can not be isolated. Further, Ampère derived both Ampère's force law describingthe force between two currents and Ampère's law which, like the Biot–Savart law, correctly described the magneticfield generated by a steady current. Also in this work, Ampère introduced the term electrodynamics to describe therelationship between electricity and magnetism.In 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic fieldgenerates an encircling electric field. He described this phenomenon in what is known as Faraday's law of induction.Later, Franz Ernst Neumann proved that, for a moving conductor in a magnetic field, induction is a consequence ofAmpère's force law .[9] In the process he introduced the magnetic vector potential which was later shown to beequivalent to the underlying mechanism proposed by Faraday.In 1850, Lord Kelvin, then known as William Thomson, distinguished between two magnetic fields now denoted Hand B. The former applied to Poisson's model and the latter to Ampère's model and induction.[10] Further, he derivedhow H and B relate to each other.

Magnetic field 118

Between 1861 and 1865, James Clerk Maxwell developed and published Maxwell's equations which explained andunited all of classical electricity and magnetism. The first set of these equations was published in a paper entitled OnPhysical Lines of Force in 1861. These equations were valid although incomplete. He completed Maxwell's set ofequations in his later 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrated the fact thatlight is an electromagnetic wave. Heinrich Hertz experimentally confirmed this fact in 1887.Although implicit in Ampère's force law the force due to a magnetic field on a moving electric charge was notcorrectly and explicitly stated until 1892 by Hendrik Lorentz who theoretically derived it from Maxwell'sequations.[11] With this last piece of the puzzle, the classical theory of electrodynamics was essentially complete.The twentieth century extended electrodynamics to include relativity and quantum mechanics. Albert Einstein, in hispaper of 1905 that established relativity, showed that both the electric and magnetic fields are part of the samephenomena viewed from different reference frames. (See moving magnet and conductor problem for details aboutthe thought experiment that eventually helped Albert Einstein to develop special relativity.) Finally, the emergentfield of quantum mechanics was merged with electrodynamics to form quantum electrodynamics (QED).

Definitions, units, and measurementThe magnetic field can be defined in several equivalent ways based on the effects it has on its environment.Often the magnetic field is defined by the force it exerts on a moving charged particle. It is known from experimentsin electrostatics that a particle of charge q in an electric field E experiences a force F = qE. However, electrostaticsalone is insufficient to explain the force a charged particle experiences in other situations, such as when it moves inthe vicinity of a current-carrying wire. In these situations, the force can be correctly accounted for if one introduces avector B and then writes down a new equation for the force, known as the Lorentz force law:

Here v is the particle's velocity and × denotes the cross product. The vector B is termed the magnetic field, and it isdefined as the vector field necessary to make the Lorentz force law correctly describe the motion of a chargedparticle. This definition allows one to determine B in the following way, as described by Purcell:[12]

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

Alternatively, the magnetic field can be defined in terms of the torque it produces on a magnetic dipole (seemagnetic torque on permanent magnets below).Devices used to measure the local magnetic field are called magnetometers. Important classes of magnetometersinclude using a rotating coil, Hall effect magnetometers, NMR magnetometers, SQUID magnetometers, and fluxgatemagnetometers. The magnetic fields of distant astronomical objects are measured through their effects on localcharged particles. For instance, electrons spiraling around a field line produce synchrotron radiation which isdetectable in radio waves.

Magnetic field 119

Alternative names for B[13]

•• Magnetic flux density•• Magnetic induction•• Magnetic field

Alternative names for H[13][14]

•• Magnetic field intensity•• Magnetic field strength•• Magnetic field•• Magnetizing field

There are two magnetic fields, H and B. In a vacuum they are indistinguishable, differing only by a multiplicativeconstant that depends on the physical units. Inside a material they are different (see H and B inside and outside ofmagnetic materials). The term magnetic field is historically reserved for H while using other terms for B. Informally,though, and formally for some recent textbooks mostly in physics, the term 'magnetic field' is used to describe B aswell as or in place of H.[15] There are many alternative names for both (see sidebar).In SI units, the B-field (magnetic flux density) is measured in teslas (symbol: T) and correspondingly ΦB (magneticflux) is measured in weber (Wb) so that a flux density of one Wb/m2 is one tesla. In Gaussian-cgs units, the B-fieldis measured in gauss (symbol: G). (The conversion is 1 T = 10,000 G.) The SI unit of tesla is equivalent to(newton·second)/(coulomb·metre).[16] The H-field is measured in ampere per metre (A/m) in SI units, and in oersteds(Oe) in cgs units.[17]

The smallest precision level for a magnetic field measurement[18] is on the order of attoteslas (10−18 tesla); thelargest magnetic field produced in a laboratory is 2.8 kT (VNIIEF in Sarov, Russia, 1998).[19] The magnetic field ofsome astronomical objects such as magnetars are much higher; magnetars range from 0.1 to 100 GT (108 to1011 T).[20] See orders of magnitude (magnetic field).

Magnetic field lines

Compasses reveal the direction ofthe local magnetic field. As seenhere, the magnetic field points

towards a magnet's south pole andaway from its north pole.

Mapping the magnetic field of an object is simple in principle. First, measure thestrength and direction of the magnetic field at a large number of locations. Then,mark each location with an arrow (called a vector) pointing in the direction of thelocal magnetic field with a length proportional to the strength of the magnetic field.

A simpler method to map the magnetic field is to 'connect' the arrows to formmagnetic field lines. On a magnetic field line diagram, the direction of themagnetic field at any point is represented by the direction of nearby field lines.Further, if drawn carefully, a higher density of nearby field lines indicates astronger magnetic field.

Magnetic field lines are like the contour lines (constant altitude) on a topographicmap in that a different mapping scale would show more or fewer lines. Anadvantage of using magnetic field lines, though, is that many laws of magnetism(and electromagnetism) can be stated completely and concisely using simpleconcepts such as the 'number' of field lines through a surface. These concepts canbe quickly 'translated' to their mathematical form. For example, the number of fieldlines through a given surface is the surface integral of the magnetic field.

Magnetic field 120

The direction of magnetic field lines representedby the alignment of iron filings sprinkled on

paper placed above a bar magnet.

Various phenomena have the effect of "displaying" magnetic field linesas though the field lines are physical phenomena. For example, ironfilings placed in a magnetic field line up to form lines that correspondto 'field lines'.[21] Magnetic fields "lines" are also visually displayed inpolar auroras, in which plasma particle dipole interactions createvisible streaks of light that line up with the local direction of Earth'smagnetic field.

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

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

Magnetic field of permanent magnetsThe magnetic field of permanent magnets can be quite complicated, especially near the magnet. The magnetic fieldof a small[22] straight magnet is proportional to the magnet's strength (called its magnetic dipole moment m). Theequations are non-trivial and also depend on the distance from the magnet and the orientation of the magnet. Forsimple magnets, m points in the direction of a line drawn from the south to the north pole of the magnet. Flipping abar magnet is equivalent to rotating its m by 180 degrees.The magnetic field of larger magnets can be obtained by modelling them as a collection of a large number of smallmagnets called dipoles each having their own m. The magnetic field produced by the magnet then is the netmagnetic field of these dipoles. And, any net force on the magnet is a result of adding up the forces on the individualdipoles.There are two competing models for the nature of these dipoles. These two models produce two different magneticfields, H and B. Outside a material, though, the two are identical (to a multiplicative constant) so that in many casesthe distinction can be ignored. This is particularly true for magnetic fields, such as those due to electric currents, thatare not generated by magnetic materials.

Magnetic field 121

Magnetic pole model and the H-field

The magnetic pole model: two opposingpoles, North (+) and South (-), separated

by a distance d produce an H-field(lines).

It is sometimes useful to model the force and torques between two magnets asdue to magnetic poles repelling or attracting each other in the same manner asthe Coulomb force between electric charges. In this model, a magnetic H-fieldis produced by magnetic charges that are 'smeared' around each pole. TheH-field, therefore, is analogous to the electric field E which starts at a positiveelectric charge and ends at a negative electric charge. Near the north pole,therefore, all H-field lines point away from the north pole (whether inside themagnet or out) while near the south pole (whether inside the magnet or out)all H-field lines point toward the south pole. A north pole, then, feels a forcein the direction of the H-field while the force on the south pole is opposite tothe H-field.

In the magnetic pole model, the elementary magnetic dipole m is formed bytwo opposite magnetic poles of pole strength qm separated by a very smalldistance vector d, such that m = qm d.

Magnetic poles cannot exist apart from each other; all magnets have north/south pairs which cannot be separatedwithout creating two magnets each having a north/south pair. The magnetic pole model does not account formagnetism that is produced by electric currents, nor the force that a magnetic field applies to moving electriccharges.

Amperian loop model and the B-field

The Amperian loop model: A current loop (ring) whichgoes into the page at the x and comes out at the dot

produces a B field (lines). The north pole is to the rightand the south to the left.

After Oersted discovered that electric currents produce a magneticfield and Ampere discovered that electric currents attracted andrepelled each other similar to magnets, it was natural tohypothesize that all magnetic fields are due to electric currentloops. In this model developed by Ampere, the elementarymagnetic dipole that makes up all magnets is a sufficiently smallAmperian loop of current I. The dipole moment of this loop is m =I A where A is the area of the loop.These magnetic dipoles produce a magnetic B field. One importantproperty of the B-field produced this way is that magnetic B fieldlines neither start nor end (mathematically, B is a solenoidal vectorfield); a field line either extends to infinity or wraps around toform a closed curve.[23] To date no exception to this rule has beenfound. (See magnetic monopole below.) Magnetic field lines exit amagnet near its north pole and enter near its south pole, but insidethe magnet B-field lines continue through the magnet from thesouth pole back to the north.[24] If a B-field line enters a magnetsomewhere it has to leave somewhere else; it is not allowed to have an end point. Magnetic poles, therefore, alwayscome in N and S pairs.

More formally, since all the magnetic field lines that enter any given region must also leave that region, subtractingthe 'number'[25] of field lines that enter the region from the number that exit gives identically zero. Mathematicallythis is equivalent to:

Magnetic field 122

where the integral is a surface integral over the closed surface S (a closed surface is one that completely surrounds aregion with no holes to let any field lines escape). Since dA points outward, the dot product in the integral is positivefor B-field pointing out and negative for B-field pointing in.There is also a corresponding differential form of this equation covered in Maxwell's equations below.

Force between magnetsThe force between two small magnets is quite complicated and depends on the strength and orientation of bothmagnets and the distance and direction of the magnets relative to each other. The force is particularly sensitive torotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and themagnetic field[26] of the other.To understand the force between magnets, it is useful to examine the magnetic pole model given above. In thismodel, the H-field of one magnet pushes and pulls on both poles of a second magnet. If this H-field is the same atboth poles of the second magnet then there is no net force on that magnet since the force is opposite for oppositepoles. If, however, the magnetic field of the first magnet is nonuniform (such as the H near one of its poles), eachpole of the second magnet sees a different field and is subject to a different force. This difference in the two forcesmoves the magnet in the direction of increasing magnetic field and may also cause a net torque.This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation ofthe magnet) into regions of higher magnetic field. Any non-uniform magnetic field whether caused by permanentmagnets or by electric currents will exert a force on a small magnet in this way.The details of the Amperian loop model are different and more complicated but yield the same result: that magneticdipoles are attracted/repelled into regions of higher magnetic field. Mathematically, the force on a small magnethaving a magnetic moment m due to a magnetic field B is:[27]

where the gradient ∇ is the change of the quantity m · B per unit distance and the direction is that of maximumincrease of m · B. To understand this equation, note that the dot product m · B = mBcos(θ), where m and B representthe magnitude of the m and B vectors and θ is the angle between them. If m is in the same direction as B then the dotproduct is positive and the gradient points 'uphill' pulling the magnet into regions of higher B-field (more strictlylarger m · B). This equation is strictly only valid for magnets of zero size, but is often a good approximation for nottoo large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions havingtheir own m then summing up the forces on each of these regions.

Magnetic torque on permanent magnetsIf two like poles of two separate magnets are brought near each other and one of the magnets is allowed to turn itwill promptly rotate to align itself with the first. In this example, the magnetic field of the stationary magnet creates amagnetic torque on the magnet that is free to rotate. This magnetic torque τ tends to align a magnet's poles with themagnetic field lines. A compass, therefore, will turn to align itself with earth's magnetic field.Magnetic torque is used to drive electric motors. In one simple motor design, a magnet is fixed to a freely rotatingshaft and subjected to a magnetic field from an array of electromagnets. By continuously switching the electriccurrent through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are keptnext to the rotor; the resultant torque is transferred to the shaft. See Rotating magnetic fields below.

Magnetic field 123

torque on a dipole: An H field (to right) causesequal but opposite forces on a N pole (+q) and a

S pole (-q) creating a torque.

As is the case for the force between magnets, the magnetic pole modelleads more readily to the correct equation. Here, two equal andopposite magnetic charges experiencing the same H also experienceequal and opposite forces. Since these equal and opposite forces are indifferent locations, this produces a torque proportional to the distance(perpendicular to the force) between them. With the definition of m asthe pole strength times the distance between the poles, this leads to τ =μ0mHsinθ, where μ0 is a constant called the magnetic constant and θ isthe angle between H and m.

The Amperian loop model also predicts the same magnetic torque. Here, it is the B field interacting with theAmperian current loop through a Lorentz force described below. Again, the results are the same although the modelsare completely different.

Cross product: |a × b| = a b sinθ.

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

where × represents the vector cross product. Note that this equationincludes all of the qualitative information included above. There is notorque on a magnet if m is in the same direction as the magnetic field.(The cross product is zero for two vectors that are in the samedirection.) Further, all other orientations feel a torque that twists themtoward the direction of magnetic field.

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

Magnetic field due to moving charges and electric currents

Right hand grip rule: a current flowing in thedirection of the white arrow produces a magnetic

field shown by the red arrows.

All moving charged particles produce magnetic fields. Moving pointcharges, such as electrons, produce complicated but well knownmagnetic fields that depend on the charge, velocity, and acceleration ofthe particles.[28]

Magnetic field lines form in concentric circles around a cylindricalcurrent-carrying conductor, such as a length of wire. The direction ofsuch a magnetic field can be determined by using the "right hand griprule" (see figure at right). The strength of the magnetic field decreaseswith distance from the wire. (For an infinite length wire the strengthdecreases inversely proportional to the distance.)

Magnetic field 124

Solenoid

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

The magnetic field generated by a steady current (a constant flow of electric charges in whichcharge is neither accumulating nor depleting at any point)[29] is described by the Biot–Savart law:

where the integral sums over the wire length where vector dℓ is the direction of the current, μ0 is the magneticconstant, r is the distance between the location of dℓ and the location at which the magnetic field is being calculated,and r̂ is a unit vector in the direction of r.A slightly more general[30][31] way of relating the current to the B-field is through Ampère's law:

where the line integral is over any arbitrary loop and enc is the current enclosed by that loop. Ampère's law isalways valid for steady currents and can be used to calculate the B-field for certain highly symmetric situations suchas an infinite wire or an infinite solenoid.In a modified form that accounts for time varying electric fields, Ampère's law is one of four Maxwell's equationsthat describe electricity and magnetism.

Magnetic field 125

Force on moving charges and current

Charged particle drifts in a magnetic field with (A) no net force, (B)an electric field, E, (C) a charge independent force, F (e.g. gravity), and

(D) an inhomogeneous magnetic field, grad H.

Force on a charged particle

A charged particle moving in a B-field experiences asideways force that is proportional to the strength ofthe magnetic field, the component of the velocity thatis perpendicular to the magnetic field and the chargeof the particle. This force is known as the Lorentzforce, and is given by

where F is the force, q is the electric charge of theparticle, v is the instantaneous velocity of theparticle, and B is the magnetic field (in teslas).

The Lorentz force is always perpendicular to both thevelocity of the particle and the magnetic field thatcreated it. When a charged particle moves in a staticmagnetic field it will trace out a helical path in whichthe helix axis is parallel to the magnetic field and inwhich the speed of the particle will remain constant.Because the magnetic force is always perpendicularto the motion, the magnetic field can do no work onan isolated charge. It can only do work indirectly, viathe electric field generated by a changing magneticfield. It is often claimed that the magnetic force cando work to a non-elementary magnetic dipole, or tocharged particles whose motion is constrained byother forces, but this is incorrect[32] because the workin those cases is performed by the electric forces of the charges deflected by the magnetic field.

Force on current-carrying wire

The force on a current carrying wire is similar to that of a moving charge as expected since a charge carrying wire isa collection of moving charges. A current carrying wire feels a force in the presence of a magnetic field. The Lorentzforce on a macroscopic current is often referred to as the Laplace force. Consider a conductor of length l, crosssection A, and charge q which is due to electric current i. If this conductor is placed in a magnetic field of inductionB which makes an angle θ (theta) with the velocity of charges in the conductor, the force exerted on a single chargeq is

,so, for N charges where

,the force exerted on the conductor is

,,

where .

Magnetic field 126

The right-hand rule: Pointing the thumb of the right hand in thedirection of the conventional current and the fingers in the direction

of the B-field the force on the current points out of the palm. Theforce is reversed for a negative charge.

Direction of force

The direction of force on a charge or a current can bedetermined by a mnemonic known as the right-handrule (see the figure). Using the right hand and pointingthe thumb in the direction of the moving positivecharge or positive current and the fingers in thedirection of the magnetic field the resulting force on thecharge points outwards from the palm. The force on anegatively charged particle is in the opposite direction.If both the speed and the charge are reversed then thedirection of the force remains the same. For that reasona magnetic field measurement (by itself) cannotdistinguish whether there is a positive charge moving tothe right or a negative charge moving to the left. (Both of these cases produce the same current.) On the other hand, amagnetic field combined with an electric field can distinguish between these, see Hall effect below.

An alternative mnemonic to the right hand rule is Fleming's left hand rule.

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

MagnetizationThe magnetization vector field M represents how strongly a region of material is magnetized. It is defined as the netmagnetic dipole moment per unit volume of that region. The magnetization of a uniform magnet, therefore, is aconstant in the material equal to its magnetic moment, m, divided by its volume. Since the SI unit of magneticmoment is ampere meter2, the SI unit of magnetization M is ampere per meter, identical to that of the H-field.The magnetization M field of a region points in the direction of the average magnetic dipole moment in that region.Magnetization field lines, therefore, begin near the magnetic south pole and ends near the magnetic north pole.(Magnetization does not exist outside of the magnet.)In the Amperian loop model, the magnetization is due to combining many tiny Amperian loops to form a resultantcurrent called bound current. This bound current, then, is the source of the magnetic B field due to the magnet. (SeeMagnetic dipoles below and magnetic poles vs. atomic currents for more information.) Given the definition of themagnetic dipole, the magnetization field follows a similar law to that of Ampere's law:[33]

where the integral is a line integral over any closed loop and Ib is the 'bound current' enclosed by that closed loop.In the magnetic pole model, magnetization begins at and ends at magnetic poles. If a given region, therefore, has anet positive 'magnetic pole strength' (corresponding to a north pole) then it will have more magnetization field linesentering it than leaving it. Mathematically this is equivalent to:

,

Magnetic field 127

where the integral is a closed surface integral over the closed surface S and qM is the 'magnetic charge' (in units ofmagnetic flux) enclosed by S. (A closed surface completely surrounds a region with no holes to let any field linesescape.) The negative sign occurs because the magnetization field moves from south to north.

H-field and magnetic materialsThe H-field is defined as:

(definition of H in SI units)

With this definition, Ampere's law becomes:

where If represents the 'free current' enclosed by the loop so that the line integral of H does not depend at all on thebound currents.[34] For the differential equivalent of this equation see Maxwell's equations. Ampere's law leads to theboundary condition

where Kf is the surface free current density.[35]

Similarly, a surface integral of H over any closed surface is independent of the free currents and picks out the'magnetic charges' within that closed surface:

which does not depend on the free currents.The H-field, therefore, can be separated into two[36] independent parts:

where H0 is the applied magnetic field due only to the free currents and Hd is the demagnetizing field due only to thebound currents.The magnetic H-field, therefore, re-factors the bound current in terms of 'magnetic charges'. The H field lines looponly around 'free current' and, unlike the magnetic B field, begins and ends near magnetic poles as well.

MagnetismMost materials respond to an applied B-field by producing their own magnetization M and therefore their ownB-field. Typically, the response is very weak and exists only when the magnetic field is applied. The term magnetismdescribes how materials respond on the microscopic level to an applied magnetic field and is used to categorize themagnetic phase of a material. Materials are divided into groups based upon their magnetic behavior:• Diamagnetic materials[37] produce a magnetization that opposes the magnetic field.• Paramagnetic materials[37] produce a magnetization in the same direction as the applied magnetic field.• Ferromagnetic materials and the closely related ferrimagnetic materials and antiferromagnetic materials[38][39] can

have a magnetization independent of an applied B-field with a complex relationship between the two fields.• Superconductors (and ferromagnetic superconductors)[40][41] are materials that are characterized by perfect

conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfectdiamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures andmagnetic fields (the so named mixed state) for which they exhibit a complex hysteretic dependence of M on B.

In the case of paramagnetism and diamagnetism, the magnetization M is often proportional to the applied magneticfield such that:

Magnetic field 128

where μ is a material dependent parameter called the permeability. In some cases the permeability may be a secondrank tensor so that H may not point in the same direction as B. These relations between B and H are examples ofconstitutive equations. However, superconductors and ferromagnets have a more complex B to H relation; seemagnetic hysteresis.

Energy stored in magnetic fieldsEnergy is needed to generate a magnetic field both to work against the electric field that a changing magnetic fieldcreates and to change the magnetization of any material within the magnetic field. For non-dispersive materials thissame energy is released when the magnetic field is destroyed so that this energy can be modeled as being stored inthe magnetic field.For linear, non-dispersive, materials (such that B = μH where μ is frequency-independent), the energy density is:

If there are no magnetic materials around then μ can be replaced by μ0. The above equation cannot be used fornonlinear materials, though; a more general expression given below must be used.In general, the incremental amount of work per unit volume δW needed to cause a small change of magnetic field δBis:

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

Electromagnetism: the relationship between magnetic and electric fields

Faraday's Law: Electric force due to a changing B-fieldA changing magnetic field, such as a magnet moving through a conducting coil, generates an electric field (andtherefore tends to drive a current in the coil). This is known as Faraday's law and forms the basis of many electricalgenerators and electric motors.Mathematically, Faraday's law is:

where is the electromotive force (or EMF, the voltage generated around a closed loop) and Φm is the magneticflux—the product of the area times the magnetic field normal to that area. (This definition of magnetic flux is why Bis often referred to as magnetic flux density.)The negative sign is necessary and represents the fact that any current generated by a changing magnetic field in acoil produces a magnetic field that opposes the change in the magnetic field that induced it. This phenomenon isknown as Lenz's Law.This integral formulation of Faraday's law can be converted[42] into a differential form, which applies under slightlydifferent conditions. This form is covered as one of Maxwell's equations below.

Magnetic field 129

Maxwell's correction to Ampère's Law: The magnetic field due to a changing electric fieldSimilar to the way that a changing magnetic field generates an electric field, a changing electric field generates amagnetic field. This fact is known as Maxwell's correction to Ampère's law. Maxwell's correction to Ampère's Lawbootstrap together with Faraday's law of induction to form electromagnetic waves, such as light. Thus, a changingelectric field generates a changing magnetic field which generates a changing electric field again.Maxwell's correction to Ampère law is applied as an additive term to Ampere's law given above. This additive termis proportional to the time rate of change of the electric flux and is similar to Faraday's law above but with a differentand positive constant out front. (The electric flux through an area is proportional to the area times the perpendicularpart of the electric field.)This full Ampère law including the correction term is known as the Maxwell–Ampère equation. It is not commonlygiven in integral form because the effect is so small that it can typically be ignored in most cases where the integralform is used. The Maxwell term is critically important in the creation and propagation of electromagnetic waves.These, though, are usually described using the differential form of this equation given below.

Maxwell's equationsLike all vector fields, magnetic field has two important mathematical properties that relates it to its sources. (For theB-field the sources are currents and changing electric fields.) These two properties, along with the twocorresponding properties of the electric field, make up Maxwell's Equations. Maxwell's Equations together with theLorentz force law form a complete description of classical electrodynamics including both electricity andmagnetism.The first property is the divergence of a vector field A, ∇ · A which represents how A 'flows' outward from a givenpoint. As discussed above, a B-field line never starts or ends at a point but instead forms a complete loop. This ismathematically equivalent to saying that the divergence of B is zero. (Such vector fields are called solenoidal vectorfields.) This property is called Gauss's law for magnetism and is equivalent to the statement that there are no isolatedmagnetic poles or magnetic monopoles. The electric field on the other hand begins and ends at electric charges sothat its divergence is non-zero and proportional to the charge density (See Gauss's law).The second mathematical property is called the curl, such that ∇ × A represents how A curls or 'circulates' around agiven point. The result of the curl is called a 'circulation source'. The equations for the curl of B and of E are calledthe Ampère–Maxwell equation and Faraday's law respectively. They represent the differential forms of the integralequations given above.The complete set of Maxwell's equations then are:

where J = complete microscopic current density and ρ is the charge density.

Magnetic field 130

Magnetic field, like all pseudovectors, changessign when reflected in a mirror: When a current

carrying loop (black) is reflected in a mirror(dotted line), its magnetic field (blue) is reflected

and reversed.

Technically, B is a pseudovector (also called an axial vector) due tobeing defined by a vector cross product. (See diagram.)

As discussed above, materials respond to an applied electric E fieldand an applied magnetic B field by producing their own internal'bound' charge and current distributions that contribute to E and B butare difficult to calculate. To circumvent this problem, H and D fieldsare used to re-factor Maxwell's equations in terms of the free currentdensity Jf and free charge density ρf:

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

Electric and magnetic fields: different aspects of the same phenomenonAccording to the special theory of relativity, the partition of the electromagnetic force into separate electric andmagnetic components is not fundamental, but varies with the observational frame of reference: An electric forceperceived by one observer may be perceived by another (in a different frame of reference) as a magnetic force, or amixture of electric and magnetic forces.Formally, special relativity combines the electric and magnetic fields into a rank-2 tensor, called theelectromagnetic tensor. Changing reference frames mixes these components. This is analogous to the way thatspecial relativity mixes space and time into spacetime, and mass, momentum and energy into four-momentum.[43]

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

The vector potential A may be interpreted as a generalized potential momentum per unit charge[44] just as φ isinterpreted as a generalized potential energy per unit charge.Maxwell's equations when expressed in terms of the potentials can be cast into a form that agrees with specialrelativity with little effort.[45] In relativity A together with φ forms the four-potential analogous to thefour-momentum which combines the momentum and energy of a particle. Using the four potential instead of theelectromagnetic tensor has the advantage of being much simpler; further it can be easily modified to work withquantum mechanics.

Magnetic field 131

Quantum electrodynamicsIn modern physics, the electromagnetic field is understood to be not a classical field, but rather a quantum field; it isrepresented not as a vector of three numbers at each point, but as a vector of three quantum operators at each point.The most accurate modern description of the electromagnetic interaction (and much else) is Quantumelectrodynamics (QED),[46] which is incorporated into a more complete theory known as the Standard Model ofparticle physics.In QED, the magnitude of the electromagnetic interactions between charged particles (and their antiparticles) iscomputed using perturbation theory; these rather complex formulas have a remarkable pictorial representation asFeynman diagrams in which virtual photons are exchanged.Predictions of QED agree with experiments to an extremely high degree of accuracy: currently about 10−12 (andlimited by experimental errors); for details see precision tests of QED. This makes QED one of the most accuratephysical theories constructed thus far.All equations in this article are in the classical approximation, which is less accurate than the quantum descriptionmentioned here. However, under most everyday circumstances, the difference between the two theories is negligible.

Important uses and examples of magnetic field

Earth's magnetic field

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

diagram, the south pole near the bottom. The south pole of thatmagnet is deep in Earth's interior below Earth's North Magnetic Pole.

The Earth's magnetic field is thought to be produced byconvection currents in the outer liquid of Earth's core.The Dynamo theory proposes that these movementsproduce electric currents which, in turn, produce themagnetic field.[47]

The presence of this field causes a compass, placedanywhere within it, to rotate so that the "north pole" ofthe magnet in the compass points roughly north,toward Earth's north magnetic pole. This is thetraditional definition of the "north pole" of a magnet,although other equivalent definitions are also possible.

One confusion that arises from this definition is that, ifEarth itself is considered as a magnet, the south pole ofthat magnet would be the one nearer the northmagnetic pole, and vice-versa. The north magneticpole is so-named not because of the polarity of thefield there but because of its geographical location.The north and south poles of a permanent magnet areso-called because they are "north-seeking" and"south-seeking", respectively.[48][49]

The figure is a sketch of Earth's magnetic field represented by field lines. For most locations, the magnetic field has asignificant up/down component in addition to the North/South component. (There is also an East/West component;Earth's magnetic poles do not coincide exactly with Earth's geological pole.) The magnetic field can be visualised asa bar magnet buried deep in Earth's interior.Earth's magnetic field is not constant—the strength of the field and the location of its poles vary. Moreover, the poles periodically reverse their orientation in a process called geomagnetic reversal. The most recent reversal occurred

Magnetic field 132

780,000 years ago.

Rotating magnetic fieldsThe rotating magnetic field is a key principle in the operation of alternating-current motors. A permanent magnet insuch a field rotates so as to maintain its alignment with the external field. This effect was conceptualized by NikolaTesla, and later utilized in his, and others', early AC (alternating-current) electric motors.A rotating magnetic field can be constructed using two orthogonal coils with 90 degrees phase difference in their ACcurrents. However, in practice such a system would be supplied through a three-wire arrangement with unequalcurrents.This inequality would cause serious problems in standardization of the conductor size and so, in order to overcomeit, three-phase systems are used where the three currents are equal in magnitude and have 120 degrees phasedifference. Three similar coils having mutual geometrical angles of 120 degrees create the rotating magnetic field inthis case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the mainreasons why three-phase systems dominate the world's electrical power supply systems.Because magnets degrade with time, synchronous motors use DC voltage fed rotor windings which allows theexcitation of the machine to be controlled and induction motors use short-circuited rotors (instead of a magnet)following the rotating magnetic field of a multicoiled stator. The short-circuited turns of the rotor develop eddycurrents in the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force.In 1882, Nikola Tesla identified the concept of the rotating magnetic field. In 1885, Galileo Ferraris independentlyresearched the concept. In 1888, Tesla gained U.S. Patent 381968 [50] for his work. Also in 1888, Ferraris publishedhis research in a paper to the Royal Academy of Sciences in Turin.

Hall effectThe charge carriers of a current carrying conductor placed in a transverse magnetic field experience a sidewaysLorentz force; this results in a charge separation in a direction perpendicular to the current and to the magnetic field.The resultant voltage in that direction is proportional to the applied magnetic field. This is known as the Hall effect.The Hall effect is often used to measure the magnitude of a magnetic field. It is used as well to find the sign of thedominant charge carriers in materials such as semiconductors (negative electrons or positive holes).

Magnetic circuitsAn important use of H is in magnetic circuits where B = μH inside a linear material. Here, μ is the magneticpermeability of the material. This result is similar in form to Ohm's law J = σE, where J is the current density, σ isthe conductance and E is the electric field. Extending this analogy, the counterpart to the macroscopic Ohm's law(I = V⁄R) is:

where is the magnetic flux in the circuit, is the magnetomotive force applied to

the circuit, and Rm is the reluctance of the circuit. Here the reluctance Rm is a quantity similar in nature to resistancefor the flux.Using this analogy it is straightforward to calculate the magnetic flux of complicated magnetic field geometries, byusing all the available techniques of circuit theory.

Magnetic field 133

Magnetic field shape descriptions

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

magnetic south poles.

• An azimuthal magnetic field is one that runs east-west.• A meridional magnetic field is one that runs north-south. In the

solar dynamo model of the Sun, differential rotation of the solarplasma causes the meridional magnetic field to stretch into anazimuthal magnetic field, a process called the omega-effect. Thereverse process is called the alpha-effect.[51]

• A dipole magnetic field is one seen around a bar magnet or around acharged elementary particle with nonzero spin.

• A quadrupole magnetic field is one seen, for example, between thepoles of four bar magnets. The field strength grows linearly with theradial distance from its longitudinal axis.

• A solenoidal magnetic field is similar to a dipole magnetic field,except that a solid bar magnet is replaced by a hollowelectromagnetic coil magnet.

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

• A poloidal magnetic field is generated by a current flowing in a ring, and is found, for example, in a tokamak.• A radial magnetic field is one in which the field lines are directed from the center outwards, similar to the spokes

in a bicycle wheel. An example can be found in a loudspeaker transducers (driver).[52]

• A helical magnetic field is corkscrew-shaped, and sometimes seen in space plasmas such as the Orion MolecularCloud.[53]

Magnetic dipoles

Magnetic field lines around a ”magnetostaticdipole” pointing to the right.

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

where the direction of m is perpendicular to the area of the loop anddepends on the direction of the current using the right-hand rule. Anideal magnetic dipole is modeled as a real magnetic dipole whose areaa has been reduced to zero and its current I increased to infinity suchthat the product m = Ia is finite. In this model it is easy to see theconnection between angular momentum and magnetic moment whichis the basis of the Einstein-de Haas effect "rotation by magnetization"and its inverse, the Barnett effect or "magnetization by rotation".[54]

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

It is sometimes useful to model the magnetic dipole similar to the electric dipole with two equal but oppositemagnetic charges (one south the other north) separated by distance d. This model produces an H-field not a B-field.Such a model is deficient, though, both in that there are no magnetic charges and in that it obscures the link betweenelectricity and magnetism. Further, as discussed above it fails to explain the inherent connection between angularmomentum and magnetism.

Magnetic field 134

Magnetic monopole (hypothetical)A magnetic monopole is a hypothetical particle (or class of particles) that has, as its name suggests, only onemagnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogousto an electric charge. Magnetic field lines would start or end on magnetic monopoles, so if they exist, they wouldgive exceptions to the rule that magnetic field lines neither start nor end.Modern interest in this concept stems from particle theories, notably Grand Unified Theories and superstringtheories, that predict either the existence, or the possibility, of magnetic monopoles. These theories and others haveinspired extensive efforts to search for monopoles. Despite these efforts, no magnetic monopole has been observedto date.[55]

In recent research, materials known as spin ices can simulate monopoles, but do not contain actual monopoles.

Notes[1] Technically, a magnetic field is a pseudo vector; pseudo-vectors, which also include torque and rotational velocity, are similar to vectors

except that they remain unchanged when the coordinates are inverted.[2] His Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete, which is often shortened to Epistola de magnete,

is dated 1269 C.E.[3][3] Whittaker 1951, p. 34[4][4] Whittaker 1951, p. 56[5][5] Whittaker 1951, p. 59[6][6] Whittaker 1951, p. 64[7][7] Whittaker 1951, p. 88[8][8] From the outside, the field of a dipole of magnetic charge has the exact same form as that of a current loop when both are sufficiently small.

Therefore, the two models differ only for magnetism inside magnetic material.[9][9] Whittaker 1951, p. 222[10][10] Whittaker 1951, p. 244[11][11] Whittaker 1951, p. 422[12] Purcell, E. (2011). Electricity and Magnetism (2nd ed.). Cambridge University Press. pp. 173–4. ISBN 1107013607.[13] Electromagnetics, by Rothwell and Cloud, p23 (http:/ / books. google. com/ books?id=jCqv1UygjA4C& pg=PA23)[14] R.P. Feynman, R.B. Leighton, M. Sands (1963). The Feynman Lectures on Physics, volume 2.[15] Edward Purcell, in Electricity and Magnetism, McGraw-Hill, 1963, writes, Even some modern writers who treat B as the primary field feel

obliged to call it the magnetic induction because the name magnetic field was historically preempted by H. This seems clumsy and pedantic. Ifyou go into the laboratory and ask a physicist what causes the pion trajectories in his bubble chamber to curve, he'll probably answer"magnetic field", not "magnetic induction." You will seldom hear a geophysicist refer to the Earth's magnetic induction, or an astrophysicisttalk about the magnetic induction of the galaxy. We propose to keep on calling B the magnetic field. As for H, although other names havebeen invented for it, we shall call it "the field H" or even "the magnetic field H." In a similar vein, M Gerloch (1983). Magnetism andLigand-field Analysis (http:/ / books. google. com/ ?id=Ovo8AAAAIAAJ& pg=PA110). Cambridge University Press. p. 110.ISBN 0-521-24939-2. . says: "So we may think of both B and H as magnetic fields, but drop the word 'magnetic' from H so as to maintain thedistinction ... As Purcell points out, 'it is only the names that give trouble, not the symbols'."

[16] This can be seen from the magnetic part of the Lorentz force law F = qvBsinθ.[17] "International system of units (SI)" (http:/ / physics. nist. gov/ cuu/ Units/ units. html). NIST reference on constants, units, and uncertainty.

National Institute of Standards and Technology. . Retrieved 9 May 2012.[18] "Gravity Probe B Executive Summary" (http:/ / www. nasa. gov/ pdf/ 168808main_gp-b_pfar_cvr-pref-execsum. pdf). pp. 10, 21. .[19] "With record magnetic fields to the 21st Century" (http:/ / ieeexplore. ieee. org/ xpl/ freeabs_all. jsp?arnumber=823621). IEEE Xplore. .[20] Kouveliotou, C.; Duncan, R. C.; Thompson, C. (February 2003). " Magnetars (http:/ / solomon. as. utexas. edu/ ~duncan/ sciam. pdf)".

Scientific American; Page 36.[21] The use of iron filings to display a field presents something of an exception to this picture; the filings alter the magnetic field so that it is

much larger along the "lines" of iron, due to the large permeability of iron relative to air.[22] Here 'small' means that the observer is sufficiently far away that it can be treated as being infinitesimally small. 'Larger' magnets need to

include more complicated terms in the expression and depend on the entire geometry of the magnet not just m.[23][23] Magnetic field lines may also wrap around and around without closing but also without ending. These more complicated non-closing

non-ending magnetic field lines are moot, though, since the magnetic field of objects that produce them are calculated by adding the magneticfields of 'elementary parts' having magnetic field lines that do form closed curves or extend to infinity.

[24][24] To see that this must be true imagine placing a compass inside a magnet. There, the north pole of the compass points toward the north poleof the magnet since magnets stacked on each other point in the same direction.

Magnetic field 135

[25][25] As discussed above, magnetic field lines are primarily a conceptual tool used to represent the mathematics behind magnetic fields. The total'number' of field lines is dependent on how the field lines are drawn. In practice, integral equations such as the one that follows in the maintext are used instead.

[26] Either B or H may be used for the magnetic field outside of the magnet.[27] See Eq. 11.42 in E. Richard Cohen, David R. Lide, George L. Trigg (2003). AIP physics desk reference (http:/ / books. google. com/

?id=JStYf6WlXpgC& pg=PA381) (3 ed.). Birkhäuser. p. 381. ISBN 0-387-98973-0. .[28][28] Griffiths 1999, p. 438[29] In practice, the Biot–Savart law and other laws of magnetostatics are often used even when the currents are changing in time as long as it is

not changing too quickly. It is often used, for instance, for standard household currents which oscillate sixty times per second.[30] Griffiths 1999, pp. 222–225[31] The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also

depends on the divergence of B being zero, which is always valid. (There are no magnetic charges.)[32] Deissler, R.J. (2008). "Dipole in a magnetic field, work, and quantum spin" (http:/ / academic. csuohio. edu/ deissler/

PhysRevE_77_036609. pdf). Physical Review E 77 (3, pt 2): 036609. Bibcode 2008PhRvE..77c6609D. doi:10.1103/PhysRevE.77.036609.PMID 18517545. .

[33] Griffiths 1999, pp. 266–268[34] John Clarke Slater, Nathaniel Herman Frank (1969). Electromagnetism (http:/ / books. google. com/ ?id=GYsphnFwUuUC& pg=PA69)

(first published in 1947 ed.). Courier Dover Publications. p. 69. ISBN 0-486-62263-0. .[35][35] Griffiths 1999, p. 332[36][36] A third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's

equations below.[37] RJD Tilley (2004). Understanding Solids (http:/ / books. google. com/ ?id=ZVgOLCXNoMoC& pg=PA368). Wiley. p. 368.

ISBN 0-470-85275-5. .[38] Sōshin Chikazumi, Chad D. Graham (1997). Physics of ferromagnetism (http:/ / books. google. com/ ?id=AZVfuxXF2GsC&

printsec=frontcover) (2 ed.). Oxford University Press. p. 118. ISBN 0-19-851776-9. .[39] Amikam Aharoni (2000). Introduction to the theory of ferromagnetism (http:/ / books. google. com/ ?id=9RvNuIDh0qMC& pg=PA27) (2

ed.). Oxford University Press. p. 27. ISBN 0-19-850808-5. .[40] M Brian Maple et al. (2008). "Unconventional superconductivity in novel materials" (http:/ / books. google. com/ ?id=PguAgEQTiQwC&

pg=PA640). In K. H. Bennemann, John B. Ketterson. Superconductivity. Springer. p. 640. ISBN 3-540-73252-7. .[41] Naoum Karchev (2003). "Itinerant ferromagnetism and superconductivity" (http:/ / books. google. com/ ?id=3AFo_yxBkD0C& pg=PA169).

In Paul S. Lewis, D. Di (CON) Castro. Superconductivity research at the leading edge. Nova Publishers. p. 169. ISBN 1-59033-861-8. .

[42] A complete expression for Faraday's law of induction in terms of the electric E and magnetic fields can be written as:

where ∂Σ(t) is the moving closed path

bounding the moving surface Σ(t), and dA is an element of surface area of Σ(t). The first integral calculates the work done moving a charge adistance dℓ based upon the Lorentz force law. In the case where the bounding surface is stationary, the Kelvin–Stokes theorem can be used toshow this equation is equivalent to the Maxwell–Faraday equation.

[43] C. Doran and A. Lasenby (2003) Geometric Algebra for Physicists, Cambridge University Press, p.233[44] E. J. Konopinski (1978). "What the electromagnetic vector potential describes". Am. J. Phys. 46 (5): 499–502.

