The most familiar effects occur inferromagnetic materials, which are strongly attracted by magnetic fields and can bemagnetized to become permanentmagnets, producing magnetic fields themselves. Demagnetizing a magnet is also possible. Only a few substances are ferromagnetic; the most common ones areiron,cobalt,nickel, and their alloys.
All substances exhibit some type of magnetism. Magnetic materials are classified according to their bulk susceptibility.[1] Ferromagnetism is responsible for most of the effects of magnetism encountered in everyday life, but there are actually several types of magnetism.Paramagnetic substances, such asaluminium andoxygen, are weakly attracted to an applied magnetic field;diamagnetic substances, such ascopper andcarbon, are weakly repelled; whileantiferromagnetic materials, such aschromium, have a more complex relationship with a magnetic field.[vague] The force of a magnet on paramagnetic, diamagnetic, and antiferromagnetic materials is usually too weak to be felt and can be detected only by laboratory instruments, so in everyday life, these substances are often described as non-magnetic.
The strength of amagnetic field always decreases with distance from the magnetic source,[2] though the exact mathematical relationship between strength and distance varies. Many factors can influence the magnetic field of an object including the magnetic moment of the material, the physical shape of the object, both the magnitude and direction of any electric current present within the object, and the temperature of the object.
Lodestone, a naturalmagnet, attracting iron nails. Ancient humans discovered the property of magnetism from lodestone.An illustration from Gilbert's 1600De Magnete showing one of the earliest methods of making a magnet. A blacksmith holds a piece of red-hot iron in a north–south direction and hammers it as it cools. The magnetic field of the Earth aligns the domains, leaving the iron a weak magnet.Drawing of a medical treatment using magnetic brushes.Charles Jacque 1843, France.
Magnetism was first discovered in the ancient world when people noticed thatlodestones, naturally magnetized pieces of the mineralmagnetite, could attract iron.[3] The wordmagnet comes from theGreek term μαγνῆτις λίθοςmagnētis lithos,[4] "the Magnesian stone, lodestone".[5] In ancient Greece,Aristotle attributed the first of what could be called a scientific discussion of magnetism to the philosopherThales ofMiletus, who lived from about 625 BCE to about 545 BCE.[6] Theancient Indian medical textSushruta Samhita describes using magnetite to remove arrows embedded in a person's body.[7]
Inancient China, the earliest literary reference to magnetism lies in a 4th-century BCE book named after its author,Guiguzi.[8]The 2nd-century BCE annals,Lüshi Chunqiu, also notes:"Thelodestone makes iron approach; some (force) is attracting it."[9] The earliest mention of the attraction of a needle is in a 1st-century workLunheng (Balanced Inquiries): "A lodestone attracts a needle."[10] The 11th-centuryChinese scientistShen Kuo was the first person to write—in theDream Pool Essays—of the magnetic needle compass and that it improved the accuracy of navigation by employing theastronomical concept oftrue north.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 spoon always pointed south.
Leonardo Garzoni's only extant work, theDue trattati sopra la natura, e le qualità della calamita (Two treatises on the nature and qualities of the magnet), is the first known example of a modern treatment of magnetic phenomena. Written in years near 1580 and never published, the treatise had a wide diffusion. In particular, Garzoni is referred to as an expert in magnetism by Niccolò Cabeo, whose Philosophia Magnetica (1629) is just a re-adjustment of Garzoni's work. Garzoni's treatise was known also toGiovanni Battista Della Porta.
In 1600,William Gilbert published hisDe Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure (On the Magnet and Magnetic Bodies, and on the Great Magnet the Earth). In this work he describes many of his experiments with his model earth called theterrella. From his experiments, he concluded that theEarth was itself magnetic and that this was the reason compasses pointed north whereas, previously, some believed that it was the pole starPolaris or a large magnetic island on the north pole that attracted the compass.
An understanding of the relationship betweenelectricity and magnetism began in 1819 with work byHans Christian Ørsted, a professor at the University of Copenhagen, who discovered, by the accidental twitching of a compass needle near a wire, that an electric current could create a magnetic field. This landmark experiment is known as Ørsted's Experiment.Jean-Baptiste Biot andFélix Savart, both of whom in 1820 came up with theBiot–Savart law giving an equation for the magnetic field from a current-carrying wire. Around the same time,André-Marie Ampère carried out numerous systematic experiments and discovered that the magnetic force between two DC current loops of any shape is equal to the sum of the individual forces that each current element of one circuit exerts on each other current element of the other circuit.