Bibcode 1978AmJPh..46..499K. doi:10.1119/1.11298.[45][45] Griffiths 1999, p. 422[46] For a good qualitative introduction see: Feynman, Richard (2006). QED: the strange theory of light and matter. Princeton University Press.

ISBN 0-691-12575-9.[47] Herbert, Yahreas (June 1954). "What makes the earth Wobble" (http:/ / books. google. com/ ?id=NiEDAAAAMBAJ& pg=PA96&

dq=What+ makes+ the+ earth+ wobble& q=What makes the earth wobble). Popular Science (New York: Godfrey Hammond): 266. .[48] Serway, Raymond A.; Chris Vuille, Jerry S. Faughn (2009). College physics (8th ed.). Belmont, CA: Brooks/Cole, Cengage Learning.

p. 628. ISBN 978-0-495-38693-3.[49] Kurtus, Ron (2004). "Magnets" (http:/ / www. school-for-champions. com/ science/ magnets. htm). School for champions: Physics topics. .

Retrieved 17 July 2010.[50] http:/ / www. google. com/ patents?vid=381968[51] The Solar Dynamo (http:/ / www. cora. nwra. com/ ~werne/ eos/ text/ dynamo. html), retrieved September 15, 2007.[52] I. S. Falconer and M. I. Large (edited by I. M. Sefton), " Magnetism: Fields and Forces (http:/ / www. physics. usyd. edu. au/ super/

life_sciences/ electricity. html)" Lecture E6, The University of Sydney, retrieved 3 October 2008[53] Robert Sanders, " Astronomers find magnetic Slinky in Orion (http:/ / berkeley. edu/ news/ media/ releases/ 2006/ 01/ 12_helical. shtml)",

12 January 2006 at UC Berkeley. Retrieved 3 October 2008[54] (See magnetic moment for further information.)

B. D. Cullity, C. D. Graham (2008). Introduction to Magnetic Materials (http:/ / books. google. com/?id=ixAe4qIGEmwC& pg=PA103) (2 ed.). Wiley-IEEE. p. 103. ISBN 0-471-47741-9. .

Magnetic field 136

[55] Two experiments produced candidate events that were initially interpreted as monopoles, but these are now regarded to be inconclusive. Fordetails and references, see magnetic monopole.

References

Further reading• Durney, Carl H. and Johnson, Curtis C. (1969). Introduction to modern 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 ed.). Springer.

ISBN 0-412-49580-5.• Kraftmakher, Yaakov (2001). "Two experiments with rotating magnetic field" (http:/ / www. iop. org/ EJ/

abstract/ 0143-0807/ 22/ 5/ 302). Eur. J. Phys. 22: 477–482.• Melle, Sonia; Rubio, Miguel A.; Fuller, Gerald G. (2000). "Structure and dynamics of magnetorheological fluids

in rotating magnetic fields" (http:/ / prola. aps. org/ abstract/ PRE/ v61/ i4/ p4111_1). Phys. Rev. E 61:4111–4117. 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 (1989). "An electron trapped in a rotating magnetic field" (http:/ / scitation. aip. org/ getabs/servlet/ GetabsServlet?prog=normal& id=JMAPAQ000030000002000537000001& idtype=cvips& gifs=yes).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 the Theories of Aether and Electricity (http:/ / www. archive. org/ details/

historyoftheorie00whitrich). Dover Publications. p. 34. ISBN 0-486-26126-3.

Magnetic field 137

External links

Information

• Crowell, B., " Electromagnetism (http:/ / www. lightandmatter. com/html_books/ 0sn/ ch11/ ch11. html)".

• Nave, R., " Magnetic Field (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/magnetic/ magfie. html)". HyperPhysics.

• "Magnetism", The Magnetic Field (http:/ / theory. uwinnipeg. ca/ physics/mag/ node2. html#SECTION00110000000000000000). theory.uwinnipeg.ca.

• Hoadley, Rick, " What do magnetic fields look like (http:/ / my. execpc. com/~rhoadley/ magfield. htm)?" 17 July 2005.

Field density

• Oppelt, Arnulf (2 November 2006). "magnetic field strength" (http:/ /searchsmb. techtarget. com/ sDefinition/ 0,290660,sid44_gci763586,00.html). Retrieved 04 June 2007.

• "magnetic field strength converter" (http:/ / www. unitconversion. org/unit_converter/ magnetic-field-strength. html). Retrieved 04 June 2007.

Rotating magnetic fields

• " Rotating magnetic fields (http:/ / www. tpub. com/ neets/book5/ 18a. htm)". Integrated Publishing.

• "Introduction to Generators and Motors", rotating magneticfield (http:/ / www. tpub. com/ content/ neets/ 14177/ css/14177_87. htm). Integrated Publishing.

• " Induction Motor – Rotating Fields (http:/ / web. archive.org/ web/ 20050929102550/ http:/ / www. egr. msu. edu/~jurkovi4/ Experiment4. pdf)".

Diagrams

• "AC Motor Theory" Figure 2 Rotating Magnetic Field (http:// www. tpub. com/ content/ doe/ h1011v4/ css/ h1011v4_23.htm). Integrated Publishing.

• "Magnetic Fields" Arc & Mitre Magnetic Field Diagrams(http:/ / www. first4magnets. com/ ekmps/ shops/ trainer27/resources/ Other/ magnetic-fields. pdf). Magnet Expert Ltd.

Magnetic monopole

It is impossible to make magnetic monopoles from abar magnet. If a bar magnet is cut in half, it is not thecase that one half has the north pole and the other half

has the south pole. Instead, each piece has its ownnorth and south poles. A magnetic monopole cannot be

created from normal matter such as atoms andelectrons, but would instead be a new elementary

particle.

A magnetic monopole is a hypothetical particle in particle physicsthat is an isolated magnet with only one magnetic pole (a northpole without a south pole or vice-versa).[1][2] In more technicalterms, a magnetic monopole would have a net "magnetic charge".Modern interest in the concept stems from particle theories,notably the grand unified and superstring theories, which predicttheir existence.[3][4]

Magnetism in bar magnets and electromagnets does not arise frommagnetic monopoles, and in fact there is no conclusiveexperimental evidence that magnetic monopoles exist at all in theuniverse.

Effective (non-isolated) magnetic monopole quasi-particles existin some condensed matter systems.

Historical background

Pre-twentieth century

Many early scientists attributed the magnetism of lodestones totwo different "magnetic fluids" ("effluvia"), a north-pole fluid at one end and a south-pole fluid at the other, whichattracted and repelled each other in analogy to positive and negative electric charge.[5][6] However, an improved

understanding of electromagnetism in the nineteenth century showed that the magnetism of lodestones was properly explained by Ampère's circuital law, not magnetic monopole fluids. It was concluded that magnetic monopoles did

Magnetic monopole 138

not exist: One of Maxwell's equations, now called Gauss's law for magnetism, is the mathematical statement thatthere are no magnetic monopoles. Nevertheless, it was pointed out by Pierre Curie in 1894[7] that magneticmonopoles could conceivably exist, despite not having been seen so far.

Twentieth centuryThe quantum theory of magnetic charge started with a paper by the physicist Paul A.M. Dirac in 1931.[8] In thispaper, Dirac showed that if any magnetic monopoles exist in the universe, then all electric charge in the universemust be quantized.[9] The electric charge is, in fact, quantized, which is consistent with (but does not prove) theexistence of monopoles.[9]

Since Dirac's paper, several systematic monopole searches have been performed. Experiments in 1975[10] and1982[11] produced candidate events that were initially interpreted as monopoles, but are now regarded asinconclusive.[12] Therefore, it remains an open question whether or not monopoles exist.Further advances in theoretical particle physics, particularly developments in grand unified theories and quantumgravity, have led to more compelling arguments that monopoles do exist. Joseph Polchinski, a string-theorist,described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen".[13]

These theories are not necessarily inconsistent with the experimental evidence. In some theoretical models, magneticmonopoles are unlikely to be observed, because they are too massive to be created in particle accelerators, and alsotoo rare in the Universe to enter a particle detector with much probability.[13]

Some condensed matter systems propose a structure superficially similar to a magnetic monopole, known as a fluxtube. The ends of a flux tube form a magnetic dipole, but since they move independently, they can be treated formany purposes as independent magnetic monopole quasiparticles. Since 2009, numerous news reports from thepopular media have incorrectly described these systems as the long-awaited discovery of the magnetic monopoles,but the two phenomena are only superficially related to one another.[14] These condensed-matter systems continue tobe an area of active research. (See "Monopoles" in condensed-matter systems below.)

Poles and magnetism in ordinary matterAll matter ever isolated to date—including every atom on the periodic table and every particle in the standardmodel—has no magnetic monopole charge. Therefore, the ordinary phenomena of magnetism and magnets havenothing to do with magnetic monopoles.Instead, magnetism in ordinary matter comes from two sources. First, electric currents create magnetic fieldsaccording to Ampère's law. Second, many elementary particles have an "intrinsic" magnetic moment, the mostimportant of which is the electron magnetic dipole moment. (This magnetism is related to quantum-mechanical"spin".)Mathematically, the magnetic field of an object is often described in terms of a multipole expansion. This is anexpression of the field as a superposition (sum) of component fields with specific mathematical forms. The first termin the expansion is called the "monopole" term, the second is called "dipole", then "quadrupole", then "octupole",and so on. Any of these terms can be present in the multipole expansion of an electric field, for example. However,in the multipole expansion of a magnetic field, the "monopole" term is always exactly zero (for ordinary matter). Amagnetic monopole, if it exists, would have the defining property of producing a magnetic field whose "monopole"term is nonzero.A magnetic dipole is something whose magnetic field is predominantly or exactly described by the magnetic dipole term of the multipole expansion. The term "dipole" means "two poles", corresponding to the fact that a dipole magnet typically contains a "north pole" on one side and a "south pole" on the other side. This is analogous to an electric dipole, which has positive charge on one side and negative charge on the other. However, an electric dipole and magnetic dipole are fundamentally quite different. In an electric dipole, the positive charge is made of protons

Magnetic monopole 139

and the negative charge is made of electrons, but a magnetic dipole does not have different types of matter creatingthe north pole and south pole. Instead, the two magnetic poles arise simultaneously from the aggregate effect of allthe currents and intrinsic moments throughout the magnet. Because of this, the two poles of a magnetic dipole mustalways have equal and opposite strength, and the two poles cannot be separated from each other.

Maxwell's equationsMaxwell's equations of electromagnetism relate the electric and magnetic fields to each other and to the motions ofelectric charges. The standard equations provide for electric charges, but they posit no magnetic charges. Except forthis difference, the equations are symmetric under the interchange of the electric and magnetic fields.[15] In fact,symmetric Maxwell's equations can be written when all charges (and hence electric currents) are zero, and this ishow the electromagnetic wave equation is derived.Fully symmetric Maxwell's equations can also be written if one allows for the possibility of "magnetic charges"analogous to electric charges.[16] With the inclusion of a variable for the density of these magnetic charges, say ρm,there will also be a "magnetic current density" variable in the equations, jm.If magnetic charges do not exist – or if they do exist but are not present in a region of space – then the new terms inMaxwell's equations are all zero, and the extended equations reduce to the conventional equations ofelectromagnetism such as ∇•B = 0 (where ∇• is divergence and B is the magnetic B field).

In Gaussian cgs unitsThe extended Maxwell's equations are as follows, in Gaussian cgs units:[17]

Maxwell's equations and Lorentz force equation with magnetic monopoles: Gaussian cgsunits

Name Without magnetic monopoles With magnetic monopoles

Gauss's law

Gauss's law for magnetism

Faraday's law of induction

Ampère's law (with Maxwell's extension)

Lorentz force law[17][18]

In these equations ρm is the magnetic charge density, jm is the magnetic current density, and qm is the magneticcharge of a test particle, all defined analogously to the related quantities of electric charge and current; v is theparticle's velocity and c is the speed of light. For all other definitions and details, see Maxwell's equations. For theequations in nondimensionalized form, remove the factors of c.

Magnetic monopole 140

In SI unitsIn SI units, there are two conflicting units in use for magnetic charge qm: webers (Wb) and ampere·meters (A·m).The conversion between them is qm(Wb) = μ0qm(A·m), since the units are 1 Wb = 1 H·A = (1 H·m−1)·(1 A·m) bydimensional analysis (H is the Henry – the SI unit of inductance).Maxwell's equations then take the following forms (using the same notation above):[19]

Maxwell's equations and Lorentz force equation with magnetic monopoles: SI units

Name Without magnetic monopoles Weber convention Ampere·meter convention

Gauss's Law

Gauss's Law for magnetism

Faraday's Law of induction

Ampère's Law (with Maxwell's extension)

Lorentz force equation

Dirac's quantizationOne of the defining advances in quantum theory was Paul Dirac's work on developing a relativistic quantumelectromagnetism. Before his formulation, the presence of electric charge was simply "inserted" into the equations ofquantum mechanics (QM), but in 1931 Dirac showed that a discrete charge naturally "falls out" of QM. That is tosay, we can maintain the form of Maxwell's equations and still have magnetic charges.Consider a system consisting of a single stationary electric monopole (an electron, say) and a single stationarymagnetic monopole. Classically, the electromagnetic field surrounding them has a momentum density given by thePoynting vector, and it also has a total angular momentum, which is proportional to the product qeqm, andindependent of the distance between them.Quantum mechanics dictates, however, that angular momentum is quantized in units of ħ, so therefore the productqeqm must also be quantized. This means that if even a single magnetic monopole existed in the universe, and theform of Maxwell's equations is valid, all electric charges would then be quantized.What are the units in which magnetic charge would be quantized? Although it would be possible simply to integrateover all space to find the total angular momentum in the above example, Dirac took a different approach. This ledhim to new ideas. He considered a point-like magnetic charge whose magnetic field behaves as qm / r 2 and isdirected in the radial direction, located at the origin. Because the divergence of B is equal to zero almost everywhere,except for the locus of the magnetic monopole at r = 0, one can locally define the vector potential such that the curlof the vector potential A equals the magnetic field B.However, the vector potential cannot be defined globally precisely because the divergence of the magnetic field isproportional to the Dirac delta function at the origin. We must define one set of functions for the vector potential onthe "northern hemisphere" (the half-space z > 0 above the particle), and another set of functions for the "southernhemisphere". These two vector potentials are matched at the "equator" (the plane z = 0 through the particle), and theydiffer by a gauge transformation. The wave function of an electrically-charged particle (a "probe charge") that orbitsthe "equator" generally changes by a phase, much like in the Aharonov–Bohm effect. This phase is proportional tothe electric charge qe of the probe, as well as to the magnetic charge qm of the source. Dirac was originallyconsidering an electron whose wave function is described by the Dirac equation.

Magnetic monopole 141

Because the electron returns to the same point after the full trip around the equator, the phase φ of its wave functionexp(iφ) must be unchanged, which implies that the phase φ added to the wave function must be a multiple of 2π:

Units Condition

Gaussian-cgs units

SI units (weber convention)[20]

SI units (ampere·meter convention)

where ε0 is the vacuum permittivity, ħ = h/2π is the reduced Planck's constant, c is the speed of light, and ℤ is the setof integers.This is known as the Dirac quantization condition. The hypothetical existence of a magnetic monopole wouldimply that the electric charge must be quantized in certain units; also, the existence of the electric charges impliesthat the magnetic charges of the hypothetical magnetic monopoles, if they exist, must be quantized in units inverselyproportional to the elementary electric charge.At the time it was not clear if such a thing existed, or even had to. After all, another theory could come along thatwould explain charge quantization without need for the monopole. The concept remained something of a curiosity.However, in the time since the publication of this seminal work, no other widely accepted explanation of chargequantization has appeared. (The concept of local gauge invariance—see gauge theory below—provides a naturalexplanation of charge quantization, without invoking the need for magnetic monopoles; but only if the U(1) gaugegroup is compact, in which case we will have magnetic monopoles anyway.)If we maximally extend the definition of the vector potential for the southern hemisphere, it will be definedeverywhere except for a semi-infinite line stretched from the origin in the direction towards the northern pole. Thissemi-infinite line is called the Dirac string and its effect on the wave function is analogous to the effect of thesolenoid in the Aharonov–Bohm effect. The quantization condition comes from the requirement that the phasesaround the Dirac string are trivial, which means that the Dirac string must be unphysical. The Dirac string is merelyan artifact of the coordinate chart used and should not be taken seriously.The Dirac monopole is a singular solution of Maxwell's equation (because it requires removing the worldline fromspacetime); in more complicated theories, it is superseded by a smooth solution such as the 't Hooft–Polyakovmonopole.

Topological interpretation

Dirac stringA gauge theory like electromagnetism is defined by a gauge field, which associates a group element to each path inspace time. For infinitesimal paths, the group element is close to the identity, while for longer paths the groupelement is the successive product of the infinitesimal group elements along the way.In electrodynamics, the group is U(1), unit complex numbers under multiplication. For infinitesimal paths, the groupelement is 1 + iAμdxμ which implies that for finite paths parametrized by s, the group element is:

The map from paths to group elements is called the Wilson loop or the holonomy, and for a U(1) gauge group it isthe phase factor which the wavefunction of a charged particle acquires as it traverses the path. For a loop:

Magnetic monopole 142

So that the phase a charged particle gets when going in a loop is the magnetic flux through the loop. When a smallsolenoid has a magnetic flux, there are interference fringes for charged particles which go around the solenoid, oraround different sides of the solenoid, which reveal its presence.But if all particle charges are integer multiples of e, solenoids with a flux of 2π/e have no interference fringes,because the phase factor for any charged particle is e2πi = 1. Such a solenoid, if thin enough, isquantum-mechanically invisible. If such a solenoid were to carry a flux of 2π/e, when the flux leaked out from oneof its ends it would be indistinguishable from a monopole.Dirac's monopole solution in fact describes an infinitesimal line solenoid ending at a point, and the location of thesolenoid is the singular part of the solution, the Dirac string. Dirac strings link monopoles and antimonopoles ofopposite magnetic charge, although in Dirac's version, the string just goes off to infinity. The string is unobservable,so you can put it anywhere, and by using two coordinate patches, the field in each patch can be made nonsingular bysliding the string to where it cannot be seen.

Grand unified theoriesIn a U(1) gauge group with quantized charge, the group is a circle of radius 2π/e. Such a U(1) gauge group is calledcompact. Any U(1) which comes from a Grand Unified Theory is compact – because only compact higher gaugegroups make sense. The size of the gauge group is a measure of the inverse coupling constant, so that in the limit of alarge-volume gauge group, the interaction of any fixed representation goes to zero.The case of the U(1) gauge group is a special case because all its irreducible representations are of the same size –the charge is bigger by an integer amount, but the field is still just a complex number – so that in U(1) gauge fieldtheory it is possible to take the decompactified limit with no contradiction. The quantum of charge becomes small,but each charged particle has a huge number of charge quanta so its charge stays finite. In a non-compact U(1) gaugegroup theory, the charges of particles are generically not integer multiples of a single unit. Since charge quantizationis an experimental certainty, it is clear that the U(1) gauge group of electromagnetism is compact.GUTs lead to compact U(1) gauge groups, so they explain charge quantization in a way that seems to be logicallyindependent from magnetic monopoles. However, the explanation is essentially the same, because in any GUTwhich breaks down into a U(1) gauge group at long distances, there are magnetic monopoles.The argument is topological:1.1. The holonomy of a gauge field maps loops to elements of the gauge group. Infinitesimal loops are mapped to

group elements infinitesimally close to the identity.2. If you imagine a big sphere in space, you can deform an infinitesimal loop which starts and ends at the north pole

as follows: stretch out the loop over the western hemisphere until it becomes a great circle (which still starts andends at the north pole) then let it shrink back to a little loop while going over the eastern hemisphere. This iscalled lassoing the sphere.

3.3. Lassoing is a sequence of loops, so the holonomy maps it to a sequence of group elements, a continuous path inthe gauge group. Since the loop at the beginning of the lassoing is the same as the loop at the end, the path in thegroup is closed.

4.4. If the group path associated to the lassoing procedure winds around the U(1), the sphere contains magneticcharge. During the lassoing, the holonomy changes by the amount of magnetic flux through the sphere.

5. Since the holonomy at the beginning and at the end is the identity, the total magnetic flux is quantized. Themagnetic charge is proportional to the number of windings N, the magnetic flux through the sphere is equal to2πN/e. This is the Dirac quantization condition, and it is a topological condition which demands that the longdistance U(1) gauge field configurations be consistent.

6. When the U(1) gauge group comes from breaking a compact Lie group, the path which winds around the U(1) group enough times is topologically trivial in the big group. In a non-U(1) compact Lie group, the covering space is a Lie group with the same Lie algebra, but where all closed loops are contractible. Lie groups are homogenous,

Magnetic monopole 143

so that any cycle in the group can be moved around so that it starts at the identity, then its lift to the coveringgroup ends at P, which is a lift of the identity. Going around the loop twice gets you to P2, three times to P3, alllifts of the identity. But there are only finitely many lifts of the identity, because the lifts can't accumulate. Thisnumber of times one has to traverse the loop to make it contractible is small, for example if the GUT group isSO(3), the covering group is SU(2), and going around any loop twice is enough.

7. This means that there is a continuous gauge-field configuration in the GUT group allows the U(1) monopoleconfiguration to unwind itself at short distances, at the cost of not staying in the U(1). In order to do this with aslittle energy as possible, you should leave only the U(1) gauge group in the neighborhood of one point, which iscalled the core of the monopole. Outside the core, the monopole has only magnetic field energy.

Hence, the Dirac monopole is a topological defect in a compact U(1) gauge theory. When there is no GUT, thedefect is a singularity — the core shrinks to a point. But when there is some sort of short-distance regulator on spacetime, the monopoles have a finite mass. Monopoles occur in lattice U(1), and there the core size is the lattice size. Ingeneral, they are expected to occur whenever there is a short-distance regulator.

String theoryIn our universe, quantum gravity provides the regulator. When gravity is included, the monopole singularity can be ablack hole, and for large magnetic charge and mass, the black hole mass is equal to the black hole charge, so that themass of the magnetic black hole is not infinite. If the black hole can decay completely by Hawking radiation, thelightest charged particles cannot be too heavy. The lightest monopole should have a mass less than or comparable toits charge in natural units.So in a consistent holographic theory, of which string theory is the only known example, there are always finite-massmonopoles. For ordinary electromagnetism, the mass bound is not very useful because it is about same size as thePlanck mass.

Mathematical formulationIn mathematics, a gauge field is defined as a connection over a principal G-bundle over spacetime. G is the gaugegroup, and it acts on each fiber of the bundle separately.A connection on a G bundle tells you how to glue F's together at nearby points of M. It starts with a continuoussymmetry group G which acts on F, and then it associates a group element with each infinitesimal path. Groupmultiplication along any path tells you how to move from one point on the bundle to another, by acting the Gelement of a path on the fiber F.In mathematics, the definition of bundle is designed to emphasize topology, so the notion of connection is added onas an afterthought. In physics, the connection is the fundamental physical object. One of the fundamentalobservations in the theory of characteristic classes in algebraic topology is that many homotopical structures ofnontrivial principal bundles may be expressed as an integral of some polynomial over any connection over it. Notethat any connection over a trivial bundle can never give us a nontrivial principal bundle.If space time has no topology, if it is R4 the space of all possible connections of the G-bundle is connected. Butconsider what happens when we remove a timelike worldline from spacetime. The resulting spacetime ishomotopically equivalent to the topological sphere S2.A principal G-bundle over S2 is defined by covering S2 by two charts, each homeomorphic to the open 2-ball suchthat their intersection is homeomorphic to the strip S1×I. 2-balls are homotopically trivial and the strip ishomotopically equivalent to the circle S1. So a topological classification of the possible connections is reduced toclassifying the transition functions. The transition function maps the strip to G, and the different ways of mapping astrip into G are given by the first homotopy group of G.

Magnetic monopole 144

So in the G-bundle formulation, a gauge theory admits Dirac monopoles provided G is not simply connected,whenever there are paths that go around the group that cannot be deformed to nothing. U(1), which has quantizedcharges, is not simply connected and can have Dirac monopoles while R, its universal covering group, is simplyconnected, doesn't have quantized charges and does not admit Dirac monopoles. The mathematical definition isequivalent to the physics definition provided that, following Dirac, gauge fields are allowed which are defined onlypatch-wise and the gauge field on different patches are glued after a gauge transformation.The total magnetic flux is none other than the first Chern number of the principal bundle, and depends only upon thechoice of the principal bundle, and not the specific connection over it. In other words, it's a topological invariant.This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to d + 1dimensions with d ≥ 2 in several ways. One way is to extend everything into the extra dimensions, so that U(1)monopoles become sheets of dimension d − 3. Another way is to examine the type of topological singularity at apoint with the homotopy group πd − 2(G).

Grand unified theoriesIn more recent years, a new class of theories has also suggested the existence of magnetic monopoles.During the early 1970s, the successes of quantum field theory and gauge theory in the development of electroweaktheory and the mathematics of the strong nuclear force led many theorists to move on to attempt to combine them ina single theory known as a Grand Unified Theory (GUT). Several GUTs were proposed, most of which had thecurious feature of implying the presence of a real magnetic monopole particle. More accurately, GUTs predicted arange of particles known as dyons, of which the most basic state was a monopole. The charge on magneticmonopoles predicted by GUTs is either 1 or 2 gD, depending on the theory.The majority of particles appearing in any quantum field theory are unstable, and they decay into other particles in avariety of reactions that must satisfy various conservation laws. Stable particles are stable because there are nolighter particles into which they can decay and still satisfy the conservation laws. For instance, the electron has alepton number of one and an electric charge of one, and there are no lighter particles that conserve these values. Onthe other hand, the muon, essentially a heavy electron, can decay into the electron plus two quanta of energy, andhence it is not stable.The dyons in these GUTs are also stable, but for an entirely different reason. The dyons are expected to exist as aside effect of the "freezing out" of the conditions of the early universe, or a symmetry breaking. In this scenario, thedyons arise due to the configuration of the vacuum in a particular area of the universe, according to the originalDirac theory. They remain stable not because of a conservation condition, but because there is no simpler topologicalstate into which they can decay.The length scale over which this special vacuum configuration exists is called the correlation length of the system. Acorrelation length cannot be larger than causality would allow, therefore the correlation length for making magneticmonopoles must be at least as big as the horizon size determined by the metric of the expanding universe. Accordingto that logic, there should be at least one magnetic monopole per horizon volume as it was when the symmetrybreaking took place. Other arguments based on the critical density of the universe indicate that monopoles should befairly common; the apparent problem of the observed scarcity of monopoles is resolved by cosmic inflation in theearly universe, which greatly reduces the expected abundance of magnetic monopoles. For these reasons, monopolesbecame a major interest in the 1970s and 80s, along with the other "approachable" predictions of GUTs such asproton decay.Many of the other particles predicted by these GUTs were beyond the abilities of current experiments to detect. Forinstance, a wide class of particles known as the X and Y bosons are predicted to mediate the coupling of theelectroweak and strong forces, but these particles are extremely heavy and well beyond the capabilities of anyreasonable particle accelerator to create.

Magnetic monopole 145

Searches for magnetic monopolesA number of attempts have been made to detect magnetic monopoles. One of the simpler ones is to use a loop ofsuperconducting wire to look for even tiny magnetic sources, a so-called "superconducting quantum interferencedevice", or SQUID. Given the predicted density, loops small enough to fit on a lab bench would expect to see aboutone monopole event per year. Although there have been tantalizing events recorded, in particular the event recordedby Blas Cabrera on the night of February 14, 1982 (thus, sometimes referred to as the "Valentine's DayMonopole"[21]), there has never been reproducible evidence for the existence of magnetic monopoles.[11] The lack ofsuch events places a limit on the number of monopoles of about one monopole per 1029 nucleons.Another experiment in 1975 resulted in the announcement of the detection of a moving magnetic monopole incosmic rays by the team led by P. Buford Price.[10] Price later retracted his claim, and a possible alternativeexplanation was offered by Alvarez.[22] In his paper it was demonstrated that the path of the cosmic ray event thatwas claimed to have been be due to a magnetic monopole could be reproduced by the path followed by a platinumnucleus decaying first to osmium, and then to tantalum.Other experiments rely on the strong coupling of monopoles with photons, as is the case for any electrically-chargedparticle as well. In experiments involving photon exchange in particle accelerators, monopoles should be produced inreasonable numbers, and detected due to their effect on the scattering of the photons. The probability of a particlebeing created in such experiments is related to their mass — with heavier particles being less likely to be created —so by examining the results of such experiments, limits on the mass of a magnetic monopole can be calculated. Themost recent such experiments suggest that monopoles with masses below 600 GeV/c2 do not exist, while upper limitson their mass due to the very existence of the universe – which would have collapsed by now if they were too heavy– are about 1017 GeV/c2.The MoEDAL experiment, installed at the Large Hadron Collider, is currently searching for magnetic monopolesand large supersymmetric particles using layers of special plastic sheets attached to the walls around LHCb's VELOdetector. The particles it is looking for will damage the sheets along their path, with various identifying features.

"Monopoles" in condensed-matter systemsWhile the (currently understood) laws of physics (specifically the law ∇⋅B = 0) forbid the existence of monopoles inB, no such restriction applies to the magnetic H field when defined macroscopically. As a result, while all knownparticles (including the protons, neutrons, and electrons that make up the periodic table) have zero magnetic charge,the phenomenon of fractionalization can lead to quasiparticles that are monopoles of H. There are indeed a numberof examples in condensed-matter physics where collective behavior leads to emergent phenomena that resemblemagnetic monopoles in certain respects,[23][24][25] including most prominently the spin ice materials.[26][27] Whilethese should not be confused with hypothetical elementary monopoles existing in the vacuum, they nonetheless havesimilar properties and can be probed using similar techniques.One example of the work on magnetic monopole quasiparticles is a paper published in the journal Science inSeptember 2009, in which researchers Jonathan Morris and Alan Tennant from the Helmholtz-Zentrum Berlin fürMaterialien und Energie (HZB) along with Santiago Grigera from Instituto de Física de Líquidos y SistemasBiológicos (IFLYSIB, CONICET) and other colleagues from Dresden University of Technology, University of St.Andrews and Oxford University described the observation of quasiparticles resembling magnetic monopoles. Asingle crystal of the spin ice material dysprosium titanate was cooled to a temperature between 0.6 kelvin and 2.0kelvin. Using observations of neutron scattering, the magnetic moments were shown to align into interwoventubelike bundles resembling Dirac strings. At the defect formed by the end of each tube, the magnetic field looks likethat of a monopole. Using an applied magnetic field to break the symmetry of the system, the researchers were ableto control the density and orientation of these strings. A contribution to the heat capacity of the system from aneffective gas of these quasiparticles was also described.[28][29]

Magnetic monopole 146

Another example is a paper in the February 11, 2011 issue of Nature Physics which describes creation andmeasurement of long-lived magnetic monopole quasiparticle currents in spin ice. By applying a magnetic-field pulseto crystal of dysprosium titanate at 0.36 K, the authors created a relaxing magnetic current that lasted for severalminutes. They measured the current by means of the electromotive force it induced in a solenoid coupled to asensitive amplifier, and quantitatively described it using a chemical kinetic model of point-like charges obeying theOnsager–Wien mechanism of carrier dissociation and recombination. They thus derived the microscopic parametersof monopole motion in spin ice and identified the distinct roles of free and bound magnetic charges.[30] This researchwent onto win the 2012 Europhysics Prize for condensed matter physics

AppendixIn physics the phrase "magnetic monopole" usually denoted a Yang–Mills potential A and Higgs field ϕ whoseequations of motion are determined by the Yang–Mills action

In mathematics, the phrase custumarily refers to a static solution to these equation in theBogomolny–Parasad–Sommerfeld limit λ → ϕ which realizes, within topological class, the absolutes minimum ofthe functional

This means that it in a connection A on a principal G-bundle over R3 (c.f. also Connections on a manifold; principalG-object) and a section ϕ of the associated adjoint bundle of Lie algebras such that the curvature FA and covariantderivative DA ϕ satisfy the Bogomolny equations

and the boundary conditions.

Pure mathematical advances in the theory of monopoles from the 1980's onwards have often proceeded on the basisof physically motived questions.The equations themselves are invariant under gauge transformation and orientation-preserving symmetries. When γis large, ϕ/||ϕ|| defines a mapping from a 2-sphere of radius γ in R3 to an adjoint orbit G/k and the homotopy class ofthis mapping is called the magnetic charge. Most work has been done in the case G = SU(2), where the charge is apositive integer k. The absolute minimum value of the functional is then 8πk and the coefficient m in the asymptoticexpansion of ϕ/||ϕ|| is k/2.The first SU(2) solution was found by E. B. Bogomolny, J. K. Parasad and C. M. Sommerfield in 1975. It isspherically symmetric of charge 1 and has the form

In 1980, C.H.Taubes[31] showed by a gluing construction that there exist solutions for all large k and soon afterexplicit axially-symmetric solutions were found. The first exact solution in the general case was given in 1981 byR.S.Ward for in terms of elliptic function.There are two ways of solving the Bogomolny equations. The first is by twistor methods. In the formulation ofN.J.Hitchin[32], an arbitrary solution corresponds to a holomorphic vector bundle over the complex surface TP1, thetangent bundle of the projective line. This is naturally isomorphic to the space of oriented straight lines in R3.

Magnetic monopole 147

The boundary condition show that the holomorphic bundle is an extension of line bundles determined by a compactalgebraic curve of genus (k − 1)2 (the spectral curve) in TP1, satisfying certain constraints.The second method, due to W.Nahm[33], involves solving an eigen value problem for the coupled Dirac operator andtransforming the equations with their boundary conditions into a system of ordinary differential equations, the Nahmequations.

where Ti(s) is a k×k -matrix valued function on (0,2).Both constructions are based on analogous procedures for instantons, the key observation due to N.S.Manton beingof the self-dual Yang–Mills equations (c.f. also Yang–Mills field) in R4.The equivalence of the two methods for SU(2) and their general applicability was established in [34] (see also [35]).Explicit formulas for A and are difficult to obtain by either method, despite some exact solutions of Nahm'sequations in symmetric situations [36].The case of a more general Lie group G, where the stabilizer of ϕ at infinity is a maximal torus, was treated byM.K.Murray [37] from the twistor point of view, where the single spectral curve of an SU(2)-monopole is replaced bya collection of curves indexed by the vortices of the Dynkin diagram of G. The corresponding Nahm constructionwas designed by J.Hustubise and Murray [38].The moduli space (c.f. also Moduli theory) of all SU(2) monopoles of charge k up to gauge equivalence was shownby Taubes[39] to be a smooth non-compact manifold fo dimension 4k − 1. Restricting to gauge transformations thatpreserve the connection at infinity gives a 4k-dimensional manifold Mk, which is a circle bundle over the true modulispace and carries a natural complete hyperKähler metric [40] (c.f. also Kähler–Einstein manifold). With suspected toany of the complex structures of the hyper-Kähler family, this manifold is holomorphically equivalent to the space ofbased rational mapping of degree k from P1 to itself [41].The metric is known in twistor terms [42], and its Kähler potential can be written using the Riemann theta functionsof the spectral curve[43], but only the case k = 2 is known in a more conventional and usable form[44] (as of 2000).This Atiyah–Hitchin manifold, the Einstein Taub-NUT metric and R4 are the only 4-dimensional completehyperKähler manifolds with a non-triholomorphic SU(2) action. Its geodesics have been studied and a programme ofManton concerning monopole dynamics put into effect. Further dynamical features have been elucidated bynumerical and analytical techniques.

A cyclic k-fold conering of Mk splits isometrically us a product , where is the space ofstrongly centred monopoles. This space features in an application of S-duality in theoretical physics, and in [45]

G.B.Segal and A.Selby studied its topology and the L2 harmonic forms defined on it, partially confirming thephysical prediction.Magnetic monopole on hyperbolic three-space were investigated from the twistor point of view b M.F.Atiyah [46]

(replacing the complex surface TP1 by the comoplement of the anti-diagonal in P1 × P1) and in terms of discreteNahm equations by Murray and M.A.Singer [47].

Magnetic monopole 148

Notes[1] Dark Cosmos: In Search of Our Universe's Missing Mass and Energy, by Dan Hooper, p192 (http:/ / books. google. com/

books?id=tGBUvLpgmUMC& pg=PA192)[2] Particle Data Group summary of magnetic monopole search (http:/ / pdg. lbl. gov/ 2004/ listings/ s028. pdf)[3] Wen, Xiao-Gang; Witten, Edward, Electric and magnetic charges in superstring models,Nuclear Physics B, Volume 261, p. 651–677[4] S. Coleman, The Magnetic Monopole 50 years Later, reprinted in Aspects of Symmetry[5] The encyclopædia britannica, Volume 17, p352 (http:/ / books. google. com/ books?id=N1YEAAAAYAAJ& pg=PA352)[6] Principles of Physics by William Francis Magie, p424 (http:/ / books. google. com/ books?id=6rYXAAAAIAAJ& pg=PA424)[7] Pierre Curie, Sur la possibilité d'existence de la conductibilité magnétique et du magnétisme libre (On the possible existence of magnetic

conductivity and free magnetism), Séances de la Société Française de Physique (Paris), p76 (1894). (French) Free access online copy (http:// www. archive. org/ stream/ sancesdelasocit19physgoog).

[8] Paul Dirac, "Quantised Singularities in the Electromagnetic Field". Proc. Roy. Soc. (London) A 133, 60 (1931). Free web link (http:/ / users.physik. fu-berlin. de/ ~kleinert/ files/ dirac1931. pdf).

[9] Lecture notes by Robert Littlejohn (http:/ / bohr. physics. berkeley. edu/ classes/ 221/ 0708/ lectures/ Lecture. 2007. 10. 11. pdf), Universityof California, Berkeley, 2007–8

[10] P. B. Price; E. K. Shirk; W. Z. Osborne; L. S. Pinsky (25 August 1975). "Evidence for Detection of a Moving Magnetic Monopole".Physical Review Letters (American Physical Society) 35 (8): 487–490. Bibcode 1975PhRvL..35..487P. doi:10.1103/PhysRevLett.35.487.

[11] Blas Cabrera (17 May 1982). "First Results from a Superconductive Detector for Moving Magnetic Monopoles". Physical Review Letters(American Physical Society) 48 (20): 1378–1381. Bibcode 1982PhRvL..48.1378C. doi:10.1103/PhysRevLett.48.1378.

[12][12] Milton p.60[13] Polchinski, arXiv 2003 (http:/ / arxiv. org/ abs/ hep-th/ 0304042)[14] Magnetic monopoles spotted in spin ices (http:/ / physicsworld. com/ cws/ article/ news/ 40302), 3 September 2009. "Oleg Tchernyshyov at

Johns Hopkins University [a researcher in this field] cautions that the theory and experiments are specific to spin ices, and are not likely toshed light on magnetic monopoles as predicted by Dirac."

[15] The fact that the electric and magnetic fields can be written in a symmetric way is specific to the fact that space is three-dimensional. Whenthe equations of electromagnetism are extrapolated to other dimensions, the magnetic field is described as being a rank-two antisymmetrictensor, whereas the electric field remains a true vector. In dimensions other than three, these two mathematical objects do not have the samenumber of components.

[16] http:/ / www. ieeeghn. org/ wiki/ index. php/ STARS:Maxwell%27s_Equations[17] F. Moulin (2001). "Magnetic monopoles and Lorentz force". Nuovo Cimento B 116 (8): 869–877. arXiv:math-ph/0203043.

Bibcode 2001NCimB.116..869M.[18] Wolfgang Rindler (November 1989). "Relativity and electromagnetism: The force on a magnetic monopole". American Journal of Physics

(American Journal of Physics) 57 (11): 993–994. Bibcode 1989AmJPh..57..993R. doi:10.1119/1.15782.[19] For the convention where magnetic charge has units of webers, see Jackson 1999. In particular, for Maxwell's equations, see section 6.11,

equation (6.150), page 273, and for the Lorentz force law, see page 290, exercise 6.17(a). For the convention where magnetic charge has unitsof ampere-meters, see (for example) arXiv:physics/0508099v1 (http:/ / arxiv. org/ abs/ physics/ 0508099v1), eqn (4).

[20][20] Jackson 1999, section 6.11, equation (6.153), page 275[21] http:/ / www. nature. com/ nature/ journal/ v429/ n6987/ full/ 429010a. html[22] Alvarez, Luis W. "Analysis of a Reported Magnetic Monopole" (http:/ / usparc. ihep. su/ spires/ find/ hep/ www?key=93726). In ed. Kirk,

W. T.. Proceedings of the 1975 international symposium on lepton and photon interactions at high energies. International symposium onlepton and photon interactions at high energies, 21 Aug 1975. pp. 967. .

[23] Zhong, Fang; Naoto Nagosa, Mei S. Takahashi, Atsushi Asamitsu, Roland Mathieu, Takeshi Ogasawara, Hiroyuki Yamada, MasashiKawasaki, Yoshinori Tokura, Kiyoyuki Terakura (October 3, 2003). "The Anomalous Hall Effect and Magnetic Monopoles in MomentumSpace". Science 302 (5642): 92–95. doi:10.1126/science.1089408. ISSN 1095-9203. http:/ / www. sciencemag. org/ cgi/ content/ abstract/302/ 5642/ 92. Retrieved on 2 August 2007.

[24] Making magnetic monopoles, and other exotica, in the lab (http:/ / www. symmetrymagazine. org/ breaking/ 2009/ 01/ 29/making-magnetic-monopoles-and-other-exotica-in-the-lab/ ), Symmetry Breaking, 29 January 2009. Retrieved 31 January 2009.

[25] Inducing a Magnetic Monopole with Topological Surface States (http:/ / www. sciencemag. org/ cgi/ content/ abstract/ 1167747), AmericanAssociation for the Advancement of Science (AAAS) Science Express magazine, Xiao-Liang Qi, Rundong Li, Jiadong Zang, Shou-ChengZhang, 29 January 2009. Retrieved 31 January 2009.

[26] Magnetic monopoles in spin ice (http:/ / dx. doi. org/ 10. 1038/ nature06433), C. Castelnovo, R. Moessner and S. L. Sondhi, Nature 451,42–45 (3 January 2008)

[27] Nature 461, 956–959 (15 October 2009); (http:/ / www. nature. com/ nature/ journal/ v461/ n7266/ abs/ nature08500. html)doi:10.1038/nature08500, Steven Bramwell et al

[28] "Magnetic Monopoles Detected In A Real Magnet For The First Time" (http:/ / www. sciencedaily. com/ releases/ 2009/ 09/ 090903163725.htm). Science Daily. 4 September 2009. . Retrieved 4 September 2009.

[29] D.J.P. Morris, D.A. Tennant, S.A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czter-nasty, M. Meissner, K.C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R.S. Perry (3 September 2009). "Dirac Strings and Magnetic Monopoles in Spin Ice

Magnetic monopole 149

Dy2Ti2O7". Science. arXiv:1011.1174. Bibcode 2009Sci...326..411M. doi:10.1126/science.1178868. PMID 19729617.[30] S. R. Giblin, S. T. Bramwell, P. C. W. Holdsworth, D. Prabhakaran & I. Terry (13 February 2011). Creation and measurement of long-lived

magnetic monopole currents in spin ice (http:/ / www. nature. com/ nphys/ journal/ v7/ n3/ full/ nphys1896. html). Nature Physics.Bibcode 2011NatPh...7..252G. doi:10.1038/nphys1896. . Retrieved 28 February 2011.

[31] A.Jaffe, C.H.Taubes (1980). Vortices and monopoles.[32] N.J.Hitchin (1982). Monopoles and geodesics.[33] W.Nahm (1982). The construction of all self-dual monopoles by the ADHM mothod.[34] N.J.Hitchin (1983). On the construction of monopoles.[35] N.J.Hitchin (1999). Integrable sustems in Riemannian geometry (K.Uhlenbeck ed.). C-L.Terng (ed.).[36] N.J.Hitchin, N.S.Manton, M.K.Murray (1995). Symmetric Monopoles.[37] M.K.Murray (1983). Monopoles and spectral curves for arbitrary Lie groups.[38] J.Hurtubise, M.K.Murray (1989). On the construction of Monopoles for the classical groups.[39] C.H.Taubes (1983). Stability in Yang–Mills theories.[40] M.F.Atiyah, N.J.Hitchin (1988). The geometry and dynamics of magnetic monopoles. Princeton Univ.Press.[41] S.K.Donaldson (1984). Nahm’s equations and the classification of monopoles.[42] M.F.Atiyah, N.J.Hitchin (1988). The geometry and dynamics of magnetic monopoles. Princeton Univ.Press.[43] N.J.Hitchin (1999). Integrable sustems in Riemannian geometry (K.Uhlenbeck ed.). C-L.Terng (ed.).[44] M.F.Atiyah, N.J.Hitchin (1988). The geometry and dynamics of magnetic monopoles. Princeton Univ.Press.[45] G.B.Segal, A.Selby (1996). The cohomology of the space of magnetic monopoles.[46] M.F.Atiyah (1987). Magnetic monopoles in hyperbolic space, Vector bundles on algebraic varieties. Oxford Univ.Press.[47] M.K.Murray (2000). On the complete integrability of the discrete Nahm equations.

• [4] N.J.Hitchin, M.K.Murray (1988). Spectral curves and the ADHM method.• [15] P.M.Sutcliffe (1997). BPS monopoles.

References• Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press.

ISBN 0-19-514665-4.• Jackson, John David (1999). Classical Electrodynamics (3rd ed.). New York: Wiley. ISBN 0-471-30932-X.• Milton, Kimball A. (June 2006). "Theoretical and experimental status of magnetic monopoles". Reports on

Progress in Physics 69 (6): 1637–1711. arXiv:hep-ex/0602040. Bibcode 2006RPPh...69.1637M.doi:10.1088/0034-4885/69/6/R02.

• Shnir, Yakov M. (2005). Magnetic Monopoles. Springer-Verlag. ISBN 3-540-25277-0.

External links• Magnetic Monopole Searches (lecture notes) (http:/ / arxiv. org/ abs/ hep-ex/ 0302011)• Particle Data Group summary of magnetic monopole search (http:/ / pdg. lbl. gov/ 2004/ listings/ s028. pdf)• 'Race for the Pole' Dr David Milstead (http:/ / www. vega. org. uk/ video/ programme/ 56) Freeview 'Snapshot'

video by the Vega Science Trust and the BBC/OU.• Interview with Jonathan Morris (http:/ / www. drillingsraum. com/ magnetic_monopole/ magnetic_monopole.

html) about magnetic monopoles and magnetic monopole quasiparticles. Drillingsraum, 16 April 2010This article incorporates material from N. Hitchin (2001), "Magnetic Monopole" (http:/ / www. encyclopediaofmath.org/ index. php?title=magnetic_monopole), in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4, which is licensed under the Creative Commons Attribution/Share-Alike License and GNUFree Documentation License.

Magnetic refrigeration 150

Magnetic refrigeration

Gadolinium alloy heats up inside the magnetic field and loses thermal energy to theenvironment, so it exits the field cooler than when it entered.

Magnetic refrigeration is a coolingtechnology based on themagnetocaloric effect. This techniquecan be used to attain extremely lowtemperatures, as well as the rangesused in common refrigerators,depending on the design of the system.

The effect was first observed by theGerman physicist Emil Warburg(1880) and the fundamental principlewas suggested by Debye (1926) andGiauque (1927).[1] The first workingmagnetic refrigerators wereconstructed by several groupsbeginning in 1933. Magneticrefrigeration was the first methoddeveloped for cooling below about0.3 K (a temperature attainable by 3He refrigeration, that is pumping on the 3He vapors).

The magnetocaloric effectThe magnetocaloric effect (MCE, from magnet and calorie) is a magneto-thermodynamic phenomenon in which areversible change in temperature of a suitable material is caused by exposing the material to a changing magneticfield. This is also known by low temperature physicists as adiabatic demagnetization, due to the application of theprocess specifically to create a temperature drop. In that part of the overall refrigeration process, a decrease in thestrength of an externally applied magnetic field allows the magnetic domains of a chosen (magnetocaloric) materialto become disoriented from the magnetic field by the agitating action of the thermal energy (phonons) present in thematerial. If the material is isolated so that no energy is allowed to (re)migrate into the material during this time, i.e.,an adiabatic process, the temperature drops as the domains absorb the thermal energy to perform their reorientation.The randomization of the domains occurs in a similar fashion to the randomization at the curie temperature of aferromagnetic material, except that magnetic dipoles overcome a decreasing external magnetic field while energyremains constant, instead of magnetic domains being disrupted from internal ferromagnetism as energy is added.One of the most notable examples of the magnetocaloric effect is in the chemical element gadolinium and some of itsalloys. Gadolinium's temperature is observed to increase when it enters certain magnetic fields. When it leaves themagnetic field, the temperature drops. The effect is considerably stronger for the gadolinium alloy Gd5(Si2Ge2).[2]

Praseodymium alloyed with nickel (PrNi5) has such a strong magnetocaloric effect that it has allowed scientists toapproach within one thousandth of a degree of absolute zero.[3]

Magnetic refrigeration 151

Thermodynamic cycle

Analogy between magnetic refrigeration and vapor cycle or conventional refrigeration. H= externally applied magnetic field; Q = heat quantity; P = pressure; ΔTad = adiabatic

temperature variation

The cycle is performed as arefrigeration cycle, analogous to theCarnot cycle, and can be described at astarting point whereby the chosenworking substance is introduced into amagnetic field, i.e., the magnetic fluxdensity is increased. The workingmaterial is the refrigerant, and starts inthermal equilibrium with therefrigerated environment.

• Adiabatic magnetization: Amagnetocaloric substance is placedin an insulated environment. Theincreasing external magnetic field(+H) causes the magnetic dipoles ofthe atoms to align, therebydecreasing the material's magneticentropy and heat capacity. Sinceoverall energy is not lost (yet) andtherefore total entropy is notreduced (according tothermodynamic laws), the net resultis that the item heats up (T + ΔTad).

• Isomagnetic enthalpic transfer: This added heat can then be removed (-Q) by a fluid or gas — gaseous or liquidhelium, for example. The magnetic field is held constant to prevent the dipoles from reabsorbing the heat. Oncesufficiently cooled, the magnetocaloric substance and the coolant are separated (H=0).

• Adiabatic demagnetization: The substance is returned to another adiabatic (insulated) condition so the totalentropy remains constant. However, this time the magnetic field is decreased, the thermal energy causes themagnetic moments to overcome the field, and thus the sample cools, i.e., an adiabatic temperature change. Energy(and entropy) transfers from thermal entropy to magnetic entropy (disorder of the magnetic dipoles).

• Isomagnetic entropic transfer: The magnetic field is held constant to prevent the material from heating back up.The material is placed in thermal contact with the environment being refrigerated. Because the working materialis cooler than the refrigerated environment (by design), heat energy migrates into the working material (+Q).

Once the refrigerant and refrigerated environment are in thermal equilibrium, the cycle begins again.

Applied techniqueThe basic operating principle of an adiabatic demagnetization refrigerator (ADR) is the use of a strong magnetic field to control the entropy of a sample of material, often called the "refrigerant". Magnetic field constrains the orientation of magnetic dipoles in the refrigerant. The stronger the magnetic field, the more aligned the dipoles are, and this corresponds to lower entropy and heat capacity because the material has (effectively) lost some of its internal degrees of freedom. If the refrigerant is kept at a constant temperature through thermal contact with a heat sink (usually liquid helium) while the magnetic field is switched on, the refrigerant must lose some energy because it is equilibrated with the heat sink. When the magnetic field is subsequently switched off, the heat capacity of the refrigerant rises again because the degrees of freedom associated with orientation of the dipoles are once again

Magnetic refrigeration 152

liberated, pulling their share of equipartitioned energy from the motion of the molecules, thereby lowering theoverall temperature of a system with decreased energy. Since the system is now insulated when the magnetic field isswitched off, the process is adiabatic, i.e., the system can no longer exchange energy with its surroundings (the heatsink), and its temperature decreases below its initial value, that of the heat sink.The operation of a standard ADR proceeds roughly as follows. First, a strong magnetic field is applied to therefrigerant, forcing its various magnetic dipoles to align and putting these degrees of freedom of the refrigerant into astate of lowered entropy. The heat sink then absorbs the heat released by the refrigerant due to its loss of entropy.Thermal contact with the heat sink is then broken so that the system is insulated, and the magnetic field is switchedoff, increasing the heat capacity of the refrigerant, thus decreasing its temperature below the temperature of thehelium heat sink. In practice, the magnetic field is decreased slowly in order to provide continuous cooling and keepthe sample at an approximately constant low temperature. Once the field falls to zero or to some low limiting valuedetermined by the properties of the refrigerant, the cooling power of the ADR vanishes, and heat leaks will cause therefrigerant to warm up.

Working materialsThe magnetocaloric effect is an intrinsic property of a magnetic solid. This thermal response of a solid to theapplication or removal of magnetic fields is maximized when the solid is near its magnetic ordering temperature.The magnitudes of the magnetic entropy and the adiabatic temperature changes are strongly dependent upon themagnetic order process: the magnitude is generally small in antiferromagnets, ferrimagnets and spin glass systems; itcan be substantial for normal ferromagnets which undergo a second order magnetic transition; and it is generally thelargest for a ferromagnet which undergoes a first order magnetic transition.Also, crystalline electric fields and pressure can have a substantial influence on magnetic entropy and adiabatictemperature changes.Currently, alloys of gadolinium producing 3 - 4 K per tesla [K/T] of change in a magnetic field can be used formagnetic refrigeration.Recent research on materials that exhibit a giant entropy change showed that Gd5(SixGe1−x)4, La(FexSi1−x)13Hx andMnFeP1−xAsx alloys, for example, are some of the most promising substitutes for gadolinium and its alloys —GdDy, GdTb, etc. These materials are called giant magnetocaloric effect (GMCE) materials.Gadolinium and its alloys are the best material available today for magnetic refrigeration near room temperaturesince they undergo second-order phase transitions which have no magnetic or thermal hysteresis involved.

Paramagnetic saltsThe originally suggested refrigerant was a paramagnetic salt, such as cerium magnesium nitrate. The active magneticdipoles in this case are those of the electron shells of the paramagnetic atoms.In a paramagnetic salt ADR, the heat sink is usually provided by a pumped 4He (about 1.2 K) or 3He (about 0.3 K)cryostat. An easily attainable 1 T magnetic field is generally required for the initial magnetization. The minimumtemperature attainable is determined by the self-magnetization tendencies of the chosen refrigerant salt, buttemperatures from 1 to 100 mK are accessible. Dilution refrigerators had for many years supplanted paramagneticsalt ADRs, but interest in space-based and simple to use lab-ADRs has remained, due to the complexity andunreliability of the dilution refrigeratorEventually paramagnetic salts become either diamagnetic or ferromagnetic, limiting the lowest temperature whichcan be reached using this method.

Magnetic refrigeration 153

Nuclear demagnetizationOne variant of adiabatic demagnetization that continues to find substantial research application is nucleardemagnetization refrigeration (NDR). NDR follows the same principle described above, but in this case the coolingpower arises from the magnetic dipoles of the nuclei of the refrigerant atoms, rather than their electronconfigurations. Since these dipoles are of much smaller magnitude, they are less prone to self-alignment and havelower intrinsic minimum fields. This allows NDR to cool the nuclear spin system to very low temperatures, often 1µK or below. Unfortunately, the small magnitudes of nuclear magnetic dipoles also makes them less inclined to alignto external fields. Magnetic fields of 3 teslas or greater are often needed for the initial magnetization step of NDR.In NDR systems, the initial heat sink must sit at very low temperatures (10–100 mK). This precooling is oftenprovided by the mixing chamber of a dilution refrigerator or a paramagnetic salt.

Commercial developmentThis refrigeration, once proven viable, could be used in any possible application where cooling, heating or powergeneration is used today. Since it is only at an early stage of development, there are several technical and efficiencyissues that should be analyzed. The magnetocaloric refrigeration system is composed of pumps, electric motors,secondary fluids, heat exchangers of different types, magnets and magnetic materials. These processes are greatlyaffected by irreversibilities and should be adequately considered.Appliances using this method could have a smaller environmental impact if the method is perfected and replaceshydrofluorocarbon (HFCs) refrigerators (some refrigerators still use HFCs which have considerable effect on theozone layer. At present, however, the superconducting magnets that are used in the process have to themselves becooled down to the temperature of liquid nitrogen, or with even colder, and relatively expensive, liquid helium.Considering these fluids have boiling points of 77.36 K and 4.22 K respectively, the technology is clearly not cost-and energy-efficient for home appliances, but for experimental, laboratory, and industrial use only.Recent research on materials that exhibit a large entropy change showed that alloys are some of the most promisingsubstitutes of gadolinium and its alloys — GdDy, GdTb, etc. Gadolinium and its alloys are the best materialavailable today for magnetic refrigeration near room temperature. There are still some thermal and magnetichysteresis problems to be solved for them to become truly useful [V. Provenzano, A.J. Shapiro, and R.D. Shull,Nature 429, 853 (2004)] and scientists are working hard to achieve this goal. Thermal hysteresis problems is solvedtherefore in adding ferrite (5:4).Research and a demonstration proof of concept in 2001 succeeded in applying commercial-grade materials andpermanent magnets at room temperatures to construct a magnetocaloric refrigerator which promises wide use.[4]

This technique has been used for many years in cryogenic systems for producing further cooling in systems alreadycooled to temperatures of 4 K and lower. In England, a company called Cambridge Magnetic Refrigeration [5]

produces cryogenic systems based on the magnetocaloric effect.On August 20, 2007, the Risø National Laboratory (Denmark) at the Technical University of Denmark, claimed tohave reached a milestone in their magnetic cooling research when they reported a temperature span of 8.7 C.[6] Theyhope to introduce the first commercial applications of the technology by 2010.

Current and future usesThere are still some thermal and magnetic hysteresis problems to be solved for these first-order phase transition materials that exhibit the GMCE to become really useful; this is a subject of current research. A useful review on magnetocaloric materials published in 2005 is entitled "Recent developments in magnetocaloric materials" by Dr. Karl A. Gschneidner, et al.[7] This effect is currently being explored to produce better refrigeration techniques, especially for use in spacecraft. This technique is already used to achieve cryogenic temperatures in the laboratory setting (below 10K). As an object displaying MCE is moved into a magnetic field, the magnetic spins align, lowering

Magnetic refrigeration 154

the entropy. Moving that object out of the field allows the object to increase its entropy by absorbing heat from theenvironment and disordering the spins. In this way, heat can be taken from one area to another. Should materials befound to display this effect near room temperature, refrigeration without the need for compression may be possible,increasing energy efficiency.The use of this technology to replace larger vapor-compression refrigeration units, which typically achieveperformance coefficients of 60% of that of a theoretical ideal Carnot cycle is unlikely in the near term. Smalldomestic refrigerators are however much less efficient. [8]

This technology could eventually compete with other cryogenic heat pumps for gas liquefaction purposes.Gschneidner stated in 1999 that: "large-scale applications using magnetic refrigeration, such as commercial airconditioning and supermarket refrigeration systems, could be available within 5–10 years. Within 10–15 years, thetechnology could be available in home refrigerators and air conditioners."[9]

HistoryThe effect was discovered in pure iron in 1880 by German physicist Emil Warburg. Originally, the cooling effectvaried between 0.5 to 2 K/T.Major advances first appeared in the late 1920s when cooling via adiabatic demagnetization was independentlyproposed by two scientists, Peter Debye in 1926 and William Giauque in 1927.This cooling technology was first demonstrated experimentally by chemist Nobel Laureate William F. Giauque andhis colleague D. P. MacDougall in 1933 for cryogenic purposes when they reached 0.25 K.[10] Between 1933 and1997, a number of advances in utilization of the MCE for cooling occurred.[11][12][13][14]

In 1997, the first near room temperature proof of concept magnetic refrigerator was demonstrated by Karl A.Gschneidner, Jr. by the Iowa State University at Ames Laboratory. This event attracted interest from scientists andcompanies worldwide who started developing new kinds of room temperature materials and magnetic refrigeratordesigns.[2] A major breakthrough came 2002 when a group at the University of Amsterdam demonstrated the giantmagnetocaloric effect in MnFe(P,As) alloys that are based on earth abundant materials.[15]

Refrigerators based on the magnetocaloric effect have been demonstrated in laboratories, using magnetic fieldsstarting at 0.6 T up to 10 T. Magnetic fields above 2 T are difficult to produce with permanent magnets and areproduced by a superconducting magnet (1 T is about 20,000 times the Earth's magnetic field).

Room temperature devicesSome recent research has focused on the use of the process to perform refrigeration near "room temperature".Constructed examples of room temperature magnetic refrigerators are listed in the table below:

Room temperature magnetic refrigerators

Institute/Company Location Announcementdate

Type Max.coolingpower(W)[1]

MaxΔT

(K)[2]

Magneticfield (T)

Solid refrigerant Quantity(kg)

AmesLaboratory/Astronautics[16]

Ames,Iowa/Madison,Wisconsin,USA

February 20,1997

Reciprocating 600 10 5 (S) Gd spheres

Mater. Science InstituteBarcelona[17]

Barcelona,Spain

May 2000 Rotary ? 5 0.95 (P) Gd foil

Magnetic refrigeration 155

Chubu Electric/Toshiba[18] Yokohama,Japan

Summer 2000 Reciprocating 100 21 4 (S) Gd spheres

University ofVictoria[19][20][21]

Victoria,BritishColumbiaCanada

July 2001 Reciprocating 2 14 2 (S) Gd & Gd1−xTbx L.B.

Astronautics[22] Madison,Wisconsin,USA

September 18,2001

Rotary 95 25 1.5 (P) Gd spheres

Sichuan Inst. Tech./NanjingUniversity[23]

Nanjing,China

23 April 2002 Reciprocating ? 23 1.4 (P) Gd spheres andGd5Si1.985Ge1.985Ga0.03powder

Chubu Electric/Toshiba[24] Yokohama,Japan

October 5, 2002 Reciprocating 40 27 0.6 (P) Gd1−xDyx L.B.

Chubu Electric/Toshiba[24] Yokohama,Japan

March 4, 2003 Rotary 60 10 0.76 (P) Gd 1−xDyx L.B. 1

Lab. d’ElectrotechniqueGrenoble[25]

Grenoble,France

April 2003 Reciprocating 8.8 4 0.8 (P) Gd foil

George WashingtonUniversity [26]

USA July 2004 Reciprocating ? 5 2 (P) Gd foil

Astronautics[27] Madison,Wisconsin,USA

2004 Rotary 95 25 1.5 (P) Gd and GdEr spheres /La(Fe0.88Si0.12)13H1.0

University of Victoria[28] Victoria,BritishColumbiaCanada

2006 Reciprocating 15 50 2 (S) Gd, Gd0.74Tb0.26 andGd0.85Er0.15 pucks

0.12

1maximum cooling power at zero temperature difference (ΔT=0); 2maximum temperature span at zero cooling capacity (W=0); L.B.= layered bed; P = permanent magnet; S = superconducting magnet

In one example, Prof. Karl A. Gschneidner, Jr. unveiled a proof of concept magnetic refrigerator near roomtemperature on February 20, 1997. He also announced the discovery of the GMCE in Gd5Si2Ge2 on June 9, 1997 [29]

(see below). Since then, hundreds of peer-reviewed articles have been written describing materials exhibitingmagnetocaloric effects.

References[1] Zemansky, Mark W. (1981). Temperatures very low and very high. New York: Dover. p. 50. ISBN 0-486-24072-X.[2] Karl Gschneidner, Jr. and Kerry Gibson (December 7, 2001). "Magnetic Refrigerator Successfully Tested" (http:/ / www. external. ameslab.

gov/ news/ release/ 01magneticrefrig. htm). Ames Laboratory News Release. Ames Laboratory. . Retrieved 2006-12-17.[3] Emsley, John (2001). Nature's Building Blocks. Oxford University Press. p. 342. ISBN 0-19-850341-5.[4] Gibson, Kerry (November 2001). "Magnetic Refrigerator Successfully Tested: Ames Laboratory develoments help push boundaries of new

refrigeration technology" (http:/ / www. ameslab. gov/ news/ ins01-11Magnetic. htm). INSIDER Newsletter for employees of AmesLaboratory. .(Vol. 112, No.10 )

[5] http:/ / www. cmr. uk. com/[6] Milestone in magnetic cooling, Risø News, August 20, 2007 (http:/ / www. risoe. dk/ News_archives/ News/ 2007/ 0820_magnetisk_koeling.

aspx). Retrieved August 28, 2007.[7] Gschneidner, Karl A., Jr.; Pecharsky, V. K. and Tsokol1, A.O. Recent developments in magnetocaloric materials (http:/ / www. iop. org/ EJ/

abstract/ 0034-4885/ 68/ 6/ R04/ ) Report on Progress in Physics. (2005) Volume 68, pages 1479–1539.[8] http:/ / www. osti. gov/ bridge/ purl. cover. jsp?purl=/ 40784-UgOxYh/ webviewable/ 40784. pdf[9] http:/ / www. ameslab. gov/ final/ News/ 1999rel/ 99crada. html

Magnetic refrigeration 156

[10] Giauque, W. F.; MacDougall, D. P. (1933). "Attainment of Temperatures Below 1° Absolute by Demagnetization of Gd2(SO4)3·8H2O".Phys. Rev. 43 (9): 768. Bibcode 1933PhRv...43..768G. doi:10.1103/PhysRev.43.768.

[11] Gschneidner, K. A. Jr.; Pecharsky, V. K. (1997). Bautista, R. G.; et al.. eds. Rare Earths: Science, Technology and Applications III.Warrendale, PA: The Minerals, Metals and Materials Society. p. 209.

[12] Pecharsky, V. K.; Gschneidner, K. A. Jr. (1999). "Magnetocaloric Effect and Magnetic Refrigeration". J. Magn. Magn. Mater. 200 (1–3):44–56. Bibcode 1999JMMM..200...44P. doi:10.1016/S0304-8853(99)00397-2.

[13] Gschneidner, K. A. Jr.; Pecharsky, V. K. (2000). "Magnetocaloric Materials". Annu. Rev. Mater. Sci. 30 (1): 387–429.Bibcode 2000AnRMS..30..387G. doi:10.1146/annurev.matsci.30.1.387.

[14] Gschneidner, K. A. Jr.; Pecharsky, V. K. (2002). Chandra, D.; Bautista, R. G.. eds. Fundamentals of Advanced Materials for EnergyConversion. Warrendale, PA: The Minerals, Metals and Materials Society. p. 9.

[15] Tegus, O.; Brück, E.; de Boer, F. R.; Buschow, K. H. J. (2002). "Transition-metal-based magnetic refrigerants for room-temperatureapplications". Nature 415 (6868): 150–152. Bibcode 2002Natur.415..150T. doi:10.1038/415150a.

[16][16] Zimm C, Jastrab A., Sternberg A., Pecharsky V.K., Gschneidner K.A. Jr., Osborne M. and Anderson I., Adv. Cryog. Eng. 43, 1759 (1998).[17][17] Bohigas X., Molins E., Roig A., Tejada J. and Zhang X.X., IEEE Trans. Magn. 36 538 (2000).[18][18] Hirano N., Nagaya S., Takahashi M., Kuriyama T., Ito K. and Nomura S. 2002 Adv. Cryog. Eng. 47 1027[19][19] Rowe A.M. and Barclay J.A., Adv. Cryog. Eng. 47 995 (2002).[20][20] Rowe A.M. and Barclay J.A., Adv. Cryog. Eng. 47 1003 (2002).[21][21] Richard M.A., Rowe A.M. and Chahine R., J. Appl. Phys. 95 2146 (2004).[22] Zimm C, Paper No K7.003 Am. Phys. Soc. Meeting, March 4, Austin, Texas (2003) (http:/ / www. aps. org/ meet/ MAR03/ baps/ tocK.

html)[23] Wu W., Paper No. K7.004 Am. Phys. Soc. Meeting, March 4, Austin, Texas (2003) (http:/ / www. aps. org/ meet/ MAR03/ baps/ tocK.

html)[24] Hirano N., Paper No. K7.002 Am. Phys. Soc. Meeting March 4, Austin, Texas, (http:/ / www. aps. org/ meet/ MAR03/ baps/ tocK. html)[25][25] Clot P., Viallet D., Allab F., Kedous-LeBouc A., Fournier J.M. and Yonnet J.P., IEEE Trans. Magn. 30 3349 (2003).[26][26] F. Shir, C. Mavriplis, L.H. Bennett, E. Della Torre, "Analysis of room temperature magnetic regenerative refrigeration," International

Journal of Refrigeration, 28, 4 (2005) 616.[27] Zimm C, Paper No. K7.003 Am. Phys. Soc. Meeting, March 4, Austin, Texas (2003) (http:/ / www. aps. org/ meet/ MAR03/ baps/ tocK.

html)[28] Rowe A.M. and Tura A., International Journal of Refrigeration 29 1286–1293 (2006).[29] http:/ / prola. aps. org/ abstract/ PRL/ v78/ i23/ p4494_1

Further reading• Lounasmaa, Experimental Principles and Methods Below 1 K, Academic Press (1974).• Richardson and Smith, Experimental Techniques in Condensed Matter Physics at Low Temperatures, Addison

Wesley (1988).• Lucia, U. General approach to obtain the magnetic refrigeretion ideal Coefficient of Performance COP, Physica

A: Statistical Mechanics and its Applications, 387/14 (2008) 3477–3479, doi:10.1016/j.physa.2008.02.026; seealso http:/ / arxiv. org/ abs/ 1011. 1684

External links• NASA – How does an Adiabatic Demagnetization Refrigerator Work ? (http:/ / imagine. gsfc. nasa. gov/ docs/

teachers/ lessons/ xray_spectra/ background-adr. html)• What is magnetocaloric effect and what materials exhibit this effect the most? (http:/ / www. physlink. com/

Education/ AskExperts/ ae488. cfm)• Magnetocaloric materials keep fridges cool by C. Wu (http:/ / www. sciencenews. org/ pages/ sn_arc98/ 3_28_98/

fob3. htm)• Ames Laboratory news release, May 25, 1999, Work begins on prototype magnetic-refrigeration unit (http:/ /

www. ameslab. gov/ News/ release/ crada. html).• Magnetic refrigerator successfully tested (http:/ / www. eurekalert. org/ features/ doe/ 2001-11/ dl-mrs062802.

php)• Refrigeration Systems (http:/ / lorien. ncl. ac. uk/ ming/ cleantech/ refrigeration. htm) Terry Heppenstall's notes,

University of Newcastle upon Tyne (November 2000)

Magnetic refrigeration 157

• XRS Adiabatic Demagnetization Refrigerator (http:/ / www. universe. nasa. gov/ xrays/ programs/ astroe/ eng/adr. html)

• Executive Summary: A Continuous Adiabatic Demagnetization Refrigerator (http:/ / www. cs. wpi. edu/ ~dfinkel/Sponsor/ PH1. doc) (.doc format) ( Google cache (http:/ / google. com/ search?q=cache:www. cs. wpi. edu/~dfinkel/ Sponsor/ PH1. doc))

• Origin and tuning of the magnetocaloric effect in the magnetic refrigerant Mn1.1Fe0.9(P0.8Ge0.2) (http:/ / link.aps. org/ doi/ 10. 1103/ PhysRevB. 79. 014435)

• Magnetic technology revolutionizes refrigeration (http:/ / www. basf. com/ group/ pressrelease/ P-09-348)• Evaluation of thermodynamic quantities in magnetic refrigeration (http:/ / arxiv. org/ abs/ 1011. 1684)

Magnetic stirrer 158

Magnetic stirrer

Magnetic stirrer

A stir bar mixing a solution on a combined hot-plate magnetic-stirrer device. The left knob controls the stirring rate and the right knob controlsheating.

Other names Magnetic mixer

Uses Liquid mixing

Inventor Arthur Rosinger

Related items Stir barVortex mixerStatic mixer

A magnetic stirrer or magnetic mixer is a laboratory device that employs a rotating magnetic field to cause a stirbar (also called "flea") immersed in a liquid to spin very quickly, thus stirring it. The rotating field may be createdeither by a rotating magnet or a set of stationary electromagnets, placed beneath the vessel with the liquid. Magneticstirrers often include a hot plate or some other means for heating the liquid.Magnetic stirrers are often used in chemistry and biology. They are preferred over gear-driven motorized stirrersbecause they are quieter, more efficient, and have no moving external parts to break or wear out (other than thesimple bar magnet itself). Due to its small size, a stirring bar is more easily cleaned and sterilized than other stirringdevices. They do not require lubricants which could contaminate the reaction vessel and the product. They can beused inside hermetically closed vessels or systems, without the need for complicated rotary seals.On the other hand, the limited size of the bar means that magnetic stirrers can only be used for relatively small(under 4 liters) experiments. They also have difficulty dealing with viscous liquids or thick suspensions.

Magnetic stirrer 159

History

Different sizes of magnetic stir bars

Arthur Rosinger of Newark, New Jersey, U.S.A. obtained US Patent2,350,534, titled Magnetic Stirrer on 6 June 1944, having filed anapplication therefor on 5 October 1942.[1] Mr. Rosinger's patentincludes a description of a coated bar magnet placed in a vessel, whichis driven by a rotating magnet in a base below the vessel. Mr. Rosingeralso explains in his patent that coating the magnet in plastic orcovering it with glass or porcelain makes it chemically inert.

The plastic-coated bar magnet was independently invented in the late1940s by Edward McLaughlin, of the Torpedo ExperimentalEstablishment (TEE), Greenock, Scotland, who named it the 'flea'because of the way it jumps about if the rotating magnet is driven too

fast.

An even earlier patent for a magnetic mixer is US 1,242,493, issued 9 October 1917 to Richard H. Stringham ofBountiful, Utah, U.S.A. Mr. Stringman's mixer used stationary electromagnets in the base, rather than a rotatingpermanent magnet, to rotate the stirrer.The first multipoint magnetic stirrer was developed and patented by Salvador Bonet of SBS Company in 1977. Healso introduced the practice of noting the denomination of stirring power in "liters of water", which is a marketstandard today.Heating elements may range from 120 W or lower to 500 W or more. The maximum reachable fluid temperaturedepends on the size of the flask, the quantity of solution to be heated, and the power of the heating element.

References[1] "MAGNETIC STIRRER Arthur Rosinger" (http:/ / www. google. com/ patents/ about?id=3CxTAAAAEBAJ& dq=US+ Patent+ 2,350,534).

Google patents. .

External links• DIY Stir plate (http:/ / brewiki. org/ StirPlate) Make your own stir plate from an old computer fan.• Short video of a home made stir plate. (http:/ / www. youtube. com/ watch?v=tdhXESny0II)• (http:/ / www. google. com/ patents/ about?id=3CxTAAAAEBAJ& dq=US+ Patent+ 2,350,534) Arthur

Rosinger's patent on "Magnetic Stirrer"

Magnetic structure 160

Magnetic structureThe term magnetic structure of a material pertains to the ordered arrangement of magnetic spins, typically withinan ordered crystallographic lattice. Its study is a branch of solid-state chemistry.

A very simple antiferromagnetic structure

Magnetic structures

A very simple ferromagnetic structure

Most solid materials are Pauli-paramagnetic. These materials either donot have electron spins or these spins are not ordered unless an externalfield induces some non-random orientation. Such materials are notconsidered to 'have' a magnetic structure. This is different for ferro-,ferri- and antiferromagnetic materials. They differ in the relativeordering of their spins within the lattice. In some ferromagnetic casesthe structure may be relatively simple in that all spins point in the samedirection, or at least that would be the case at very low temperatures.

Towards higher temperatures there will be more and more 'rebellious' spins defying the order of the magneticstructure and at a certain temperature the order will break down and the spins will point in random directions. Insome materials the pattern in which the spins order is much more complicated[1]. In antiferromagnetic materialsspins point in opposite directions so that the overall magnetic moment is zero. However, this is not necessarilyachieved by a simple up and down pattern. Much more complicated structures can arise. At times one can recognizelayers in which all spins point in one direction (as in a ferromagnet) but in the next layer they point in the oppositedirection giving an overall antiferromagnetic arrangement. The possible number of arrangements is very large andcan include spirals, clusters, tetrahedra etc.

A different simpleantiferromagnetic arrangement in

2D

Techniques to study them

Such ordering can be studied by observing the magnetic susceptibility as afunction of temperature and/or the size of the applied magnetic field, but a trulythree-dimensional picture of the arrangement of the spins is best obtained bymeans of neutron diffraction[2][3]. Neutrons are primarily scattered by the nuclei ofthe atoms in the structure. At a temperature above the ordering point of themagnetic moments, where the material behaves as a paramagnetic one, neutrondiffraction will therefore give a picture of the crystallographic structure only.Below the ordering point, e.g. the Néel temperature of an antiferromagnet or theCurie-point of a ferromagnet the neutrons will also experience scattering from the

magnetic moments because they themselves possess spin. The intensities of the Bragg reflections will thereforechange. In fact in some cases entirely new Bragg-reflections will occur if the unit cell of the ordering is larger than

Magnetic structure 161

that of the crystallographic structure. This is a form of superstructure formation. Thus the symmetry of the totalstructure may well differ from the crystallographic substructure. It needs to be described by one of the 1651magnetic (Shubnikov) groups[4] rather than one of the non-magnetic space groups.Although ordinary X-ray diffraction is 'blind' to the arrangement of the spins, it has become possible to use a specialform of X-ray diffraction to study magnetic structure. If a wavelength is selected that is close to an absorption edgeof one of elements contained in the materials the scattering becomes anomalous and this component to the scatteringis (somewhat) sensitive to the non-spherical shape of the outer electrons of an atom with an unpaired spin. Thismeans that this type of anomalous X-ray diffraction does contain information of the desired type.

References[1] an example (http:/ / www. ftj. agh. edu. pl/ ~Pytlik/ msdb/ magngif. htm)[2][2] Neutron diffraction of magnetic materials / Yu. A. Izyumov, V.E. Naish, and R.P. Ozerov ; translated from Russian by Joachim Büchner.

New York : Consultants Bureau, c1991.ISBN 030611030X[3] A demonstration by Brian Toby (http:/ / www. aps. anl. gov/ Xray_Science_Division/ Powder_Diffraction_Crystallography/

2006ACNSmagnetGSAS/ YBAFEOexampleMovie/ YBAFEOexample. html)[4][4] p.428 Group Theoretical Methods and Applications to Molecules and Crystals. By Shoon Kyung Kim.1999. Cambridge University.

Press.ISBN 0521640628

MagnetismMagnetism is a property of materials that respond to an applied magnetic field. Permanent magnets have persistentmagnetic fields caused by ferromagnetism. That is the strongest and most familiar type of magnetism. However, allmaterials are influenced varyingly by the presence of a magnetic field. Some are attracted to a magnetic field(paramagnetism); others are repulsed by a magnetic field (diamagnetism); others have a much more complexrelationship with an applied magnetic field (spin glass behavior and antiferromagnetism). Substances that arenegligibly affected by magnetic fields are known as non-magnetic substances. They include copper, aluminium,gases, and plastic. Pure oxygen exhibits magnetic properties when cooled to a liquid state.The magnetic state (or phase) of a material depends on temperature (and other variables such as pressure and appliedmagnetic field) so that a material may exhibit more than one form of magnetism depending on its temperature, etc.

HistoryAristotle attributed the first of what could be called a scientific discussion on magnetism to Thales of Miletus, wholived from about 625 BC to about 545 BC.[1] Around the same time, in ancient India, the Indian surgeon, Sushruta,was the first to make use of the magnet for surgical purposes.[2]

In ancient China, the earliest literary reference to magnetism lies in a 4th century BC book named after its author,The Master of Demon Valley (鬼 谷 子): "The lodestone makes iron come or it attracts it."[3] The earliest mentionof the attraction of a needle appears in a work composed between AD 20 and 100 (Louen-heng): "A lodestoneattracts a needle."[4] The ancient Chinese scientist Shen Kuo (1031–1095) was the first person to write of themagnetic needle compass and that it improved the accuracy of navigation by employing the astronomical concept oftrue north (Dream Pool Essays, AD 1088), and by the 12th century the Chinese were known to use the lodestonecompass for navigation. They sculpted a directional spoon from lodestone in such a way that the handle of the spoonalways pointed south.Alexander Neckham, by 1187, was the first in Europe to describe the compass and its use for navigation. In 1269,Peter Peregrinus de Maricourt wrote the Epistola de magnete, the first extant treatise describing the properties ofmagnets. In 1282, the properties of magnets and the dry compass were discussed by Al-Ashraf, a Yemeni physicist,astronomer, and geographer.[5]

Magnetism 162

Michael Faraday, 1842

In 1600, William Gilbert published his De Magnete, MagneticisqueCorporibus, et de Magno Magnete Tellure (On the Magnet and MagneticBodies, and on the Great Magnet the Earth). In this work he describesmany of his experiments with his model earth called the terrella. From hisexperiments, he concluded that the Earth was itself magnetic and that thiswas the reason compasses pointed north (previously, some believed that itwas the pole star (Polaris) or a large magnetic island on the north pole thatattracted the compass).

An understanding of the relationship between electricity and magnetismbegan in 1819 with work by Hans Christian Oersted, a professor at theUniversity of Copenhagen, who discovered more or less by accident that anelectric current could influence a compass needle. This landmarkexperiment is known as Oersted's Experiment. Several other experimentsfollowed, with André-Marie Ampère, who in 1820 discovered that themagnetic field circulating in a closed-path was related to the currentflowing through the perimeter of the path; Carl Friedrich Gauss; Jean-Baptiste Biot and Félix Savart, both of whichin 1820 came up with the Biot-Savart Law giving an equation for the magnetic field from a current-carrying wire;Michael Faraday, who in 1831 found that a time-varying magnetic flux through a loop of wire induced a voltage, andothers finding further links between magnetism and electricity. James Clerk Maxwell synthesized and expandedthese insights into Maxwell's equations, unifying electricity, magnetism, and optics into the field ofelectromagnetism. In 1905, Einstein used these laws in motivating his theory of special relativity,[6] requiring thatthe laws held true in all inertial reference frames.

Electromagnetism has continued to develop into the 21st century, being incorporated into the more fundamentaltheories of gauge theory, quantum electrodynamics, electroweak theory, and finally the standard model.

Sources of magnetismMagnetism, at its root, arises from two sources:1. Electric currents or more generally, moving electric charges create magnetic fields (see Maxwell's Equations).2. Many particles have nonzero "intrinsic" (or "spin") magnetic moments. Just as each particle, by its nature, has a

certain mass and charge, each has a certain magnetic moment, possibly zero.It was found hundreds of years ago that certain materials have a tendency to orient in a particular direction. Forexample ancient people knew that "lodestones," when suspended from a string and allowed to freely rotate, come torest horizontally in the North-South direction. Ancient Mariners used lodestones for navigational purposes.In magnetic materials, sources of magnetization are the electrons' orbital angular motion around the nucleus, and theelectrons' intrinsic magnetic moment (see electron magnetic dipole moment). The other sources of magnetism are thenuclear magnetic moments of the nuclei in the material which are typically thousands of times smaller than theelectrons' magnetic moments, so they are negligible in the context of the magnetization of materials. Nuclearmagnetic moments are important in other contexts, particularly in nuclear magnetic resonance (NMR) and magneticresonance imaging (MRI).Ordinarily, the enormous number of electrons in a material are arranged such that their magnetic moments (both orbital and intrinsic) cancel out. This is due, to some extent, to electrons combining into pairs with opposite intrinsic magnetic moments as a result of the Pauli exclusion principle (see electron configuration), or combining into filled subshells with zero net orbital motion. In both cases, the electron arrangement is so as to exactly cancel the magnetic moments from each electron. Moreover, even when the electron configuration is such that there are unpaired electrons and/or non-filled subshells, it is often the case that the various electrons in the solid will contribute

Magnetism 163

magnetic moments that point in different, random directions, so that the material will not be magnetic.However, sometimes — either spontaneously, or owing to an applied external magnetic field — each of the electronmagnetic moments will be, on average, lined up. Then the material can produce a net total magnetic field, which canpotentially be quite strong.The magnetic behavior of a material depends on its structure, particularly its electron configuration, for the reasonsmentioned above, and also on the temperature. At high temperatures, random thermal motion makes it more difficultfor the electrons to maintain alignment.