From around 1861,James Clerk Maxwell synthesized and expanded many of these insights intoMaxwell's equations, unifying electricity, magnetism, andoptics into the field ofelectromagnetism. However, Gauss's interpretation of magnetism is not fully compatible with Maxwell's electrodynamics. In 1905,Albert Einstein used Maxwell's equations in motivating his theory ofspecial relativity,[13] requiring that the laws held true in allinertial reference frames. Gauss's approach of interpreting the magnetic force as a mere effect of relative velocities thus found its way back into electrodynamics to some extent.
The magnetic properties of materials are mainly due to the magnetic moments of theiratoms' orbitingelectrons. The magnetic moments of the nuclei of atoms are typically thousands of times smaller than the electrons' magnetic moments, so they are negligible in the context of the magnetization of materials. Nuclear magnetic moments are nevertheless very important in other contexts, particularly innuclear magnetic resonance (NMR) andmagnetic resonance 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 thePauli exclusion principle (seeelectron configuration), and combining into filledsubshells with zero net orbital motion. In both cases, the electrons preferentially adopt arrangements in which the magnetic moment of each electron is canceled by the opposite moment of another electron. Moreover, even when theelectron configurationis 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 magnetic moments that point in different, random directions so that the material will not be magnetic.
Sometimes—either spontaneously, or owing to an applied external magnetic field—each of the electron magnetic moments will be, on average, lined up. A suitable material can then produce a strong net magnetic field.
The magnetic behavior of a material depends on its structure, particularly itselectron configuration, for the reasons mentioned above, and also on the temperature. At high temperatures, randomthermal motion makes it more difficult for the electrons to maintain alignment.
Diamagnetism appears in all materials and is the tendency of a material to oppose an applied magnetic field, and therefore, to be repelled by a magnetic field. However, in a material with paramagnetic properties (that is, with a tendency to enhance an external magnetic field), the paramagnetic behavior dominates.[15] Thus, despite its universal occurrence, diamagnetic behavior is observed only in a purely diamagnetic material. In a diamagnetic material, there are no unpaired electrons, so the intrinsic electron magnetic moments cannot produce any bulk effect. In these cases, the magnetization arises from the electrons' orbital motions, which can be understoodclassically as follows:
When a material is put in a magnetic field, the electrons circling the nucleus will experience, in addition to theirCoulomb attraction to the nucleus, aLorentz force from the magnetic field. Depending on which direction the electron is orbiting, this force may increase thecentripetal force on the electrons, pulling them in towards the nucleus, or it may decrease the force, pulling them away from the nucleus. This effect systematically increases the orbital magnetic moments that were aligned opposite the field and decreases the ones aligned parallel to the field (in accordance withLenz's law). This results in a small bulk magnetic moment, with an opposite direction to the applied field.
This description is meant only as aheuristic; theBohr–Van Leeuwen theorem shows that diamagnetism is impossible according to classical physics, and that a proper understanding requires aquantum-mechanical description.
All materials undergo this orbital response. However, in paramagnetic and ferromagnetic substances, the diamagnetic effect is overwhelmed by the much stronger effects caused by the unpaired electrons.
In a paramagnetic material there are unpaired electrons; i.e.,atomic ormolecular orbitals with exactly one electron in them. While paired electrons are required by thePauli exclusion principle to have their intrinsic ('spin') magnetic moments pointing in opposite directions, causing their magnetic fields to cancel out, an unpaired electron is free to align its magnetic moment in any direction. When an external magnetic field is applied, these magnetic moments will tend to align themselves in the same direction as the applied field, thus reinforcing it.
A ferromagnet, like a paramagnetic substance, has unpaired electrons. However, in addition to the electrons' intrinsic magnetic moment's tendency to be parallel to an applied field, there is also in these materials a tendency for these magnetic moments to orient parallel to each other to maintain a lowered-energy state. Thus, even in the absence of an applied field, the magnetic moments of the electrons in the material spontaneously line up parallel to one another.