Topics

Hierarchy of types of magnetism.[7]

Diamagnetism

Diamagnetism appears in all materials,and is the tendency of a material tooppose an applied magnetic field, andtherefore, to be repelled by a magneticfield. However, in a material withparamagnetic properties (that is, with atendency to enhance an externalmagnetic field), the paramagneticbehavior dominates.[8] Thus, despite itsuniversal occurrence, diamagneticbehavior is observed only in a purelydiamagnetic material. In a diamagneticmaterial, there are no unpairedelectrons, so the intrinsic electron magnetic moments cannot produce any bulk effect. In these cases, themagnetization arises from the electrons' orbital motions, which can be understood classically as follows:

When a material is put in a magnetic field, the electrons circling the nucleus will experience, in addition totheir Coulomb attraction to the nucleus, a Lorentz force from the magnetic field. Depending on whichdirection the electron is orbiting, this force may increase the centripetal force on the electrons, pulling them intowards the nucleus, or it may decrease the force, pulling them away from the nucleus. This effectsystematically increases the orbital magnetic moments that were aligned opposite the field, and decreases theones aligned parallel to the field (in accordance with Lenz's law). This results in a small bulk magneticmoment, with an opposite direction to the applied field.

Note that this description is meant only as an heuristic; a proper understanding requires a quantum-mechanicaldescription.Note that all materials undergo this orbital response. However, in paramagnetic and ferromagnetic substances, thediamagnetic effect is overwhelmed by the much stronger effects caused by the unpaired electrons.

ParamagnetismIn a paramagnetic material there are unpaired electrons, i.e. atomic or molecular orbitals with exactly one electron inthem. While paired electrons are required by the Pauli exclusion principle to have their intrinsic ('spin') magneticmoments pointing in opposite directions, causing their magnetic fields to cancel out, an unpaired electron is free toalign its magnetic moment in any direction. When an external magnetic field is applied, these magnetic momentswill tend to align themselves in the same direction as the applied field, thus reinforcing it.

Magnetism 164

Ferromagnetism

A permanent magnet holding up several coins

A ferromagnet, like a paramagnetic substance, has unpairedelectrons. However, in addition to the electrons' intrinsic magneticmoment's tendency to be parallel to an applied field, there is alsoin these materials a tendency for these magnetic moments to orientparallel to each other to maintain a lowered-energy state. Thus,even when the applied field is removed, the electrons in thematerial maintain a parallel orientation.

Every ferromagnetic substance has its own individual temperature,called the Curie temperature, or Curie point, above which it losesits ferromagnetic properties. This is because the thermal tendencyto disorder overwhelms the energy-lowering due to ferromagneticorder.

Some well-known ferromagnetic materials that exhibit easily detectable magnetic properties (to form magnets) arenickel, iron, cobalt, gadolinium and their alloys.

Magnetic domains

Magnetic domains in ferromagneticmaterial.

The magnetic moment of atoms in a ferromagnetic material cause them tobehave something like tiny permanent magnets. They stick together and alignthemselves into small regions of more or less uniform alignment calledmagnetic domains or Weiss domains. Magnetic domains can be observedwith a magnetic force microscope to reveal magnetic domain boundaries thatresemble white lines in the sketch. There are many scientific experiments thatcan physically show magnetic fields.

Effect of a magnet on the domains.

When a domain contains too many molecules, it becomes unstableand divides into two domains aligned in opposite directions so thatthey stick together more stably as shown at the right.When exposed to a magnetic field, the domain boundaries move sothat the domains aligned with the magnetic field grow anddominate the structure as shown at the left. When the magnetizingfield is removed, the domains may not return to an unmagnetizedstate. This results in the ferromagnetic material's beingmagnetized, forming a permanent magnet.When magnetized strongly enough that the prevailing domainoverruns all others to result in only one single domain, the materialis magnetically saturated. When a magnetized ferromagneticmaterial is heated to the Curie point temperature, the molecules are

Magnetism 165

agitated to the point that the magnetic domains lose the organization and the magnetic properties they cause cease.When the material is cooled, this domain alignment structure spontaneously returns, in a manner roughly analogousto how a liquid can freeze into a crystalline solid.

Antiferromagnetism

Antiferromagnetic ordering

In an antiferromagnet, unlike a ferromagnet, there is a tendency for theintrinsic magnetic moments of neighboring valence electrons to pointin opposite directions. When all atoms are arranged in a substance sothat each neighbor is 'anti-aligned', the substance isantiferromagnetic. Antiferromagnets have a zero net magneticmoment, meaning no field is produced by them. Antiferromagnets areless common compared to the other types of behaviors, and are mostlyobserved at low temperatures. In varying temperatures,antiferromagnets can be seen to exhibit diamagnetic and ferrimagnetic properties.

In some materials, neighboring electrons want to point in opposite directions, but there is no geometricalarrangement in which each pair of neighbors is anti-aligned. This is called a spin glass, and is an example ofgeometrical frustration.

Ferrimagnetism

Ferrimagnetic ordering

Like ferromagnetism, ferrimagnets retain their magnetization in theabsence of a field. However, like antiferromagnets, neighboring pairsof electron spins like to point in opposite directions. These twoproperties are not contradictory, because in the optimal geometricalarrangement, there is more magnetic moment from the sublattice ofelectrons that point in one direction, than from the sublattice that pointsin the opposite direction.

Most ferrites are ferrimagnetic. The first discovered magneticsubstance, magnetite, is a ferrite and was originally believed to be a ferromagnet; Louis Néel disproved this,however, after discovering ferrimagnetism.

Magnetism 166

SuperparamagnetismWhen a ferromagnet or ferrimagnet is sufficiently small, it acts like a single magnetic spin that is subject toBrownian motion. Its response to a magnetic field is qualitatively similar to the response of a paramagnet, but muchlarger.

ElectromagnetAn electromagnet is a type of magnet whose magnetism is produced by the flow of electric current. The magneticfield disappears when the current ceases.

Electromagnets attracts paper clips when currentis applied creating a magnetic field. The

electromagnet loses them when current andmagnetic field are removed.

Other types of magnetism

•• Molecular magnet•• Metamagnetism•• Molecule-based magnet•• Spin glass

Magnetism, electricity, and special relativity

As a consequence of Einstein's theory of special relativity, electricityand magnetism are fundamentally interlinked. Both magnetism lackingelectricity, and electricity without magnetism, are inconsistent withspecial relativity, due to such effects as length contraction, timedilation, and the fact that the magnetic force is velocity-dependent.

However, when both electricity and magnetism are taken into account, the resulting theory (electromagnetism) isfully consistent with special relativity.[6][9] In particular, a phenomenon that appears purely electric to one observermay be purely magnetic to another, or more generally the relative contributions of electricity and magnetism aredependent on the frame of reference. Thus, special relativity "mixes" electricity and magnetism into a single,inseparable phenomenon called electromagnetism, analogous to how relativity "mixes" space and time intospacetime.

Magnetic fields in a materialIn a vacuum,

where μ0 is the vacuum permeability.In a material,

The quantity μ0M is called magnetic polarization.If the field H is small, the response of the magnetization M in a diamagnet or paramagnet is approximately linear:

the constant of proportionality being called the magnetic susceptibility. If so,

In a hard magnet such as a ferromagnet, M is not proportional to the field and is generally nonzero even when H iszero (see Remanence).

Magnetism 167

Force due to magnetic field - The magnetic force

Magnetic lines of force of a bar magnet shown byiron filings on paper

The phenomenon of magnetism is "mediated" by the magnetic field.An electric current or magnetic dipole creates a magnetic field, andthat field, in turn, imparts magnetic forces on other particles that are inthe fields.Maxwell's equations, which simplify to the Biot-Savart law in the caseof steady currents, describe the origin and behavior of the fields thatgovern these forces. Therefore magnetism is seen whenever electricallycharged particles are in motion---for example, from movement ofelectrons in an electric current, or in certain cases from the orbitalmotion of electrons around an atom's nucleus. They also arise from"intrinsic" magnetic dipoles arising from quantum-mechanical spin.

The same situations that create magnetic fields — charge moving in a current or in an atom, and intrinsic magneticdipoles — are also the situations in which a magnetic field has an effect, creating a force. Following is the formulafor moving charge; for the forces on an intrinsic dipole, see magnetic dipole.When a charged particle moves through a magnetic field B, it feels a Lorentz force F given by the cross product:[10]

whereis the electric charge of the particle, and

v is the velocity vector of the particleBecause this is a cross product, the force is perpendicular to both the motion of the particle and the magnetic field. Itfollows that the magnetic force does no work on the particle; it may change the direction of the particle's movement,but it cannot cause it to speed up or slow down. The magnitude of the force is

where is the angle between v and B.One tool for determining the direction of the velocity vector of a moving charge, the magnetic field, and the forceexerted is labeling the index finger "V", the middle finger "B", and the thumb "F" with your right hand. Whenmaking a gun-like configuration, with the middle finger crossing under the index finger, the fingers represent thevelocity vector, magnetic field vector, and force vector, respectively. See also right hand rule.

Magnetic dipolesA very common source of magnetic field shown in nature is a dipole, with a "South pole" and a "North pole", termsdating back to the use of magnets as compasses, interacting with the Earth's magnetic field to indicate North andSouth on the globe. Since opposite ends of magnets are attracted, the north pole of a magnet is attracted to the southpole of another magnet. The Earth's North Magnetic Pole (currently in the Arctic Ocean, north of Canada) isphysically a south pole, as it attracts the north pole of a compass.A magnetic field contains energy, and physical systems move toward configurations with lower energy. Whendiamagnetic material is placed in a magnetic field, a magnetic dipole tends to align itself in opposed polarity to thatfield, thereby lowering the net field strength. When ferromagnetic material is placed within a magnetic field, themagnetic dipoles align to the applied field, thus expanding the domain walls of the magnetic domains.

Magnetism 168

Magnetic monopolesSince a bar magnet gets its ferromagnetism from electrons distributed evenly throughout the bar, when a bar magnetis cut in half, each of the resulting pieces is a smaller bar magnet. Even though a magnet is said to have a north poleand a south pole, these two poles cannot be separated from each other. A monopole — if such a thing exists —would be a new and fundamentally different kind of magnetic object. It would act as an isolated north pole, notattached to a south pole, or vice versa. Monopoles would carry "magnetic charge" analogous to electric charge.Despite systematic searches since 1931, as of 2010, they have never been observed, and could very well not exist.[11]

Nevertheless, some theoretical physics models predict the existence of these magnetic monopoles. Paul Diracobserved in 1931 that, because electricity and magnetism show a certain symmetry, just as quantum theory predictsthat individual positive or negative electric charges can be observed without the opposing charge, isolated South orNorth magnetic poles should be observable. Using quantum theory Dirac showed that if magnetic monopoles exist,then one could explain the quantization of electric charge---that is, why the observed elementary particles carrycharges that are multiples of the charge of the electron.Certain grand unified theories predict the existence of monopoles which, unlike elementary particles, are solitons(localized energy packets). The initial results of using these models to estimate the number of monopoles created inthe big bang contradicted cosmological observations — the monopoles would have been so plentiful and massivethat they would have long since halted the expansion of the universe. However, the idea of inflation (for which thisproblem served as a partial motivation) was successful in solving this problem, creating models in which monopolesexisted but were rare enough to be consistent with current observations.[12]

Quantum-mechanical origin of magnetismIn principle all kinds of magnetism originate (similar to Superconductivity) from specific quantum-mechanicalphenomena (e.g. Mathematical formulation of quantum mechanics, in particular the chapters on spin and on the Pauliprinciple). A successful model was developed already in 1927, by Walter Heitler and Fritz London, who derivedquantum-mechanically, how hydrogen molecules are formed from hydrogen atoms, i.e. from the atomic hydrogenorbitals and centered at the nuclei A and B, see below. That this leads to magnetism, is not at all obvious,but will be explained in the following.According the Heitler-London theory, so-called two-body molecular -orbitals are formed, namely the resultingorbital is:

Here the last product means that a first electron, r1, is in an atomic hydrogen-orbital centered at the second nucleus,whereas the second electron runs around the first nucleus. This "exchange" phenomenon is an expression for thequantum-mechanical property that particles with identical properties cannot be distinguished. It is specific not onlyfor the formation of chemical bonds, but as we will see, also for magnetism, i.e. in this connection the term exchangeinteraction arises, a term which is essential for the origin of magnetism, and which is stronger, roughly by factors100 and even by 1000, than the energies arising from the electrodynamic dipole-dipole interaction.

As for the spin function , which is responsible for the magnetism, we have the already mentioned Pauli'sprinciple, namely that a symmetric orbital (i.e. with the + sign as above) must be multiplied with an antisymmetricspin function (i.e. with a - sign), and vice versa. Thus:

,

I.e., not only and must be substituted by α and β, respectively (the first entity means "spin up", the second one "spin down"), but also the sign + by the − sign, and finally ri by the discrete values si (= ±½); thereby we have

and . The "singlet state", i.e. the - sign, means: the

Magnetism 169

spins are antiparallel, i.e. for the solid we have antiferromagnetism, and for two-atomic molecules one hasdiamagnetism. The tendency to form a (homoeopolar) chemical bond (this means: the formation of a symmetricmolecular orbital, i.e. with the + sign) results through the Pauli principle automatically in an antisymmetric spin state(i.e. with the - sign). In contrast, the Coulomb repulsion of the electrons, i.e. the tendency that they try to avoid eachother by this repulsion, would lead to an antisymmetric orbital function (i.e. with the - sign) of these two particles,and complementary to a symmetric spin function (i.e. with the + sign, one of the so-called "triplet functions"). Thus,now the spins would be parallel (ferromagnetism in a solid, paramagnetism in two-atomic gases).The last-mentioned tendency dominates in the metals iron, cobalt and nickel, and in some rare earths, which areferromagnetic. Most of the other metals, where the first-mentioned tendency dominates, are nonmagnetic (e.g.sodium, aluminium, and magnesium) or antiferromagnetic (e.g. manganese). Diatomic gases are also almostexclusively diamagnetic, and not paramagnetic. However, the oxygen molecule, because of the involvement ofπ-orbitals, is an exception important for the life-sciences.The Heitler-London considerations can be generalized to the Heisenberg model of magnetism (Heisenberg 1928).The explanation of the phenomena is thus essentially based on all subtleties of quantum mechanics, whereas theelectrodynamics covers mainly the phenomenology.

Units of electromagnetism

SI units related to magnetism

SI electromagnetism units

Symbol[13] Name of Quantity Derived Units Conversion of International to SI base units

Electric current ampere (SI base unit)

Electric charge coulomb

Potential difference; Electromotive force volt

Electric resistance; Impedance; Reactance ohm

Resistivity ohm metre

Electric power watt

Capacitance farad

Electric field strength volt per metre

Electric displacement field Coulomb per square metre

Permittivity farad per metre

Electric susceptibility Dimensionless

Conductance; Admittance; Susceptance siemens

Conductivity siemens per metre

Magnetic flux density, Magnetic induction tesla

Magnetic flux weber

Magnetic field strength ampere per metre

Inductance henry

Permeability henry per metre

Magnetic susceptibility Dimensionless

Magnetism 170

Other units• gauss — The gauss is the centimeter-gram-second (CGS) unit of magnetic field (denoted B).• oersted — The oersted is the CGS unit of magnetizing field (denoted H).• maxwell — The maxwell is the CGS unit for magnetic flux.• gamma — is a unit of magnetic flux density that was commonly used before the tesla came into use (1.0 gamma

= 1.0 nanotesla)• μ0 — common symbol for the permeability of free space (4π×10−7 newton/(ampere-turn)2).

Living thingsSome organisms can detect magnetic fields, a phenomenon known as magnetoception. Magnetobiology studiesmagnetic fields as a medical treatment; fields naturally produced by an organism are known as biomagnetism.

References[1] Fowler, Michael (1997). "Historical Beginnings of Theories of Electricity and Magnetism" (http:/ / galileoandeinstein. physics. virginia. edu/

more_stuff/ E& M_Hist. html). . Retrieved 2008-04-02.[2] Vowles, Hugh P. (1932). "Early Evolution of Power Engineering". Isis (University of Chicago Press) 17 (2): 412–420 [419–20].

doi:10.1086/346662.[3] Li Shu-hua, “Origine de la Boussole 11. Aimant et Boussole,” Isis, Vol. 45, No. 2. (Jul., 1954), p.175[4] Li Shu-hua, “Origine de la Boussole 11. Aimant et Boussole,” Isis, Vol. 45, No. 2. (Jul., 1954), p.176[5] Schmidl, Petra G. (1996–1997). "Two Early Arabic Sources On The Magnetic Compass". Journal of Arabic and Islamic Studies 1: 81–132.[6] A. Einstein: "On the Electrodynamics of Moving Bodies" (http:/ / www. fourmilab. ch/ etexts/ einstein/ specrel/ www/ ), June 30, 1905.[7] HP Meyers (1997). Introductory solid state physics (http:/ / books. google. com/ ?id=Uc1pCo5TrYUC& pg=PA322) (2 ed.). CRC Press.

p. 362; Figure 11.1. ISBN 0-7484-0660-3. .[8] Catherine Westbrook, Carolyn Kaut, Carolyn Kaut-Roth (1998). MRI (Magnetic Resonance Imaging) in practice (http:/ / books. google. com/

?id=Qq1SHDtS2G8C& pg=PA217) (2 ed.). Wiley-Blackwell. p. 217. ISBN 0-632-04205-2. .[9][9] Griffiths 1998, chapter 12[10] Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 0-471-30932-X[11] Milton mentions some inconclusive events (p.60) and still concludes that "no evidence at all of magnetic monopoles has survived" (p.3).

Milton, Kimball A. (June 2006). "Theoretical and experimental status of magnetic monopoles". Reports on Progress in Physics 69 (6):1637–1711. arXiv:hep-ex/0602040. Bibcode 2006RPPh...69.1637M. doi:10.1088/0034-4885/69/6/R02..

[12] Guth, Alan (1997). The Inflationary Universe: The Quest for a New Theory of Cosmic Origins. Perseus. ISBN 0-201-32840-2.OCLC 38941224..

[13] International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford:Blackwell Science. ISBN 0-632-03583-8. pp. 14–15. Electronic version. (http:/ / old. iupac. org/ publications/ books/ gbook/ green_book_2ed.pdf)

Further reading• Furlani, Edward P. (2001). Permanent Magnet and Electromechanical Devices: Materials, Analysis and

Applications. Academic Press. ISBN 0-12-269951-3. OCLC 162129430.• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.

OCLC 40251748.• Kronmüller, Helmut. (2007). Handbook of Magnetism and Advanced Magnetic Materials, 5 Volume Set. John

Wiley & Sons. ISBN 978-0-470-02217-7. OCLC 124165851.• 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.• David K. Cheng (1992). Field and Wave Electromagnetics. Addison-Wesley Publishing Company, Inc..

ISBN 0-201-12819-5.

Magnetism 171

External links• Magnetism (http:/ / www. bbc. co. uk/ programmes/ p003k9dd) on In Our Time at the BBC. ( listen now (http:/ /

www. bbc. co. uk/ iplayer/ console/ p003k9dd/ In_Our_Time_Magnetism))• The Exploratorium Science Snacks – Snacks about Magnetism (http:/ / www. exploratorium. edu/ snacks/

iconmagnetism. html)• Electromagnetism (http:/ / www. lightandmatter. com/ html_books/ 0sn/ ch11/ ch11. html) - a chapter from an

online textbook• Video: The physicist Richard Feynman answers the question, Why do bar magnets attract or repel each other?

(http:/ / www. youtube. com/ watch?v=wMFPe-DwULM)• On the Magnet, 1600 (http:/ / www. antiquebooks. net/ readpage. html#gilbert) First scientific book on magnetism

by the father of electrical engineering. Full English text, full text search.

MetamagnetismMetamagnetism is a blanket term used loosely in physics to describe a sudden (often, dramatic) increase in themagnetization of a material with a small change in an externally applied magnetic field. The metamagnetic behaviormay have quite different physical causes for different types of metamagnets. Some examples of physicalmechanisms leading to metamagnetic behavior are:1. Itinerant Metamagnetism - Exchange splitting of the Fermi surface in a paramagnetic system of itinerant electrons

causes an energetically favorable transition to bulk magnetization near the transition to a ferromagnet or othermagnetically ordered state.[1][2]

2. Antiferromagnetic Transition - Field-induced spin flips in antiferromagnets cascade at a critical energydetermined by the applied magnetic field.[3]

Depending on the material and experimental conditions, metamagnetism may be associated with a first-order phasetransition, a continuous phase transition at a critical point (classical or quantum), or crossovers beyond a criticalpoint that do not involve a phase transition at all. These wildly different physical explanations sometimes lead toconfusion as to what the term "metamagnetic" is referring in specific cases.

References[1] EP. Wohlfarth and P. Rhodes. "Collective Electron Metamagnetism" Philos Mag 7, 1817 (1962).[2] R. Z. Levitin and A. S. Markosyan. "Itinerant Metamagnetism" Usp. Fiz. Nauk 155, 623-657 (1988)[3] E. Stryjewski and N. Giordano. "Metamagnetism" Advances in Physics 26, 487-650 (1977).

Micromagnetics 172

MicromagneticsMicromagnetics deals with the interactions between magnetic moments on sub-micrometre length scales. These aregoverned by several competing energy terms. Dipolar energy is the energy which causes magnets to align north tosouth pole. Exchange energy will attempt to make the magnetic moments in the immediately surrounding space lieparallel to one another (if the material is ferromagnetic) or antiparallel to one another (if antiferromagnetic).Anisotropy energy is low when the magnetic moments are aligned along a particular crystal direction. Zeemanenergy is at its lowest when magnetic moments lie parallel to an external magnetic field.Since the most efficient magnetic alignment (also known as a configuration) is the one in which the energy is lowest,the sum of these four energy terms will attempt to become as small as possible at the expense of the others, yieldingcomplex physical interactions.The competition of these interactions under different conditions is responsible for the overall behavior of a magneticsystem.

HistoryMicromagnetics as a field (i.e. that which deals specifically with the behaviour of (ferro)magnetic materials atsub-micrometer length scales) was introduced in 1963 when William Fuller Brown, Jr. published a paper onantiparallel domain wall structures. Until comparatively recently computation micromagnetics has been prohibitivelyexpensive in terms of computational power, but smaller problems are now solveable on a modern desktop PC.

Landau-Lifshitz-Gilbert equationGenerally, a form[1] of the Landau-Lifshitz-Gilbert equation:

is used to solve time-dependent micromagnetic problems, where is the magnetic moment per unit volume, is the effective magnetic field, is the Gilbert phenomenological damping parameter and is the electrongyromagnetic ratio. Furthermore, is the magnitude of the magnetization vector Equation (1) can be shown to be equivalent to the more complicated form

Originally, in 1935, Landau and Lifshitz used this expression without the denominator , which arosefrom Gilbert's modification in 1955.

Landau-Lifshitz equationIf in (1) we put the Gilbert damping parameter , then we get the famous, damping-free, Landau-Lifshitzequation (LLE)

The effective fieldAn essential merit of the micromagnetic theory concerns the answer on the question, how the effective field depends on the relevant interactions, namely, (i), on the exchange interaction; (ii), on the so-called anisotropy interaction; (iii), on the magnetic dipole-dipole interaction; and, (iv), on the external field (the so-called "Zeeman

Micromagnetics 173

field").The answer is somewhat involved: let the energies corresponding to (i) and (ii) be given by

and

Here we use the decomposition of the magnetization vector into its magnitude MS and the directionvector while A is the so-called exchange constant. V is the magnetic volume.

Then we have:

 [2][3][4]

Here the third term on the r.h.s. is the internal field produced at the position by the dipole-dipole interaction,whereas the fourth term is the external field, also called "Zeeman field". Usually the first and the third term play thedominating role, usually a competing one, in this complicated sum. In particular: due to the third term the effectivefield is a nonlocal function of the magnetization, i.e. although the Landau-Lifshitz-Gilbert equation looks relativelyharmless, one is actually dealing with a complicated nonlinear set of integro-differential equations.

ApplicationsApart from "conventional" magnetic domains and domain-walls, the theory also treats the statics and dynamics oftopological "line" and "point" configurations, e.g. magnetic vortex and antivortex states [5]  or even 3d-"Blochpoints" [6][7] , where, for example, the magnetization leads radially into all directions from the origin, or intotopologically equivalent configurations. Thus in space, and also in time, nano- (and even pico-)scales are used.The corresponding topological quantum numbers[7] are thought to be used as information carriers, to apply the mostrecent, and already studied, propositions in information technology.

Footnotes and References[1][1] There are different (equivalent) forms of the Landau-Lifshitz-Gilbert equation.[2] Here the minus sign at the second place on the r.h.s. is obvious: the magnetization chooses that direction which is lowest in energy. .

[3] We use the cgs system of units. In the SI system, in the third term on the r.h.s. an additional factor appears.

[4] Note that certain transformations of are always allowed, e.g. one can add any modification parallel to since this does

not change [5] S. Komineas, N. Papanicolaou: Dynamics of vortex-antivortex pairs in ferromagnets, in: arXiv:0712.3684v1, (2007)[6] A. Thiaville et al., Micromagnetic study of Bloch-point-mediated vortex core reversal, in: Phys. Rev. B, vol. 67 (9), 094410 (2003),

doi:10.1103/PhysRevB.67.094410[7] W. Döring, Point singularities in micromagnetism, J. Appl. Phys. 39, 1006 (1968), (http:/ / scitation. aip. org/ getabs/ servlet/

GetabsServlet?prog=normal& id=JAPIAU000039000002001006000001& idtype=cvips& gifs=yes)

Micromagnetics 174

Literature• Brown, William Fuller, Jr. (1963). Micromagnetics. New York: Wiley. ISBN 0-88275-665-6.• Gilbert, Thomas L. (2004). "A Phenomenological Theory of Damping in Ferromagnetic Materials". IEEE

Transactions on Magnetics 40 (6): 3443–3449. Bibcode 2004ITM....40.3443G.doi:10.1109/TMAG.2004.836740. ISSN 0018-9464.

External links• µMAG -- Micromagnetic Modeling Activity Group (http:/ / www. ctcms. nist. gov/ mumag/ mumag. org. html).• Magnetization dynamics applet (http:/ / www. bama. ua. edu/ ~tmewes/ Java/ dynamics/ MagnetizationDynamics.

shtml).• OOMMF - The Object-Oriented Micromagnetic Framework (http:/ / math. nist. gov/ oommf/ ) - a popular free

micromagnetic simulation tool using finite difference lattice discretisations of space and FFT.• MuMax — a GPU-based, open-source micromagnetic simulation code. (http:/ / arxiv. org/ pdf/ 1102. 3069)• Magpar (http:/ / www. cwscholz. net/ Main/ MagparProject) - a parallelizable, finite element based, free

micromagnetic simulation package.• Nmag (http:/ / nmag. soton. ac. uk/ ) - a parallelizable, finite element based, free micromagnetic simulator that is

scriptable in Python.• FEMME -- [[Finite element (http:/ / www. suessco. com/ simulations)] based micromagnetic package,

commercial].• LLGMicromagnetics -- [[Finite difference (http:/ / llgmicro. home. mindspring. com/ )] based micromagnetic

package, commercial].• Magsimus Deluxe -- [[Finite difference (http:/ / www. magoasis. com/ )], Multiphysics based micromagnetic

package, commercial].

Molecule-based magnets 175

Molecule-based magnetsMolecule-based magnets are a class of materials capable of displaying ferromagnetism. This class expands thematerials properties typically associated with magnets to include low density, transparency, electrical insulation, andlow-temperature fabrication, as well as combine magnetic ordering with other properties such asphotoresponsiveness. Essentially all of the common magnetic phenomena associated with conventionaltransition-metal and rare-earth-based magnets can be found in molecule-based magnets.[1]

BackgroundMolecule-based magnets comprise a class of materials which differ from conventional magnets in one of severalways. Most traditional magnetic materials are comprised purely of metals (Fe, Co, Ni) or metal oxides (CrO2) inwhich the unpaired electrons spins that contribute to the net magnetic moment reside only on metal atoms in d- orf-type orbitals.In molecule-based magnets, the structural building blocks are molecular in nature. These building blocks are eitherpurely organic molecules, coordination compounds or a combination of both. In this case, the unpaired electrons mayreside in d or f orbitals on isolated metal atoms, but may also reside in highly localized s and p orbitals as well on thepurely organic species. Like conventional magnets, they may be classified as hard or soft, depending on themagnitude of the coercive field.Another distinguishing feature is that molecule-based magnets are prepared via low-temperature solution-basedtechniques, versus high-temperature metallurgical processing or electroplating (in the case of magnetic thin films).This enables a chemical tailoring of the molecular building blocks to tune the magnetic properties.Specific materials include purely organic magnets made of organic radicals for example p-nitrophenyl nitronylnitroxides [2], decamethylferrocenium tetracyanoethenide[3], mixed coordination compounds with bridging organicradicals [4], Prussian blue related compounds [5], and charge transfer complexes [6].Molecule-based magnets derive their net moment from the cooperative effect of the spin-bearing molecular entities,and can display bulk ferromagnetic and ferrimagnetic behavior with a true critical temperature. In this regard, theyare contrasted with single-molecule magnets, which are essentially superparamagnets (displaying a blockingtemperature versus a true critical temperature). This critical temperature represents the point at which the materialsswitches from a simple paramagnet to a bulk magnet, and can be detected by ac susceptibility and specific heatmeasurements.

HistoryThe first synthesis and characterization of molecule-based magnets was accomplished by Wickman and co-workers.This was a diethyldithiocarbamate-Fe(III) chloride compound.[7][8]

TheoryThe mechanism by which molecule-based magnets stabilize and display a net magnetic moment is quite differentthan that present in traditional metal- and ceramic-based magnets. For metallic magnets, the unpaired electrons alignthrough quantum mechanical effects (termed exchange) by virtue of the way in which the electrons fill the orbitals ofthe conductive band. For most oxide-based ceramic magnets, the unpaired electrons on the metal centers align viathe intervening diamagnetic bridging oxide (termed superexchange). The magnetic moment in molecule-basedmagnets is typically stabilized by one or more of three main mechanisms:•• Through space or dipolar coupling•• Exchange between orthogonal (non-overlapping) orbitals in the same spatial region

Molecule-based magnets 176

• Net moment via antiferromagnetic coupling of non-equal spin centers (ferrimagnetism)In general, molecule-based magnets tend to be of low dimensionality. Classic magnetic alloys based on iron andother ferromangetic materials feature metallic bonding, with all atoms essentially bonded to all nearest neighbors inthe crystal lattice. Thus, critical temperatures at which point these classical magnets cross over to the orderedmagnetic state tend to be high, since interactions between spin centers is strong. Molecule-based magnets, however,have spin bearing units on molecular entities, often with highly directional bonding. In some cases, chemicalbonding is restricted to one dimension (chains). Thus, interactions between spin centers are also limited toone-dimension, and ordering temperatures are much lower than metal/alloy-type magnets. Also, large parts of themagnetic material are essentially diamagnetic, and contribute nothing to the net magnetic moment.These aspects of molecule-based magnets present significant challenges toward reaching the ultimate goal of "roomtemperature" molecule-based magnets. Low-dimensional materials, however, can provide valuable experimental datafor validating physics models of magnetism (which are often of low dimension, to simplify calculations).

ApplicationsMolecule-based magnets currently remain laboratory curiosities with no real world applications. As indicated, this islargely due to the very low critical temperature at which these materials become magnetic. This is related to themagnitude of the magnetic coupling, which is very weak in these materials. In this regard, they are similar tosuperconductors, which require cooling for use.

References[1] Molecule-Based Magnets Materials Research Society (http:/ / www. mrs. org/ s_mrs/ doc. asp?CID=9554& DID=200481) Retrieved on 20

December 2007[2] Bulk ferromagnetism in the β-phase crystal of the p-nitrophenyl nitronyl nitroxide radical Chemical Physics Letters, Volume 186, Issues 4-5,

15 November 1991, Pages 401-404 Masafumi Tamura, Yasuhiro Nakazawa, Daisuke Shiomi, Kiyokazu Nozawa, Yuko Hosokoshi, MasayasuIshikawa, Minuro Takahashi, Minoru Kinoshita doi:10.1016/0009-2614(91)90198-I

[3] Sailesh Chittipeddi K. R. Cromack Joel S. Miller A. J. Epstein Phys. Rev. Lett. 58, 2695–2698 (1987) Ferromagnetism in moleculardecamethylferrocenium tetracyanoethenide (DMeFc TCNE)

[4] Caneschi A., et al. Acc. Chem. Res. 22, 392 (1989)[5] S. Ferlay, et al. Nature 378, 701 (1995)[6] Miller J.S., et al. Chem. Rev. 88, 201 (1988)[7] Wickman, H.H., et al. Phys. Rev. 155, 563 (1967).[8] Wickman, H.H., et al. Phys. Rev. 163, 526 (1967).

Neodymium magnet 177

Neodymium magnet

Nickel plated neodymium magnet on a bracketfrom a hard drive.

Nickel-plated neodymium magnet cubes

Left: High-resolution transmission electronmicroscopy image of Nd2Fe14B; right: chemical

schema

A neodymium magnet (also known as NdFeB, NIB, or Neo magnet),the most widely-used type of rare-earth magnet, is a permanent magnetmade from an alloy of neodymium, iron, and boron to form theNd2Fe14B tetragonal crystalline structure. Developed in 1982 byGeneral Motors and Sumitomo Special Metals, neodymium magnetsare the strongest type of permanent magnet made. They have replacedother types of magnet in the many applications in modern products thatrequire strong permanent magnets, such as motors in cordless tools,hard disk drives, and magnetic fasteners.

Description

The tetragonal Nd2Fe14B crystal structure has exceptionally highuniaxial magnetocrystalline anisotropy (HA~7 teslas). This gives thecompound the potential to have high coercivity (i.e., resistance to beingdemagnetized). The compound also has a high saturationmagnetization (Js ~1.6 T or 16 kG) and typically 1.3 tesla. Therefore,as the maximum energy density is proportional to Js

2, this magneticphase has the potential for storing large amounts of magnetic energy(BHmax ~ 512 kJ/m3 or 64 MG·Oe), considerably more than samariumcobalt (SmCo) magnets, which were the first type of rare earth magnetto be commercialized. In practice, the magnetic properties ofneodymium magnets depend on the alloy composition, microstructure,and manufacturing technique employed.

History and manufacturing techniques

In 1982, General Motors and Sumitomo Special Metals discovered theNd2Fe14B compound. The effort was principally driven by the highmaterial cost of the SmCo permanent magnets, which had beendeveloped earlier. General Motors focused on the development ofmelt-spun nanocrystalline Nd2Fe14B magnets, while Sumitomodeveloped full density sintered Nd2Fe14B magnets. General MotorsCorporation commercialized its inventions of isotropic Neo powder, bonded Neo magnets and the related productionprocesses by founding Magnequench in 1986. Magnequench is now part of the Neo Materials Technology Inc. andsupplies melt spun Nd2Fe14B powder to bonded magnet manufacturers. The Sumitomo facility has become part ofthe Hitachi corporation and currently manufactures and licenses other companies to produce sintered Nd2Fe14Bmagnets. Hitachi holds more than 600 patents covering Neodymium magnets.[1]

Sintered Nd2Fe14B tends to be vulnerable to corrosion. In particular, corrosion along grain boundaries may causedeterioration of a sintered magnet. This problem is addressed in many commercial products by providing a protectivecoating. Nickel plating or two layered copper nickel plating is used as a standard method, although plating with othermetals or polymer and lacquer protective coatings are also in use.[2]

Neodymium magnet 178

ProductionThere are two principal neodymium magnet manufacturing routes:•• The classical powder metallurgy or sintered magnet process•• The rapid solidification or bonded magnet processSintered Nd-magnets are prepared by the raw materials being melted in a furnace, cast into a mold and cooled toform ingots. The ingots are pulverized and milled to tiny particles. This undergoes a process of liquid-phase sinteringwhereby the powder is magnetically aligned into dense blocks which are then heat-treated, cut to shape, surfacetreated and magnetized. Currently, between 45,000 and 50,000 tons of sintered neodymium magnets are producedeach year, mainly in China and Japan. As of 2011, China produces more than 95% of rare earth elements, andproduces 76% of the world's total rare earth magnets.[1]

Bonded Nd-magnets are prepared by melt spinning a thin ribbon of the Nd-Fe-B alloy. The ribbon containsrandomly oriented Nd2Fe14B nano-scale grains. This ribbon is then pulverized into particles, mixed with a polymerand either compression or injection molded into bonded magnets. Bonded magnets offer less flux than sinteredmagnets but can be net-shape formed into intricately shaped parts and do not suffer significant eddy current losses.There are approximately 5,500 tons of Neo bonded magnets produced each year. In addition, it is possible tohot-press the melt spun nanocrystalline particles into fully dense isotropic magnets, and thenupset-forge/back-extrude these into high-energy anisotropic magnets.

Properties

Magnetic propertiesSome important properties used to compare permanent magnets are: remanence (Mr), which measures the strength ofthe magnetic field; coercivity (Hci), the material's resistance to becoming demagnetized; energy product (BHmax), thedensity of magnetic energy; and Curie temperature (TC), the temperature at which the material loses its magnetism.Neodymium magnets have higher remanence, much higher coercivity and energy product, but often lower Curietemperature than other types. Neodymium is alloyed with terbium and dysprosium in order to preserve its magneticproperties at high temperatures.[3] The table below compares the magnetic performance of neodymium magnets withother types of permanent magnets.

Magnet Mr

(T) Hci

(kA/m) BHmax

(kJ/m3) TC

(°C)

Nd2Fe14B (sintered) 1.0–1.4 750–2000 200–440 310–400

Nd2Fe14B (bonded) 0.6–0.7 600–1200 60–100 310–400

SmCo5 (sintered) 0.8–1.1 600–2000 120–200 720

Sm(Co, Fe, Cu, Zr)7 (sintered) 0.9–1.15 450–1300 150–240 800

Alnico (sintered) 0.6–1.4 275 10–88 700–860

Sr-ferrite (sintered) 0.2–0.4 100–300 10–40 450

Neodymium magnet 179

Physical and mechanical properties

Comparison of physical properties of sintered neodymium and Sm-Co magnets[4]

Property Neodymium Sm-Co

Remanence (T) 1–1.3 0.82–1.16

Coercivity (MA/m) 0.875–1.99 0.493–1.59

Relative permeability 1.05 1.05

Temperature coefficient of remanence (%/K) −0.12 −0.03

Temperature coefficient of coercivity (%/K) −0.55..–0.65 −0.15..–0.30

Curie temperature (°C) 320 800

Density (g/cm3) 7.3–7.5 8.2–8.4

CTE, magnetizing direction (1/K) 5.2×10−6 5.2×10−6

CTE, normal to magnetizing direction (1/K) −0.8×10−6 11×10−6

Flexural strength (N/mm2) 250 150

Compressive strength (N/mm2) 1100 800

Tensile strength (N/mm2) 75 35

Vickers hardness (HV) 550–650 500–550

Electrical resistivity (Ω·cm) (110–170)×10−6 86×10−6

HazardsThe greater force exerted by rare earth magnets creates hazards that are not seen with other types of magnet.Neodymium magnets larger than a few cubic centimeters are strong enough to cause injuries to body parts pinchedbetween two magnets, or a magnet and a metal surface, even causing broken bones.[5]

Magnets allowed to get too near each other can strike each other with enough force to chip and shatter the brittlematerial, and the flying chips can cause injuries. There have even been cases where young children who haveswallowed several magnets have had a fold of the digestive tract pinched between the magnets, causing injury ordeath.[6] The stronger magnetic fields can be hazardous to mechanical and electronic devices, as they can erasemagnetic media such as floppy disks and credit cards, and magnetize watches and other clockwork mechanisms andthe shadow masks of CRT type monitors at a significant distance.

Neodymium magnet 180

Applications

In technology

Ring magnets

Hard disk drive

Neodymium magnets have replaced alnico and ferrite magnets in manyof the myriad applications in modern technology where strongpermanent magnets are required, because their greater strength allowsthe use of smaller, lighter magnets for a given application. Someexamples are:

• Head actuators for computer hard disks• Magnetic resonance imaging (MRI)• Magnetic guitar pickups• Loudspeakers and headphones• Magnetic bearings and couplings• Electric motors:

• cordless tools• Servo motors• Lifting and compressor motors• Synchronous motors• Spindle and stepper motors• Electrical power steering• Drive motors for hybrid and electric vehicles. The electric

motors of each Toyota Prius require 1 kilogram (2.2 pounds) ofneodymium.[3]

•• Actuators• Electric generators for wind turbines; up to 600 kg of PM material per megawatt (Neodymium content is

estimated to be 31% of magnet weight).[1]

Demand for neodymium in electric vehicles is estimated to be 5 times larger than that in wind turbines.[1]

Other applicationsIn addition, the greater strength of neodymium magnets has inspired new applications in areas where magnets werenot used before, such as magnetic jewelry clasps, children's magnetic building sets (and other neodymium magnettoys) and as part of the closing mechanism of modern sport parachute equipment.[7] The strength and magnetic fieldhomogeneity on neodymium magnets has also opened new applications in the medical field with the introduction ofopen magnetic resonance imaging (MRI) scanners used to image the body in radiology departments as an alternativeto superconducting magnets that use a coil of superconducting wire to produce the magnetic field. As with mostsolid-based magnets, the magnetic field gradient of neodymium magnets decreases towards the centers of theirsurfaces, thus there is a force that attracts metallic objects to the edges.

Neodymium magnet 181

References[1] Chu, Steven. Critical Materials Strategy (http:/ / energy. gov/ sites/ prod/ files/ DOE_CMS_2011. pdf) United States Department of Energy,

December 2011. Accessed: 23 December 2011.[2] Drak, M.; Dobrzanski, L.A. (2007). "Corrosion of Nd-Fe-B permanent magnets" (http:/ / www. journalamme. org/ papers_vol20/ 1369S. pdf).

Journal of Achievements in Materials and Manufacturing Engineering 20 (1–2). .[3] As hybrid cars gobble rare metals, shortage looms (http:/ / www. reuters. com/ article/ newsOne/ idUSTRE57U02B20090831), Reuters,

August 31, 2009.[4] Juha Pyrhönen, Tapani Jokinen, Valéria Hrabovcová (2009). Design of Rotating Electrical Machines (http:/ / books. google. com/

?id=_y3LSh1XTJYC& pg=PT232). John Wiley and Sons. p. 232. ISBN 0-470-69516-1. .[5] Swain, Frank (March 6, 2009). "How to remove a finger with two super magnets" (http:/ / scienceblogs. com/ sciencepunk/ 2009/ 03/

how_to_remove_a_finger_with_tw. php). The Sciencepunk Blog. Seed Media Group LLC. . Retrieved 2009-06-28.[6] "Magnet Safety Alert" (http:/ / www. cpsc. gov/ CPSCPUB/ PUBS/ magnet. pdf). U.S. Consumer Product Safety Commission. . Retrieved 7

August 2009.[7] United Parachute Technologies Options Guide: http:/ / www. unitedparachutetechnologies. com/ index. php?option=com_content&

task=view& id=22

Further reading• MMPA 0100-00, Standard Specifications for Permanent Magnet Materials (http:/ / www. intl-magnetics. org/

pdfs/ 0100-00. pdf)• K.H.J. Buschow (1998) Permanent-Magnet Materials and their Applications, Trans Tech Publications Ltd.,

Switzerland, ISBN 0-87849-796-X• Campbell, Peter (1994). Permanent Magnet Materials and their Application. New York: Cambridge University

Press. ISBN 0-521-24996-1.• Furlani, Edward P. (2001). Permanent Magnet and Electromechanical Devices: Materials, Analysis and

Applications. London: Academic Press. ISBN 0-12-269951-3.• Brown, D (2002). "Developments in the processing and properties of NdFeB-type permanent magnets". Journal

of Magnetism and Magnetic Materials 248 (3): 432–440. Bibcode 2002JMMM..248..432B.doi:10.1016/S0304-8853(02)00334-7.

• The Dependence of Magnetic Properties and Hot Workability of Rare Earth-Iron-Boride Magnets UponComposition (http:/ / www. magnequench. com/ assets/ content/ magnequench/ mag_ref/ mag_pps/ pps_040701/IEEE2004_vMAG40. pdf/ ).

External links• Magnet Man (http:/ / www. coolmagnetman. com/ magindex. htm) Cool experiments with magnets• Geeky Rare-Earth Magnets Repel Sharks, Genevieve Rajewski, 05.15.07 , wired.com (http:/ / www. wired. com/

gadgets/ miscellaneous/ news/ 2007/ 05/ sharkmagnets)• Concern as China clamps down on rare earth exports, Cahal Milmo, 01.02.10, independent.co.uk (http:/ / www.

independent. co. uk/ news/ world/ asia/ concern-as-china-clamps-down-on-rare-earth-exports-1855387. html)

Paramagnetism 182

Paramagnetism

Simple illustration of a paramagnetic probe madeup from miniature magnets.

A trickle of liquid oxygen is deflected by amagnetic field, illustrating its paramagnetic

property

Paramagnetism is a form of magnetism whereby the paramagneticmaterial is only attracted when in the presence of an externally appliedmagnetic field. In contrast with this behavior, diamagnetic materialsare repelled by magnetic fields.[1] Paramagnetic materials have arelative magnetic permeability greater or equal to unity (i.e., a positivemagnetic susceptibility) and hence are attracted to magnetic fields. Themagnetic moment induced by the applied field is linear in the fieldstrength and rather weak. It typically requires a sensitive analyticalbalance to detect the effect and modern measurements on paramagneticmaterials are often conducted with a SQUID magnetometer.

Paramagnetic materials have a small, positive susceptibility tomagnetic fields. These materials are slightly attracted by a magneticfield and the material does not retain the magnetic properties when theexternal field is removed. Paramagnetic properties are due to thepresence of some unpaired electrons, and from the realignment of theelectron paths caused by the external magnetic field. Paramagneticmaterials include magnesium, molybdenum, lithium, and tantalum.Unlike ferromagnets, paramagnets do not retain any magnetization inthe absence of an externally applied magnetic field, because thermalmotion randomizes the spin orientations. Some paramagnetic materialsretain spin disorder at absolute zero, meaning they are paramagnetic inthe ground state. Thus the total magnetization drops to zero when theapplied field is removed. Even in the presence of the field there is only a small induced magnetization because only asmall fraction of the spins will be oriented by the field. This fraction is proportional to the field strength and thisexplains the linear dependency. The attraction experienced by ferromagnetic materials is non-linear and muchstronger, so that it is easily observed, for instance, by the attraction between a refrigerator magnet and the iron of therefrigerator itself.

Relation to electron spinsConstituent atoms or molecules of paramagnetic materials have permanent magnetic moments (dipoles), even in theabsence of an applied field. The permanent moment generally is due to the spin of unpaired electrons in atomic ormolecular electron orbitals (see Magnetic moment). In pure paramagnetism, the dipoles do not interact with oneanother and are randomly oriented in the absence of an external field due to thermal agitation, resulting in zero netmagnetic moment. When a magnetic field is applied, the dipoles will tend to align with the applied field, resulting ina net magnetic moment in the direction of the applied field. In the classical description, this alignment can beunderstood to occur due to a torque being provided on the magnetic moments by an applied field, which tries to alignthe dipoles parallel to the applied field. However, the true origins of the alignment can only be understood via thequantum-mechanical properties of spin and angular momentum.If there is sufficient energy exchange between neighbouring dipoles they will interact, and may spontaneously alignor anti-align and form magnetic domains, resulting in ferromagnetism (permanent magnets) or antiferromagnetism,respectively. Paramagnetic behavior can also be observed in ferromagnetic materials that are above their Curietemperature, and in antiferromagnets above their Néel temperature. At these temperatures the available thermalenergy simply overcomes the interaction energy between the spins.

Paramagnetism 183

In general paramagnetic effects are quite small: the magnetic susceptibility is of the order of 10−3 to 10−5 for mostparamagnets, but may be as high as 10−1 for synthetic paramagnets such as ferrofluids.

Delocalization

Selected Pauli-paramagnetic metals[2]

Material Magnetic susceptibility (×10−5)

Tungsten 6.8

Cesium 5.1

Aluminium 2.2

Lithium 1.4

Magnesium 1.2

Sodium 0.72

In many metallic materials the electrons are itinerant, i.e. they travel through the solid more or less as an electrongas. This behavior results from strong interactions (good orbital overlap in a chemist's vocabulary) between the wavefunctions of neighboring atoms in the extended lattice structure. The wave functions of the valence electrons thusform a band with equal numbers of spins up and down. When exposed to an external field only those electrons closeto the Fermi level will respond and a small surplus of one type of spins will result. This effect is a weak form ofparamagnetism known as Pauli-paramagnetism. The effect always competes with a diamagnetic response of oppositesign due to all the core electrons of the atoms. Stronger forms of magnetism usually require localized rather thanitinerant electrons. However in some cases a bandstructure can result in which there are two delocalized sub-bandswith states of opposite spins that have different energies. If one subband is preferentially filled over the other, onecan have itinerant ferromagnetic order. This situation usually only occurs in relatively narrow (d-)bands, which arepoorly delocalized.

s and p electrons

Generally, strong delocalization in a solid due to large overlap with neighboring wave functions tends to lead topairing of spins (quenching) and thus weak magnetism. This is why s- and p-type metals are typically eitherPauli-paramagnetic or as in the case of gold even diamagnetic. In the latter case the diamagnetic contribution fromthe closed shell inner electrons simply wins from the weak paramagnetic term of the almost free electrons.

d and f electrons

Stronger magnetic effects are typically only observed when d- or f-electrons are involved. Particularly the latter areusually strongly localized. Moreover the size of the magnetic moment on a lanthanide atom can be quite large as itcan carry up to 7 unpaired electrons in the case of gadolinium(III) (hence its use in MRI). This high magneticmoments associated with lanthanides is one reason why superstrong magnets are typically based on elements likeneodymium or samarium.

Molecular localization

Of course the above picture is a generalization as it pertains to materials with an extended lattice rather than a molecular structure. Molecular structure can also lead to localization of electrons. Although there are usually energetic reasons why a molecular structure results such that it does not exhibit partly filled orbitals (i.e. unpaired spins), some non-closed shell moieties do occur in nature. Molecular oxygen is a good example. Even in the frozen solid it contains di-radical molecules resulting in paramagnetic behavior. The unpaired spins reside in orbitals

Paramagnetism 184

derived from oxygen p wave functions, but the overlap is limited to the one neighbor in the O2 molecules. Thedistances to other oxygen atoms in the lattice remain too large to lead to delocalization and the magnetic momentsremain unpaired.

Curie's lawFor low levels of magnetization, the magnetization of paramagnets follows what is known as Curie's law, at leastapproximately. This law indicates that the susceptibility of paramagnetic materials is inversely proportional totheir temperature, i.e. that materials become more magnetic at lower temperatures. The mathematical expression is:

where:is the resulting magnetization

is the magnetic susceptibilityis the auxiliary magnetic field, measured in amperes/meter

is absolute temperature, measured in kelvinsis a material-specific Curie constant

Curie's law is valid under the commonly encountered conditions of low magnetization (μBH ≲ kBT), but does notapply in the high-field/low-temperature regime where saturation of magnetization occurs (μBH ≳ kBT) and magneticdipoles are all aligned with the applied field. When the dipoles are aligned, increasing the external field will notincrease the total magnetization since there can be no further alignment.For a paramagnetic ion with noninteracting magnetic moments with angular momentum J, the Curie constant isrelated the individual ions' magnetic moments,

.

The parameter μeff is interpreted as the effective magnetic moment per paramagnetic ion. If one uses a classicaltreatment with molecular magnetic moments represented as discrete magnetic dipoles, μ, a Curie Law expression ofthe same form will emerge with μ appearing in place of μeff.

Click "show" to see a derivation of this law:

Curie's Law can be derived by considering a substance with noninteracting magnetic moments with angularmomentum J. If orbital contributions to the magnetic moment are negligible (a common case), then in what follows J= S. If we apply a magnetic field along what we choose to call the z-axis, the energy levels of each paramagneticcenter will experience Zeeman splitting of its energy levels, each with a z-component labeled by MJ (or just MS forthe spin-only magnetic case). Applying semiclassical Boltzmann statistics, the molar magnetization of such asubstance is

.

Where is the z-component of the magnetic moment for each Zeeman level, so – μB iscalled the Bohr Magneton and gJ is the Landé g-factor, which reduces to the free-electron g-factor, gS when J = S. (inthis treatment, we assume that the x- and y-components of the magnetization, averaged over all molecules, cancelout because the field applied along the z-axis leave them randomly oriented.) The energy of each Zeeman level is

. For temperatures over a few K, , and we can apply theapproximation :

Paramagnetism 185

,

which yields:

. The molar bulk magnetization is then

,

and the molar susceptibility is given by

.

When orbital angular momentum contributions to the magnetic moment are small, as occurs for most organicradicals or for octahedral transition metal complexes with d3 or high-spin d5 configurations, the effective magneticmoment takes the form (ge = 2.0023... ≈ 2),

, where n is the number of unpaired electrons. In othertransition metal complexes this yields a useful, if somewhat cruder, estimate.

Examples of paramagnetsMaterials that are called 'paramagnets' are most often those that exhibit, at least over an appreciable temperaturerange, magnetic susceptibilities that adhere to the Curie or Curie–Weiss laws. In principle any system that containsatoms, ions, or molecules with unpaired spins can be called a paramagnet, but the interactions between them need tobe carefully considered.

Systems with minimal interactionsThe narrowest definition would be: a system with unpaired spins that do not interact with each other. In thisnarrowest sense, the only pure paramagnet is a dilute gas of monatomic hydrogen atoms. Each atom has onenon-interacting unpaired electron. Of course, the latter could be said about a gas of lithium atoms but these alreadypossess two paired core electrons that produce a diamagnetic response of opposite sign. Strictly speaking Li is amixed system therefore, although admittedly the diamagnetic component is weak and often neglected. In the case ofheavier elements the diamagnetic contribution becomes more important and in the case of metallic gold it dominatesthe properties. Of course, the element hydrogen is virtually never called 'paramagnetic' because the monatomic gas isstable only at extremely high temperature; H atoms combine to form molecular H2 and in so doing, the magneticmoments are lost (quenched), because the spins pair. Hydrogen is therefore diamagnetic and the same holds true formost elements. Although the electronic configuration of the individual atoms (and ions) of most elements containunpaired spins, it is not correct to call these elements 'paramagnets' because at ambient temperature quenching isvery much the rule rather than the exception. However, the quenching tendency is weakest for f-electrons because f(especially 4f) orbitals are radially contracted and they overlap only weakly with orbitals on adjacent atoms.Consequently, the lanthanide elements with incompletely filled 4f-orbitals are paramagnetic or magneticallyordered.[3]

Paramagnetism 186

μeff values for typical d3 and d5 transition metal complexes.[4]

Material μeff

/μB

[Cr(NH3)6]Br3 3.77

K3[Cr(CN)6] 3.87

K3[MoCl6] 3.79

K4[V(CN)6] 3.78

[Mn(NH3)6]Cl2 5.92

(NH4)2[Mn(SO4)2]·6H2O 5.92

NH4[Fe(SO4)2]·12H2O 5.89

Thus, condensed phase paramagnets are only possible if the interactions of the spins that lead either to quenching orto ordering are kept at bay by structural isolation of the magnetic centers. There are two classes of materials forwhich this holds:•• Molecular materials with a (isolated) paramagnetic center.

• Good examples are coordination complexes of d- or f-metals or proteins with such centers, e.g. myoglobin. Insuch materials the organic part of the molecule acts as an envelope shielding the spins from their neighbors.

• Small molecules can be stable in radical form, oxygen O2 is a good example. Such systems are quite rarebecause they tend to be rather reactive.

•• Dilute systems.• Dissolving a paramagnetic species in a diamagnetic lattice at small concentrations, e.g. Nd3+ in CaCl2 will

separate the neodymium ions at large enough distances that they do not interact. Such systems are of primeimportance for what can be considered the most sensitive method to study paramagnetic systems: EPR.

Systems with interactions

Idealized Curie–Weiss behavior; N.B. TC=θ, but TN is not θ. Paramagneticregimes are denoted by solid lines. Close to TN or TC the behavior usually deviates

from ideal.

As stated above many materials that containd- or f-elements do retain unquenched spins.Salts of such elements often showparamagnetic behavior but at low enoughtemperatures the magnetic moments mayorder. It is not uncommon to call suchmaterials 'paramagnets', when referring totheir paramagnetic behavior above theirCurie or Néel-points, particularly if suchtemperatures are very low or have neverbeen properly measured. Even for iron it isnot uncommon to say that iron becomes aparamagnet above its relatively highCurie-point. In that case the Curie-point isseen as a phase transition between aferromagnet and a 'paramagnet'. The wordparamagnet now merely refers to the linearresponse of the system to an applied field, the temperature dependence of which requires an amended version ofCurie's law, known as the Curie–Weiss law:

Paramagnetism 187

This amended law includes a term θ that describes the exchange interaction that is present albeit overcome bythermal motion. The sign of θ depends on whether ferro- or antiferromagnetic interactions dominate and it is seldomexactly zero, except in the dilute, isolated cases mentioned above.Obviously, the paramagnetic Curie–Weiss description above TN or TC is a rather different interpretation of the word'paramagnet' as it does not imply the absence of interactions, but rather that the magnetic structure is random in theabsence of an external field at these sufficiently high temperatures. Even if θ is close to zero this does not mean thatthere are no interactions, just that the aligning ferro- and the anti-aligning antiferromagnetic ones cancel. Anadditional complication is that the interactions are often different in different directions of the crystalline lattice(anisotropy), leading to complicated magnetic structures once ordered.Randomness of the structure also applies to the many metals that show a net paramagnetic response over a broadtemperature range. They do not follow a Curie type law as function of temperature however, often they are more orless temperature independent. This type of behavior is of an itinerant nature and better called Pauli-paramagnetism,but it is not unusual to see e.g. the metal aluminium called a 'paramagnet', even though interactions are strongenough to give this element very good electrical conductivity.

SuperparamagnetsSome materials show induced magnetic behavior that follows a Curie type law but with exceptionally large valuesfor the Curie constants. These materials are known as superparamagnets. They are characterized by a strongferromagnetic or ferrimagnetic type of coupling into domains of a limited size that behave independently from oneanother. The bulk properties of such a system resembles that of a paramagnet, but on a microscopic level they areordered. The materials do show an ordering temperature above which the behavior reverts to ordinaryparamagnetism (with interaction). Ferrofluids are a good example, but the phenomenon can also occur inside solids,e.g., when dilute paramagnetic centers are introduced in a strong itinerant medium of ferromagnetic coupling such aswhen Fe is substituted in TlCu2Se2 or the alloy AuFe. Such systems contain ferromagnetically coupled clusters thatfreeze out at lower temperatures. They are also called mictomagnets.

References[1] G. L. Miessler and D. A. Tarr “Inorganic Chemistry” 3rd Ed, Pearson/Prentice Hall publisher, ISBN 0-13-035471-6.[2] Nave, Carl L. "Magnetic Properties of Solids" (http:/ / hyperphysics. phy-astr. gsu. edu/ Hbase/ tables/ magprop. html). HyperPhysics. .

Retrieved 2008-11-09.[3] J. Jensen and A. R. MacKintosh, "Rare Earth Magnetism" (http:/ / www2. nbi. ku. dk/ page40667. htm). . Retrieved 2009-07-12., (Clarendon

Press, Oxford: 1991).[4] A. F. Orchard, Magnetochemistry, (Oxford University Press: 2003).

General reference texts• Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).• John David Jackson, Classical Electrodynamics (Wiley: New York, 1999).

Paramagnetism 188

External linkshttp:/ / www. ndt-ed. org/ EducationResources/ CommunityCollege/ MagParticle/ Physics/ MagneticMatls. htm

Plastic magnetA plastic magnet is a non-metallic magnet made from an organic polymer. One example is PANiCNQ, which is acombination of emeraldine-based polyaniline (PANi) and tetracyanoquinodimethane (TCNQ). When it was createdby researchers Dr. Naveed Zaidi, at the University of Durham in 2004 it was the first magnetic polymer to functionat room temperature.[1]

PANi is a conductive polymer that is stable in air. When combined with the free radical-forming TCNQ as anacceptor molecule, it can mimic the mechanism of metallic magnets. The magnetic properties arise from the fullypi-conjugated nitrogen-containing backbone combined with molecular charge transfer side groups. These propertiescause the molecule to have a high density of localized spins that can give rise to coupling of their magnetic fields.When this polymer magnet is synthesized, the polymer chains need 3 months to line up before displaying anynotable magnetism.Plastic magnets could have uses in computer hardware, for example as disc drives and in medical devices such aspacemakers and cochlear implants where the organic material is more likely to be biocompatible than its metalliccounterpart.In February 2002, researchers from Ohio State University & University of Utah developed the world's firstlight-tunable plastic magnet.[2] The plastic material became 1.5 times more magnetic when blue light shines on it.Green laser light reversed the effect somewhat, by decreasing the material's magnetism to 60 percent of its normallevel. The plastic magnet was made from a polymer made of tetracyanoethylene (TCNE) combined with manganese(Mn) ions -- atoms of the metal manganese with electrons removed. The magnet functioned up to a temperature of75 K (about -200ºC, or -325ºF).

Notes[1] Naveed A. Zaidi; S.R. Giblin; I. Terry; A.P. Monkman (2004). "Room temperature magnetic order in an organic magnet derived from

polyaniline" (https:/ / www. dur. ac. uk/ ian. terry/ teaching/ Level4Projects/ Polymer_45_5683. pdf). Polymer 45 (16): 5683-5689. . Retrieved2012-04-02.

[2] Pam Frost Gorder (Feb 1, 2002). "Researchers Develop World's First Light-tunable 'Plastic' Magnet" (http:/ / researchnews. osu. edu/ archive/magnetun. htm). Ohio State University. . Retrieved 2012-04-02.

External links• Matthew Killeya (30 August 2004). "First practical plastic magnets created" (http:/ / www. newscientist. com/

article. ns?id=dn6326). New Scientist. Retrieved 2012-04-02.• Dušan A. Pejaković; Chitoshi Kitamura; Joel S. Miller; Arthur J. Epstein (2002). "Photoinduced Magnetization in

the Organic-Based Magnet Mn(TCNE)x•y(CH2Cl2)". Physical Review Letters 88 (5): 057202.Bibcode 2002PhRvL..88e7202P. doi:10.1103/PhysRevLett.88.057202.

Rare-earth magnet 189

Rare-earth magnet

Ferrofluid on glass, with a rare-earth magnet underneath.

Rare-earth magnets are strong permanent magnetsmade from alloys of rare earth elements. Developed inthe 1970s and 80s, rare-earth magnets are the strongesttype of permanent magnets made, producingsignificantly stronger magnetic fields than other typessuch as ferrite or alnico magnets. The magnetic fieldtypically produced by rare-earth magnets can be inexcess of 1.4 teslas, whereas ferrite or ceramic magnetstypically exhibit fields of 0.5 to 1 tesla. There are twotypes: neodymium magnets and samarium-cobaltmagnets. Rare earth magnets are extremely brittle andalso vulnerable to corrosion, so they are usually platedor coated to protect them from breaking and chipping.

The term "rare earth" can be misleading as these metals are not particularly rare or precious;[1][2] they are about asabundant as tin or lead.[3] The development of rare earth magnets began around 1966, when K. J. Strnat and G.Hoffer of the US Air Force Materials Laboratory discovered that an alloy of yttrium and cobalt, YCo5, had by far thelargest magnetic anisotropy constant of any material then known.[4][5]

Explanation of strengthThe rare earth (lanthanide) elements are metals that are ferromagnetic, meaning that like iron they can bemagnetized, but their Curie temperatures are below room temperature, so in pure form their magnetism only appearsat low temperatures. However, they form compounds with the transition metals such as iron, nickel, and cobalt, andsome of these have Curie temperatures well above room temperature. Rare earth magnets are made from thesecompounds.The advantage of the rare earth compounds over other magnets is that their crystalline structures have very highmagnetic anisotropy. This means that a crystal of the material is easy to magnetize in one particular direction, butresists being magnetized in any other direction.Atoms of rare earth elements can retain high magnetic moments in the solid state. This is a consequence ofincomplete filling of the f-shell, which can contain up to 7 unpaired electrons with aligned spins. Electrons in suchorbitals are strongly localized and therefore easily retain their magnetic moments and function as paramagneticcenters. Magnetic moments in other orbitals are often lost due to strong overlap with the neighbors; for example,electrons participating in covalent bonds form pairs with zero net spin.High magnetic moments at the atomic level in combination with a stable alignment (high anisotropy) results in highstrength.

Magnetic propertiesSome important properties used to compare permanent magnets are: remanence (Br), which measures the strength ofthe magnetic field; coercivity (Hci), the material's resistance to becoming demagnetized; energy product (BHmax), thedensity of magnetic energy; and Curie temperature (Tc), the temperature at which the material loses its magnetism.Rare earth magnets have higher remanence, much higher coercivity and energy product, but (for neodymium) lowerCurie temperature than other types. The table below compares the magnetic performance of the two types of rareearth magnet, neodymium (Nd2Fe14B) and samarium-cobalt (SmCo5), with other types of permanent magnets.

Rare-earth magnet 190

Magnet Br

(T) Hci

(kA/m) (BH)max

(kJ/m3) Tc

(°C)

Nd2Fe14B (sintered) 1.0–1.4 750–2000 200–440 310–400

Nd2Fe14B (bonded) 0.6–0.7 600–1200 60–100 310–400

SmCo5 (sintered) 0.8–1.1 600–2000 120–200 720

Sm(Co,Fe,Cu,Zr)7 (sintered) 0.9–1.15 450–1300 150–240 800

Alnico (sintered) 0.6–1.4 275 10–88 700–860

Sr-ferrite (sintered) 0.2–0.4 100–300 10–40 450

Types

Samarium-cobaltSamarium-cobalt magnets (chemical formula: SmCo5), the first family of rare earth magnets invented, are less usedthan neodymium magnets because of their higher cost and weaker magnetic field strength. However,samarium-cobalt has a higher Curie temperature, creating a niche for these magnets in applications where high fieldstrength is needed at high operating temperatures. They are highly resistant to oxidation, but sinteredsamarium-cobalt magnets are brittle and prone to chipping and cracking and may fracture when subjected to thermalshock.

Neodymium

Neodymium magnet with nickel plate mostlyremoved

Neodymium magnets, invented in the 1980s, are the strongest and mostaffordable type of rare-earth magnet. They are made of an alloy ofneodymium, iron and boron: (Nd2Fe14B) Neodymium magnets areused in numerous applications requiring strong, compact permanentmagnets, such as electric motors for cordless tools, hard drives, andmagnetic holddowns and jewelry clasps. They have the highestmagnetic field strength and have a higher coercivity (which makesthem magnetically stable), but have lower Curie temperature and aremore vulnerable to oxidation than samarium-cobalt magnets. Use ofprotective surface treatments such as gold, nickel, zinc and tin platingand epoxy resin coating can provide corrosion protection whererequired.

Originally, the high cost of these magnets limited their use to applications requiring compactness together with highfield strength. Both raw materials and patent licenses were expensive. Beginning in the 1990s, NIB magnets havebecome steadily less expensive, and the low cost has inspired new uses such as magnetic building toys.

HazardsThe greater force exerted by rare earth magnets creates hazards that are not seen with other types of magnet. Magnetslarger than a few centimeters are strong enough to cause injuries to body parts pinched between two magnets, or amagnet and a metal surface, even causing broken bones.[6] Magnets allowed to get too near each other can strikeeach other with enough force to chip and shatter the brittle material, and the flying chips can cause injuries. Therehave even been cases where young children who have swallowed several magnets have had a fold of the digestivetract pinched between the magnets, causing injury and in one case intestines perforations, sepsis and death.[7]

Rare-earth magnet 191

ApplicationsSince their prices became competitive in the 1990s, neodymium magnets have been replacing Alnico and ferritemagnets in the many applications in modern technology requiring powerful magnets. Their greater strength allowssmaller and lighter magnets to be used for a given application.

Common applicationsCommon applications of rare-earth magnets include:• computer hard drives• wind turbine generators• audio speakers / headphones• bicycle dynamos• fishing reel brakes• permanent magnet motors in cordless tools•• self-powered flashlights, employing rare earth magnets for generating electricity in a shaking motion

Other applicationsOther applications of rare-earth magnets include:• Linear motors (used in Mag-lev trains, etc.)• Stop motion animation as tie-downs when the use of traditional screw and nut tie-downs is impractical• Diamagnetic levitation experimentation, the study of magnetic field dynamics and superconductor levitation•• Electrodynamic bearings• Launched roller coaster technology found on roller coaster and other thrill rides• LED throwies, small LEDs attached to a coin battery and a rare earth magnet• Electric guitar pickups• Miniature figures, in particular Warhammer 40,000 and Warhammer Fantasy Battle, for which rare-earth magnets

have gained popularity in the miniatures gaming community for their small size and relative strength assisting inswapping weapons between models

• Windbelts for electricity generation through electromagnetic induction and aeroelastic flutter principles

References[1] McCaig, Malcolm (1977). Permanent Magnets in Theory and Practice. USA: Wiley. pp. 123. ISBN 0-7273-1604-4.[2] Sigel, Astrid; Helmut Sigel (2003). The lanthanides and their interrelations with biosystems. USA: CRC Press. pp. v. ISBN 0-8247-4245-1.[3] Bobber, Robert J. (1981). "New types of transducers". Underwater acoustics and signal processing: proceedings of the NATO Advanced

Study Institute held at Kollekolle, Copenhagen, Denmark, August 18–29, 1980. USA: Springer. pp. 251–252.[4] Cullity, B. D.; C. D. Graham (2008). Introduction to Magnetic Materials (http:/ / books. google. com/ books?id=ixAe4qIGEmwC&

pg=PA489). Wiley-IEEE. pp. 489. ISBN 0-471-47741-9. .[5] Lovelace, Alan M. (March-April 1971). "More Mileage Than Programmed From Military R&D" (http:/ / www. airpower. au. af. mil/

airchronicles/ aureview/ 1971/ mar-apr/ Lovelace. html). Air University Review (US Air Force) 22 (3): 14-23. . Retrieved July 4, 2012.[6] Swain, Frank (March 6, 2009). "How to remove a finger with two super magnets" (http:/ / scienceblogs. com/ sciencepunk/ 2009/ 03/

how_to_remove_a_finger_with_tw. php). The Sciencepunk Blog. Seed Media Group LLC. . Retrieved 2009-06-28.[7] "Magnet Safety Alert" (http:/ / www. cpsc. gov/ CPSCPUB/ PUBS/ magnet. pdf). U.S. Consumer Product Safety Commission. . Retrieved 7

August 2009.

Rare-earth magnet 192

Further reading•• Edward P. Furlani, "Permanent Magnet and Electromechanical Devices: Materials, Analysis and Applications",

Academic Press Series in Electromagnetism (2001). ISBN 0-12-269951-3.•• Peter Campbell, "Permanent Magnet Materials and their Application" (Cambridge Studies in Magnetism)(1996).

ISBN 978-0-521-56688-9.• Brown, D.N.; B. Smith, B.M. Ma, P. Campbell (2004). "The Dependence of Magnetic Properties and Hot

Workability of Rare Earth-Iron-Boride Magnets Upon Composition" (http:/ / www. magnequench. com/ assets/content/ magnequench/ mag_ref/ mag_pps/ pps_040701/ IEEE2004_vMAG40. pdf). IEEE Transactions onMagnetics 40 (4): 2895–2897. Bibcode 2004ITM....40.2895B. doi:10.1109/TMAG.2004.832240.ISSN 0018-9464.

External links• MMPA 0100-00, Standard Specifications for Permanent Magnet Materials (http:/ / www. intl-magnetics. org/

pdfs/ 0100-00. pdf)• Edwards, Lin (22 March 2010). "Iron-nitrogen compound forms strongest magnet known" (http:/ / www. physorg.

com/ news188458077. html). PhysOrg.

Single-molecule magnetSingle-molecule magnets or SMMs are a class of metalorganic compounds, that show superparamagnetic behaviorbelow a certain blocking temperature at the molecular scale. In this temperature range, SMMs exhibit magnetichysteresis of purely molecular origin.[1] Contrary to conventional bulk magnets and molecule-based magnets,collective long-range magnetic ordering of magnetic moments is not necessary.[1]

Intramolecular couplingThe magnetic coupling between the spins of the metal ions is mediated via superexchange interactions and can bedescribed by the following isotropic Heisenberg Hamiltonian:

where is the coupling constant between spin i (operator ) and spin j (operator ). For positive J thecoupling is called ferromagnetic (parallel alignment of spins) and for negative J the coupling is calledantiferromagnetic (antiparallel alignment of spins).• a high spin ground state,• a high zero-field-splitting (due to high magnetic anisotropy), and•• negligible magnetic interaction between molecules.The combination of these properties can lead to an energy barrier so that, at low temperatures, the system can betrapped in one of the high-spin energy wells.[1]

"These molecules contain a finite number of interacting spin centers (e.g. paramagnetic ions) and thus provide ideal opportunities to study basic concepts of magnetism. Some of them possess magnetic ground states and give rise to hysteresis effects and metastable magnetic phases. They may show quantum tunneling of the magnetization which raises the question of coherent dynamics in such systems. Other types of molecules exhibit pronounced frustration effects[2], whereas so-called spin crossover substances can switch their magnetic ground state and related properties such as color under irradiation of laser light, pressure or heat. Scientists from various fields – chemistry, physics; theory and experiment – have joined the research on molecular magnetism in order to explore the unprecedented

Single-molecule magnet 193

properties of these new compounds."[3]

"Single-molecule magnets (SMMs) have many important advantages over conventional nanoscale magnetic particlescomposed of metals, metal alloys or metal oxides. These advantages include uniform size, solubility in organicsolvents, and readily alterable peripheral ligands, among others."[4]

"A single molecule magnet is an example of a macroscopic quantum system. [...] If we could detect spin flips in asingle atom or molecule, we could use the spin to store information. This would enable us to increase the storagecapacity of computer hard disks. [...] A good starting point for trying to detect spin flips is to find a molecule with aspin of several Bohr magnetons. [An electron has an intrinsic magnetic dipole moment of approximately one Bohrmagneton.] There is a very well studied molecular magnet, Mn12-acetate, which has a spin S = 10 (Figure 3). Thismolecule is a disc-shaped organic molecule in which twelve Mn ions are embedded. Eight of these form a ring, eachhaving a charge of +3 and a spin S = 2. The other four form a tetrahedron, each having a charge of +4 and a spin S =3/2. The exchange interactions within the molecule are such that the spins of the ring align themselves in oppositionto the spins of the tetrahedron, giving the molecule a total net spin S = 10."[5]

Blocking temperatureMeasurements take place at very low temperatures. The so-called blocking temperature is defined as the temperaturebelow which the relaxation of the magnetisation becomes slow compared to the time scale of a particularinvestigation technique.[6] A molecule magnetised at 2 K will keep 40% of its magnetisation after 2 months and bylowering the temperature to 1.5 K this will take 40 years.[6]

Future applicationsAs of 2008 there are many discovered types and potential uses. "Single molecule magnets (SMM) are a class ofmolecules exhibiting magnetic properties similar to those observed in conventional bulk magnets, but of molecularorigin. SMMs have been proposed as potential candidates for several technological applications that require highlycontrolled thin films and patterns."[7]

"The ability of a single molecule to behave like a tiny magnet (single molecular magnets, SMMs) has seen a rapidgrowth in research over the last few years. SMMs represent the smallest possible magnetic devices and are acontrollable, bottom-up approach to nanoscale magnetism. Potential applications of SMMs include quantumcomputing, high-density information storage and magnetic refrigeration."[8]

One possible use of SMMs is superior magneticthin films to coat hard disks.

"A single molecule magnet is an example of a macroscopic quantumsystem. [...] If we could detect spin flips in a single atom or molecule,we could use the spin to store information. This would enable us toincrease the storage capacity of computer hard disks. [...] A goodstarting point for trying to detect spin flips is to find a molecule with aspin of several Bohr magnetons. [An electron has an intrinsic magneticdipole moment of approximately one Bohr magneton.] There is a verywell studied molecular magnet, Mn12-acetate, which has a spin S = 10(Figure 3). This molecule is a disc-shaped organic molecule in whichtwelve Mn ions are embedded. Eight of these form a ring, each havinga charge of +3 and a spin S = 2. The other four form a tetrahedron,each having a charge of +4 and a spin S = 3/2. The exchangeinteractions within the molecule are such that the spins of the ring align themselves in opposition to the spins of thetetrahedron, giving the molecule a total net spin S = 10."[9]

Single-molecule magnet 194

Types

Ferritin

The archetype of single-molecule magnets is called "Mn12". It is apolymetallic manganese (Mn) complex having the formula[Mn12O12(OAc)16(H2O)4], where OAc stands for acetate. It has theremarkable property of showing an extremely slow relaxation of theirmagnetization below a blocking temperature.[10]

[Mn12O12(OAc)16(H2O)4]·4H2O·2AcOH which is called"Mn12-acetate" is a common form of this used in research.

"Mn4" is another researched type single-molecule magnet. Three ofthese are:[11]

• [Mn4(hmp)6(NO3)2(MeCN)2](ClO4)2·2MeCN• [Mn4(hmp)6(NO3)4]·(MeCN)• [Mn4(hmp)4(acac)2(MeO)2](ClO4)2·2MeOHIn each of these Mn4 complexes "there is a planar diamond core ofMnIII

2MnII2 ions. An analysis of the variable-temperature and variable-field magnetization data indicate that all three

molecules have intramolecular ferromagnetic coupling and a S = 9 ground state. The presence of afrequency-dependent alternating current susceptibility signal indicates a significant energy barrier between thespin-up and spin-down states for each of these three MnIII

2MnII2 complexes."[11]

Single-molecule magnets are also based on iron clusters[6] because they potentially have large spin states. In additionthe biomolecule ferritin is also considered a nanomagnet. In the cluster Fe8Br the cation Fe8 stands for[Fe8O2(OH)12(tacn)6]8+ with tacn representing 1,4,7-triazacyclononane.

HistoryAlthough the term "single-molecule magnet" was first employed by David Hendrickson, a chemist at the Universityof California, San Diego and George Christou (Indiana University) in 1996,[12] the first single-molecule magnetreported dates back to 1991.[13] The European researchers discovered that a Mn12O12(MeCO2)16(H2O)4 complex(Mn12Ac16) first synthesized in 1980[14] exhibits slow relaxation of the magnetization at low temperatures. Thismanganese oxide compound is composed of a central Mn(IV)4O4 cube surrounded by a ring of 8 Mn(III) unitsconnected through bridging oxo ligands. In addition, it has 16 acetate and 4 water ligands.[15]

It was known in 2006 that the "deliberate structural distortion of a Mn6 compound via the use of a bulkysalicylaldoxime derivative switches the intra-triangular magnetic exchange from antiferromagnetic to ferromagneticresulting in an S = 12 ground state.[16]

A record magnetization was reported in 2007 for a compound related to MnAc12 ([Mn(III)

6O2(sao)6(O2CPh)2(EtOH)4]) with S = 12, D = -0.43 cm−1 and hence U = 62 cm−1 or 86 K[17] at a blockingtemperature of 4.3 K. This was accomplished by replacing acetate ligands by the bulkier salicylaldoxime thusdistorting the manganese ligand sphere. It is prepared by mixing the perchlorate of manganese, the sodium salt ofbenzoic acid, a salicylaldoxime derivate and tetramethylammonium hydroxide in water and collecting the filtrate.

Single-molecule magnet 195

Detailed behaviorMolecular magnets exhibit an increasing product (magnetic susceptibility times temperature) with decreasingtemperature, and can be characterized by a shift both in position and intensity of the a.c. magnetic susceptibility.Single-molecule magnets represent a molecular approach to nanomagnets (nanoscale magnetic particles). Inaddition, single-molecule magnets have provided physicists with useful test-beds for the study of quantummechanics. Macroscopic quantum tunneling of the magnetization was first observed in Mn12O12, characterized byevenly-spaced steps in the hysteresis curve. The periodic quenching of this tunneling rate in the compound Fe8 hasbeen observed and explained with geometric phases.Due to the typically large, bi-stable spin anisotropy, single-molecule magnets promise the realization of perhaps thesmallest practical unit for magnetic memory, and thus are possible building blocks for a quantum computer.Consequently, many groups have devoted great efforts into synthesis of additional single molecule magnets;however, the Mn12O12 complex and analogous complexes remain the canonical single molecule magnet with a50 cm−1 spin anisotropy.The spin anisotropy manifests itself as an energy barrier that spins must overcome when they switch from parallelalignment to antiparallel alignment. This barrier (U) is defined as:

where S is the dimensionless total spin state and D the zero-field splitting parameter (in cm−1); D can be negative butonly its absolute value is considered in the equation. The barrier U is generally reported in cm−1 units or in units ofKelvin (see: electronvolt). The higher the barrier the longer a material remains magnetized and a high barrier isobtained when the molecule contains many unpaired electrons and when its zero field splitting value is large. Forexample, the MnAc12 cluster the spin state is 10 (involving 20 unpaired electrons) and D = -0.5 cm−1 resulting in abarrier of 50 cm−1 (equivalent to 60 K).[18]

The effect is also observed by hysteresis experienced when magnetization is measured in a magnetic field sweep: onlowering the magnetic field again after reaching the maximum magnetization the magnetization remains at highlevels and it requires a reversed field to bring magnetization back to zero.Recently, it has been reported that the energy barrier, U, is slightly dependent on Mn12 crystal size/morphology, aswell as the magnetization relaxation times, which varies as function of particle size and size distributions .[19]

References[1] Introduction to Molecular Magnetism by Dr. Joris van Slageren (http:/ / obelix. physik. uni-bielefeld. de/ ~schnack/ molmag/ material/ 123.

pdf)[2] Frustrated Magnets (http:/ / www. ifw-dresden. de/ institutes/ iff/ research/ TMO/ frustrated-magnets), Leibniz Institute for Solid State and

Materials Research, Dresden, Germany[3] Molecular Magnetism Web (http:/ / obelix. physik. uni-bielefeld. de/ ~schnack/ molmag/ introduction. html) Introduction page[4] ScienceDaily (Mar. 27, 2000) (http:/ / www. sciencedaily. com/ releases/ 2000/ 03/ 000327084104. htm) article Several New Single-Molecule

Magnets Discovered[5] National Physical Laboratory (UK) (http:/ / www. npl. co. uk/ server. php?show=ConWebDoc. 1175) Home > Science + Technology >

Quantum Phenomena > Nanophysics > Research – article Molecular Magnets[6] Single-molecule magnets based on iron(III) oxo clusters Dante Gatteschi, Roberta Sessoli and Andrea Cornia Chem. Commun., 2000, 725 –

732, doi:10.1039/a908254i[7] Cavallini, Massimiliano; Facchini, Massimo; Albonetti, Cristiano; Biscarini, Fabio (2008). "Single molecule magnets: from thin films to

nano-patterns". Physical Chemistry Chemical Physics 10 (6): 784. Bibcode 2008PCCP...10..784C. doi:10.1039/b711677b. PMID 18231680.[8] Beautiful new single molecule magnets (http:/ / www. rsc. org/ Publishing/ Journals/ dt/ News/ b716355jpersp. asp), 26 March 2008 –

summary of the article Milios, Constantinos J.; Piligkos, Stergios; Brechin, Euan K. (2008). "Ground state spin-switching via targetedstructural distortion: twisted single-molecule magnets from derivatised salicylaldoximes". Dalton Transactions (14): 1809.doi:10.1039/b716355j.