Every ferromagnetic substance has its own individual temperature, called theCurie temperature, or Curie point, above which it loses its ferromagnetic properties. This is because the thermal tendency to disorder overwhelms the energy-lowering due to ferromagnetic order.
Ferromagnetism only occurs in a few substances; common ones areiron,nickel,cobalt, theiralloys, and some alloys ofrare-earth metals.
Magnetic domains boundaries (white lines) in ferromagnetic material (black rectangle)
Effect of a magnet on the domains
The magnetic moments of atoms in aferromagnetic material cause them to behave something like tiny permanent magnets. They stick together and align themselves into small regions of more or less uniform alignment calledmagnetic domains orWeiss domains. Magnetic domains can be observed with amagnetic force microscope to reveal magnetic domain boundaries that resemble white lines in the sketch. There are many scientific experiments that can physically show magnetic fields.
When a domain contains too many molecules, it becomes unstable and divides into two domains aligned in opposite directions so that they stick together more stably.
When exposed to a magnetic field, the domain boundaries move, so that the domains aligned with the magnetic field grow and dominate the structure (dotted yellow area), as shown at the left. When the magnetizing field is removed, the domains may not return to an unmagnetized state. This results in the ferromagnetic material's being magnetized, forming a permanent magnet.
When magnetized strongly enough that the prevailing domain overruns all others to result in only one single domain, the material ismagnetically saturated. When a magnetized ferromagnetic material is heated to theCurie point temperature, the molecules are 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 analogous to how a liquid canfreeze into a crystalline solid.
In anantiferromagnet, unlike a ferromagnet, there is a tendency for the intrinsic magnetic moments of neighboring valence electrons to point inopposite directions. When all atoms are arranged in a substance so that each neighbor is anti-parallel, the substance isantiferromagnetic. Antiferromagnets have a zero net magnetic moment because adjacent opposite moment cancels out, meaning that no field is produced by them. Antiferromagnets are less common compared to the other types of behaviors and are mostly observed at low temperatures. In varying temperatures, antiferromagnets can be seen to exhibit diamagnetic and ferromagnetic properties.
In some materials, neighboring electrons prefer to point in opposite directions, but there is no geometrical arrangement in whicheach pair of neighbors is anti-aligned. This is called acanted antiferromagnet orspin ice and is an example ofgeometrical frustration.
Like ferromagnetism,ferrimagnets retain their magnetization in the absence of a field. However, like antiferromagnets, neighboring pairs of electron spins tend to point in opposite directions. These two properties are not contradictory, because in the optimal geometrical arrangement, there is more magnetic moment from the sublattice of electrons that point in one direction, than from the sublattice that points in the opposite direction.
Mostferrites are ferrimagnetic. The first discovered magnetic substance,magnetite, is a ferrite and was originally believed to be a ferromagnet;Louis Néel disproved this, however, after discovering ferrimagnetism.
When 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 much larger.
Japanese physicist Yosuke Nagaoka conceived of a type of magnetism in a square, two-dimensional lattice where every lattice node had one electron. If one electron was removed under specific conditions, the lattice's energy would be minimal only when all electrons' spins were parallel.
A variation on this was achieved experimentally by arranging the atoms in a triangularmoiré lattice ofmolybdenum diselenide andtungsten disulfide monolayers. Applying a weak magnetic field and a voltage led to ferromagnetic behavior when 100–150% more electrons than lattice nodes were present. The extra electrons delocalized and paired with lattice electrons to form doublons. Delocalization was prevented unless the lattice electrons had aligned spins. The doublons thus created localized ferromagnetic regions. The phenomenon took place at 140 millikelvins.[16]
An electromagnet attracts paper clips when current is applied, creating a magnetic field. The electromagnet loses them when current and magnetic field are removed.
Anelectromagnet is a type ofmagnet in which themagnetic field is produced by anelectric current.[17] The magnetic field disappears when the current is turned off. Electromagnets usually consist of a large number of closely spaced turns of wire that create the magnetic field. The wire turns are often wound around amagnetic core made from aferromagnetic orferrimagnetic material such asiron; the magnetic core concentrates themagnetic flux and makes a more powerful magnet.
The main advantage of an electromagnet over apermanent magnet is that the magnetic field can be quickly changed by controlling the amount of electric current in the winding. However, unlike a permanent magnet that needs no power, an electromagnet requires a continuous supply of current to maintain the magnetic field.