[9] National Physical Laboratory (UK) (http:/ / www. npl. co. uk/ server. php?show=ConWebDoc. 1175) Home > Science + Technology >Quantum Phenomena > Nanophysics > Research – article Molecular Magnets

Single-molecule magnet 196

[10] IPCMS Liquid-crystalline Single Molecule Magnets (http:/ / www-ipcms. u-strasbg. fr/ spip. php?article1341) – summary of the articleTerazzi, Emmanuel; Bourgogne, Cyril; Welter, Richard; Gallani, Jean-Louis; Guillon, Daniel; Rogez, Guillaume; Donnio, Bertrand (2008)."Single-Molecule Magnets with Mesomorphic Lamellar Ordering". Angew. Chem. Int. Ed. 47 (3): 490–495. doi:10.1002/anie.200704460.

[11] Yang, E (2003). "Mn4 single-molecule magnets with a planar diamond core and S=9". Polyhedron 22 (14–17): 1857.doi:10.1016/S0277-5387(03)00173-6.

[12] Aubin, Sheila M. J.; Wemple, Michael W.; Adams, David M.; Tsai, Hui-Lien; Christou, George; Hendrickson, David N. (1996). "DistortedMnIVMnIII3Cubane Complexes as Single-Molecule Magnets". Journal of the American Chemical Society 118 (33): 7746.doi:10.1021/ja960970f.

[13] Caneschi, Andrea; Gatteschi, Dante; Sessoli, Roberta; Barra, Anne Laure; Brunel, Louis Claude; Guillot, Maurice (1991). "Alternatingcurrent susceptibility, high field magnetization, and millimeter band EPR evidence for a ground S = 10 state in[Mn12O12(Ch3COO)16(H2O)4].2CH3COOH.4H2O". Journal of the American Chemical Society 113 (15): 5873. doi:10.1021/ja00015a057.

[14] Lis, T. (1980). "Preparation, structure, and magnetic properties of a dodecanuclear mixed-valence manganese carboxylate". ActaCrystallographica Section B Structural Crystallography and Crystal Chemistry 36 (9): 2042. doi:10.1107/S0567740880007893.

[15] Chemistry of Nanostructured Materials; Yang, P., Ed.; World Scientific Publishing: Hong Kong, 2003.[16] Milios, Constantinos J.; Vinslava, Alina; Wood, Peter A.; Parsons, Simon; Wernsdorfer, Wolfgang; Christou, George; Perlepes, Spyros P.;

Brechin, Euan K. (2007). "A Single-Molecule Magnet with a “Twist”". Journal of the American Chemical Society 129 (1): 8.doi:10.1021/ja0666755. PMID 17199262.

[17] Milios, Constantinos J.; Vinslava, Alina; Wernsdorfer, Wolfgang; Moggach, Stephen; Parsons, Simon; Perlepes, Spyros P.; Christou,George; Brechin, Euan K. (2007). "A Record Anisotropy Barrier for a Single-Molecule Magnet". Journal of the American Chemical Society129 (10): 2754. doi:10.1021/ja068961m. PMID 17309264.

[18] del Barco, E.; Kent, A. D.; Hill, S.; North, J. M.; Dalal, N. S.; Rumberger, E. M.; Hendrickson, D. N.; Chakov, N. et al. (2005). "MagneticQuantum Tunneling in the Single-Molecule Magnet Mn12-Acetate". Journal of Low Temperature Physics 140 (1/2): 119.Bibcode 2005JLTP..140..119B. doi:10.1007/s10909-005-6016-3.

[19] Muntó, María; Gómez-Segura, Jordi; Campo, Javier; Nakano, Motohiro; Ventosa, Nora; Ruiz-Molina, Daniel; Veciana, Jaume (2006)."Controlled crystallization of Mn12 single-molecule magnets by compressed CO2 and its influence on the magnetization relaxation". Journal ofMaterials Chemistry 16 (26): 2612. doi:10.1039/b603497g.

External links• European Institute of Molecular Magnetism EIMM (http:/ / www. eimm. eu/ )• MAGMANet (Molecular Approach to Nanomagnets and Multifunctional Materials) (http:/ / www. unizar. es/

magmanet/ magmanet-eu/ ), a Network of centres of Excellence, coordinated by the INSTM – ConsorzioInteruniversitario Nazionale per la Scienza e la Tecnologia dei Materiali

• Molecular Magnetism Web (http:/ / www. molmag. de/ ), Jürgen Schnack

Spin glass 197

Spin glass

Schematic representation of the random spinstructure of a spin glass (top) and the ordered

one of a ferromagnet (bottom)

Glass (amorphous SiO2)

Quartz (crystalline SiO2)

A spin glass is a disordered magnet with frustrated interactions, augmented by stochastic positions of the spins,where conflicting interactions, namely both ferromagnetic and also antiferromagnetic bonds, are randomlydistributed with comparable frequency. The term "glass" in the bold-printed title name refers to the fact that itsmagnetic disorder brings to mind the positional disorder of a conventional, chemical glass, e.g., a window glass.Whereas, however, these are typically nonmagnetic, here the "glass property" refers essentially to the magneticproperties only, i.e. to the spin structure only.Spin glasses display many metastable structures leading to a plenitude of time scales which are difficult to exploreexperimentally or in simulations.

Spin glass 198

Magnetic behaviorIt is the time dependence which distinguishes spin glasses from other magnetic systems.Above the spin glass transition temperature, Tc,

[1] the spin glass exhibits typical magnetic behaviour (such asparamagnetism).If a magnetic field is applied as the sample is cooled to the transition temperature, magnetization of the sampleincreases as described by the Curie law. Upon reaching Tc, the sample becomes a spin glass and further coolingresults in little change in magnetization. This is referred to as the field-cooled magnetization.When the external magnetic field is removed, the magnetization of the spin glass falls rapidly to a lower valueknown as the remanent magnetization.Magnetization then decays slowly as it approaches zero (or some small fraction of the original value—this remainsunknown). This decay is non-exponential and no simple function can fit the curve of magnetization versus timeadequately. This slow decay is particular to spin glasses. Experimental measurements on the order of days haveshown continual changes above the noise level of instrumentation.Spin glasses differ from ferromagnetic materials by the fact that after the external magnetic field is removed from aferromagnetic substance, the magnetization remains indefinitely at the remanent value. Paramagnetic materials differfrom spin glasses by the fact that, after the external magnetic field is removed, the magnetization rapidly falls tozero, with no remanent magnetization. In each case the decay is rapid and exponential.If the sample is cooled below Tc in the absence of an external magnetic field and a magnetic field is applied after thetransition to the spin glass phase, there is a rapid initial increase to a value called the zero-field-cooledmagnetization. A slow upward drift then occurs toward the field-cooled magnetization.Surprisingly, the sum of the two complex functions of time (the zero-field-cooled and remanent magnetizations) is aconstant, namely the field-cooled value, and thus both share identical functional forms with time (Nordblad et al.), atleast in the limit of very small external fields.

Edwards–Anderson modelIn this model, we have spins arranged on a -dimensional lattice with only nearest neighbor interactions similar tothe Ising model. This model can be solved exactly for the critical temperatures and a glassy phase is observed toexist at low temperatures.[2] The Hamiltonian for this spin system is given by:

where refers to the Pauli spin matrix for the spin-half particle at lattice point . A negative value of denotesan antiferromagnetic type interaction between spins at points and . The sum runs over all nearest neighborpositions on a lattice, of any dimension. The variables magnetic nature of the spin-spin interactions are calledbond or link variables. In order to determine the partition function for this system, one needs to average the free

energy where , over all possible values of . The

distribution of values of is taken to be a gaussian with a mean and a variance :

Solving for the free energy using the replica method, below a certain temperature, a new magnetic phase called the spin glass phase (or glassy phase) of the system is found to exist which is characterized by a vanishing magnetization

along with a non-vanishing value of the two point correlation function between spins at the same lattice

point but at two different replicas: , where are replica incides. The order parameter for

Spin glass 199

the ferromagnetic to spin glass phase transition is therefore , and that for paramagnetic to spin glass is again . Hencethe new set of order parameters describing the three magnetic phases constitutes of both and . Free energy of thissystem can be found, both under assumptions of replica symmetry as well as considering replica symmetry breaking.Under the assumption of replica symmetry, the free energy is given by the expression:

The model of Sherrington and KirkpatrickIn addition to unusual experimental properties, spin glasses are the subject of extensive theoretical andcomputational investigations. A substantial part of early theoretical work on spin glasses dealt with a form of meanfield theory based on a set of replicas of the partition function of the system.An important, exactly solvable model of a spin glass was introduced by D. Sherrington and S. Kirkpatrick in 1975. Itis an Ising model with long range frustrated ferro- as well as antiferromagnetic couplings. It corresponds to a meanfield approximation of spin glasses describing the slow dynamics of the magnetization and the complex non-ergodicequilibrium state.Unlike the Edwards–Anderson (EA) model, in the system though only two spins interactions are considered, therange of each interaction can be potentially infinite (of the order of the size of the lattice). Therefore we see that anytwo spins can be lined with a ferromagnetic or an antiferromagnetic bond and the distribution of these is givenexactly as in the case of Edwards–Anderson model. The Hamiltonian for SK model is very similar to the EA model:

where have same meanings as in the EA model. The equilibrium solution of the model, after someinitial attempts by Sherrington, Kirkpatrick and others, was found by Giorgio Parisi in 1979 within the replicamethod. The subsequent work of interpretation of the Parisi solution—by M. Mezard, G. Parisi, M.A. Virasoro andmany others—revealed the complex nature of a glassy low temperature phase characterized by ergodicity breaking,ultrametricity and non-selfaverageness. Further developments led to the creation of the cavity method, whichallowed study of the low temperature phase without replicas. A rigorous proof of the Parisi solution has beenprovided in the work of Francesco Guerra and Michel Talagrand.The formalism of replica mean field theory has also been applied in the study of neural networks, where it hasenabled calculations of properties such as the storage capacity of simple neural network architectures withoutrequiring a training algorithm (such as backpropagation) to be designed or implemented.More realistic spin glass models with short range frustrated interactions and disorder, like the Gaussian model wherethe couplings between neighboring spins follow a Gaussian distribution, have been studied extensively as well,especially using Monte Carlo simulations. These models display spin glass phases bordered by sharp phasetransitions.Besides its relevance in condensed matter physics, spin glass theory has acquired a strongly interdisciplinarycharacter, with applications to neural network theory, computer science, theoretical biology, econophysics etc.

Spin glass 200

Infinite-range modelThe infinite-range model is a generalization of the Sherrington–Kirkpatrik model where we not only consider twospin interactions but -spin interactions, where and is the total number of spins. Unlike theEdwards–Anderson model, similar to the SK model, the interaction range is still infinite. The Hamiltonian for thismodel is described by:

where have similar meanings as in the EA model. The limit of this model is knownas the Random energy model. In this limit, it can be seen that the probability of the spin glass existing in a particularstate, depends only on the energy of that state and not on the individual spin configurations in it. A gaussiandistribution of magnetic bonds across the lattice is assumed usually to solve this model. Any other distribution isexpected to give the same result, as a consequence of the central limit theorem. The gaussian distribution function,

with mean and variance , is given as:

The order parameters for this system are given by the magnetization and the two point spin correlation betweenspins at the same site , in two different replicas, which are the same as for the SK model. This infinite rangemodel can be solved explicitly for the free energy[2] in terms of and , under the assumption of replicasymmetry as well as 1-Replica Symmetry Breaking.[2]

Non-ergodic behavior and applicationsA so-called non-ergodic behavior happens in spin glasses below the freezing temperature , since below thattemperature the system cannot escape from the ultradeep minima of the hierarchically-disordered energylandscape.[3] Although the freezing temperature is typically as low as 30 kelvin (−240 degrees Celsius), so that thespin glass magnetism appears to be practically without applications in daily life, there are applications in differentcontexts, e.g. in the already mentioned theory of neural networks, i.e. in theoretical brain research, and in themathematical-economical theory of optimization.

Notes[1] is identical with the so-called "freezing temperature" [2] Nishimori, Hidetoshi (2001). Statistical Physics of Spin Glasses and Information Processing: An Introduction (http:/ / preterhuman. net/ texts/

science_and_technology/ physics/ Statistical_physics/ Statistical physics of spin glasses and information processing an introduction -Nishimori H. . pdf). Oxford: Oxford University Press. pp. 243. ISBN 0-19-850940-5, 9780198509400. .

[3][3] The hierarchical disorder of the energy landscape may be verbally characterized by a single sentence: in this landscape there are "(random)valleys within still deeper (random) valleys within still deeper (random) valleys, ..., etc,"

Spin glass 201

References

Literature• Sherrington, David; Kirkpatrick, Scott (1975), "Solvable model of a spin-glass", Physics Review Letters 35 (26):

1792–1796, Bibcode 1975PhRvL..35.1792S, doi:10.1103/PhysRevLett.35.1792. Papercore Summary http:/ /papercore. org/ Sherrington1975 (http:/ / papercore. org/ Sherrington1975)

• Nordblad, P.; Lundgren, L.; Sandlund, L. (1986), "A link between the relaxation of the zero field cooled and thethermoremanent magnetizations in spin glasses", Journal of Magnetism and Magnetic Materials 54: 185–186,Bibcode 1986JMMM...54..185N, doi:10.1016/0304-8853(86)90543-3.

• Binder, K.; Young, A. P. (1986), "Spin glasses: Experimental facts, theoretical concepts, and open questions",Reviews of Modern Physics 58: 801–976, Bibcode 1986RvMP...58..801B, doi:10.1103/RevModPhys.58.801.

• Bryngelson, Joseph D.; Wolynes, Peter G. (1987), "Spin glasses and the statistical mechanics of protein folding",Proceedings of the National Academy of Sciences 84: 7524–7528, Bibcode 1987PNAS...84.7524B,doi:10.1073/pnas.84.21.7524.

• Fischer, K. H.; Hertz, J. A. (1991), Spin Glasses, Cambridge University Press.• Mezard, Marc; Parisi, Giorgio; Virasoro, Miguel Angel (1987), Spin glass theory and beyond, Singapore: World

Scientific, ISBN 9971-5-0115-5.• Mydosh, J. A. (1995), Spin Glasses, Taylor & Francis.• Parisi, G. (1980), "The order parameter for spin glasses: a function on the interval 0-1", J. Phys. A: Math. Gen.

13: 1101-1112, Bibcode 1980JPhA...13.1101P, doi:10.1088/0305-4470/13/3/042 Papercore Summary http:/ /papercore. org/ Parisi1980 (http:/ / papercore. org/ Parisi1980).

• Talagrand, Michel (2000), "Replica symmetry breaking and exponential inequalities for theSherrington–Kirkpatrick model", Annals of Probability 28 (3): 1018–1062, JSTOR 2652978.

• Guerra, F.; Toninelli, F. L. (2002), "The thermodynamic limit in mean field spin glass models", Communicationsin Mathematical Physics 230 (1): 71–79, arXiv:cond-mat/0204280, Bibcode 2002CMaPh.230...71G,doi:10.1007/s00220-002-0699-y.

External links• Papercore summary of seminal Sherrington/Kirkpatrick paper (http:/ / papercore. org/ summaries/

solvable-model-of-a-spin-glass)• Statistics of frequency of the term "Spin glass" in arxiv.org (http:/ / xstructure. inr. ac. ru/ x-bin/ theme3.

py?level=2& index1=125728)

Spin wave 202

Spin waveSpin waves are propagating disturbances in the ordering of magnetic materials. These low-lying collectiveexcitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view,spin waves are known as magnons, which are boson modes of the spin lattice that correspond roughly to the phononexcitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces aferromagnet's spontaneous magnetization. The energies of spin waves are typically only μeV in keeping with typicalCurie points at room temperature and below. The discussion of spin waves in antiferromagnets is presently beyondthe scope of this article.

Theory

An illustration of the precession of a spin wave about an applied magnetic fieldwith a wavevector that is eleven times the lattice constant.

The projection of the magnetization of the samespin wave along the chain direction as a function

of distance along the spin chain.

The simplest way of understanding spinwaves is to consider the Hamiltonian forthe Heisenberg ferromagnet:

where is the exchange energy, theoperators represent the spins at Bravaislattice points, is the Landé g-factor, is the Bohr magneton and is the internalfield which includes the external field plusany "molecular" field. Note that in theclassical continuum case and in 1+1dimensions Heisenberg ferromagnetequation has the form

In 1+1, 2+1 and 3+1 dimensions thisequation admits several integrable andnon-integrable extensions like theLandau-Lifshitz equation, the Ishimoriequation and so on. For a ferromagnet >0 and the ground state of the Hamiltonian

is that in which all spins are alignedparallel with the field . That is an eigenstate of can be verified by rewriting it in terms of thespin-raising and -lowering operators given by:

resulting in

where has been taken as the direction of the magnetic field. The spin-lowering operator annihilates the statewith minimum projection of spin along the z-axis, while the spin-raising operator annihilates the ground statewith maximum spin projection along the -axis. Since for the maximally aligned state, we find

Spin wave 203

where N is the total number of Bravais lattice sites. The proposition that the ground state is an eigenstate of theHamiltonian is confirmed.One might guess that the first excited state of the Hamiltonian has one randomly selected spin at position rotatedso that , but in fact this arrangement of spins is not an eigenstate. The reason is that such astate is transformed by the spin raising and lowering operators. The operator will increase the z-projection of thespin at position back to its low-energy orientation, but the operator will lower the z-projection of the spin at

position . The combined effect of the two operators is therefore to propagate the rotated spin to a new position,which is a hint that the correct eigenstate is a spin wave, namely a superposition of states with one reduced spin. Theexchange energy penalty associated with changing the orientation of one spin is reduced by spreading thedisturbance over a long wavelength. The degree of misorientation of any two near-neighbor spins is therebyminimized. From this explanation one can see why the Ising model magnet with discrete symmetry has no spinwaves: the notion of spreading a disturbance in the spin lattice over a long wavelength makes no sense when spinshave only two possible orientations. The existence of low-energy excitations is related to the fact that in the absenceof an external field, the spin system has an infinite number of degenerate ground states with infinitesimally differentspin orientations. That these ground states exist can be seen from the fact that the state does not have the fullrotational symmetry of the Hamiltonian , a phenomenon which is called spontaneous symmetry breaking.In this model the magnetization where is the volume. The propagation of spin waves is

described by the Landau-Lifzhitz equation of motion:

where is the gyromagnetic ratio and is the damping constant. The cross-products in this forbidding-lookingequation show that the propagation of spin waves is governed by the torques generated by internal and externalfields. (An equivalent form is the Landau-Lifshitz-Gilbert equation, which replaces the final term by a more "simplylooking" equivalent one.)The first term on the r.h.s. describes the precession of the magnetization under the influence of the applied field,while the above-mentioned final term describes how the magnetization vector "spirals in" towards the field directionas time progresses. In metals the damping forces described by the constant are in many cases dominated by theeddy currents.One important difference between phonons and magnons lies in their dispersion relations. The dispersion relation forphonons is to first order linear in wavevector : , where is frequency, and is the velocity of sound.Magnons have a parabolic dispersion relation: where the parameter represents a "spin stiffness." The

form is the third term of a Taylor expansion of a cosine term in the energy expression originating from thedot-product.The underlying reason for the difference in dispersion relation is that ferromagnets violate

time-reversal symmetry. Two adjacent spins in a solid with lattice constant that participate in a mode withwavevector have an angle between them equal to .

Experimental observationSpin waves are observed through four experimental methods: inelastic neutron scattering, inelastic light scattering (Brillouin scattering, Raman scattering and inelastic X-ray scattering), inelastic electron scattering (spin-resolved electron energy loss spectroscopy), and spin-wave resonance (ferromagnetic resonance). In the first method the energy loss of a beam of neutrons that excite a magnon is measured, typically as a function of scattering vector (or equivalently momentum transfer), temperature and external magnetic field. Inelastic neutron scattering measurements can determine the dispersion curve for magnons just as they can for phonons. Important inelastic neutron scattering facilities are present at the ISIS neutron source in Oxfordshire, UK, the Institut Laue-Langevin in Grenoble, France, the High Flux Isotope Reactor at Oak Ridge National Laboratory in Tennessee, USA, and at the

Spin wave 204

National Institute of Standards and Technology in Maryland, USA. Brillouin scattering similarly measures theenergy loss of photons (usually at a convenient visible wavelength) reflected from or transmitted through a magneticmaterial. Brillouin spectroscopy is similar to the more widely known Raman scattering but probes a lower energyand has a higher energy resolution in order to be able to detect the meV energy of magnons. Ferromagnetic (orantiferromagnetic) resonance instead measures the absorption of microwaves, incident on a magnetic material, byspin waves, typically as a function of angle, temperature and applied field. Ferromagnetic resonance is a convenientlaboratory method for determining the effect of magnetocrystalline anisotropy on the dispersion of spin waves. Veryrecently, one group in Max Planck Institute for Microstructure Physics in Halle Germany proved that by using spinpolarized electron energy loss spectroscopy (SPEELS), very high energy surface magnons can be exited. Thistechnique allows people first time to probe the magnons and its dispersion in the ultrathin magnetical system. Thefirst experiment was successful done in 5 ML Fe film by SPEELS, the signature of magnons were revealed. Later,with momentum resolution, magnon dispersion and full peak was explored in 8 ML fcc Co film on Cu(001) and 8ML hcp Co on W(110), respectively. Those magnons are obtained up to the SBZ at the energy range about fewhundreds meV.

Practical significanceWhen magnetoelectronic devices are operated at high frequencies, the generation of spin waves can be an importantenergy loss mechanism. Spin wave generation limits the linewidths and therefore the quality factors Q of ferritecomponents used in microwave devices. The reciprocal of the lowest frequency of the characteristic spin waves of amagnetic material gives a time scale for the switching of a device based on that material.

References• List of labs (http:/ / www. icmm. csic. es/ brillouin/ BrillouinEN. htm) performing Brillouin scattering

measurements.• P.W. Anderson, Concepts in Solids, ISBN 981-02-3231-4; Basic Notions of *Condensed Matter Physics, ISBN

0-201-32830-5• N.W. Ashcroft and N.D. Mermin, Solid-State Physics, ISBN 0-03-083993-9.• S. Chikazumi and S.H. Charap, Physics of Magnetism, ASIN B0007DODNA (out of print).•• M.Plihal, D.L.Mills, and J.Kirschner, " Spin wave signature in the spin polarized electron energy loss spectrum in

ultrathin Fe film: theory and experiment"• Phys. Rev. Lett., 82, 2579,(1999)• Phys. Rev. Lett., 91, 147201,(2003)•• R.Vollmer, M.Etzkorn, P.S.Anil Kumar, H.lbach, and J.Kirschner, "Spin polarized electron energy loss

spectroscopy of high energy, large wave vector spin waves in fcc Co films on Cu(001)"• A.T.Costa, R. B. Muniz and D. L. Mills, "Theory of spin waves in ultrathin ferromagnetic films: the case of Co

on Cu(100)", Phys. Rev. B 69, 064413 (2004)• A.T.Costa, R. B. Muniz and D. L. Mills, "Theory of large wave-vector spin waves in ferromagnetic films:

sensitivity to electronic structure", Phys. Rev. B 70, 54406 (2004)

Spontaneous magnetization 205

Spontaneous magnetizationSpontaneous magnetization is the appearance of an ordered spin state (magnetization) at zero applied magneticfield in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or T

C.

OverviewHeated to temperatures above T

C, ferromagnetic materials become paramagnetic and their magnetic behavior is

dominated by spin waves or magnons, which are boson collective excitations with energies in the meV range. Themagnetization that occurs below T

C is a famous example of the "spontaneous" breaking of a global symmetry, a

phenomenon that is described by Goldstone's theorem. The term "symmetry breaking" refers to the choice of amagnetization direction by the spins, which have spherical symmetry above T

C, but a preferred axis (the

magnetization direction) below TC.

Temperature dependenceTo first order, the temperature dependence of spontaneous magnetization at low temperatures is given by Bloch'sLaw: [1]

where M(0) is the spontaneous magnetization at absolute zero. The decrease in spontaneous magnetization at highertemperatures is caused by the increasing excitation of spin waves. In a particle description, the spin wavescorrespond to magnons, which are the massless Goldstone bosons corresponding to the broken symmetry. This isexactly true for an isotropic magnet.Magnetic anisotropy, that is the existence of a easy direction along which the moments align spontaneously in thecrystal, corresponds however to "massive" magnons. This is a way of saying that they cost a minimum amount ofenergy to excite, hence they are very unlikely to be excited as . Hence the magnetization of an anisotropicmagnet is harder to destroy at low temperature and the temperature dependence of the magnetization deviatesaccordingly from the Bloch's law. All real magnets are anisotropic to some extent.Near the Curie temperature,

where β is a critical exponent that depends on composition. The exponent is 0.34 for Fe and 0.51 for Ni.[2]

An empirical interpolation of the two regimes is given by

it is easy to check two limits of this interpolation that follow laws similar to the Bloch law, for , and thecritical behavior, for , respectively.

Spontaneous magnetization 206

Notes and references[1] Ashcroft & Mermin 1976, p. 708[2] Chikazumi 1997, pp. 128–129

Further reading• Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Holt, Rinehart and Winston.

ISBN 0-03-083993-9.• Chikazumi, Sōshin (1997). Physics of Ferromagnetism. Clarendon Press. ISBN 0-19-851776-9.

SuperparamagnetismSuperparamagnetism is a form of magnetism, which appears in small ferromagnetic or ferrimagnetic nanoparticles.In sufficiently small nanoparticles, magnetization can randomly flip direction under the influence of temperature.The typical time between two flips is called the Néel relaxation time. In the absence of external magnetic field, whenthe time used to measure the magnetization of the nanoparticles is much longer than the Néel relaxation time, theirmagnetization appears to be in average zero: they are said to be in the superparamagnetic state. In this state, anexternal magnetic field is able to magnetize the nanoparticles, similarly to a paramagnet. However, their magneticsusceptibility is much larger than the one of paramagnets.

The Néel relaxation in the absence of magnetic fieldNormally, any ferromagnetic or ferrimagnetic material undergoes a transition to a paramagnetic state above its Curietemperature. Superparamagnetism is different from this standard transition since it occurs below the Curietemperature of the material.Superparamagnetism occurs in nanoparticles which are single-domain, i.e. composed of a single magnetic domain.This is possible when their diameter is below 3–50 nm, depending on the materials. In this condition, it is consideredthat the magnetization of the nanoparticles is a single giant magnetic moment, sum of all the individual magneticmoments carried by the atoms of the nanoparticle. Those in the field of superparamagnetism call this “macro-spinapproximation”.Because of the nanoparticle’s magnetic anisotropy, the magnetic moment has usually only two stable orientationsantiparallel to each other, separated by an energy barrier. The stable orientations define the nanoparticle’s so called“easy axis”. At finite temperature, there is a finite probability for the magnetization to flip and reverse its direction.The mean time between two flips is called the Néel relaxation time and is given by the following Néel-Arrheniusequation:[1]

,

where:• is thus the average length of time that it takes for the nanoparticle’s magnetization to randomly flip as a result

of thermal fluctuations.• is a length of time, characteristic of the material, called the attempt time or attempt period (its reciprocal is

called the attempt frequency); its typical value is 10−9–10−10 second.• K is the nanoparticle’s magnetic anisotropy energy density and V its volume. KV is therefore the energy barrier

associated with the magnetization moving from its initial easy axis direction, through a “hard plane”, to the othereasy axis direction.

• kB is the Boltzmann constant.

Superparamagnetism 207

• T is the temperature.This length of time can be anywhere from a few nanoseconds to years or much longer. In particular, it can be seenthat the Néel relaxation time is an exponential function of the grain volume, which explains why the flippingprobability becomes rapidly negligible for bulk materials or large nanoparticles.

Blocking temperatureLet us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us define as the measurement time. If , the nanoparticle magnetization will flip several times during themeasurement, then the measured magnetization will average to zero. If , the magnetization will not flipduring the measurement, so the measured magnetization will be what the instantaneous magnetization was at thebeginning of the measurement. In the former case, the nanoparticle will appear to be in the superparamagnetic statewhereas in the latter case it will appear to be “blocked” in its initial state. The state of the nanoparticle(superparamagnetic or blocked) depends on the measurement time. A transition between superparamagnetismand blocked state occurs when . In several experiments, the measurement time is kept constant but thetemperature is varied, so the transition between superparamagnetism and blocked state is seen as a function of thetemperature. The temperature for which is called the blocking temperature:

For typical laboratory measurements, the value of the logarithm in the previous equation is in the order of 20–25.

Effect of a magnetic field

Langevin function (red line), compared with (blue line).

When an external magnetic field is appliedto an assembly of superparamagneticnanoparticles, their magnetic moments tendto align along the applied field, leading to anet magnetization. The magnetization curveof the assembly, i.e. the magnetization as afunction of the applied field, is a reversibleS-shaped increasing function. This functionis quite complicated but for some simplecases:

1. If all the particles are identical (sameenergy barrier and same magneticmoment), their easy axes are all orientedparallel to the applied field and thetemperature is low enough(TB < T ≲ KV/(10 kB)), then themagnetization of the assembly is

.

2. If all the particles are identical and the temperature is high enough (T ≳ KV/kB), then, irrespective of theorientations of the easy axes:

In the above equations:

Superparamagnetism 208

• n in the density of nanoparticles in the sample• is the magnetic permeability of vacuum• is the magnetic moment of a nanoparticle• is the Langevin functionThe initial slope of the function is the magnetic susceptibility of the sample :

in the first case

in the second case.

The later susceptibility is also valid for all temperatures if the easy axes of the nanoparticles are randomlyoriented.It can be seen from these equations that large nanoparticles have a larger µ and so a larger susceptibility. Thisexplains why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: theybehave exactly as a paramagnet with a huge magnetic moment.

Time dependence of the magnetization

There is no time-dependence of the magnetization when the nanoparticles are either completely blocked ( ) or completely superparamagnetic ( ). There is, however, a narrow window around where themeasurement time and the relaxation time have comparable magnitude. In this case, a frequency-dependence of thesusceptibility can be observed. For a randomly-oriented sample, the complex susceptibility[2] is:

where

• is the frequency of the applied field• is the susceptibility in the superparamagnetic state

• is the susceptibility in the blocked state• is the relaxation time of the assemblyFrom this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can bederived:

MeasurementsA superparamagnetic system can be measured with AC susceptibility measurements, where an applied magneticfield varies in time, and the magnetic response of the system is measured. A superparamagnetic system will show acharacteristic frequency dependence: When the frequency is much higher than 1/τN, there will be a differentmagnetic response than when the frequency is much lower than 1/τN, since in the latter case, but not the former, theferromagnetic clusters will have time to respond to the field by flipping their magnetization.[3] The precisedependence can be calculated from the Néel-Arrhenius equation, assuming that the neighboring clusters behaveindependently of one another (if clusters interact, their behavior becomes more complicated).

Superparamagnetism 209

Effect on hard drivesSuperparamagnetism sets a limit on the storage density of hard disk drives due to the minimum size of particles thatcan be used. This limit is known as the superparamagnetic limit.• Older hard disk technology uses longitudinal recording. It has an estimated limit of 100 to 200 Gbit/in²[4]

• Current hard disk technology uses perpendicular recording. As of August 2010 drives with densities of 667Gb/in2

are available commercially. Perpendicular recording is predicted to allow information densities of up to around 1Tbit/in² (1024 Gbit/in²).[5]

• Future hard disk technologies currently in development include: heat-assisted magnetic recording (HAMR),which use materials that are stable at much smaller sizes. They require heating before the magnetic orientation ofa bit can be changed; and bit-patterned recording (BPR).[6]

Applications of superparamagnetism

General Applications• Ferrofluid: tunable viscosity• Data analysis: superparamagnetic clustering [7] (SPC) and its extension global SPC [8] (gSPC)

Biomedical applications• Imaging: Contrast agents in Magnetic Resonance Imaging (MRI)•• Magnetic separation: cell-, DNA-, protein- separation, RNA fishing• Treatments: targeted drug delivery, magnetic hyperthermia, magnetofection

References[1] Néel, L. (1949). "Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites". Ann. Géophys 5:

pp. 99–136. (in French; an English translation is available in "Selected Works of Louis Néel". Gordon and Breach. 1988. pp. 407–427.ISBN 2-88124-300-2.).

[2] Gittleman, J. I.; Abeles, B.; Bozowski, S. (1974). "Superparamagnetism and relaxation effects in granular Ni-SiO2 and Ni-Al2O3 films" (http:// link. aps. org/ doi/ 10. 1103/ PhysRevB. 9. 3891). Physical Review B 9: 3891–3897. Bibcode 1974PhRvB...9.3891G.doi:10.1103/PhysRevB.9.3891. .

[3] Martien, Dinesh. "Introduction to: AC susceptibility" (http:/ / www. qdusa. com/ resources/ pdf/ 1078-201. pdf) (pdf). Quantum Design. .Retrieved September 2011.

[4][4] Kryder, M. H.. "Magnetic recording beyond the superparamagnetic limit". pp. 575. doi:10.1109/INTMAG.2000.872350.[5] "Hitachi achieves nanotechnology milestone for quadrupling terabyte hard drive" (http:/ / www. hitachi. com/ New/ cnews/ 071015a. html)

(Press release). Hitachi. October 15, 2007. . Retrieved September 2011.[6] Murray, Matthew (2010-08-19). "Will Toshiba's Bit-Patterned Drives Change the HDD Landscape?" (http:/ / www. pcmag. com/ article2/

0,2817,2368023,00. asp). PC Magazine. . Retrieved 2010-08-21.[7] http:/ / ctwc. weizmann. ac. il/ spc. html[8] http:/ / vcclab. org/ lab/ spc

• Néel, L. (1949). "Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terrescuites" (in French). Ann. Géophys. 5: 99–136. An English translation is available in Kurti, N., ed. (1988). SelectedWorks of Louis Néel. New York: Gordon and Breach. pp. 407–427. ISBN 2-88124-300-2.

• Weller, D.; Moser, A. (1999). "Thermal Effect Limits in Ultrahigh Density Magnetic Recording" (http:/ / dx. doi.org/ 10. 1109/ 20. 809134). IEEE Transactions on Magnetics 35: 4423–4439. Bibcode 1999ITM....35.4423W.doi:10.1109/20.809134.

Superparamagnetism 210

External links• Superparamagnetism of Co-Ferrite Nanoparticles (http:/ / www. mff. cuni. cz/ veda/ konference/ wds/ contents/

pdf05/ WDS05_090_f3_Vejpravova. pdf)• Powerpoint presentation on Superparamagnetism in pdf (http:/ / lmis1. epfl. ch/ webdav/ site/ lmis1/ shared/ Files/

Lectures/ Nanotechnology for engineers/ Archives/ 2004_05/ Superparamagnetism. pdf)

Vibrating sample magnetometer

Vibrating Sample Magnetometer - sample holderand detection mechanism

A vibrating sample magnetometer or VSM is a scientific instrumentthat measures magnetic properties, invented in 1955 by Simon Foner atLincoln Laboratory MIT. The paper about his work was publishedshortly afterward in 1959[1] A sample is placed inside a uniformmagnetic field to magnetize the sample. The sample is then physicallyvibrated sinusoidally, typically through the use of a piezoelectricmaterial. Commercial systems use linear actuators of some form, andhistorically the development of these systems was done using modifiedaudio speakers, though this approach was dropped due to theinterference through the in-phase magnetic noise produced, as themagnetic flux through a nearby pickup coil varies sinusoidally. Theinduced voltage in the pickup coil is proportional to the sample'smagnetic moment, but does not depend on the strength of the appliedmagnetic field. In a typical setup, the induced voltage is measuredthrough the use of a lock-in amplifier using the piezoelectric signal asits reference signal. By measuring in the field of an externalelectromagnet, it is possible to obtain the hysteresis curve of a material.

References[1] .Foner, S. "Versatile and Sensitive Vibrating-Sample Magnetometer". Rev. Sci. Instrum 30 (7): 548–557.

Article Sources and Contributors 211

Article Sources and ContributorsAntiferromagnetism  Source: http://en.wikipedia.org/w/index.php?oldid=502540012  Contributors: Aka042, Andre Engels, Anterior1, Bjf, Carcharoth, Chaiken, Connormah, Daneshvar,DragonflySixtyseven, Emperorbma, Felix0411, Floorsheim, Freddy78, Guillom, Headbomb, Heron, Icairns, Isheden, Jag123, JaredAllred, Jcwf, Kmarinas86, Kusma, LarRan, Materialscientist,Mboverload, Michael Hardy, Mlpearc, Mnmngb, Niceguyedc, Phys, Pieter Kuiper, Piil, Quibik, Rg998, RockMagnetist, Rod57, Salsb, Savidan, Srnec, Stevvers, Tamtamar, Timo Honkasalo,Timothykinney, Toh, Tone, V8rik, Venny85, Vorpal blade, Wasell, Xram, Zahid Abdassabur, 50 anonymous edits

Biot–Savart law  Source: http://en.wikipedia.org/w/index.php?oldid=505992395  Contributors: 124Nick, Andres Agudelo, Antixt, Arc-, AugPi, Bender235, Burn, Caliston, Charles Matthews,Charlym, Choster, Complexica, Craig Pemberton, Crowsnest, DJIndica, Daniel.Cardenas, DavidLevinson, Deb, Decumanus, Deflective, Dicklyon, Dilipmeena22, Dixtosa, Dolphin51, Drrngrvy,Ed Poor, Eliz81, Enormousdude, F=q(E+v^B), Ferengi, FyzixFighter, Gabridelca, Gaius Cornelius, George Smyth XI, Giftlite, Grebaldar, H2g2bob, Headbomb, Icairns, Ioverka, JabberWok,JayEsJay, Jojalozzo, Khashishi, Klunk6, Kwamikagami, Laurascudder, Lejarrag, Linuxlad, MC10, MFNickster, Martynas Patasius, Mboverload, Mebden, Metacomet, Michael Hardy, Mild BillHiccup, Mjohnrussell, Mtodorov 69, Muu-karhu, Mythealias, Onco p53, Oobayly, Paolo.dL, Petwil, Pfalstad, Qxz, Revolver, RockMagnetist, Roo72, Rtdrury, Salsb, Sbyrnes321, Sheliak,StradivariusTV, Svick, TStein, The wub, Tim Shuba, Tim Starling, Tobiasgt79, Toby Bartels, Toolnut, User A1, Vinograd19, Weialawaga, Wik, Wolfkeeper, Wrude bouie, Zvn, 老 陳, 霧 木

諒 二, 142 anonymous edits

Classical electromagnetism and special relativity  Source: http://en.wikipedia.org/w/index.php?oldid=510148120  Contributors: Brews ohare, DS1000, DVdm, Dario Gnani, F=q(E+v^B),Headbomb, J04n, JacobTrue, Magioladitis, Sbyrnes321, Stevenj, TStein, Teply, Wavelength, Woohookitty, Ywaz, 16 anonymous edits

Coercivity  Source: http://en.wikipedia.org/w/index.php?oldid=503149937  Contributors: Ahecht, Andrewwall, Bissinger, Brouhaha, Chaiken, Chetvorno, CosineKitty, Dogcow, Electron9,Emes, Eric.weigle, Ferengi, Genghiskhanviet, Genish, Icairns, JEBrown87544, Jag123, Jwagner61, Kingturtle, Kirkmeister, Kjkolb, Kurgus, Magnetix1, Materialscientist, Maximus Rex, MeganReyes, MichaelBillington, Mnmngb, Omegatron, Paolo.dL, Planetscared, RockMagnetist, Rostislav Lapshin, Rostislav V. Lapshin, Salsb, Venny85, Voodoom, WBardwin, Wegsjac, Wolfkeeper,Zocky, Zureks, റസൽ ഗോപിനാഥൻ, 29 anonymous edits

Diamagnetism  Source: http://en.wikipedia.org/w/index.php?oldid=507480124  Contributors: 213.253.39.xxx, 24.1.200.xxx, AJim, ALACE, Aaagmnr, Acroterion, Adashiel, Arkadipta banerjee,Bakuryuu, Beland, Belg4mit, Bluefalcon07, Bodnotbod, Brews ohare, Bryan Derksen, Busukxuan, Campuzano85, Candleknight, Casey boy, Cesiumfrog, CharlesC, Cheeseifyer, Constructiveeditor, Conversion script, Cp111, Cquan, DARTH SIDIOUS 2, Darekun, DarkHorse, Deepnightblue, Deglr6328, Dfinkel, Dimwitt Flathead, Dirac1933, Don4of4, DragonflySixtyseven,Dwmyers, EbedYahweh, Eigenpirate, Embrittled, Favonian, Foobar, Gaius Cornelius, Gene Nygaard, Georgelazenby, Giftlite, Glacialfox, Graham87, Gudeldar, Guswandhi, Hans Dunkelberg,Headbomb, Hede2000, Henrygb, Heron, Hesperian, Horkana, Icairns, Iliev, Inter rest, Jaapkroe, Jackelfive, Jafet, Jcline1, Jcwf, Jinxed, Jkeohane, Joanjoc, JustAddPeter, K Eliza Coyne, Kaifeng,Karol Langner, KasugaHuang, Katalaveno, Kmarinas86, L'Aquatique, Leobh, Lfh, LogaRhythm, Looxix, Lumrs, Macderv15h, Mbweissman, Mech Aaron, Midgrid, Mike Rosoft, MisterSheik,Mmm, Modify, Moemin05, Netscott, OlEnglish, Oli Filth, Omegatron, Paul venter, Pearle, Peterburton, Petergans, Pharaoh of the Wizards, Phoenix79, Phys, Piil, Planetscared, Poisonmilk,Prikryl, Rage, Rememberway, Rifleman 82, Roadrunner, Robin Whittle, RockMagnetist, Salsb, Sappe, Scott Dial, Serverxeon, Sibian, Sikkema, SilentOpen, Silly rabbit, Slakr, Smalljim,Smokefoot, Snigbrook, Splarka, Stokerm, Suffusion of Yellow, Tarotcards, TedPavlic, Tim Starling, Tmadge, Tomothy, Troyrock, Vanderdecken, Vanished user, Vrenator, Vsmith, WLU,Waleswatcher, Whitepaw, Wolfkeeper, Xanzzibar, Xompanthy, Yakiniku, Zamirm, Zereshk, Zinger0, خالقیان, 老 陳, 183 anonymous edits

Electromagnet  Source: http://en.wikipedia.org/w/index.php?oldid=505846963  Contributors: 2help, 5 albert square, 90 Auto, A little insignificant, A.K.Karthikeyan, A3RO, Adambro, Adashiel,Addshore, Adrianwadey, Aitias, Alansohn, Alkoury, Allstarecho, Andeveron, Andonic, Andy Dingley, Animum, Anna Lincoln, Antandrus, Antikon, Anyeverybody, Ashleano,Athenaabc123drm, AtiwH, Atlant, Avenged Eightfold, Avoided, Baa, Babartown, Badgernet, Banaticus, Bart133, Bennybp, Bentogoa, Bert Hickman, Bhadani, BillyWHU, Bobo192, BorisBarowski, Bradcallaghan23, Branchc, Bruno Roso, CJLL Wright, Cadby Waydell Bainbrydge, Caiaffa, Can't sleep, clown will eat me, Cardboardbox, Chamal N, Chetvorno, Chrislk02,Christendom, Chua, Clarince63, Closedmouth, Constructive editor, Cool3, Cowardly Lion, Cquan, Cynical, D. Recorder, DARTH SIDIOUS 2, DMacks, DV8 2XL, DanBoy96, Daniel5127,Davidprior, Dawn Bard, Deconstructhis, Delicious boy, DerHexer, Dfgarcia, Dgw, Dhp1080, Diberri, Dijiyd, Dina, Dirkbb, Disavian, Discospinster, DivineAlpha, Djwelshy, Dolazflow,Dougofborg, Doyley, Dpotter, Draconis Neurocam, Dreadstar, Dureo, Dwmyers, ESkog, Earlypsychosis, Edderso, Edivorce, Ekkert, Electricmoose, Electron9, Eliz81, Elron WolfBane, Emmaozzy, Emote, Enigmatarius, Epbr123, EugeniaxD, EvelinaB, Everyking, Excirial, FT2, Falcon8765, Faradayplank, Favonian, Ferritecore, Fieldday-sunday, Fireball, Firsfron, FletcherD, Floatjon,Flubbit, Fluffernutter, Frencheigh, Future02, Gail, Gary63, Gcm, Geek1337, Gene Nygaard, Geologyguy, Giftlite, Gilliam, GinaDana, Glane23, Gogo Dodo, Goodturtle, Goudzovski, GroveGuy,Gtstricky, Gurch, Gwernol, Hadal, Happy-melon, Hdt83, Heron, HexaChord, Hooperbloob, Hulagutten, Hut 8.5, IRP, Inter, Ixfd64, J.delanoy, J04n, J9mosely, JForget, JFreeman, JaGa,Jargon777, JavierMC, Jd Tendril, Jean-François Clet, JesseW, JinJ, Jklin, Joebob1234567, John of Reading, John254, Juliancolton, Jusdafax, K Eliza Coyne, Karada, Kaztom, Keith D, KeithB,KevinTR, King Lopez, King of Hearts, Kingpin13, KitemanSA, KnowledgeOfSelf, Kon michael, Krich, Kukini, Kuru, Lambrosus, Lancevortex, Lazulilasher, Light current, Little Mountain 5,Lommer, Lradrama, LuYiSi, Lumos3, Luna Santin, MC MasterChef, MMAfrk1988, Maniac98, MarkS, Martial75, MartinHarper, Martinmdp, Matanbz, Matdrodes, Materialscientist, Mav,Mdwyer, Meggar, Mercury, Miremare, Mnmngb, Monty845, Morton devonshire, MrFish, Msiddalingaiah, Mugunth Kumar, Mumia-w-18, Munford, Mygerardromance, NAHID, NawlinWiki,Neilw90, Nene1993, Nerd bzh, Neverquick, NewEnglandYankee, Nick1nildram, Nicolae Coman, Nishkid64, Nivix, Northumbrian, Nsaa, Nubscaper, NuclearWarfare, Nunquam Dormio,Nuttycoconut, Oda Mari, Ojigiri, Oli Filth, Openstrings, Oscarthecat, Oskay, OverlordQ, Owen, Owendude1210, Oxymoron83, PMDrive1061, Paige12345678910, Panoptical, Patrick, Pdcook,Peruvianllama, Pgk, Philip Trueman, Phoebe, Piano non troppo, Pigsonthewing, Pinethicket, Pinkit1, Playerlooter, PleaseStand, Pnavijay, Postglock, PrestonH, Prodego, Pyrrhus16, QubitOtaku,Quest442, Quibik, QuiteUnusual, R'n'B, RFerreira, RJaguar3, RSM, Radon210, Random user 39849958, RayAYang, Rbeas, Reddi, RexNL, Robert Foley, RobertG, RobertReg, Ronhjones,RoyBoy, Rugbyboywill, SD5, Salsb, Sanbeg, SarahStierch, Scateswung1, Sceptre, Scjessey, Scottfisher, Sderose, Sevtoah, Shithead111111, Shoeofdeath, Shoessss, Sidney, SimonP, Skarebo,Skyezx, Slartibartfast1992, Smack, Smalljim, Smiller933, Snowolf, Sotaru, Soulkeeper, SparrowsWing, Special-T, Spitfire19, Spongebob-05, StaticGull, Stephenb, Stroppolo, Stubblyhead,Sunrain, Sunray, Supaman89, Superbeecat, Syrthiss, THEN WHO WAS PHONE?, Tangotango, Tannkremen, Tarquin, Tbhotch, TeaDrinker, Tearlach, Teddks, Teslaton, Tex, That Guy, FromThat Show!, The Rambling Man, The Thing That Should Not Be, TheFearow, Thebug44, Thuktun, Thw1309, Tide rolls, Tim122911, Tim1357, Tomdabomb3010, Tommy2010, Tpk5010,Triddle, Uncle Dick, Vanished user 39948282, Versus22, Vinhtantran, Vipinhari, Vivio Testarossa, Vsmith, Wapcaplet, Wenli, Wexcan, Whale plane, Wikieditor06, William Avery, Willpower,Woohookitty, Worm 19, Wtshymanski, Xiahou, Xxagile, Zed toocool, Zerbinos, Zimbardo Cookie Experiment, Zoicon5, రవిచంద్ర, ಠ ಠ, 1229 anonymous edits

Ferrimagnetism  Source: http://en.wikipedia.org/w/index.php?oldid=495589894  Contributors: Avicennasis, Benbest, Capricorn42, Catslash, Dan100, DragonflySixtyseven, Fascinet, Freddy78,Furrykef, GregVolk, Guillom, HorsePunchKid, Icairns, Iliev, Jag123, Jmrowland, Larryisgood, Leaky Lens, Logicwax, Lovecz, Mimihitam, Mnmngb, Petergans, Petteri Aimonen, PhilipTrueman, Phys, Pieter Kuiper, Piil, Prime Entelechy, Ratzd'mishukribo, Rifleman 82, RockMagnetist, Rod57, Ryan858, Salsb, Sanku p, Signalhead, Slaweks, Srleffler, Stevenj, Teammm, TimStarling, TimBentley, Tone, Vivacissamamente, WaveJones, Wavgfkl, Wikfr, Yaroslav Blanter, Zetawoof, Zhangzhe0101, 46 anonymous edits

Ferromagnetism  Source: http://en.wikipedia.org/w/index.php?oldid=507530766  Contributors: 16@r, 8472, Abiermans, Anarcho hipster, Astronouth7303, Bad ideas, Bantman, Bduke,Bemoeial, BenFrantzDale, Benbest, Bert Hickman, Blainster, Bped1985, Breno, BriEnBest, Cacycle, Can't sleep, clown will eat me, Canoe1967, Capricorn42, Charles Gaudette, CharlesC,Chetvorno, Circeus, ClickRick, Complexica, Conversion script, Cophus, Dan Austin, Dan100, DardanAeneas, Darrien, Davidmack, Dgrant, Dimody, Don4of4, Donarreiskoffer,DragonflySixtyseven, DÅ‚ugosz, EEMIV, Eequor, Ekilfeather, Electron9, Eras-mus, FSHero, Fangfufu, Felix0411, Fieldday-sunday, Foobar, Freddy78, Furmanj, Gbravenclaw, Gene Nygaard,Giftlite, Headbomb, Hede2000, Heimstern, HermCain, HorsePunchKid, Icairns, Iliev, Infinoid, Itub, JerzyTarasiuk, Jessetjenkins, Joaosampaio, John, JohnOwens, Jonon, Jrockley, Jschnur,Klidge, Knowwayfarer, Kopovoi, Leandros, Light current, LilHelpa, Macronyx, MaratIk, Marshallsumter, Materialscientist, MementoVivere, Mild Bill Hiccup, Morelight, MrXow, Mtd2006,Nabla, Nimur, Northumbrian, Novel-Technology, Omegatron, Pakaran, Paolo.dL, Payonel, Pcordes, Peterwhy, Philopp, Phys, PierreAbbat, Piil, Pollinator, Res2216firestar, Richard75, Rifleman82, Rob Hooft, RockMagnetist, Rossami, Rracecarr, Rrburke, Sagaciousuk, Salsb, Salvor, Sam Hocevar, Sesquiculus1, Shlomke, Sripad94, Srleffler, Stevenj, Stevenmitchell, Stokerm, SunCreator, Tamtamar, Tgiebult, The real dan, Tide rolls, Tim Starling, Twang, Vary, Wapcaplet, Wavgfkl, Wdanbae, West London Dweller, Wiki me, Wikibob, Wizard191, Woodstocksabird,Xaonon, Xxanthippe, Zereshk, 179 ,کاشف عقیل anonymous edits

History of electromagnetic theory  Source: http://en.wikipedia.org/w/index.php?oldid=510039008  Contributors: 3rdAlcove, 5theye, ARTE, Abbyratsolee, Adavidb, Alan Liefting, Ambermist,Andonic, Andy85719, Arjen Dijksman, Arpingstone, BD2412, Balrore, Barticus88, Bartledan, Beland, BillC, Billinghurst, Carcharoth, Cenarium, Charles Matthews, Chris the speller, Cladist,CommonsDelinker, Cyhawk, D.H, DarkFalls, Davidiad, Dbachmann, Denver26, Dingy, Dirac66, Dougweller, EJohn59, ELApro, Electriccatfish2, Elmer Clark, Emerald Melios, Emj, Erzahler,Eugene-elgato, Evrik, Fr33Lanc3r, Gaius Cornelius, Gene Nygaard, Gerry Ashton, GhostPirate, Goldenrowley, Gracenotes, Grievous Angel, Gtxfrance, Gurch, HardBoiledEggs, Harryboyles,Headbomb, Hmains, Hobartimus, Hulagutten, Hut 8.5, Ignoranteconomist, Igoldste, Insanity Incarnate, InternetMeme, Ivc392, J.M.Domingo, J8079s, JMilty, JRSpriggs, Jagged 85, Jcarroll, Johnof Reading, K.C. Tang, KConWiki, KNHaw, Kanags, Keilana, King of Hearts, Kkmurray, Knight1993, Korg, Kuru, Letter Ezh, LindsayH, Lupo, Madcoverboy, Mais oui!, Mallocks,Manishearth, Martin Hogbin, Materialscientist, Matthew Fennell, McGeddon, Meshed Gears, Middayexpress, Mikespedia, Mild Bill Hiccup, Misedo, Mitch Ames, Mkch, Movementarian,Muhandes, Muhends, Netrapt, Never give in, NewEnglandYankee, Nick Cooper, Nickbeland, OliverBBurke, Omnipaedista, Ottawa4ever, Palthrow, PearlSt82, Pharaoh of the Wizards,Phirosiberia, PieterJanR, Pixelface, Plenumchamber, Qniemiec, R'n'B, RandomCritic, Rcarey1, Reddi, RekishiEJ, Rich Farmbrough, Rjwilmsi, RockMagnetist, Ruslik0, Ruud Koot,SchreiberBike, ScienceApologist, Sesel, Simetrical, Sj, Sluzzelin, SmallRepair, Ssilvers, Stemonitis, Stevertigo, SummerWithMorons, Surferboy244, Sven Manguard, Sxoa, TStein, Tanthalas39,Tassedethe, Tiggerjay, Tim Shuba, TimVickers, Tom Pippens, Tomasz Prochownik, Topbanana, Valenciano, Welsh, William Avery, Woohookitty, Work permit, Wtshymanski, Xantan5, 122anonymous edits

Article Sources and Contributors 212

Lorentz force  Source: http://en.wikipedia.org/w/index.php?oldid=507534194  Contributors: Acmedogs, Alexcalamaro, Alfredo, Alousybum, Ambros-aba, Amicus of borg, Ancheta Wis,Andres, BenRG, Bender235, Boethius65, Brews ohare, Bryan Derksen, CUSENZA Mario, Capricorn42, Chris Howard, Complexica, Conversion script, Cpiral, D-Kuru, D.keenan, DJIndica,DVdm, Deans-nl, Decltype, Dgrant, Dicklyon, DrBob, Drkirkby, Edmundo ba, El C, Electron9, F=q(E+v^B), FDT, Falcon8765, Frobnitzem, Fuhghettaboutit, FyzixFighter, Gene Nygaard,Geoffrey.landis, George Smyth XI, Giftlite, Headbomb, Heron, HolIgor, Inbamkumar86, InverseHypercube, JNW, JRSpriggs, JabberWok, Jaro.p, Jauhienij, Jcc77, Jdcanfield, Jjalexand,Jkeohane, JohnBlackburne, Jradavenport, K Eliza Coyne, Khazar2, Khunglongcon, Kieff, Kiyabg, Kwamikagami, Laurascudder, Leonard G., Lerdthenerd, Logichulk, Looxix, LtPowers,Lwiniarski, MFNickster, Maschen, Masgatotkaca, Metacomet, Michael C Price, Michael Devore, Mihaip, Mikeblas, Mikiemike, Modeha, Monica.alonso.UEM, Morning277, Mpatel, Mrdice,Myasuda, Neptune5000, Nick, Nmnogueira, Orderud, Paclopes, Paolo.dL, Petri Krohn, Philip Trueman, Qwasty, Rbj, Reach Out to the Truth, Rich Farmbrough, RobertG, RockMagnetist, Rror,Rtdrury, Sadi Carnot, Salsb, Sankalpdravid, Sbyrnes321, SebastianHelm, Sfu, Sheliak, Smb1001, Spartaz, StaticGull, Sun Creator, Sunnysite, TStein, Tetracube, Tharunsr121, That Guy, FromThat Show!, The Anome, The mexican boodle, TheBFG, Thurth, Ti89TProgrammer, Tim Shuba, Tim Starling, Treisijs, Tttrung, Uncle Milty, Utcursch, Wavgfkl, Werdna, Wessmaniac,WikHead, Yakeyglee, Yevgeny Kats, Zueignung, ธวัชชัย, 215 anonymous edits

Magnet  Source: http://en.wikipedia.org/w/index.php?oldid=509525314  Contributors: 06RBambe, 2over0, 4I7.4I7, 4lex, 4twenty42o, 5 albert square, 7, 75oharas, A More Perfect Onion, A. diM., A8UDI, ABF, ARTE, Abb615, Abdaal, Accurizer, Ace Frahm, Adam Riggall, Adavidb, Aghost, Ahoerstemeier, Ahudson, Aidarzver, Aitias, Aka042, Akabla, Akriasas, Alan Liefting,Alanmp, Alansohn, Alaphent, Aldnonymous, Aleichem, Alex43223, Ali@gwc.org.uk, Alphachimp, Amcfreely, AnOddName, Anarchist42, Andonee, Andrewpayneaqa, Andy M. Wang,Andy120290, Anetode, Ann Stouter, Antandrus, Antonio Lopez, Antrikshy, Archimerged, ArielGold, ArmadilloFromHell, Ashleano, Ashthewise, Atomicthumbs, AttendonsGodot, Augen Zu,Avicennasis, AxelBoldt, Azhyd, BPositive, BW95, Badcc, BarretB, Bassbonerocks, Beland, Belindajade, BenFrantzDale, Bento00, Bert Hickman, Betterusername, Bidgee, Bignchunky,Bill52270, Biodragon, Bissinger, Blanchardb, BlckKnght, Bluefalcon07, Bluerasberry, Bobo192, Bogey97, Bonadea, Bongwarrior, BostonMA, Bpringlemeir, Braken, Brews ohare, Brianga,Brossmac, Brumski, Bryce es joto, Bsadowski1, Bubba hotep, Bucklau, C.Fred, C1k3, CWii, CYD, Caltrop, Cameron168, Can't sleep, clown will eat me, Candleknight, Cantiorix, Cap601,Capricorn42, Captain-tucker, Carmichael, Casper2k3, Catgut, Cathardic, Catherineyronwode, Catslash, Cdang, Cessator, Chase me ladies, I'm the Cavalry, Chase8662, Cheeseifyer,ChemGardener, Chetvorno, Chris G, Chris83, Chrisbil09, Chriswiki, Chucksez, Chuggoman123456789, Church of emacs, Ckatz, Clansmugglar, Closedmouth, Clovervidia, Cmdrjameson,Cometstyles, Condem, Corpx, Courcelles, Crazynas, Cryptic, Cst17, D, DARTH SIDIOUS 2, DMacks, Daniel.Cardenas, Dansiman, Dantadd, Dantheman 360, Dantheok, Dar-Ape,DardanAeneas, Dark sav, Darkdeva92, Darth Panda, David0811, Davin, Dawn Bard, Dbachmann, Dbennetts, Dbfirs, Demi, Deor, DerHexer, Derppred, Derumi, Dethme0w, Dicklyon,DieSwartzPunkt, Digon3, Dino, Dirkbb, Discuss-Dubious, Dispenser, Dobz116, Doczilla, Dogsgomoo, Dougofborg, Doulos Christos, Dputig07, Dragon-w77, DuncanB, DurinsBane87,Dwmyers, Dylan620, E. Wayne, ELI BRICKO, ESkog, EastwickEducation, Ebaumswarrior, Edward321, Egil, Eidako, Eitheladar, Ejay, Elektrik Shoos, Elvis cow 123, Emesee, Emperorbma,EnJx, Enigmaman, Enviroboy, Epbr123, Ephebi, Eras-mus, Escape Orbit, Etz Haim, Eubulides, Euyyn, Everbeen, Everyking, Extra999, FF2010, Farquaadhnchmn, Femto, Fentener vanVlissingen, Fieldday-sunday, Fir0002, Fireice, Fons22, Fragma08, FrankCostanza, Frankenpuppy, FreplySpang, Freshacconci, Froid, Fuhghettaboutit, Funitude, Furrykef, Fæ, Gaius Cornelius,Gastopyra, Gdavidp, Geek1337, Gene Nygaard, George The Dragon, Georgeperez, Giftlite, Giree, Gladiator777, Glane23, Glass Sword, Gmpiglet, Go229, Gogo Dodo, Goluckyryan,Goodnightmush, GorillaWarfare, Goudzovski, Gracenotes, Graibeard, Gravecat, Gravitan, GreenSpigot, Greew, GregorB, Grubber, Gublefink, Gumbos, Gurch, Gökhan, HCA, Hackmun, Hadal,Haham hanuka, Hallenrm, HamburgerRadio, Hammer Raccoon, Hankjweb, HappyInGeneral, HenryLi, Henrygb, Heron, HeroofTime55, Hersfold, Herwest299, Hfoltz, Hgrobe, HiDrNick,Hibernian, Hires an editor, Hobojoebob, Hongooi, HowMagnetsWork, Huntthetroll, Hydrogen Iodide, Hyperwolf, II MusLiM HyBRiD II, IanOfNorwich, Icairns, Ikh, Imnotminkus, Improve,Incompetence, Inklein, Inky, Ioscius, Iridescent, IronGargoyle, Isaac, Ixfd64, J.delanoy, JA.Davidson, JEB90, JForget, JHunterJ, JaGa, Jackfork, Jagged 85, Jake Wartenberg, Jakhai1000,JamesBWatson, Jamesontai, Jamessugrono, January, Jauhienij, JavierMC, Jayyub, Jbusenitz, Jeff G., Jerzy, Jh51681, Jhvbhbigbv, Jivecat, Jjron, Jmjanzen, JoJan, JoanneB, JoaoRicardo, John,John254, JohnClarknew, Johnstoner, Jose Ramos, Joyous!, Jozue, Jsnydr, Julesd, Jusdafax, Just James, Justin W Smith, Jéské Couriano, K Eliza Coyne, Ka Faraq Gatri, Kakila, Karen Johnson,Katieh5584, Katrielalex, Kayau, Keegan, Keenan Pepper, Kelisi, Kelly Martin, Kf4bdy, Killer Deer, Kilo-Lima, Kingfish, Kingpin13, Kmarinas86, KnowledgeOfSelf, KuduIO, Kukini,Kungfuactionjesus1111, Kurt Shaped Box, Kuru, Kusma, Kweeket, L Kensington, L33tminion, LAX, La goutte de pluie, Landon1980, Languagehat, LeaveSleaves, Letstalk, LibLord, Lichenfrom Hell, Light current, Linnell, Lisatwo, Llywrch, Lol191, Loldogz, Lolipr0nz, Lord RTL, Lovecz, Lowellian, LukeScalone, Lummn, Lyncas, M3tainfo, MASFROG, MC10, MOO, Mac, MacDavis, Macedonian, Macy, Madchester, Maddie!, Madhero88, Magnet4less, Magnus Manske, Mahewa, Mahmud Halimi Wardag, Majorly, Mandarax, Mannymoon, Manway, Marco Polo,Mareino, Marek69, MarkSutton, Markussep, Masonm95, Materialscientist, Mato, MattieTK, Mattimole, MatzMiller, Mauler90, Maxus96, Meggar, Meisam.fa, Meno25, Merovingian, MessageFrom Xenu, Mgmirkin, Michael Devore, Michael Hardy, Micogobodo, Mike Rosoft, MikeyMouse10, Mild Bill Hiccup, Mildred36, Minimac, Minna Sora no Shita, Mladjo1988, Mmoneypenny,Moarlurking, Moogle10000, Moogyboy, Moomoomoo, Mr Stephen, Mr. Vernon, MrElijinkins, MrFish, MrXow, Mrllama123, Mrpage4life, Mufka, Mullet, Multichill, Mwanner, Mxn, Naddy,Naidim, Nakon, Nal101, Namayan, Natelewis, Nathanael Bar-Aur L., NawlinWiki, Nayvik, NellieBly, Nevadacall, Nia2hip, Nishkid64, Njaelkies Lea, Nnnniiiiinnnnnjjjjaaa, NorsemanII, Nsaa,Nubiatech, NuclearWarfare, Nueroma, Numbo3, OEP, Octahedron80, Oda Mari, Ohnoitsjamie, OisinisiO, Ojigiri, Ojones1, Olivier, Omegatron, Omicronpersei8, Omnipaedista, Onopearls,Opelio, Opspin, Optakeover, Orange Suede Sofa, Otisjimmy1, OverlordQ, Oxymoron83, PACO, Pablo125, Paddy-B-Jr, Pakaran, Panarchy, Paolo.dL, Patar knight, Patrick, Patrick Schwemmer,PatrickHenryPDX, Pawsrent, Pb30, Pcordes, PeaceNT, Persian Poet Gal, PeterBrooks, PeterCanthropus, Peterlin, Pgan002, Phantomsteve, Philip Trueman, Phoenixrod, Piano non troppo,Pichpich, Pietrow, Pinethicket, Pink!Teen, Pipexlul, Pkeck, Polyamorph, Ponser, Prari, PrestonH, Private Smith, Private ham, Proofreader77, Protonk, Punkletom, PÆon, Qst, Quadell,QueenCake, Quintote, Qxz, RA0808, RAClarke, RB972, RJASE1, Radon210, RainbowOfLight, Ravenhull, RayAYang, Razer Montzerro, Razorflame, Reasonit, Recardome, Redblack305,Reddi, Rememberway, Renee45almond, Rettetast, Reverting, RexNL, Rhobite, Rich Farmbrough, Richard W.M. Jones, Riventree, Rjwilmsi, Roastytoast, Robbie098, Robertvan1, Robinh,Robma, RockMagnetist, Ronhjones, Rory096, RoyBoy, Rrburke, Rror, Rtdrury, Ryulong, SDC, SGS CTS, SWAdair, Saimhe, Salsb, Samtheboy, SamuelRiv, Sango123, Sarahisbi, Sbyrnes321,Scalene, Scientizzle, Sciurinæ, Scoolfire, Scottfisher, Sean D Martin, Seanwal111111, Seapal, Sephiroth BCR, Setreset, Shadowjams, Shalom Yechiel, Shanes, Shaneybouy, Sharkie3000,Shearonink, Shimaspawn, Shirik, Shoeofdeath, SimpsonDG, Sjö, Skizzik, Smalljim, Smeira, Smiggen, SmilesALot, Smite-Meister, Smokizzy, Snori, Snowolf, Soap, Sokari, Solarra, Soledude,Some jerk on the Internet, Sonicfreak360, Soxixos, Spinningspark, Splash, SpuriousQ, Ste b, Stephenb, Stevey7788, Stizz, Sunray, Supermaster2011, Sweeneysarah08, THEN WHO WASPHONE?, TStein, Tadnel, Tamachao, Tangotango, Tbhotch, Tcncv, Tcsetattr, Teapeat, Techman224, Technopat, Th1rt3en, Thabin, The Rambling Man, The Thing That Should Not Be, Thepenfool, The wub, TheDuckLair, TheGrimReaper NS, Thingg, Thumperward, TicketMan, Tiddly Tom, Tide rolls, Tim Starling, Tiptoety, Tom harrison, Tombomp, Tommo5678, Tommy2010,Treeko master, Truco, Trusilver, Tryson, TungstenCarbide, Tvdm, U.S.Vevek, UMADDUDE, Uncle Dick, Uncloudedvision, UniQue tree, Urbandarkness, Usnerd, Ute in DC, Uusitunnus,VASANTH S.N., Vbnvnbvbvcbv, Vegas Bleeds Neon, Veinor, Vidarlo, Vinsfan368, Vipinhari, Vishnu2011, Voltorb, Vortexrealm, WMSwiki, WMarsh, Wandering Ghost, WarwickAllison,WatermelonPotion, Wavelength, Wayne Slam, WaysToEscape, Wayward, Wesley8631, West.andrew.g, Whitemilo1, Why Not A Duck, Wiggles, WikHead, Wikfr, Wiki alf, WikiMarshall,Wikiborg, Wikizenmagman, Wimt, Wizard191, Wjejskenewr, Wolfkeeper, Woohookitty, Woombi, Wysprgr2005, Xaosflux, Xezbeth, Xihix, Xymmax, YUL89YYZ, Yacov29, Yaysatan101,Yekrats, YellowFives, Yoduh2007, Yungszen, YuriLandman, ZenerV, Zotel, Zundark, Zureks, Zvika, Zzuuzz, Zzzzzzus, 1923 anonymous edits

Magnetic bearing  Source: http://en.wikipedia.org/w/index.php?oldid=501980169  Contributors: Angellcruz, Bobo192, Bryan Derksen, ChrisHodgesUK, Cyrius, Dheeraj4ever, Efcmagnew,Eidako, Embrittled, Fosnez, Fribb, Gigabobs, Halpaugh, Hamiltondaniel, Henning Makholm, Hoo man, Hooperbloob, Incompetence, JobanWiki, Jpc4031, Kosiakk, Light current, Mac Davis,MagnInd, MagnetalWiki, Mohammedrasi, Netscott, PMLawrence, Pahazzard, Paul Allaire, Pekaje, Pjbarraud, Planetscared, Ray Van De Walker, Rememberway, Rjwilmsi, Robbins, Sillybilly,Skarkkai, Thejapanesegeek, Veinor, Velvetfog49, WHdeGroot, Wbrameld, WikHead, Wilbern Cobb, Wizard191, Wolfkeeper, Xantolus, Zejrus, 153 anonymous edits

Magnetic circuit  Source: http://en.wikipedia.org/w/index.php?oldid=504270261  Contributors: Angstorm, BenFrantzDale, Brossow, Capricorn42, Ccrrccrr, Chetvorno, CyrilB, Dah31,Dicklyon, Dvorak729, Edward, Elagatis, Gene Nygaard, GibsonDavidBCRA, Glenn, Heron, Hooperbloob, Icairns, J04n, Jag123, James086, Jfraser, John of Reading, Joramar, Jp in England, KEliza Coyne, Khazar2, Kmarinas86, Kushkul, Leandropls, MFNickster, Mel Etitis, MovGP0, Msiddalingaiah, Myasuda, Nabla, Omegatron, Open2universe, Ospalh, PV=nRT, Paverider, Pharaohof the Wizards, Rainald62, Rubber hound, SCEhardt, Salsb, Spinningspark, Ssilvers, Syd1435, TStein, The Berzerk Dragon, TheAMmollusc, Ulric1313, Wood Thrush, YUL89YYZ, ZoftWhere,Zundark, 38 anonymous edits

Magnetic dipole  Source: http://en.wikipedia.org/w/index.php?oldid=510618279  Contributors: Bearcat, Brews ohare, Davius, Drphysics, Facegarden, Grandfatherclok, Jojalozzo, Mark91,Maschen, Oliver Jennrich, Redirect fixer, RockMagnetist, Tim Starling, Wickey-nl, 2 anonymous edits

Magnetic domain  Source: http://en.wikipedia.org/w/index.php?oldid=507584892  Contributors: 7daysahead, AVand, Basawala, Bobo192, Bwrs, Caliston, Captin Shmit, Chetvorno,Diagonalfish, Dolphin51, Eio, Elephant in a tornado, GoOhm, Gorchy, Gracefool, Headbomb, JPFen, Julius Sahara, K Eliza Coyne, Kezz88, Kibethblade, Kjkolb, LilHelpa, Llgc, Matesy,Mhesselb, Mygerardromance, NerdyScienceDude, ROSHINIk7, Radagast83, Res2216firestar, RockMagnetist, Roulianne, Sfrabble, SkyWalker, Squiddy, Vsmith, Wikfr, Wikieditoroftoday,Wnzrf, Wwoods, Zhangzhe0101, Zureks, 68 anonymous edits

Magnetic field  Source: http://en.wikipedia.org/w/index.php?oldid=510180286  Contributors: 1994bhaskar, 1howardsr1, 2001:610:1908:1202:649A:1B84:2A86:B0D0, 213.253.39.xxx, 23790AD, 2D, 2over0, 4twenty42o, @pple, Abductive, Af648, Ahoerstemeier, Aitias, Aka042, Alansohn, Alex Klotz, Alfred Centauri, Alphachimp, Ambros-aba, Andy Dingley, Ankid, Anna512, Anterior1, Antixt, Arch dude, Armius, Arthena, Ascidian, Ashill, Aulis Eskola, B21O303V3941W42371, BSTR, Bachrach44, Barneca, Bart133, Bekus, Beland, BenFrantzDale, Bender235, Bishoppowell, Black Shadow, Bobyorox, Bookandcoffee, Brews ohare, BrianWilloughby, Brichcja, Brigman, Bryan Derksen, Buster79, C14, CUSENZA Mario, Calvin 1998, Cantiorix, Capricorn42, Captain Yankee, CaptinJohn, Catalanhost, Catslash, Ccrrccrr, Celebere, Cessator, CharlesChandler, Chetvorno, Chris the speller, Chrislk02, Christian75, Chrsschm, Citeseer, Claus Ableiter, Clicketyclack, CliffC, Clw, Cocytus, Coldwarrier, Cometstyles, Complexica, CosineKitty, Creidieki, Curps, Cxz111, CyrilB, D6, DARTH SIDIOUS 2, DJIndica, DMahalko, Da Joe, Daf, Damo0078, Daniel.Cardenas, DeMk9D76, Deadlyops, Defender of torch, Delirium, DemonThing, Dennis Brown, DerHexer, DesertAngel, Dgmyer, Dgroseth, Diannaa, Dicklyon, Dino, Direvus, Discospinster, Djr32, Dmn, Doc aberdeen, DocWatson42, DomenicDenicola, DoubleBlue, Download, Doyley, Dr. Seaweed, DrBob, Dreadengineer, Drrngrvy, Dynaflow, Edivorce, Egmontaz, Ekkert, El C, El estremeñu, Enchanter, Enormousdude, Ethan, Ettrig, Eudoxie, Evgeny, Excirial, Explodinglam, FDT, FelisLeo, Femto, Filemon, Filippopoulos, Floorsheim, Fongs, Fpahl, Freelance Physicist, Fresheneesz, From-cary, FrstFrs, Fyyer, FyzixFighter, Fæ, G-W, GDonato, GRB, Gaius Cornelius, Gary King, Gatoatigrado, Geek1337, Gene Nygaard, Gfoley4, Giftlite, Giggy12345, Gilliam, Gingavitus777, Gits (Neo), Glenn, Glmory, Goudzovski, Greenpowered, Grstain, Gryllida, Gunnergrady, Gökhan, H0dges, H2g2bob, HEL, Harry, Headbomb, Helix84, Hellbus, Helohe, HenryLi, Herbee, Heron, HexaChord, Hommadi2001, Hongooi, Hqb, Hunter360x, Hurricane Angel, Hydrogen Iodide, ICE77, IVAN3MAN, Iantresman, Icairns, Icep, Imafjbks, Imrankhan85, Incompetence, Inquisitus, Intangir, InverseHypercube, Iridescent, Isdarts222, Ixfd64, J.delanoy, JD554, JForget, JJ Harrison, JSquish, JaGa,

Article Sources and Contributors 213

JabberWok, Jackelfive, JackyR, Jacobymathews99, Jaganath, Jagun, Jakebarrington, Janolaf30, JasonSaulG, Jauhienij, Javalenok, Jaxl, JerrySteal, Jfx319, Jim1138, JoanneB, John of Reading,John254, JohnBlackburne, Jojalozzo, Justanyone, Jvansanten, KJS77, Kafka Liz, KasugaHuang, Katalaveno, Katieh5584, Kenshin9554, Kesac, Khashishi, Kimse, Kingpin13, Kku, Kmarinas86,Kri, Kurt.hewett, Kurzon, Laurascudder, LeCire, LeilaniLad, Lenko, Lichen from Hell, Light current, LilHelpa, Lindberg G Williams Jr, Locriani, Lookang, Lseixas, Luke490, Luna Santin, M CY 1008, MC10, MER-C, MacedonianBoy, Magnetic7, Magog the Ogre, Manscher, MarcoLittel, Marmzok, MarsRover, Maschen, Matdrodes, Materialscientist, Mattbr, Maxhutch,Mcmonkeyburger, Mdmilagre, Mebden, Melchoir, Mentifisto, Merseyless, Metacomet, Mh liv01, Michael Hardy, Michi zh, Mihail Vasiliev, Mikiemike, Mjpieters, Mni9791, Modulatum, Momosan, Mossd, Msiddalingaiah, Mstyne, Myasuda, Mygerardromance, NOrbeck, Nabla, Nageh, Nakon, NatureA16, Ndhuang, Neko-chan, Netheril96, NewEnglandYankee, Nicolharper, Nielchiano,Nigilan, Nmnogueira, Noommos, Notinasnaid, Ocolon, Oda Mari, Ohnoitsjamie, Old Moonraker, Olivier, Omegatron, Onco p53, Opticron, Oreo Priest, Owlbuster, Oxymoron83, Palle.haastrup,Paolo.dL, Papa November, Patrick, Paul venter, Paverider, Pax:Vobiscum, Pdn, Pearle, Pedantik, Pedro, Pekinensis, Penubag, Persian Poet Gal, Pfalstad, Pgadfor, Pgosta, Phasespace, PhilBoswell, Philip Trueman, Photodude, Phynicen, Physis, Pi zza314159, Pinethicket, Pissipo, Pjacobi, Pokipsy76, Pol098, Pollinator, Postglock, Prodego, Qniemiec, Quantpole, Quibik,QuiteUnusual, Quondum, RG2, RJHall, Racko94, RadioFan, Radon210, Rafonseca, Ral315, Razimantv, Rbj, Rdsmith4, Reddi, Regig, Rich Farmbrough, Richerman, Rico402, Rjstott, Rjwilmsi,Robinh, RockMagnetist, Ross Burgess, Rossami, Rpf, Rrburke, Rundquist, Ryan032, SCZenz, SDC, SJP, Salsb, Salt Yeung, Sam Korn, Santista1982, Sbharris, Sbyrnes321, Scarian, Scohoust,Scooter, Scott Illini, Sfu, Shawn81, Sheliak, Signalhead, Sintau.tayua, Sir48, SirEditALot, Sjakkalle, Smack, Smark33021, Smile a While, Sneller2, Snigbrook, Sole Soul, Some jerk on theInternet, SomeUsr, Southen, Speedevil, Srleffler, Ssilvers, Stan Sykora, Stannered, Starwed, Starwiz, Steel, Steve Quinn, SteveBaker, Stevenj, Strait, Sublimedragon28, TStein, Tagray, TalonArtaine, Tarquin, TedPavlic, Telanis, Telpardec, Template namespace initialisation script, Tempodivalse, Tgoyen, Thatguyflint, The Earwig, The Thing That Should Not Be, The wub,The-G-Unit-Boss, TheOldJacobite, Thric3, Tide rolls, Tim Shuba, Tim Starling, Time3000, Tkirkman, Tlabshier, Tonyalfrey, Treisijs, Trovatore, Trusilver, Twsx, UnknownForEver, Useight,Utcursch, Vamaviscool123, Van der Hoorn, Van helsing, Vary, Vcelloho, Veinor, Venom4u 31, Verdi1, Versus22, Vlus, Wahying, Wavelength, Wavgfkl, Wayward, Whitepaw,Whydowefallbruce?, WikHead, WikiDao, William Avery, Wizard191, Wogboy52, Wolfkeeper, Woohookitty, Woseph, Xclassmechluv, Ximenes Resende, Xtremepunker, Yevgeny Kats,Yill577, Yoduh2007, Yurei-eggtart, ZodTron, Zoicon5, Zueignung, محبوب عالم, સતિષચંદ્ર, 1088 anonymous edits

Magnetic monopole  Source: http://en.wikipedia.org/w/index.php?oldid=510890046  Contributors: 2bithacker, 2over0, 84user, Aarchiba, Aarghdvaark, Achoo5000, Adarsh116098,Ahoerstemeier, Alansohn, Alex Fix, Andre Engels, Andrija radovic, Antixt, Aoosten, Ap, ArnoldReinhold, BD2412, Bakken, Balashpersia, Barak Sh, Barraki, Beland, BenRG, Bryan Derksen,C.R.Selvakumar, Camembert, Capefeather, CatherS, Catslash, Charles Matthews, CharlesC, Charvest, Chetvorno, Congruence, ConradPino, Courcelles, Crumley, Cutler, Cwedhrin, Cyan,DVdm, David Thorne, Dawright12, Dchoulette, Deanmullen09, Dickontoo, Difty, Disambiguator, Dominus, DonSiano, Dorftrottel, Dougweller, DragonHawk, Drrngrvy, Długosz, ESkog,EddEdmondson, El C, Elektron, Emerson7, Enochlau, Enyokoyama, Erkcan, Eyu100, F=q(E+v^B), FDominec, FKLS, Falcorian, Floquenbeam, Flying hazard, Fru1tbat, GRB, Gaius Cornelius,Gareth McCaughan, Giftlite, Gillis, Giscard2, Goodbye Galaxy, Gorog, GregorB, Gzuckier, Headbomb, Henke37, Henrygb, Heron, HolIgor, HonoluluMan, Icairns, Igodard, Ixfd64,JA.Davidson, JRYon, JabberWok, JarlaxleArtemis, Jeffq, Jerzy, Jkl, John of Reading, Jonathan Karlsson, JorisvS, Jstrater, Karl Dickman, Karl-H, Kbodouhi, Khukri, KingCarrot, Kjoonlee,Likebox, Linas, Lisatwo, Lixo2, Loadmaster, LonelyBeacon, Loohcsnuf, Looxix, Lumidek, Luna Santin, MFNickster, MarSch, MarcelB612, Maschen, Mathfreak11235, Maury Markowitz,Maxime.Debosschere, Melchoir, Michael C Price, Michael Hardy, Mike Rosoft, Mild Bill Hiccup, Mintleaf, Mkweise, Mohseng, Morphotomy, Moyerjax, Mpatel, MuDavid, Munkel Davidson,N0RND123, Nagualdesign, Nat2, NerdyNSK, Nick Mks, Nlalic, Nova77, Nsande01, Octahedron80, Oldnoah, Particle hep, Patrick, Pauli133, PearlSt82, Peter Ellis, Pharotic, Phys, Piccor,Pit-trout, Pjacobi, Plasticup, PoorLeno, Q0k, QFT, Qaswqaswgd, Quibik, Quondum, R.e.b., Randall Nortman, Rapjo, RaseaC, Rasmus Faber, Razimantv, Relke, Renaissancee, Rich Farmbrough,Rjwilmsi, Roadrunner, Rock4arolla, Ross Fraser, Ruud Koot, SamuelRiv, Sasquatch, Sbyrnes321, Seraphimblade, Skatche, Skeptical scientist, Skiminki, Skippy le Grand Gourou, Skysmith,Spartaz, Splartmaggot, Stephen B Streater, Stevenj, Stevvers, Stigin, StringTheory11, Tabletop, Tardis, Tarotcards, TedPavlic, The Anome, TheAlphaWolf, Thranduil, Thuktun, Tim Starling,Timwi, Trmatthe, Urvabara, V1adis1av, WLU, Waltpohl, Wiki alf, Wikipelli, Wingedsubmariner, Wrotesolid, Wrsh11, Xerxes314, Xihr, YURI-21century, Yevgeny Kats, Zoooooooooaa, Мыша,318 anonymous edits