Electromagnets are widely used as components of other electrical devices, such asmotors,generators,relays, solenoids,loudspeakers,hard disks,MRI machines, scientific instruments, andmagnetic separation equipment. Electromagnets are also employed in industry for picking up and moving heavy iron objects such as scrap iron and steel.[18] Electromagnetism was discovered in 1820.[19]
As a consequence of Einstein's theory ofspecial relativity, electricity and magnetism are fundamentally interlinked. Both magnetism lacking electricity, and electricity without magnetism, are inconsistent with special relativity, due to such effects aslength contraction,time dilation, and the fact that themagnetic force is velocity-dependent. However, when both electricity and magnetism are taken into account, the resulting theory (electromagnetism) is fully consistent with special relativity.[13][20] In particular, a phenomenon that appears purely electric or purely magnetic to one observer may be a mix of both to another, or more generally the relative contributions of electricity and magnetism are dependent on the frame of reference. Thus, special relativity "mixes" electricity and magnetism into a single, inseparable phenomenon calledelectromagnetism, analogous to how general relativity "mixes" space and time intospacetime.
All observations onelectromagnetism apply to what might be considered to be primarily magnetism, e.g. perturbations in the magnetic field are necessarily accompanied by a nonzero electric field, and propagate at thespeed of light.[21]
The phenomenon of magnetism is "mediated" by the magnetic field. An electric current or magnetic dipole creates a magnetic field, and that field, in turn, imparts magnetic forces on other particles that are in the fields.
Maxwell's equations, which simplify to theBiot–Savart law in the case of steady currents, describe the origin and behavior of the fields that govern these forces. Therefore, magnetism is seen whenever electricallycharged particles are inmotion—for example, from movement of electrons in anelectric current, or in certain cases from the orbital motion of electrons around an atom's nucleus. They also arise from "intrinsic"magnetic dipoles arising from quantum-mechanicalspin.
The same situations that create magnetic fields—charge moving in a current or in an atom, and intrinsic magnetic dipoles—are also the situations in which a magnetic field has an effect, creating a force. Following is the formula for moving charge; for the forces on an intrinsic dipole, see magnetic dipole.
Because this is across product, the force isperpendicular to both the motion of the particle and the magnetic field. The magnitude of the force is
where is the angle betweenv andB.
One tool for determining the direction of the velocity vector of a moving charge, the magnetic field, and the force exerted is labeling theindex finger "V"[dubious –discuss], themiddle finger "B", and thethumb "F" with your right hand. When making a gun-like configuration, with the middle finger crossing under the index finger, the fingers represent the velocity vector, magnetic field vector, and force vector, respectively. See alsoright-hand rule.
A very common source of magnetic field found in nature is adipole, with a "South pole" and a "North pole", terms dating back to the use of magnets as compasses, interacting with theEarth's magnetic field to indicate North and South on theglobe. Since opposite ends of magnets are attracted, the north pole of a magnet is attracted to the south pole of another magnet. The Earth'sNorth Magnetic Pole (currently in the Arctic Ocean, north of Canada) is physically a south pole, as it attracts the north pole of a compass.A magnetic field containsenergy, and physical systems move toward configurations with lower energy. When diamagnetic material is placed in a magnetic field, amagnetic dipole tends to align itself in opposed polarity to that field, thereby lowering the net field strength. When ferromagnetic material is placed within a magnetic field, the magnetic dipoles align to the applied field, thus expanding the domain walls of the magnetic domains.
Since a bar magnet gets its ferromagnetism from electrons distributed evenly throughout the bar, when a bar magnet is cut in half, each of the resulting pieces is a smaller bar magnet. Even though a magnet is said to have a north pole and 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, not attached to a south pole, or vice versa. Monopoles would carry "magnetic charge" analogous to electric charge. Despite systematic searches since 1931, as of 2010[update], they have never been observed, and could very well not exist.[23]
Nevertheless, sometheoretical physics models predict the existence of thesemagnetic monopoles.Paul Dirac observed in 1931 that, because electricity and magnetism show a certainsymmetry, just asquantum theory predicts that individualpositive ornegative electric charges can be observed without the opposing charge, isolated South or North 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 observedelementary particles carry charges that are multiples of the charge of the electron.