Magnetic refrigeration  Source: http://en.wikipedia.org/w/index.php?oldid=495824580  Contributors: AdamW, Adwaele, Andonic, AndrooUK, Antilived, Apple2, Benbest, Bender235,Bobblewik, Boing! said Zebedee, Buster2058, CharlesC, Chem-awb, Chevinbrown, ChrisGualtieri, ChrisHodgesUK, CleanCoolingSolutions, DMahalko, Dakott, Dave souza, Droll, Electron9,Gadfium, Gene Nygaard, Gigs, Glenn, Grandonia, Grj23, Heron, Hydrargyrum, Ipigott, Irate, Jaraalbe, Jesuitson, John of Reading, Jw2034, KVDP, Knightofdark, Mechatronik, MightyWarrior,Minesweeper, Mortense, Mtodorov 69, Omegatron, Pbroks13, Petrsuhaj, Poppafuze, Quibik, Razorflame, Rdshull, Re-mark, Rich Farmbrough, Rjwilmsi, Rootbeerinacan, S Levchenkov, SDC,Saimhe, Saq!b, Seb az86556, Shoeofdeath, Shuikouhw, Speedevil, Spiel496, Tamer Abdel Wahab, Tarquin, Tetracube, Tom harrison, Trojancowboy, Typ932, Virtualerian, Vsmith, WISo,Well.caffeinated, XJamRastafire, Xerxesnine, Yellowdesk, Zundark, 89 anonymous edits

Magnetic stirrer  Source: http://en.wikipedia.org/w/index.php?oldid=500844899  Contributors: 224238scott, Adaptron, Biglama, Bryan Derksen, Clearly kefir, Collabi, Danielle dk,Davidswanepoel, Debresser, Dismas, Email4mobile, Ewen, Femto, Gaius Cornelius, Gcm, Groyolo, Ikaproduct, JVinocur, Jorge Stolfi, Kerttie, Kevinb, Kkmurray, Liangren3, Masur, Mav,Megamix, Minored, Of, Puffin, Rifleman 82, Ruhrfisch, Sam Hocevar, Sardanaphalus, Seanghosh, Stefan da, Thargrav, Thumperward, Tim Starling, Tonei, Towerman86, Veinor, Zheisey, قلیanonymous edits 70 ,زادگان

Magnetic structure  Source: http://en.wikipedia.org/w/index.php?oldid=457385028  Contributors: EoGuy, GAMerritt, Jcwf, Johntinker, Judge Nutmeg, Kannie, RockMagnetist, 16 anonymousedits

Magnetism  Source: http://en.wikipedia.org/w/index.php?oldid=507774092  Contributors: 16@r, 2over0, 4lex, 5 albert square, A Mom, A8UDI, AL2TB, Abce2, Adawg117, Addshore,Ahoerstemeier, Akendall, Akriasas, Aksi great, Al.locke, Alansohn, Ale jrb, Alex-engraver, Aliwikii, Allstarecho, American Eagle, Amog, Andonic, Andy Dingley, Anonymous Dissident,Antandrus, Apocalypse2009, Bassbonerocks, Bcfootball, Bchaplucian, Beland, Benbest, Bensaccount, Blizzarex, Bob sagget jr., Bobo192, Bongwarrior, Boredzo, Brews ohare, Brianga, Brockert,Bryan Derksen, C d h, CDN99, CYD, Caknuck, CalebNoble, Caltas, Calvin 1998, Can't sleep, clown will eat me, Capricorn42, Capybara123, Casull, Catgut, Caturdayz, Celarnor, Chaiken,Changer-guy, Charivari, Charles Matthews, Chaser, Chetvorno, Chrislk02, ChucksGay123456789, Chzz, Clark89, ClarkLewis, Closedmouth, Cmandouble3, Cmichael, Cntras,CommonsDelinker, Conversion script, Craig Pemberton, Crawlbeforeiwalk, Cryptoid, Ctachme, D6, DJIndica, DMacks, DV8 2XL, Dalegudmunsen, Dan Austin, Dan kelley90, DanMatan,Daniel.Cardenas, Davdclehn, David.Mestel, Davidhorman, Davidkazuhiro, Dbtfz, Delirium, DerHexer, Dfrg.msc, Dina, Discospinster, Donarreiskoffer, Doodoobutter, Doulos Christos, Dr.Sunglasses, Drostie, Drrngrvy, Dubbin, Duncan.france, Duncharris, DÅ‚ugosz, Długosz, E23, E2eamon, ESkog, EWikist, Ed Poor, Edderso, Edinborgarstefan, Edison, Edward, Edward Z. Yang,Eeekster, Egil, Egmontaz, El aprendelenguas, ElTyrant, Electrodynamicist, ElectronicsEnthusiast, Elipongo, Ellywa, Enormousdude, Epbr123, EquinoX, Erniesaurus, Eubulides, FJPB, Fang Aili,Feline1, Femto, Ferix, Firozmusthafa, Fizped, Flubber88, Fluidchameleon, Flutterman, Fredrik, Froid, Fuhghettaboutit, Gail, Gaius Cornelius, Gangasudhan, Gene Nygaard, Geologyguy,George2001hi, GhostPirate, Giftlite, Gioto, Glacialfox, Glenn, Gnowor, Gogo Dodo, GoldenTorc, GraemeL, Granito diaz, Grapetonix, Griffin5, Gscshoyru, Guanaco, Gurch, Gzuckier, Hahamhanuka, Hairy Dude, Halpaugh, Harold f, Headbomb, Hellbus, HereToHelp, Heron, HexaChord, Hmains, Hobartimus, Hongooi, Hu, Hv, Hydrogen Iodide, II MusLiM HyBRiD II,IMNOTARETARDATALL, IW.HG, Icairns, IceUnshattered, Immunize, IndulgentReader, Insanity Incarnate, Inthepink, InverseHypercube, Ipigott, Iridescent, Isis, Ixfd64, J. Spencer, J.delanoy,JA.Davidson, JCPH, JDspeeder1, JForget, JabberWok, JacobTrue, Jaeger5432, Jagged 85, Jakhai1000, Jamesontai, Jauhienij, Jcwf, Jeffhoy, Jeremy Visser, Jesse V., Jfdwolff, Jim.belk, Jmabel,Jmundo, Joanjoc, Jocelyne Heys-Gerard, John Aplessed, JorisvS, JulianB12, Junglecat, Jusdafax, JustAddPeter, Karch, Karcih, Karl-Henner, Karol Langner, Karteek987, Kazvorpal, Keegan,Keilana, Kerotan, Khyranleander, King of Hearts, Kingpin13, Kmarinas86, Knutsi, Koavf, Kostisl, Ktsquare, KyleCardoza, Lahiru k, Lankiveil, Leif27, Light current, Lightmouse, Lir, Llywrch,Lone Skeptic, Lordyou, LorenzoB, M412k, MELISASIMPSONS, MER-C, MZMcBride, Mac, Magister Mathematicae, MarcoAurelio, Martarius, Masonm95, Masonprof, Materialscientist,Maxellus, Maxrokatanski, Mdanziger, Meggar, Melchoir, Mentifisto, Mephistophelian, Metacomet, Michael Hardy, Mike Dill, Mike by, Mistercow, Mjspe1, Mkch, Modemac, Mp50967, MrStephen, Mr. Trustegious, Mspraveen, Mufka, MusicMaker5376, Mxn, Mygerardromance, NHRHS2010, Nabla, Natalie Erin, Nauticashades, NawlinWiki, NewEnglandYankee, Nick Number,Nicktfx, Nihiltres, Ninjackster, Nmgrad, Nmnogueira, Noor Qasmieh, Nsaa, Octaazacubane, Octahedron80, Oliviosu, Omicronpersei8, Opelio, Otolemur crassicaudatus, Ottawa4ever,Ownedestroy, Ozone77, Ozuma, Paddy-B-Jr, Pak21, Paolo.dL, ParkerHiggins, Party, Pedant17, Peterlin, Petr Kopač, Petrb, Pgk, Pharaoh of the Wizards, Philip Trueman, Physchim62, Piano nontroppo, Pigman, Pilotguy, Pinestone, Piotrus, Pjbcool103, Pmbeck, Pointillist, PoolDoc, Postscript07, PrincessofLlyr, ProDigit, Proficient, Proofreader77, Puchiko, Pudgy78685, PwncakesNbacon, Qui-Gon Jinn, Quickbeam, Quintote, RDR, RG2, Radon210, Rafonseca, Raminmahpour, RaulRavndra, Reconsider the static, Reddi, Redfarmer, Renesis, RexNL, Rifleman 82,Rising*From*Ashes, Roberdor, Robma, Robo Cop, RockMagnetist, Rolinator, Ron B. Thomson, Rossf18, Royboycrashfan, Rpb01r, Rrburke, Russell4, Ryalisaivamsi, Ryne1, S colligan, S ortiz,SGMD1, Sadi Carnot, Saideepak.budaraju, Sammyk214, Samsee, Sandstein, Satanael, Sbyrnes321, Schultz.Ryan, Science5, ScienceApologist, Scientizzle, Scott3, Shadowjams, Shotwell,Shrigley, Shuipzv3, Sietse Snel, Silent78, Sinewalker, Sionus, Skew-t, Skizzik, Slazenger, Slgrandson, Slon02, Sluzzelin, Smack, Smilesfozwood, Snowmanradio, Snowolf, Snoyes, Some jerk onthe Internet, Someguy1221, Spangineer, Special-T, Spencer, Spinningspark, Splartmaggot, Srikar33, Ssilvers, St. Hubert, Stanlste, Stephenb, SteveBaker, Stevertigo, Stokerm, Stone, Strait,Stuart07, Suffusion of Yellow, Sunray, Syrthiss, THEN WHO WAS PHONE?, TStein, Teapeat, Techguru, Techman224, Technobebop, TehBrandon, Template namespace initialisation script,Tempodivalse, Texture, Tgv8925, That Guy, From That Show!, The Rambling Man, The Thing That Should Not Be, The Troll lolololololololol, TheBendster, TheFronze05, TheMolecularMan,Thekillerpenguin, Theresa knott, Thingg, This user has left wikipedia, Thisisborin9, Tiddly Tom, Tide rolls, Tim Shuba, Tim Starling, Tobby72, Tobias Bergemann, Tommy2010, Trojancowboy,Tyler Matthews, Ulric1313, Unyoyega, Usp, Uvainio, VASANTH S.N., Vadim Makarov, Vanilluv30, Vanished user 39948282, Versus22, Viriditas, Vrenator, Vsmith, Vssun, WOSlinker, Warut,Waveguy, Wayne Slam, Wenli, Where, Whereizben, WikHead, Wiki alf, Wikier.ko, William S. Saturn, Willking1979, Wolfkeeper, Woudloper, Wtmitchell, Xenonice, Xxanthippe, YamamotoIchiro, Yammer68, Yevgeny Kats, Ykral, Yoduh2007, Yoyo2222, Yuwangswisscom, Z.E.R.O., Zbxgscqf, Zoragotcha, Zundark, Zvika, 老 陳, 1225 anonymous edits

Metamagnetism  Source: http://en.wikipedia.org/w/index.php?oldid=452952435  Contributors: Arthena, Ashley Pomeroy, DragonflySixtyseven, Jmnbatista, Robb The Physicist, RockMagnetist,TheTito, Tone, Vortmeester, 7 anonymous edits

Micromagnetics  Source: http://en.wikipedia.org/w/index.php?oldid=500030003  Contributors: ACrush, Bobblewik, Giftlite, Ixfd64, Joaosampaio, Johnoti, Magnetix1, Pearle, Rettetast,Rjwilmsi, RockMagnetist, Rpb01r, Sheepe2004, WestwoodMatt, Wnzrf, Zureks, 38 anonymous edits

Article Sources and Contributors 214

Molecule-based magnets  Source: http://en.wikipedia.org/w/index.php?oldid=491418719  Contributors: Barkeep, Frap, Giraffedata, GoingBatty, Gueneverey, HonorTheKing, JorisvS, Thatjenn,The Thing That Should Not Be, Thomasnet, V8rik, 10 anonymous edits

Neodymium magnet  Source: http://en.wikipedia.org/w/index.php?oldid=510085390  Contributors: 2over0, A Mom, A More Perfect Onion, Abce2, Alansohn, Aldoaldoz, Anarchemitis,Andrewpmk, Andrés D., Anjouan, AnotherWikiGuy, Arcette, AtOMiCNebula, Beetstra, Benbest, Bender235, Bert Hickman, Bloodshedder, Bob.os, Borislav Dopudja, BuickCenturyDriver,Buuneko, Can't sleep, clown will eat me, Candleknight, Chaiken, Charlesrkiss, Chaz 11, Cheasy123, Chetvorno, CommonsDelinker, CoolMike, DMacks, Danielphin, Darkthunderz, Darrien,DarwinPeacock, Dave6, Dega180, Deglr6328, Deli nk, DerHexer, Derumi, Di-gata, Discospinster, DivineAlpha, EGoodier, EarthCom1000, Earthlyreason, Emre D., Eteq, Evans1982, Everbeen,Ewlyahoocom, Exeunt, Farkas2029, Femto, FrozenMan, Fungicord, Funnyjunkftw, Gene Nygaard, Geoffrey.landis, Georgy90, Gogo Dodo, Goudron, Granito diaz, GregorB, HappyCamper,Harris7, Hellno2, Heron, I DONT CARE, Icairns, Igoldste, Infynyxx, Jacob C Jordan, JamesBWatson, Jbusenitz, Jcwf, Jivecat, Jmoorhouse, Julesd, Jumping cheese, Jyril, KKPie, Karn, KayDekker, Krj373, LFaraone, Laager, Lamro, Lawyer2b, Leuk he, Lmatt, Lmhill, LorenzoB, Luis Dantas, M0M3NTUM, MC10, MPF, Magnequench, Makgraf, Malcolma, Manscher, Marek69,Materialscientist, MathStuf, Maximus Rex, Mayfly may fly, Mazroxz, Mboverload, Mc013579, Mdekok3000, Mebob123, Mikael Häggström, Mikiemike, Mmj, MrJones, Nakon, Ncurses,NewEnglandYankee, Ngchen, Nikevich, Nmnogueira, Osmodiar, Ourai, Ozkidzez91, Pakaran, Patentmat, Peter bertok, Philip.marshall, Pimlottc, Pinethicket, Polyparadigm, Pschemp, Quietust,QuiteUnusual, Remember the dot, Rgephart, Rjwilmsi, Ronhjones, Roo72, Russoc4, Rwalker, SCEhardt, Salsb, Schneelocke, SchuminWeb, SciberDoc, Shaddack, Shii, SlipperyHippo, Smelybrando, Specter01010, Spike Wilbury, Splarka, Spliffy, Split, Stevey7788, Stone, Suradnik50, TGCP, THEN WHO WAS PHONE?, Techbert, The Letter J, Thingg, Thorwald, Thumperward,Tim Chambers, Timtrent, Vedantm, Veeeeeeini, Vranak, Walksonwalls, Warut, Whitepaw, WikHead, Winndm31, Wolfkeeper, Xi311, Yoduh2007, Zekozo, Ásgeir IV., 324 anonymous edits

Paramagnetism  Source: http://en.wikipedia.org/w/index.php?oldid=507370107  Contributors: Aaagmnr, Admiral Norton, Ahoerstemeier, AlexGWU, Anchananatarajan, Andre Engels, Art andMuscle, Bandy, Bduke, Beetstra, Benbest, Berland, Brews ohare, Busukxuan, CRON, Cmcnicoll, Complexica, Conversion script, Cyanoir, DanielRigal, Dcoetzee, DeadEyeArrow, Derschueler,Dina, Eborreson, Electricmic, Electron9, Eric Kvaalen, F-402, Freddy78, Gadolinist, Gamera2, Gene Nygaard, Giftlite, Hammersbach, Heron, HopeChrist, Humanist, Icairns, Inquisitus,IronGargoyle, IvanLanin, Jcwf, Jdedmond, Joanjoc, John, JorisvS, KathrynLybarger, Kbrose, Koweja, Lfh, Light current, Looxix, Ls1955, Luckas Blade, Lugh23, Mac Davis, Markjoseph125,Materialscientist, Mdsam2, Medeis, Mercurywoodrose, Mgiganteus1, Michael Hardy, Murukesh mohanan, Mwhiz, NCurse, Nick Pisarro, Jr., No1lakersfan, Omegatron, Oxymoron83, Pde,Peter.C, Petergans, Phys, Piano non troppo, Piil, Polyamorph, Potatoswatter, Ppxatc, Profero, Quantum7, Quibik, Rakista, Rangergordon, Rebroad, Rifleman 82, Robinsoncrusoe, RockMagnetist,RocketDavid, Salsb, Samaritan13, Sbharris, Schneelocke, Silenced, Silverplasma, SimonArlott, Slakr, Smokefoot, Someones life, Sonicology, Stevenj, Stokerm, Tassedethe, Taw, Thevenin77,Thumperward, Tim Starling, Troyrock, Turbos10, WhiteDragon, Zero sharp, 183 ,زرشک anonymous edits

Plastic magnet  Source: http://en.wikipedia.org/w/index.php?oldid=508183577  Contributors: Antonrojo, Bobo192, ChemGardener, Closedmouth, GargoyleMT, Giftlite, Giraffedata,Grendelkhan, Icairns, J04n, Jag123, JeanJPoirier, Kbh3rd, Oxymoron83, Qaz, Rjwilmsi, Robma, Salsb, Tone, Vanischenu, Yakushima, 26 anonymous edits

Rare-earth magnet  Source: http://en.wikipedia.org/w/index.php?oldid=504741289  Contributors: 2over0, Aaagmnr, Alan Liefting, Alansohn, BCube, Backpackadam, Bert Hickman, Brim,Brz7, Can't sleep, clown will eat me, Charlieb000, Chetvorno, Chris the speller, ChrisGualtieri, D6, DMacks, Dismas, Dr.Jeschu, Dv82matt, EEPROM Eagle, Embrittled, Enviroboy, Fastplanet,Femto, Fr4an1s, Fubar Obfusco, Gene Nygaard, Granito diaz, Gyre86, HappyCamper, Headbomb, Headybrew, Heirpixel, Ixnayonthetimmay, J.delanoy, JamesBurns, Jbusenitz, Jcbarr, Jcwf,Jezza333, Jimmilu, Julesd, Kittybrewster, Kraftlos, Lee Carre, Materialscientist, Michalis Famelis, Mindmatrix, Mohhingman, Moletrouser, Netscott, Oknazevad, Opelio, Panzer V Panther, PaulAugust, Paulezra, Poi0987654321, Polyamorph, Polyparadigm, Pro crast in a tor, Redrose64, Rjwilmsi, SCEhardt, Sbharris, Schmloof, Shaddack, Shingra, Shirulashem, Suradnik50, Teapeat,Tetracube, Theottovonbismark, Tide rolls, Tremaster, Trojancowboy, Twang, UAwiki, Virpik, Voltaic, Weareryan, Whytecypress, Wintorez, Wolfkeeper, YUL89YYZ, Yngvarr, Yoduh2007, 138anonymous edits

Single-molecule magnet  Source: http://en.wikipedia.org/w/index.php?oldid=507378164  Contributors: Awickert, Barras, Bloomy717, Buzz-tardis, Cdion, Christian75, Cimorcus, Daisystanton,Dougszathkey, DragonflySixtyseven, Dreamer08, Embec, Enric Naval, Giraffedata, Gobonobo, Headbomb, Heron, Hollyev, Intgr, Lantonov, MagnInd, Materialscientist, Mereda, Nickptar, Rayc,Rbrausse, Reedijk, RockMagnetist, Takaaki, Thatjenn, Tiglet, Tomgally, V8rik, Venny85, WAS 4.250, 46 anonymous edits

Spin glass  Source: http://en.wikipedia.org/w/index.php?oldid=505112908  Contributors: 4lex, AHM, Afluegel, AmarChandra, Antoni Barau, Aranel, Avicennasis, Baxxterr, BenFrantzDale,Brammers, Caiaffa, Compsim, David Eppstein, Edward, Electricmic, Emperorbma, Eric Shalov, Frank Shearar, Gala.martin, Gene Nygaard, Giftlite, Gingekerr, Headbomb, Hovnatan, Icairns,Javirl, Jcoetzee, Knotwork, Koumz, Lfh, Lgstarn, Linas, Man It's So Loud In Here, Marie Poise, Meier99, Michael Hardy, Mlaffs, Modeha, Niteowlneils, Pamputt, Pasteurizer, PatrickFisher,Pavithransiyer, Philopedia, Phys, Pjvpjv, Qwfp, RSRScrooge, Rjwilmsi, RockMagnetist, Rwp, SPat, Salsb, Shaddack, Sodin, Stillnotelf, Svick, Unara, Venny85, Waltpohl, Zureks, 47 anonymousedits

Spin wave  Source: http://en.wikipedia.org/w/index.php?oldid=494846256  Contributors: Anlace, BullRangifer, Chaiken, Charles Matthews, ChrisGualtieri, Docu, Freddy78, Happy-melon,Headbomb, Jeff G., JorisvS, Lovecz, Lupin, Matthias Buchmeier, Mild Bill Hiccup, Nielswalet, PhiMAP, QuantumSquirrel, Rafaelgr, RockMagnetist, Tabletop, Topbanana, UnHoly, Venny85,Xerxes314, Xxanthippe, 58 anonymous edits

Spontaneous magnetization  Source: http://en.wikipedia.org/w/index.php?oldid=509684734  Contributors: Aeusoes1, Chaiken, CiaPan, Keoki, RDR, RockMagnetist, Venny85, Yevgeny Kats, 2anonymous edits

Superparamagnetism  Source: http://en.wikipedia.org/w/index.php?oldid=496014290  Contributors: AvicAWB, Cinnamon colbert, Conversion script, DARTH SIDIOUS 2,DragonflySixtyseven, EH101, Edgar.bonet, GregorB, Gurch, HighKing, IRbaboon, Icairns, Keoki, LilHelpa, LorenzoB, MartinSpacek, MasterCheese, Nate Silva, Nostraticispeak, Op47, Quibik,Qutezuce, Rifleman 82, Rjwilmsi, RockMagnetist, Roulianne, Salsb, Sbyrnes321, ScienceGuy5555, Stevenj, Stokerm, Tim Starling, Tone, Tothwolf, Unyoyega, WAS 4.250, Wafulz, Wang lvan,52 anonymous edits

Vibrating sample magnetometer  Source: http://en.wikipedia.org/w/index.php?oldid=472689537  Contributors: Dchristle, La Pianista, MagnInd, Qwyrxian, R0oland, RHaworth, Rcsprinter123,WvEngen, 19 anonymous edits

Image Sources, Licenses and Contributors 215

Image Sources, Licenses and ContributorsImage:Antiferromagnetic ordering.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Antiferromagnetic_ordering.svg  License: Creative Commons Attribution-ShareAlike 3.0Unported  Contributors: Michael SchmidFile:Loudspeaker.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Loudspeaker.svg  License: Public Domain  Contributors: Bayo, Gmaxwell, Husky, Iamunknown, Mirithing,Myself488, Nethac DIU, Omegatron, Rocket000, The Evil IP address, Wouterhagens, 20 anonymous editsFile:Vortex filament (Biot-Savart law illustration).png  Source: http://en.wikipedia.org/w/index.php?title=File:Vortex_filament_(Biot-Savart_law_illustration).png  License: Public Domain Contributors: mythImage:B-H loop.png  Source: http://en.wikipedia.org/w/index.php?title=File:B-H_loop.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Alno, Chaiken,Inductiveload, Loxosceles Laeta, Omegatron, Pieter Kuiper, Sautoir, WikipediaMaster, Zureks, 3 anonymous editsImage:Coercivity.png  Source: http://en.wikipedia.org/w/index.php?title=File:Coercivity.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: User:ChaikenImage:Diamagnetic graphite levitation.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Diamagnetic_graphite_levitation.jpg  License: Public domain  Contributors: en:User:SplarkaImage:Superconductor.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:Superconductor.GIF  License: Public Domain  Contributors: David Meeker wrote FEMM 4.2Image:Frog diamagnetic levitation.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Frog_diamagnetic_levitation.jpg  License: GNU Free Documentation License  Contributors:Lijnis NelemansFile:Simple electromagnet2.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Simple_electromagnet2.gif  License: Public Domain  Contributors: Simple_electromagnet.gif: Originaluploader was Berserkerus at ru.wikipedia derivative work: Chetvorno (talk) Alterations to source image: Rotated CCW 90° and lightened to bring out detail.File:Electromagnetism.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Electromagnetism.svg  License: GNU Free Documentation License  Contributors: User:StanneredFile:VFPt Solenoid correct.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VFPt_Solenoid_correct.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Geek3Image:Fermilab - 400 MeV Lambertson Magnet.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Fermilab_-_400_MeV_Lambertson_Magnet.jpg  License: Public Domain Contributors: Avron, Bomazi, NH2501, TeslatonImage:Laboratory electromagnet.png  Source: http://en.wikipedia.org/w/index.php?title=File:Laboratory_electromagnet.png  License: Public Domain  Contributors: John Ambrose FlemingImage:ICP-SFMS Magnet 1.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:ICP-SFMS_Magnet_1.JPG  License: Creative Commons Attribution-Sharealike 3.0  Contributors:SuperchilumImage:Stator eines Universalmotor.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Stator_eines_Universalmotor.JPG  License: Creative Commons Attribution-Sharealike 3.0 Contributors: MarrrciImage:DoorBell 001.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:DoorBell_001.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: HNHFile:Sturgeon electromagnet.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sturgeon_electromagnet.png  License: Public Domain  Contributors: William SturgeonFile:Industrial lifting magnet.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Industrial_lifting_magnet.jpg  License: Public Domain  Contributors: Cyril Methodius JanskyImage:Electromagnet with gap.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Electromagnet_with_gap.svg  License: Public Domain  Contributors: ChetvornoFile:Lifting electromagnet cross section.png  Source: http://en.wikipedia.org/w/index.php?title=File:Lifting_electromagnet_cross_section.png  License: Public Domain  Contributors: CyrilMethodius JanskyImage:Current_carrying_busbars_at_the_LNCMI.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Current_carrying_busbars_at_the_LNCMI.jpg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: Nerd bzhImage:Small small IMG 0836.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Small_small_IMG_0836.jpg  License: GNU Free Documentation License  Contributors: Avron,KasugaHuang, NYCRuss, Pieter Kuiper, Superm401Image:Ferrimagnetic ordering.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Ferrimagnetic_ordering.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: Michael SchmidImage:Ferrimagnetism - magnetic moment as a function of temperature.svg  Source:http://en.wikipedia.org/w/index.php?title=File:Ferrimagnetism_-_magnetic_moment_as_a_function_of_temperature.svg  License: Public Domain  Contributors: Petteri AimonenImage:MagnetEZ.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:MagnetEZ.jpg  License: Creative Commons Attribution-Sharealike 2.0  Contributors: Eurico Zimbres FGEL/UERJImage:Weiss-Bezirke1.png  Source: http://en.wikipedia.org/w/index.php?title=File:Weiss-Bezirke1.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Ra'ike(see also: de:Benutzer:Ra'ike)File:Lodestone attracting nails.png  Source: http://en.wikipedia.org/w/index.php?title=File:Lodestone_attracting_nails.png  License: Public Domain  Contributors: Fred Anzley AnnetFile:Malapterurus electricus 1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Malapterurus_electricus_1.jpg  License: Attribution-ShareAlike 3.0 Unported  Contributors:User:Stan ShebsFile:Shen Kua.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Shen_Kua.JPG  License: Creative Commons Attribution-Sharealike 2.5  Contributors: Original uploader wasWikimachine at en.wikipediaFile:Robert Boyle 0001.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Robert_Boyle_0001.jpg  License: Public Domain  Contributors: Johann KerseboomFile:Hauksbee Generator.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Hauksbee_Generator.JPG  License: Public Domain  Contributors: Aushulz, Enomil, Gerben49, VericaAtrebatumFile:Pieter van Musschenbroek.jpeg  Source: http://en.wikipedia.org/w/index.php?title=File:Pieter_van_Musschenbroek.jpeg  License: Public Domain  Contributors: User Dr. Manuel onde.wikipediaFile:Franklin-Benjamin-LOC.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Franklin-Benjamin-LOC.jpg  License: Public Domain  Contributors: Androstachys, Davepape, Editorat Large, Eubulides, Jengod, Jh12, Kilom691, Mschel, Nagy, Nonenmac, Raymond, Shizhao, 3 anonymous editsFile:Bcoulomb.png  Source: http://en.wikipedia.org/w/index.php?title=File:Bcoulomb.png  License: Public Domain  Contributors: Chanchocan, Lmbuga, Mutter Erde, N.borisenkov, PieterKuiper, Sertion, WikipediaMaster, 1 anonymous editsFile:Volta A.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Volta_A.jpg  License: Public Domain  Contributors: Garavaglia, Giovita , 1790 - 1835File:Ørsted.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Ørsted.jpg  License: Public Domain  Contributors: Aastrup, Anne-Sophie Ofrim, JdH, Joonasl, Smeira, Thue, 1anonymous editsFile:Ohm3.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Ohm3.gif  License: Public Domain  Contributors: ABF, Grayshi, Magog the Ogre, Shizhao, Spiderwoman, Texnik, 1anonymous editsFile:Jospeh Henry (1879).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Jospeh_Henry_(1879).jpg  License: unknown  Contributors: Henry Ulke (1821-1910)File:Faraday-Millikan-Gale-1913.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Faraday-Millikan-Gale-1913.jpg  License: Public Domain  Contributors: Probably albumencarte-de-visite by John WatkinsFile:Lord Kelvin photograph.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Lord_Kelvin_photograph.jpg  License: Public Domain  Contributors: Fastfission, Kalki, LobStoR,Pieter Kuiper, QuibikFile:James Clerk Maxwell.png  Source: http://en.wikipedia.org/w/index.php?title=File:James_Clerk_Maxwell.png  License: Public Domain  Contributors: G. J. StodartFile:N.Tesla.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:N.Tesla.JPG  License: Public Domain  Contributors: Blacklake, CLI, Choihei, DIREKTOR, Emerson7, GreenStork,Kilom691, Nikola Smolenski, PRODUCER, Rainman, Veliki Kategorizator, 3 anonymous editsFile:Crookes William.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Crookes_William.jpg  License: Public Domain  Contributors: Henry Smith WilliamsFile:Oliver Heaviside2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Oliver_Heaviside2.jpg  License: Public Domain  Contributors: Original uploader was SuperGirl aten.wikipediaFile:Jj-thomson3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Jj-thomson3.jpg  License: Public Domain  Contributors: Original uploader was Bletchley at en.wikipediaFile:WorldsFairTeslaPresentation.png  Source: http://en.wikipedia.org/w/index.php?title=File:WorldsFairTeslaPresentation.png  License: Public Domain  Contributors: Original uploader wasReddi at en.wikipedia

Image Sources, Licenses and Contributors 216

File:Thomas Edison.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Thomas_Edison.jpg  License: Public Domain  Contributors: Amano1, Jmabel, John Vandenberg, Makthorpe,Martin H., Shizhao, TheCuriousGnome, Tony Wills, VizuFile:Charlesproteussteinmetz.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Charlesproteussteinmetz.jpg  License: Public Domain  Contributors: Orgullomoore, PieterJanRFile:Hendrik Antoon Lorentz.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Hendrik_Antoon_Lorentz.jpg  License: Public Domain  Contributors: The website of the RoyalLibrary shows a picture from the same photosession that is attributed to Museum Boerhaave. The website of the Museum states "vrij beschikbaar voor publicatie" (freely available forpublication).File:JH Poincare.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:JH_Poincare.jpg  License: Public Domain  Contributors: Dabomb87, Mdd, ХацкерFile:Einstein patentoffice.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Einstein_patentoffice.jpg  License: Public Domain  Contributors: Lucien ChavanUNIQ-ref-2-4f5ac45699a2e89b-QINU (1868 - 1942), a friend of Einstein's when he was living in Berne.File:Dirac_4.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Dirac_4.jpg  License: unknown  Contributors: Nobel FoundationFile:Feynman at Los Alamos.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Feynman_at_Los_Alamos.jpg  License: Public Domain  Contributors:Feynman_and_Oppenheimer_at_Los_Alamos.jpg: unknown derivative work: Materialscientist (talk)File:Lorentz force.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Lorentz_force.svg  License: GNU Free Documentation License  Contributors: User:Jaro.pFile:Cyclotron motion.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Cyclotron_motion.jpg  License: Creative Commons Attribution-Share Alike  Contributors: Marcin BiałekFile:charged-particle-drifts.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Charged-particle-drifts.svg  License: Creative Commons Attribution 2.5  Contributors: User:StanneredFile:Regla mano derecha Laplace.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Regla_mano_derecha_Laplace.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Jfmelerofile:MagnetEZ.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:MagnetEZ.jpg  License: Creative Commons Attribution-Sharealike 2.0  Contributors: Eurico Zimbres FGEL/UERJfile:Magnet0873.png  Source: http://en.wikipedia.org/w/index.php?title=File:Magnet0873.png  License: Public Domain  Contributors: Newton Henry Blackfile:Solenoid Rotated.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Solenoid_Rotated.svg  License: Public Domain  Contributors: Nuno Nogueira. Original uploader was Andoneeat en.wikipediafile:The Effects of Magnetism.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:The_Effects_of_Magnetism.JPG  License: unknown  Contributors: “Jon Zander (Digon3)"file:VFPt cylindrical magnet thumb.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VFPt_cylindrical_magnet_thumb.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Geek3file:Hard disk.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Hard_disk.jpg  License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0  Contributors: Inkleinfile:Magnetic separator hg.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_separator_hg.jpg  License: Creative Commons Attribution-Sharealike 2.5  Contributors: HannesGrobe 19:04, 3 September 2006 (UTC)file:M tic.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:M_tic.jpg  License: unknown  Contributors: ARTE, Fir0002, INVERTED, Pieter Kuiper, Solbrisfile:Ceramic magnets.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Ceramic_magnets.jpg  License: GNU Free Documentation License  Contributors: Omegatron, Pieter KuiperFile:Magneticbearings.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Magneticbearings.jpg  License: Public Domain  Contributors: NASAFile:amb2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Amb2.svg  License: Public Domain  Contributors: HalpaughFile:Magnetic Mirroring.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_Mirroring.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: JobanWikiFile:Magnetischer Kreis.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetischer_Kreis.svg  License: Creative Commons Attribution-Sharealike 2.0  Contributors: — MovGP0.Original uploader was MovGP0 at de.wikipediaImage:Magnetic field due to dipole moment.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_field_due_to_dipole_moment.svg  License: Public Domain  Contributors:User:MaschenImage:Magnetic field due to current.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_field_due_to_current.svg  License: Public Domain  Contributors: User:MaschenImage:VFPt dipole electric.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VFPt_dipole_electric.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Geek3Image:VFPt dipole magnetic3.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VFPt_dipole_magnetic3.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:Geek3File:NdFeB-Domains.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:NdFeB-Domains.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: GorchyFile:Powstawanie domen by Zureks.png  Source: http://en.wikipedia.org/w/index.php?title=File:Powstawanie_domen_by_Zureks.png  License: Public Domain  Contributors: ZureksFile:Weiss-Bezirke1.png  Source: http://en.wikipedia.org/w/index.php?title=File:Weiss-Bezirke1.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Ra'ike(see also: de:Benutzer:Ra'ike)Image:Magnetostriction by Zureks.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetostriction_by_Zureks.gif  License: Public Domain  Contributors: ZureksFile:Moving magnetic domains by Zureks.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Moving_magnetic_domains_by_Zureks.gif  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: Zureks, Chris VardonFile:Dominios.png  Source: http://en.wikipedia.org/w/index.php?title=File:Dominios.png  License: GNU Free Documentation License  Contributors: Original uploader was 4lex at es.wikipediaImage:CMOS Domänen Formgedächtnislegierung.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:CMOS_Domänen_Formgedächtnislegierung.jpg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: User:MatesyImage:CMOS Mäanderdomänen.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:CMOS_Mäanderdomänen.jpg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:MatesyImage:CMOS magnetische Blasendomänen.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:CMOS_magnetische_Blasendomänen.jpg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: User:MatesyFile:VFPt cylindrical magnet thumb.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VFPt_cylindrical_magnet_thumb.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Geek3Image:Descartes magnetic field.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Descartes_magnetic_field.jpg  License: Public Domain  Contributors: René DescartesImage:Magnetic field near pole.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_field_near_pole.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:TSteinImage:Magnet0873.png  Source: http://en.wikipedia.org/w/index.php?title=File:Magnet0873.png  License: Public Domain  Contributors: Newton Henry Blackimage:Dipole in uniform H field.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Dipole_in_uniform_H_field.svg  License: Creative Commons Zero  Contributors: User:Fred theOysterImage:cross parallelogram.png  Source: http://en.wikipedia.org/w/index.php?title=File:Cross_parallelogram.png  License: Public Domain  Contributors: Oleg AlexandrovImage:Manoderecha.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Manoderecha.svg  License: GNU Free Documentation License  Contributors: JfmeleroImage:Solenoid-1 (vertical).png  Source: http://en.wikipedia.org/w/index.php?title=File:Solenoid-1_(vertical).png  License: Public Domain  Contributors: ZureksImage:Regla mano derecha Laplace.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Regla_mano_derecha_Laplace.svg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: JfmeleroFile:BIsAPseudovector.svg  Source: http://en.wikipedia.org/w/index.php?title=File:BIsAPseudovector.svg  License: Public Domain  Contributors: Sbyrnes321Image:Earths Magnetic Field Confusion.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Earths_Magnetic_Field_Confusion.svg  License: Creative Commons Attribution-Sharealike3.0  Contributors: TSteinImage:Magnetic quadrupole moment.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_quadrupole_moment.svg  License: Public Domain  Contributors: Original uploaderwas K. Aainsqatsi at en.wikipediaFile:CuttingABarMagnet.svg  Source: http://en.wikipedia.org/w/index.php?title=File:CuttingABarMagnet.svg  License: Creative Commons Zero  Contributors: User:Sbyrnes321Image:Magnetocaloric effect1 04a.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetocaloric_effect1_04a.svg  License: unknown  Contributors: Mtodorov 69 (original);Pbroks13 Original uploader was Pbroks13 at en.wikipediaImage:MCE.gif  Source: http://en.wikipedia.org/w/index.php?title=File:MCE.gif  License: Public Domain  Contributors: Yurij Mozharivskyj. Original uploader was Grandonia at en.wikipedia

Image Sources, Licenses and Contributors 217

Image:Magnetic Stirrer.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_Stirrer.JPG  License: GNU Free Documentation License  Contributors: User:RuhrfischImage:Magnetic stirring bars aligned.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetic_stirring_bars_aligned.jpg  License: Public Domain  Contributors: MasurImage:Ferromagnetic ordering.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Ferromagnetic_ordering.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: Michael SchmidImage:Antiferro2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Antiferro2.jpg  License: Creative Commons Attribution-Share Alike  Contributors: NlJcwfFile:M Faraday Th Phillips oil 1842.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:M_Faraday_Th_Phillips_oil_1842.jpg  License: Public Domain  Contributors: Thomas PhillipsFile:Magnetism.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Magnetism.svg  License: Public Domain  Contributors: User:John AplessedFile:Ferromagneses penzermek 1.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Ferromagneses_penzermek_1.jpg  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Zátonyi Sándor, (ifj.) FizpedImage:Ferromag Matl Sketch.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Ferromag_Matl_Sketch.JPG  License: Creative Commons Attribution 2.5  Contributors: Originaluploader was JA.Davidson at en.wikipediaImage:Ferromag Matl Magnetized.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Ferromag_Matl_Magnetized.JPG  License: Creative Commons Attribution 2.5  Contributors:Original uploader was JA.Davidson at en.wikipediaFile:Electromagnet.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Electromagnet.gif  License: unknown  Contributors: AnynobodyFile:Neodymag.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Neodymag.jpg  License: Public domain  Contributors: Bloodshedder at en.wikipediaFile:Nd-magnet.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Nd-magnet.jpg  License: Creative Commons Attribution  Contributors: unknownFile:Neodymium Crystal Structure Nd2Fe14B.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Neodymium_Crystal_Structure_Nd2Fe14B.jpg  License: Creative Commons Zero Contributors: UAwikifile:2 Ferrite ring_magnets.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:2_Ferrite_ring_magnets.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:Magnequenchfile:Hdd od srodka.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Hdd_od_srodka.jpg  License: Public Domain  Contributors: EugeneZelenko, Qurren, TOR, Tothwolf, Yann, 3anonymous editsImage:Paramagnetic probe without magnetic field.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Paramagnetic_probe_without_magnetic_field.svg  License: Public Domain Contributors: Jens Böning (Jensel)File:Paramagnetism of liquid oxygen.jpeg  Source: http://en.wikipedia.org/w/index.php?title=File:Paramagnetism_of_liquid_oxygen.jpeg  License: Public Domain  Contributors: Pieter KuiperImage:Para-ferro-anti.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Para-ferro-anti.jpg  License: GNU Free Documentation License  Contributors: NlJcwfFile:Ferrofluid Magnet under glass edit.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Ferrofluid_Magnet_under_glass_edit.jpg  License: GNU Free Documentation License Contributors: Gregory F. Maxwell < gmaxwell@gmail.com>File:Neodymium_magnet_-_19-11-2010.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Neodymium_magnet_-_19-11-2010.JPG  License: Public domain  Contributors: Tremasterat en.wikipediaFile:Hard disk.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Hard_disk.jpg  License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0  Contributors: InkleinFile:Ferritin.png  Source: http://en.wikipedia.org/w/index.php?title=File:Ferritin.png  License: GNU General Public License  Contributors: Simonxag, Torsch, VossmanFile:Spin glass by Zureks.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Spin_glass_by_Zureks.svg  License: Creative Commons Zero  Contributors: ZureksImage:SiO2_-_Glas_-_2D.png  Source: http://en.wikipedia.org/w/index.php?title=File:SiO2_-_Glas_-_2D.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors:User:127.0.0.lImage:SiO2_-_Quarz_-_2D.png  Source: http://en.wikipedia.org/w/index.php?title=File:SiO2_-_Quarz_-_2D.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: 127.0.0.l, Matthias M.Image:Precession2.png  Source: http://en.wikipedia.org/w/index.php?title=File:Precession2.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Chaiken,Newone, Pieter KuiperImage:Precessionplot.png  Source: http://en.wikipedia.org/w/index.php?title=File:Precessionplot.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Chaiken,Pieter KuiperImage:Langevin function.png  Source: http://en.wikipedia.org/w/index.php?title=File:Langevin_function.png  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Ddcampayoat en.wikipediaImage:VSM_en.svg  Source: http://en.wikipedia.org/w/index.php?title=File:VSM_en.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: R0oland

License 218

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/