Certaingrand unified theories predict the existence of monopoles which, unlike elementary particles, aresolitons (localized energy packets). The initial results of using these models to estimate the number of monopoles created in theBig Bang contradicted cosmological observations—the monopoles would have been so plentiful and massive that they would have long since halted the expansion of the universe. However, the idea ofinflation (for which this problem served as a partial motivation) was successful in solving this problem, creating models in which monopoles existed but were rare enough to be consistent with current observations.[24]
A live frog levitates inside a 32mmdiameter vertical bore of aBitter solenoid in a very strong magnetic field—about 16teslas
Someorganisms can detect magnetic fields, a phenomenon known asmagnetoception. Some materials in living things are ferromagnetic, though it is unclear if the magnetic properties serve a special function or are merely a byproduct of containing iron. For instance,chitons, a type of marine mollusk, produce magnetite to harden their teeth, and even humans producemagnetite in bodily tissue.[26]
Magnetobiology studies the effects of magnetic fields on living organisms; fields naturally produced by an organism are known asbiomagnetism. Many biological organisms are mostly made of water, and because water isdiamagnetic, extremely strong magnetic fields can repel these living things.
Interpretation of magnetism by means of relative velocities
In the years after 1820,André-Marie Ampère carried out numerous experiments in which he measured the forces between direct currents. In particular, he also studied the magnetic forces between non-parallel wires.[27] The final result of his work was a force law that is now named after him. In 1835,Carl Friedrich Gauss realized[12] thatAmpere's force law in its original form can be explained by a generalization ofCoulomb's law.
Gauss's force law states that the electromagnetic force experienced by a point charge, with trajectory, in the vicinity of another point charge, with trajectory, in a vacuum is equal to thecentral force
,
where is the distance between the charges and is the relative velocity.Wilhelm Eduard Weber confirmed Gauss's hypothesis in numerous experiments.[28][29][30] By means ofWeber electrodynamics it is possible to explain the static and quasi-static effects in the non-relativistic regime of classical electrodynamics withoutmagnetic field andLorentz force.
Since 1870,Maxwell electrodynamics has been developed, which postulates that electric and magnetic fields exist. In Maxwell's electrodynamics, the actual electromagnetic force can be calculated using the Lorentz force, which, like the Weber force, is speed-dependent. However, Maxwell's electrodynamics is not fully compatible with the work of Ampère, Gauss and Weber in the quasi-static regime. In particular, Ampère's original force law and theBiot-Savart law are only equivalent if the field-generating conductor loop is closed.[31] Maxwell's electrodynamics therefore represents a break with the interpretation of magnetism by Gauss and Weber, since in Maxwell's electrodynamics it is no longer possible to deduce the magnetic force from a central force.
While heuristic explanations based on classical physics can be formulated, diamagnetism, paramagnetism and ferromagnetism can be fully explained only using quantum theory.[32][33]A successful model was developed already in 1927, byWalter Heitler andFritz London, who derived, quantum-mechanically, how hydrogen molecules are formed from hydrogen atoms, i.e. from the atomic hydrogen orbitals and centered at the nucleiA andB, see below. That this leads to magnetism is not at all obvious, but will be explained in the following.
According to the Heitler–London theory, so-called two-body molecular-orbitals are formed, namely the resulting orbital 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 the quantum-mechanical property that particles with identical properties cannot be distinguished. It is specific not only for the formation ofchemical bonds, but also for magnetism. That is, in this connection the termexchange interaction arises, a term which is essential for the origin of magnetism, and which is stronger, roughly by factors 100 and even by 1000, than the energies arising from the electrodynamic dipole-dipole interaction.
As for thespin function, which is responsible for the magnetism, we have the already mentioned Pauli's principle, namely that a symmetric orbital (i.e. with the + sign as above) must be multiplied with an antisymmetric spin function (i.e. with a − sign), andvice versa. Thus:
,
I.e., not only and must be substituted byα andβ, respectively (the first entity means "spin up", the second one "spin down"), but also the sign + by the − sign, and finallyri by the discrete valuessi (= ±1⁄2); thereby we have and. The "singlet state", i.e. the − sign, means: the spins areantiparallel, i.e. for the solid we haveantiferromagnetism, and for two-atomic molecules one hasdiamagnetism. The tendency to form a (homoeopolar) chemical bond (this means: the formation of asymmetric molecular orbital, i.e. with the + sign) results through the Pauli principle automatically in anantisymmetric spin state (i.e. with the − sign). In contrast, the Coulomb repulsion of the electrons, i.e. the tendency that they try to avoid each other by this repulsion, would lead to anantisymmetric orbital function (i.e. with the − sign) of these two particles, and complementary to asymmetric spin function (i.e. with the + sign, one of the so-called "triplet functions"). Thus, now the spins would beparallel (ferromagnetism in a solid,paramagnetism in two-atomic gases).
The last-mentioned tendency dominates in the metalsiron,cobalt andnickel, and in some rare earths, which areferromagnetic. Most of the other metals, where the first-mentioned tendency dominates, arenonmagnetic (e.g.sodium,aluminium, andmagnesium) orantiferromagnetic (e.g.manganese). Diatomic gases are also almost exclusively diamagnetic, and not paramagnetic. However, the oxygen molecule, because of the involvement of π-orbitals, is an exception important for the life-sciences.
The Heitler-London considerations can be generalized to theHeisenberg model of magnetism (Heisenberg 1928).
The explanation of the phenomena is thus essentially based on all subtleties of quantum mechanics, whereas the electrodynamics covers mainly the phenomenology.
^Li, Shu-hua (1954). "Origine de la Boussole II. Aimant et Boussole".Isis (in French).45 (2):175–196.doi:10.1086/348315.JSTOR227361.S2CID143585290.un passage dans leLiu-che-tch'ouen-ts'ieou [...]: "La pierre d'aimant fait venir le fer ou elle l'attire." From the section "Jingtong" (精通) of the "Almanac of the Last Autumn Month" (季秋紀): "慈石召鐵，或引之也]"
^In the section "A Last Word on Dragons" (亂龍篇Luanlong) of theLunheng: "Amber takes up straws, and a load-stone attracts needles" (頓牟掇芥，磁石引針).
^Schmidl, Petra G. (1996–1997). "Two Early Arabic Sources On The Magnetic Compass".Journal of Arabic and Islamic Studies.1:81–132.
^abGauss, Carl Friedrich (1867).Carl Friedrich Gauss Werke. Fünfter Band. Königliche Gesellschaft der Wissenschaften zu Göttingen. p. 617.
^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.S2CID119061150..
^Assis, A. K. T.; J. P. M. C. Chaib (2015).Ampère's electrodynamics: Analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience. C. Roy Keys Inc.ISBN978-1-987980-03-5.
^Wilhelm Weber (2021). Andre Koch Torres Assis (ed.).Wilhelm Weber's Main Works in Electrodynamics Translated into English. Volume I: Gauss und Weber's Absolute System of Units. Apeiron Montreal.
^Wilhelm Weber (2021). Andre Koch Torres Assis (ed.).Wilhelm Weber's Main Works in Electrodynamics Translated into English. Volume II: Weber's Fundamental Force and the Unification of the Laws of Coulomb, Ampere and Faraday. Apeiron Montreal.
^Wilhelm Weber (2021). Andre Koch Torres Assis (ed.).Wilhelm Weber's Main Works in Electrodynamics Translated into English. Volume III: Measurement of Weber's Constant c, Diamagnetism, the Telegraph Equation and the Propagation of Electric Waves at Light Velocity. Apeiron Montreal.
^Maxwell, James Clerk (1881).Treatise on Electricity and Magnetism. Volume 2. Vol. 2 (2 ed.). The Clarendon Press, Oxdord. p. 162.
Kronmüller, Helmut. (2007).Handbook of Magnetism and Advanced Magnetic Materials, 5 Volume Set. John Wiley & Sons.ISBN978-0-470-02217-7.OCLC124165851.
Purcell, Edward M. (2012).Electricity and magnetism (3rd ed.). Cambridge: Cambridge Univ. Press.ISBN9781-10701-4022.
Tipler, Paul (2004).Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W.H. Freeman.ISBN978-0-7167-0810-0.OCLC51095685.
Coey, J. M. D. (2019).Magnetism and Magnetic Materials. Cambridge University Press.ISBN978-1108717519.
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